Dependence of spin susceptibility of a two-dimensional electron system on the valley degree of freedom

Spontaneous Valley Polarization of Itinerant Electrons

Memory or transistor devices based on an electron's spin rather than its charge degree of freedom offer certain distinct advantages and comprise a cornerstone of spintronics. Recent years have witnessed the emergence of a new field, valleytronics, which seeks to exploit an electron's valley index rather than its spin. An important component in this quest would be the ability to control the valley index in a convenient fashion. Here we show that the valley polarization can be switched from zero to 1 by a small reduction in density, simply tuned by a gate bias, in a two-dimensional electron system. This phenomenon, which is akin to Bloch spin ferromagnetism, arises fundamentally as a result of electron-electron interaction in an itinerant, dilute electron system. Essentially, the kinetic energy favors an equal distribution of electrons over the available valleys, whereas the interaction between electrons prefers single-valley occupancy below a critical density. The gate-bias-tuned transition we observe is accompanied by a sudden, twofold change in sample resistance, making the phenomenon of interest for potential valleytronic transistor device applications. Our observation constitutes a quintessential demonstration of valleytronics in a very simple experiment.

Observation of spontaneous ferromagnetism in a two-dimensional electron system

What are the ground states of an interacting, low-density electron system? In the absence of disorder, it has long been expected that as the electron density is lowered, the exchange energy gained by aligning the electron spins should exceed the enhancement in the kinetic (Fermi) energy, leading to a (Bloch) ferromagnetic transition. At even lower densities, another transition to a (Wigner) solid, an ordered array of electrons, should occur. Experimental access to these regimes, however, has been limited because of the absence of a material platform that supports an electron system with very high quality (low disorder) and low density simultaneously. Here we explore the ground states of interacting electrons in an exceptionally clean, two-dimensional electron system confined to a modulation-doped AlAs quantum well. The large electron effective mass in this system allows us to reach very large values of the interaction parameter [Formula: see text], defined as the ratio of the Coulomb to Fermi energies. As we lower the electron density via gate bias, we find a sequence of phases, qualitatively consistent with the above scenario: a paramagnetic phase at large densities, a spontaneous transition to a ferromagnetic state when [Formula: see text] surpasses 35, and then a phase with strongly nonlinear current-voltage characteristics, suggestive of a pinned Wigner solid, when [Formula: see text] exceeds [Formula: see text] However, our sample makes a transition to an insulating state at [Formula: see text], preceding the onset of the spontaneous ferromagnetism, implying that besides interaction, the role of disorder must also be taken into account in understanding the different phases of a realistic dilute electron system.

Quantum Hall valley nematics

Two-dimensional electron gases in strong magnetic fields provide a canonical platform for realizing a variety of electronic ordering phenomena. Here we review the physics of one intriguing class of interaction-driven quantum Hall states: quantum Hall valley nematics. These phases of matter emerge when the formation of a topologically insulating quantum Hall state is accompanied by the spontaneous breaking of a point-group symmetry that combines a spatial rotation with a permutation of valley indices. The resulting orientational order is particularly sensitive to quenched disorder, while quantum Hall physics links charge conduction to topological defects. We discuss how these combine to yield a rich phase structure, and their implications for transport and spectroscopy measurements. In parallel, we discuss relevant experimental systems. We close with an outlook on future directions.

Realization of a Valley Superlattice

In a number of widely studied materials, such as Si, AlAs, Bi, graphene, MoS_{2}, and many transition metal dichalcogenide monolayers, electrons acquire an additional, spinlike degree of freedom at the degenerate conduction band minima, also known as "valleys." External symmetry-breaking fields such as mechanical strain, or electric or magnetic fields, can tune the valley polarization of these materials, making them suitable candidates for "valleytronics." Here we study a quantum well of AlAs, where the two-dimensional electrons reside in two energetically degenerate valleys. By fabricating a strain-inducing grating on the sample surface, we engineer a spatial modulation of the electron population in different valleys, i.e., a "valley superlattice" in the quantum well plane. Our results establish a novel manipulation technique of the valley degree of freedom, paving the way to realizing a valley-selective layered structure in multivalley materials, with potential application in valleytronics.

Strongly Interaction-Enhanced Valley Magnetic Response in Monolayer WSe_{2}

We measure the doping dependence of the valley Zeeman splitting of the fundamental optical transitions in monolayer WSe_{2} under an out-of-plane magnetic field by optical reflection contrast and photoluminescence spectroscopy. A nonlinear valley Zeeman effect, correlated with an over fourfold enhancement in the g factor, is observed. The effect occurs when the Fermi level crosses the spin-split upper conduction band, corresponding to a change of the spin-valley degeneracy from two to four. The enhancement increases and shows no sign of saturation as the sample temperature decreases. Our result demonstrates the importance of the Coulomb interactions in the valley magnetic response of two-dimensional transition metal dichalcogenide semiconductors.

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.

Valley polarization assisted spin polarization in two dimensions

Valleytronics is rapidly emerging as an exciting area of basic and applied research. In two-dimensional systems, valley polarization can dramatically modify physical properties through electron-electron interactions as demonstrated by such phenomena as the fractional quantum Hall effect and the metal-insulator transition. Here, we address the electrons' spin alignment in a magnetic field in silicon-on-insulator quantum wells under valley polarization. In stark contrast to expectations from a non-interacting model, we show experimentally that less magnetic field can be required to fully spin polarize a valley-polarized system than a valley-degenerate one. Furthermore, we show that these observations are quantitatively described by parameter-free ab initio quantum Monte Carlo simulations. We interpret the results as a manifestation of the greater stability of the spin- and valley-degenerate system against ferromagnetic instability and Wigner crystalization, which in turn suggests the existence of a new strongly correlated electron liquid at low electron densities.

Metallic behaviour in SOI quantum wells with strong intervalley scattering

The fundamental properties of valleys are recently attracting growing attention due to electrons in new and topical materials possessing this degree-of-freedom and recent proposals for valleytronics devices. In silicon MOSFETs, the interest has a longer history since the valley degree of freedom had been identified as a key parameter in the observation of the controversial "metallic behaviour" in two dimensions. However, while it has been recently demonstrated that lifting valley degeneracy can destroy the metallic behaviour, little is known about the role of intervalley scattering. Here, we show that the metallic behaviour can be observed in the presence of strong intervalley scattering in silicon on insulator (SOI) quantum wells. Analysis of the conductivity in terms of quantum corrections reveals that interactions are much stronger in SOI than in conventional MOSFETs, leading to the metallic behaviour despite the strong intervalley scattering.

Valley polarization and susceptibility of composite fermions around a filling factor nu=3/2

We report magnetotransport measurements of fractional quantum Hall states in an AlAs quantum well around a Landau level filling factor nu=3/2, demonstrating that the quasiparticles are composite fermions (CFs) with a valley degree of freedom. By monitoring the valley level crossings for these states as a function of applied symmetry-breaking strain, we determine the CF valley susceptibility and polarization. The data can be explained well by a simple Landau level fan diagram for CFs, and are in nearly quantitative agreement with the results reported for CF spin polarization.

Valley susceptibility of an interacting two-dimensional electron system

We report direct measurements of the valley susceptibility, the change of valley population in response to an applied symmetry-breaking strain, in an AlAs two-dimensional electron system. As the two-dimensional density is reduced, the valley susceptibility dramatically increases relative to its band value, reflecting the system's strong electron-electron interaction. The increase has a remarkable resemblance to the enhancement of the spin susceptibility and establishes the analogy between the spin and valley degrees of freedom.

Dependence of persistent gaps at landau level crossings on relative spin

We report measurements of the quantum Hall state energy gap at avoided crossings between Landau levels originating from different conduction band valleys in AlAs quantum wells. These gaps exhibit an approximately linear dependence on the magnetic field over a wide range of fields and filling factors. More remarkably, we observe an unexpected dependence of the gap size on the relative spin orientation of the crossing levels, with parallel-spin crossings exhibiting larger gaps than antiparallel-spin crossings.

Spin-independent effective mass in a valley-degenerate electron system

In a generic spin-polarized Fermi liquid, the masses of spin-up and spin-down electrons are expected to be different and to depend on the degree of polarization. This expectation is not confirmed by the experiments on two-dimensional heterostructures. We consider a model of an N-fold degenerate electron gas. It is shown that in the large- limit, the mass is enhanced via a polaronic mechanism of emission or absorption of virtual plasmons. As plasmons are classical collective excitations, the resulting mass does not depend on , and thus on polarization, to the leading order in 1/N. We evaluate the 1/N corrections and show that they are small even for N = 2.

Observation of quantum Hall "valley Skyrmions"

We report measurements of the interaction-induced quantum Hall effect in a spin-polarized AlAs two-dimensional electron system where the electrons occupy two in-plane conduction band valleys. Via the application of in-plane strain, we tune the energies of these valleys and measure the energy gap of the quantum Hall state at filling factor nu = 1. The gap has a finite value even at zero strain and, with strain, rises much faster than expected from a single-particle picture, suggesting that the lowest energy charged excitations at nu = 1 are "valley Skyrmions."

Effects of thickness on the spin susceptibility of the two dimensional electron gas

Using available quantum Monte Carlo predictions for a strictly 2D electron gas, we estimate the spin susceptibility of electrons in actual devices taking into account the effect of the finite transverse thickness and finding very good agreement with experiments. A weak disorder, as found in very clean devices and/or at densities not too low, just brings about a minor enhancement of the susceptibility.

Spin-dependent resistivity at transitions between integer quantum hall states

The longitudinal resistivity at transitions between integer quantum Hall states in two-dimensional electrons confined to AlAs quantum wells is found to depend on the spin orientation of the partially filled Landau level in which the Fermi energy resides. The resistivity can be enhanced by an order of magnitude as the spin orientation of this energy level is aligned with the majority spin. We discuss possible causes and suggest a new explanation for the spikelike features observed at the edges of quantum Hall minima near Landau level crossings.