Conductivity Tensor of Striped Quantum Hall Phases

Abstract

We study the transport properties of pinned striped quantum Hall phases. We show that under quite general assumptions, the macroscopic conductivity tensor satisfies a semicircle law. In particular, this result is valid for both smectic and nematic stripe phases, independent of the presence of topological and orientational defects such as dislocations and grain boundaries. As a special case, our results explain the experimental validity of a product rule for the dissipative part of the resistivity tensor, which was previously derived by MacDonald and Fisher for a perfect stripe structure.PACS numbers: 73.40. Hm, 73.20.Dx, Recent experiments [1,2] have revealed strikingly anisotropic dc transport properties of very clean two dimensional (2D) electron systems when the Landau level filling factor is close to ν = N + 1/2, and N ≥ 4 is an integer. It is believed that this is related to previous theoretical proposals [3,4] that Coulomb interactions would lead to an instability towards charge-density wave (CDW) formation in high, spin resolved, Landau levels (LL). Specifically, formation of a striped phase of the uppermost Landau level was predicted when it is close to half filling, while a "bubble phase" should be favorable further away from half filling. The striped phase consists of one-dimensional stripes alternating between the integer filling factors N and N + 1 with period of order of the cyclotron radius R c . In the bubble phase, clusters of minority filling factor with size R c order in a triangular lattice. These predictions, obtained within the HartreeFock (HF) approximation, have also been supported by numerical exact diagonalization studies [5].At finite temperature and/or in the presence of disorder, the perfect stripe ordering predicted by HF calculations will presumably be destroyed. The unidirectional CDW shares the symmetries of 2D smectic liquid crystals [6]. This implies that if there is no external force tending to align the stripes, then a dislocation will cost only a finite amount of energy and thus there will be a finite density of dislocations at non-zero temperatures. This is expected to destroy translational long-range order, except at zero temperature, but preserve quasi-longrange orientational order of the remaining stripe segments, characteristic of a 2D nematic phase. The orientational order would be effectively locked in by any small added anisotropy. As the temperature becomes large enough, there will be a Kosterlitz-Thouless transition to an isotropic state in which the stripe segments lose their orientational order. Short-range stripe order should disappear completely only around the presumably much higher HF transition temperature.Transport properties of the striped phases should be affected by even small amounts of disorder on the substrate, which will pin the stripe positions at low temperatures. Disorder should also lead to a finite density of dislocations, even at zero temperature. Moreover, since the forces aligning the stripes are believed to be very weak, steps or other large-...

Elastic theory of quantum Hall smectics: Effects of disorder

“…It is believed that this is a manifestation of the formation of striped quantum Hall states whose existence had previously been predicted based on Hartree-Fock (HF) calculations [2] (for a recent review, see [3]). The existence of striped states in higher Landau levels has also been supported by exact numerical diagonalization studies [4] and by the experimental confirmation [1] of some explicit predictions [5,6] for their transport properties.The HF calculations predict that the ground state near half filling of higher Landau levels is a unidirectional charge density wave (CDW) whose period a is of the order of the cyclotron radius [7]. In addition to orientational order, this state also breaks translational symmetry in one direction.…”

Highly anisotropic transport in the integer quantum Hall effect

“…It is believed that there the electron system spontaneously breaks into striped domains of alternating filling factors such as ν = 4 and ν = 5 around ν = 9/2 [9][10][11][12][13][14][15][16][17][18][19][20][21]. Given the similarity of the observed properties of the anisotropic phases around ν = 9/2 and ν = 6 one might speculate on a similar underlying striped geometry.…”

“…At large B ip the easy-axis of anisotropy in the plane of the sample (the direction of minimum resistance) is always perpendicular to B ip [5,6]. Although the nature of this anisotropy remains uncertain, experimental data [2][3][4][5][6][7][8] and theoretical models [9][10][11][12][13][14][15][16][17][18][19][20][21] point to the formation of a unidirectional charge density wave, often referred to as the "stripe phase", or to a state akin to a liquid crystal phase [11]. A very similar anisotropy is also observed at ν = 5/2 and ν = 7/2 in the second Landau level under large B ip [5,6].…”

Reorientation of Anisotropy in a Square Well Quantum Hall Sample

“…The most appealing interpretation suggests that the 2D electron gas spontaneously breaks the translational symmetry by forming a unidirectional charge density wave (UCDW), as predicted by Hartree-Fock theory [13,14]. This idea has spurred much theoretical interest [15][16][17][18][19][20][21][22][23][24][25][26][27][28]. Because of uncertainty about the reliability of this Hartree-Fock prediction, there has been a special emphasis [19,20] placed on tests of its ability to explain experimental results on "stripe" orientation in tilted magnetic fields.…”