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CRC Press eBooks

Quantum Phase Transitions in Two-Dimensional Electron

Systems Alexander Shashkin, Sergey Kravchenko

Published Nov 2, 2010• 1 citation
Publisher

Abstract

Alexander Shashkin Institute of Solid State Physics, Chernogolovka, Moscow District 142432, Russia Sergey Kravchenko Physics Department, Northeastern University, Boston, Massachusetts 02115, U.S.A. Two-dimensional (2D) electron systems are realized when the electrons are free to move in a plane but their motion perpendicular to the plane is quantized in a confining potential well. Quantum phase transitions realized experimentally in such systems so far include metal-insulator transitions in perpendicular magnetic fields, metal-insulator transition in zero magnetic field, and a possible transition to a Wigner crystal. The first transition is governed by the externally controlled electron density or magnetic field, while the other two are governed by the electron density. At low electron densities in 2D systems, the strongly-interacting limit is reached because the kinetic energy is overwhelmed by the energy of electron-electron interactions. The interaction strength is characterized by the ratio between the Coulomb energy and the Fermi energy, r∗s = Eee/EF . Assuming that the effective electron mass is equal to the band mass, the interaction parameter r∗s in the single-valley case reduces to the Wigner-Seitz radius, rs = 1/(πns)1/2aB, and therefore increases as the electron density, ns, decreases (here aB is the Bohr radius in the semiconductor). Possible candidates for the ground state of the system include a Wigner crystal characterized by spatial and spin ordering [1], a ferromagnetic Fermi liquid with spontaneous spin ordering [2], a paramagnetic Fermi liquid [3], etc. In the strongly-interacting limit (rs 1), no analytical theory has been developed to date. According to numerical simulations [4], Wigner crystallization is expected in a very dilute regime, when rs reaches approximately 35. Refined numerical simulations [5] have predicted that prior to the crystallization, in the range of the interaction parameter 25 ≤ rs ≤ 35, the ground state of the system is a strongly correlated ferromagnetic Fermi liquid. At higher electron densities, rs ∼ 1, the electron liquid is expected to be paramagnetic, with the effective mass, m, and Lande´ g factor renormalized by interactions. Apart from the ferromagnetic Fermi liquid, other intermediate phases between the Wigner crystal and the paramagnetic Fermi liquid may also exist.

DOI
10.1201/b10273-23

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Alexander Shashkin Institute of Solid State Physics, Chernogolovka, Moscow District 142432, Russia Sergey Kravchenko Physics Department, Northeastern University, Boston, Massachusetts 02115, U.S.A. Two-dimensional (2D) electron systems are realized when the electrons are free to move in a plane but their motion perpendicular to the plane is quantized in a confining potential well. Quantum phase transitions realized experimentally in such systems so far include metal-insulator transitions in perpendicular magnetic fields, metal-insulator transition in zero magnetic field, and a possible transition to a Wigner crystal. The first transition is governed by the externally controlled electron density or magnetic field, while the other two are governed by the electron density. At low electron densities in 2D systems, the strongly-interacting limit is reached because the kinetic energy is overwhelmed by the energy of electron-electron interactions. The interaction strength is characterized by the ratio between the Coulomb energy and the Fermi energy, r∗s = Eee/EF . Assuming that the effective electron mass is equal to the band mass, the interaction parameter r∗s in the single-valley case reduces to the Wigner-Seitz radius, rs = 1/(πns)1/2aB, and therefore increases as the electron density, ns, decreases (here aB is the Bohr radius in the semiconductor). Possible candidates for the ground state of the system include a Wigner crystal characterized by spatial and spin ordering [1], a ferromagnetic Fermi liquid with spontaneous spin ordering [2], a paramagnetic Fermi liquid [3], etc. In the strongly-interacting limit (rs 1), no analytical theory has been developed to date. According to numerical simulations [4], Wigner crystallization is expected in a very dilute regime, when rs reaches approximately 35. Refined numerical simulations [5] have predicted that prior to the crystallization, in the range of the interaction parameter 25 ≤ rs ≤ 35, the ground state of the system is a strongly correlated ferromagnetic Fermi liquid. At higher electron densities, rs ∼ 1, the electron liquid is expected to be paramagnetic, with the effective mass, m, and Lande´ g factor renormalized by interactions. Apart from the ferromagnetic Fermi liquid, other intermediate phases between the Wigner crystal and the paramagnetic Fermi liquid may also exist.
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.
The nature of the ground state of degenerate two dimensional (2D) fermions at zero magnetic field is an outstanding open problem, which has not been deciphered despite decades of research. In the absence of disorder, the ground state is believed to be determined by the interplay between the kinetic energy, E f, and the inter-particle interaction energy, Ec = e2/ica, where a = (nn)~1^ 2 is the typical inter-particle distance, n is the areal particle density, and k is the host dielectric constant. The relative importance of the two energy scales is characterized by rs = a/ao, with ao being the Bohr radius. For electrons in a single band, rs = Ec/ E f while for the (100) surface of silicon rs = Ec/2E p , due to the two-fold valley degeneracy. At very high densities (rs <£ 1) a 2D system approaches the paramagnetic limit of a non-interacting degenerate gas characterized by the Pauli susceptibility, xo • As the density is reduced, the growing ferromagnetic correlations lead to substantial enhancement of the spin susceptibility, The system is predicted to remain paramagnetic up to rs « 20, where numerical calculations [1] find a quantum phase transition to a ferromagnetic liquid phase [2]. At lower density, rs & 37 [3], the Coulomb correlations are predicted to lead to a second phase transition to an anti-ferromagnetic quantum Wigner crystal. The energy balance between the ferro and paramagnetic states is very subtle, therefore the density window where ferromagnetism can take place is small [1].
Electrons in two-dimensional (2D) layers at semiconductor interfaces may be driven to a solid phase by reducing their density. Understandmg of this electron liquid-solid transition at mK-temperatures and the origin of the 2D electron solid phase remain challenging. The experimental system under study is subject to strong interactions and disorder. Charge transport in the electron solid phase demonstrates many features anticipated for the pinned crystal, though no direct imaging of a spatial charge- or magnetic-ordering is available by now. Rich information has been recently obtained for dilute electron liquid phase on strong renormalization of the Pauli spin susceptibility, magnetization, and the effective mass. These data provide a set of the Fermi-liquid coupling parameters for densities down to and at the transition. The steep density dependence of the susceptibility is intriguing because it might indicate a developing magnetic transition, though it is more likely reflects a tendency to crystallization.
The two-dimensional system of electrons in a high magnetic field offers an opportunity to investigate a phase transition from a quantum liquid into a Wigner solid. Recent experiments have revealed an incipient composite fermion liquid in a parameter range where theory and many experiments had previously suggested the Wigner crystal phase, thus calling into question our current understanding. This Letter shows how very small quantitative corrections (<1%) in the energy due to the weak interaction between composite fermions can cause a fundamental change in the nature of the ground state, thus providing insight into the puzzling experimental results.
The electrons in inversion or accumulation layers of MOSFETS or in semiconductor-semiconductor interfaces show prominent many-body effects. A new theory of the correlation energy is developed for these electrons such that the results are applicable to a wide density range. The ground-state energy is obtained for all densities. It is continuous and convergent, but changes its analytical form at rs = 1.414. Associated with this change is a divergence of the compressibility that occurs at rs = 1.989. Hence, around this rs, the system can be considered to be in a liquidlike state. In the [001] direction of silicon inversion layers, the two valleys, occupied equally by the electrons at high densities, may be populated unevenly due to electron correlations. This unbalance will cause a valley occupancy phase transition at rs = 8.011, in close agreement with a recent experiment. Under a magnetic field, the susceptibility is enhanced nonlinearly when the density of electrons is reduced. This enhancement becomes very strong toward rs = 13.0, beyond which a spin-polarized state is favored.
(received; accepted) PACS. 71.30+h – Metal-insulator transitions and other electronic transitions. PACS. 72.15Rn – Quantum localization. PACS. 71.27+a – Strongly correlated electron systems. Abstract. – Extending finite size scaling theory to the many body ground state, one finds that Coulomb repulsion can drive a system of spinless fermions in a random potential from the Anderson insulator (Fermi glass of localized states) towards a new extended phase in dimension d = 2. The transition occurs at a Coulomb energy to Fermi energy ratio rs ≈ 4, where a change in the nature of the persistent currents has been previously observed. Relevance to the recently observed 2d metallic phase is suggested. Following the scaling theory of localization [1], all 2d systems of electrons (or holes) are localized when electron-electron interaction is negligible. This absence of metals in two dimensions is nowadays questioned after the observation of a transition towards a metallic behavior in silicon based 2d electron gases [2, 3, 4] when the carrier density is varied. This unexpected phenomenon occurs also for hole gases in GaAs heterostructures [5, 6], SiGe [7]