Cyclotron Gap Research Articles

Density-controlled quantum Hall ferromagnetic transition in a two-dimensional hole system

Quantum Hall ferromagnetic transitions are typically achieved by increasing the Zeeman energy through in-situ sample rotation, while transitions in systems with pseudo-spin indices can be induced by gate control. We report here a gate-controlled quantum Hall ferromagnetic transition between two real spin states in a conventional two-dimensional system without any in-plane magnetic field. We show that the ratio of the Zeeman splitting to the cyclotron gap in a Ge two-dimensional hole system increases with decreasing density owing to inter-carrier interactions. Below a critical density of ~2.4 × 1010 cm−2, this ratio grows greater than 1, resulting in a ferromagnetic ground state at filling factor ν = 2. At the critical density, a resistance peak due to the formation of microscopic domains of opposite spin orientations is observed. Such gate-controlled spin-polarizations in the quantum Hall regime opens the door to realizing Majorana modes using two-dimensional systems in conventional, low-spin-orbit-coupling semiconductors.

Phase Diagram of theν=5/2Fractional Quantum Hall Effect: Effects of Landau-Level Mixing and Nonzero Width

Interesting non-Abelian states, e.g., the Moore-Read Pfaffian and the anti-Pfaffian, offer candidate descriptions of the $\nu = 5/2$ fractional quantum Hall state. But the significant controversy surrounding the nature of the $\nu = 5/2$ state has been hampered by the fact that the competition between these and other states is affected by small parameter changes. To study the phase diagram of the $\nu = 5/2$ state we numerically diagonalize a comprehensive effective Hamiltonian describing the fractional quantum Hall effect of electrons under realistic conditions in GaAs semiconductors. The effective Hamiltonian takes Landau level mixing into account to lowest-order perturbatively in $\kappa$, the ratio of the Coulomb energy scale to the cyclotron gap. We also incorporate non-zero width $w$ of the quantum well and sub-band mixing. We find the ground state in both the torus and spherical geometries as a function of $\kappa$ and $w$. To sort out the non-trivial competition between candidate ground states we analyze the following 4 criteria: its overlap with trial wave functions; the magnitude of energy gaps; the sign of the expectation value of an order parameter for particle-hole symmetry breaking; and the entanglement spectrum. We conclude that the ground state is in the universality class of the Moore-Read Pfaffian state, rather than the anti-Pfaffian, for $\kappa < {\kappa_c}(w)$, where ${\kappa_c}(w)$ is a $w$-dependent critical value $0.6 \lesssim{\kappa_c}(w)\lesssim 1$. We observe that both Landau level mixing and non-zero width suppress the excitation gap, but Landau level mixing has a larger effect in this regard. Our findings have important implications for the identification of non-Abelian fractional quantum Hall states.

Probing the coexistence of semiclassical transport and localization in a two-dimensional electron gas using microwave radiation

We have performed magnetoresistance measurements on a GaAs/AlGaAs two-dimensional electron gas under microwave heating. Both the magneto-oscillations and magnetoresistance minima are used as electron thermometers to determine the electron effective temperature Te. When there is no clear cyclotron gap, it is found that Te determined from these two methods can be substantially different such that we can still distinguish localized electrons from the semiclassical conducting ones. The almost constant Te obtained from the magnetoresistance minima reveals the survived equivalence between different Landau-band tails under insufficient localization.

Probing semiclassical magneto-oscillations in the low-field quantum Hall effect

The low-field quantum Hall effect is investigated on a two-dimensional electron system in an AlGaAs/GaAs heterostructure. Magneto-oscillations following the semiclassical Shubnikov-de Haas formula are observed even when the emergence of the mobility gap shows the importance of quantum localization effects. Moreover, the Lifshitz-Kosevich formula can survive as the oscillating amplitude becomes large enough for the deviation to the Dingle factor. The crossover from the semiclassical transport to the description of quantum diffusion is discussed. From our study, the difference between the mobility and cyclotron gaps indicates that some electron states away from the Landau-band tails can be responsible for the semiclassical behaviors under low-field Landau quantization.

Double-exciton component of the cyclotron spin-flip mode in a quantum Hall ferromagnet

We report on the study of the cyclotron spin-flip excitation (CSFE) in a quantum Hall system at unit filling, where both the cyclotron quantum number and the spin number are changed by one compared to the ground state. The CSFE composite mode has a double-exciton component, which, even within the first-order approximation in terms of the interaction energy, contributes to the CSFE correlation shift measured from the combined cyclotron and Zeeman gap. This component cannot be accounted for in the single-mode approach, and the problem effectively becomes the quantum four-body one. The final result is obtained for zero total momentum. Since in that case, the excitation is optically active, the result is compared with available experimental data.

Spin Phase Transition in a Correlated Composite Fermion Liquid

The composite fermion (CF) model [1] has been very successful in the explanation of incompressibility of a whole family of fractional quantum Hall (FQH) states [2] at the “Jain series” of the Landau level (LL) filling factors νe = n/(2np ± 1). At these fillings, the CF transformation involving attachment of an even number 2p of magnetic flux quanta φ0 = hc/e to each electron converts a partially filled LL of electrons with strong (Coulomb) interaction into a small number νCF ≡ n of completely filled LL’s of nearly noninteracting CF’s (the general relation between electron and CF filling factors being ν−1 CF = ν −1 e − 2p). In filled shells, the weak residual CF–CF interactions play no role and the incompressibility of νe = n/(2np ± 1) states is said to result from a single-particle cyclotron gap of the CF’s in the reduced magnetic field B∗ = B − 2pφ0%, where % is the 2D electron/CF concentration (connected to νe = 2π%λ via magnetic length λ = √ hc/eB). Recently, Pan et al. [3] have observed the FQH effect in a spin-polarized electron gas at a series of filling factors between two neighboring Jain states, i.e. at 1/3 < νe < 2/5, and thus corresponding to the fractional CF filling factors, 1 < νCF < 2. In particular, the incompressible states observed at νe = 4/11, 3/8, 5/13 correspond to νCF = 4/3, 3/2, and 5/3, respectively. Clearly, incompressibility of these states must depend on the CF–CF interaction within a partially filled second CF LL.

Vanishing cyclotron gaps in a two-dimensional electron system with a strong short-period modulation

Magnetotransport in a density-tunable two-dimensional electron gas that is modulated with a strong, atomically precise superlattice potential of a $15\phantom{\rule{0.3em}{0ex}}\mathrm{nm}$ period is investigated. For low densities, the system shows typical two-dimensional behavior. In the regime of large densities, however, a quasi-one-dimensional state is entered, in which the cyclotron gaps vanish and the quantum Hall effect breaks down. Both regimes, and the transition between them, are theoretically described in the framework of a quantum-mechanical tight-binding model.

Interaction Effects and Pseudogap in Two-Dimensional Lateral Tunnel Junctions

Tunneling characteristics of a two-dimensional lateral tunnel junction are reported. A pseudogap on the order of Coulomb energy is detected in the tunneling density of states (TDOS) when two identical two-dimensional electron systems are laterally separated by a thin energy barrier. The Coulombic pseudogap remains robust well into the quantum Hall regime until it is overshadowed by the cyclotron gap in the TDOS. The pseudogap is modified by the in-plane magnetic field, demonstrating a nontrivial effect of the in-plane magnetic field on the electron-electron interaction.

Cyclotron spin-flip mode as the lowest-energy excitation of unpolarized integer quantum Hall states

The cyclotron spin-flip modes of spin unpolarized integer quantum Hall states ($\nu =2,4$) have been studied with inelastic light scattering. The energy of these modes is significantly smaller compared to the bare cyclotron gap. Second order exchange corrections are held responsible for a negative energy contribution and render these modes the lowest energy excitations of unpolarized integer quantum Hall states.

Zero-momentum cyclotron spin-flip mode in a spin-unpolarized quantum Hall system

We report on a study of the zero-momentum cyclotron spin-flip excitation in the $\mathcal{V}=2$ quantum Hall regime. Using the excitonic representation the excitation energy is calculated up to the second-order Coulomb corrections. A considerable negative exchange shift relative to the cyclotron gap is established for cyclotron spin-flip excitations in the spin-unpolarized electronic system. Under these conditions this type of state presents the lowest-energy excitations. For a fixed filling factor $(\mathcal{V}=2)$ the energy shift is independent of the magnetic field, which is in agreement with recent experimental observations.

High magnetic field studies of AlGaN/GaN heterostructures grown on bulk GaN

We present transport properties of AlGaN/GaN heterostructures grown over high-pressure bulk GaN substrates. The experimental results include the conductivity tensor measurements in a magnetic field up to 23 T in a wide temperature range 2 K–300 K for Hall bar samples. The room temperature high field data allow us to clearly separate contributions of a parasitic parallel conduction from 2DEG conduction in all investigated heterostructures. The room temperature mobility limit for 2D electrons in GaN/AlGaN heterojunctions grown on defect free GaN bulk substrates is around 2400 cm2/Vs. The Quantum Hall Effect studies are performed in the magnetic fields up to 23 T and temperatures between 1.6 K and 15 K This high magnetic field in combination with very high mobility (over 60000 cm2/Vs) in the sample grown on the bulk GaN substrate allow us to determine the activation energy in cyclotron gap from longitudinal magnetoresistance. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

Effects of Zeeman spin splitting on the modular symmetry in the quantum Hall effect

Magnetic-field-induced phase transitions in the integer quantum Hall effect are studied under the formation of paired Landau bands arising from Zeeman spin splitting. By investigating features of modular symmetry, we showed that modifications to the particle-hole transformation should be considered under the coupling between the paired Landau bands. Our study indicates that such a transformation should be modified either when the Zeeman gap is much smaller than the cyclotron gap, or when these two gaps are comparable.

Spin and interaction effects in Shubnikov–de Haas oscillations and the quantum Hall effectin GaN/AlGaN heterostructures

We present the results of high magnetic field (up to 30 T) and temperature (50 mK–80 K) dependenttransport measurements on a 2DEG in GaN/AlGaN heterojunctions. A high mobility (above60 000 cm2 V−1 s−1 at 4 K) 2DEG was obtained by MBE growth of dislocation free GaN and AlGaN layers onsemi-insulating bulk GaN substrates. A cyclotron gap and spin splitting are observed.Results from two studies are reported: (i) a tilted field experiment determining the 2DEGg*-factor from the angular modulation of the amplitude of SdH oscillations; (ii) quantum Halleffect measurements determining the activation energies for spin and cyclotron energygaps at even and odd filling factors. The observed ‘cyclotron gap’ enhancementis attributed to the effect of electron–electron interaction and it is estimatedusing the model of a 2D-screened Coulomb potential. The analytic result for theenhancement of the ‘cyclotron gap’ yields an addition to the activation energy,, , which is proportional to the magnetic field and resembles the mass renormalization.

Direct measurements of the spin and the cyclotron gaps in a 2D electron system in silicon.

Using magnetocapacitance data in tilted magnetic fields, we directly determine the chemical potential jump in a strongly correlated two-dimensional electron system in silicon when the filling factor traverses the spin and the cyclotron gaps. The data yield an effective g factor that is close to its value in bulk silicon and does not depend on the filling factor. The cyclotron splitting corresponds to the effective mass that is strongly enhanced at low electron densities.

Integer and fractional quantum Hall effects in bilayer electron systems

Integer and fractional quantum Hall (QH) effects are studied in bilayer electron systems both theoretically and experimentally, especially, at ν=2 and 2/3. Due to the spin and layer degrees of freedom, the SU(4) symmetry underlies the integer QH states, where quantum coherence develops spontaneously and quasiparticles are coherent excitations. It is intriguing that a pair of skyrmions makes one quasiparticle at ν=2. In the fractional QH regime, on the other hand, the composite-fermion cyclotron gap competes with the Zeeman and tunneling gaps, bringing in new phases and excitations. At ν=2/3 our experimental data suggest that a quasiparticle is not a coherent excitation but simply a composite fermion.

Magnetocapacitance studiesof two-dimensional electron systemswith long-range potential fluctuations

We report on magnetocapacitance study of thequantum Hall effect (QHE) states. Capacitance minima widthwas found to be independent of magnetic field and to be the samefor even, odd and fractional QHE states when measured as afunction of the average electron density. This result indicatesthat the width of capacitance minima in the samples investigated are governed by long-range carrier density fluctuations.At low temperatures, the amplitudes of the minima decreaselinearly with the temperature increase. All our experimentalresults for the integer QHE states are quantitatively explainedby introducing unbroadened magnetic levels and dispersion ofthe electron density along the sample. The energy gaps at evenfilling factors obtained from fitting the experimental data arefound to be close to the known cyclotron gaps. At odd fillings v = 1, 3, and 5, the energy gaps appear to be enhanced incomparison with the Zeeman splitting, with the enhancementdecreasing with filling factor.The capacitance minima are argued to originate from themotion of incompressible regions along a sample caused by thegate voltage variation. We derive the condition for the appearance and motion of such regions for the case of gated sampleswith long-range fluctuations of density of charged donors.The appearance of narrow magnetocapacitance peaks whena dc current is passed through the sample is reported. Wehypothesize that these peaks are due to the current percolationalong incompressible regions.

Imaging of low-compressibility strips in the quantum Hall liquid

Using Subsurface Charge Accumulation scanning microscopy we image strips of low compressibility corresponding to several integer Quantum Hall filling factors. We study in detail the strips at Landau level filling factors $\nu =$ 2 and 4. The observed strips appear significantly wider than predicted by theory. We present a model accounting for the discrepancy by considering a disorder-induced nonzero density of states in the cyclotron gap.

Magnetization and dissipation measurements in the quantum Hall regime using an integrated micromechanical magnetometer

We present low-temperature (365 mK) magnetization measurements of 40×100 μm2 mesas of two dimensional electron gases (2DEGs) integrated into micromechanical cantilever magnetometers. Over a wide range of applied magnetic field, the cantilever resonance frequency reveals the thermodynamic magnetization of the 2DEG. Upon illumination of the sample, we observe the appearance of both cyclotron and Zeeman gaps in the density of states. We attribute this to the narrowing of the disorder-broadened Landau levels as the carrier concentration is increased. Additionally, we observe strong peaks in the dissipation of the system at small integer filling factors which we associate with eddy currents excited by the cantilever motion.

Imaging the low compressibility strips formed by the quantum hall liquid in a smooth potential gradient

We locally image the low-compressibility strip corresponding to integer Quantum Hall filling factor ν=2 and find that its width appears significantly larger than that predicted by theory. We explain this result by a disorder-induced finite density of states in the cyclotron gap.

Spatial structure of an incompressible quantum Hall strip

The incompressible quantum Hall strip is sensitive to charging of localized states in the cyclotron gap. We study the effect of localized states by a density functional approach and find electron density and the strip width as a function of the density of states in the gap. Another important effect is electron exchange. By using a model density functional which accounts for negative compressibility of the QH state, we find electron density around the strip. At large exchange, the density profile becomes nonmonotonic. Both effects, localized states and exchange, lead to a substantial increase of the strip width.