Composite Fermion Crystals in the Lowest Landau Level

Open Access
Archer, Alex Charles
Graduate Program:
Doctor of Philosophy
Document Type:
Date of Defense:
August 27, 2014
Committee Members:
  • Jainendra Jain, Dissertation Advisor
  • Jainendra Jain, Committee Chair
  • Gerald Dennis Mahan, Committee Member
  • John C Collins, Committee Member
  • Stephane Coutu, Committee Member
  • Akhlesh Lakhtakia, Committee Member
  • FQHE
  • fractional quantum Hall effect
  • ionic crystal
  • 2DES
  • two dimensional electron system
Two-dimensional electrons systems (2DES) exposed to strong magnetic fields at electron den- sities above 10*10 cm^{−2}, such as those realized in AlGaAs/GaAs semiconductor heterostructures and quantum wells, exhibit a host of fascinating quantum phenomena. The most notable of these, the integer quantum Hall effect and fractional quantum Hall effect, continue to be a source of intense theoretical and experimental scrutiny. It has been hypothesized that, at the large magnetic fields where the fractional quantum Hall effect manifests, the ground state consists mostly of a liquid state, with a small fraction of the system forming a crystal. At even larger magnetic fields, the fractional quantum Hall effect no longer manifests and the two-dimensional electron system becomes insulating. It is suspected that this insulating phase is a consequence of the formation of a crystal state. Despite more than two decades of experimental work, the question as to whether the 2DES at strong magnetic fields forms crystalline phases remains unanswered. While it has long been understood that the ground state of the 2DES at exactly the rational filling factors \nu = n/(2pn +/- 1), where n and p are both integers, consists entirely of a liquid state, the hypothesis that a small fraction of the 2DES forms an ordered crystalline state when the filling factor \nu is in the vicinity of (but not exactly equal to) the rational values n/(2pn +/- 1) has received support from only a few experiments, largely due to a dearth of both theoretical predictions and experimental studies. The low filling factor insulating phase of the 2DES has been subject to much greater scrutiny, and the consensus is that it possesses some type of long-range order and that its properties are a consequence of electron-electron interactions, and not disorder. However, no experiment offers direct evidence of crystalline ordering, such as the observation of Bragg diffraction peaks or the k^{3/2} dispersion (where k is the magnitude of the wavevector \vec{k}) characteristic of a 2D crystal, nor is the importance of quantum effects clear. Support of the proposal that the insulating phase is crystalline has only emerged by considering all of the experimental evidence that has accumulated as a whole over the last two plus decades. The goal of this dissertation is to establish theoretically some physical signatures of a crystalline ground state in a 2DES exposed to a strong magnetic field that would enable experimentalists to conclusively identify the presence of crystalline states. To this end, I, in collaboration with my advisor J. K. Jain, have proposed two classes of ansatzes for the 2DES ground state. The first, which we refer to as a `type-1 composite fermion crystals’, are strongly correlated hexagonal crystals of electrons and are our model for the low-filling-factor insulating state. We solve a longstanding experimental puzzle (nearly 25 years old) by show- ing that a type-1 composite fermion crystal is responsible for the reentrant insulating phase between the \nu = 1/5 and 2/9 fractional quantum Hall phases. We predict that, at low filling factors, there is a series of phase transitions between different composite fermion crystals and that an unambiguous signature of these phase transitions would be the observation of an abrupt change in the shear modulus of the 2DES ground state, which magnetoabsorption experiments may be able to measure. We also calculate the energies of defects in the type-1 composite fermion crystals. We show that a defect of non-classical origin, the `bubble interstitial’, has the lowest energy, the low defect energy of these crystal states is a unique quantum signature of the ground state, and that the defect energies are comparable to the activation energies measured in transport experiments, thus providing further support for the notion that the insulating phase is crystalline. The second class of ansatzes, referred to as `type-2 composite fermion crystals’, is a model for the ground state of the 2DES in the vicinity of the filling factors \nu = n/(2pn +/- 1). These ansatzes consist of placing most of the 2DES into a liquid state, while taking the remaining fraction and forming a crystalline state, i.e., a type-2 composite fermion crystal. One characteristic of these ansatzes is that the density of the type-2 composite fermion crystal goes to zero as the filling factor approaches the rational values \nu \rightarrow n/(2pn +/- 1). These wavefunctions enables us, for the first time, to calculate the energy of fractional quantum Hall states as a continuous function of filling factor. We predict that, in the filling-factor range 1/3 \leq \nu \leq 4/9, a series of spin polarization phase transitions will occur as a continuous function of filling factor. We also predict that these type-2 composite fermion crystals will be stable only in a narrow range around the filling factors \nu = n/(2n + 1), for n = 1, 2, and 3.