Unidirectional Charge Density Wave Research Articles

Photoemission signatures of valence-bond stripes in cuprates: Long-range vs. short-range order

Recent experiments indicate that the tendency toward the formation of unidirectional charge density waves (“stripes”) is common to various underdoped cuprates. We discuss momentum-resolved spectral properties of valence-bond stripes, comparing the situations of ideal and short-range stripe order, the latter being relevant for weak and/or disorder-pinned stripes. We find clear signatures of ordered stripes, although matrix element effects suppress most shadow band features. With decreasing stripe correlation length, stripe signatures are quickly washed out, the only remaining effect being a broadening of antinodal quasiparticles. This insensitivity of photoemission to short-range stripe order may be employed to distinguish it from nematic order, e.g. in underdoped YBa2Cu3O6+δ.

STM Studies ofTbTe3: Evidence for a Fully Incommensurate Charge Density Wave

We observe unidirectional charge density wave (CDW) ordering on the quasi-2D material TbTe3 with a scanning tunneling microscope at approximately 6 K. Our analysis indicates that the CDW is fully incommensurate, with wave vector qCDW approximately 0.71x2pi/c. By imaging at various tip-sample voltages, we highlight effects of the subsurface layer and its effect on the CDW. We also observe an additional (possibly surface) dimerization and approximately 0.68x2pi/a ordering perpendicular to the CDW.

From stripe to checkerboard ordering of charge-density waves on the square lattice in the presence of quenched disorder

We discuss the effects of quenched disorder on a model of charge density wave (CDW) ordering on the square lattice. Our model may be applicable to the cuprate superconductors, where a random electrostatic potential exists in the $\mathrm{Cu}{\mathrm{O}}_{2}$ planes as a result of the presence of charged dopants. We argue that the presence of a random potential can affect the unidirectionality of the CDW order, characterized by an Ising order parameter. Coupling to a unidirectional CDW, the random potential can lead to the formation of domains with 90\ifmmode^\circ\else\textdegree\fi{} relative orientation, thus tending to restore the rotational symmetry of the underlying lattice. We find that the correlation length of the Ising order can be significantly larger than the CDW correlation length. For a checkerboard CDW on the other hand, disorder generates spatial anisotropies on short length scales and, thus, some degree of unidirectionality. We quantify these disorder effects and suggest techniques for analyzing the spatially dependent local density of states data measured in scanning tunneling microscopy experiments.

Anisotropic conductivity tensor on a half-filled high Landau level

We study two-dimensional interacting electrons in a weak perpendicular magnetic field with the filling factor $\ensuremath{\nu}⪢1$ and in the presence of a quenched disorder. As it is known, the unidirectional charge density wave state can exist near a half-filled high Landau level at low temperatures if disorder is weak enough. We show that the existence of the unidirectional charge density wave state at temperature $T<{T}_{c}$ where ${T}_{c}$ is the transition temperature leads to the anisotropic conductivity tensor. We find that the anisotropic part of conductivity tensor is proportional to $({T}_{c}\ensuremath{-}T)∕{T}_{c}$ below the transition. The order parameter fluctuations wash out the mean-field cusp at $T={T}_{c}$ and the conductivity tensor becomes anisotropic even above the mean-field transition temperature ${T}_{c}$.

Competition between quantum-liquid and electron-solid phases in intermediate Landau levels

On the basis of energy calculations we investigate the competition between quantum-liquid and electron-solid phases in the Landau levels n=1,2, and 3 as a function of their partial filling factor. Whereas the quantum-liquid phases are stable only in the vicinity of quantized values 1/(2s+1) of the partial filling factor, an electron solid in the form of a triangular lattice of clusters with a few number of electrons (bubble phase) is energetically favorable between these fillings. This alternation of electron-solid phases, which are insulating because they are pinned by the residual impurities in the sample, and quantum liquids displaying the fractional quantum Hall effect explains a recently observed reentrance of the integral quantum Hall effect in the Landau levels n=1 and 2. Around half-filling of the last Landau level, a uni-directional charge density wave (stripe phase) has a lower energy than the bubble phase.

The anisotropic conductivity of two-dimensional electrons on a half-filled high Landau evel

We study the conductivity of two-dimensional interacting electrons on the half-filled $N$th Landau level with $N \gg 1$ in the presence of the quenched disorder. The existence of the unidirectional charge-density wave state at temperature $T<T_c$, where $T_c$ is the transition temperature, leads to the anisotropic conductivity tensor. We find that the leading anisotropic corrections are proportional to $(T_c-T)/T_c$ just below the transition in accordance with the experimental findings. Above $T_c$ the correlations corresponding to the unidirectional charge-density wave state below $T_c$ result in the corrections to the conductivity proportional to $\sqrt{T_c/(T-T_c)}$.

Striped states in quantum Hall effect: Deriving a low-energy theory from Hartree-Fock

There is growing experimental and theoretical evidence that very clean two dimensional electron systems form unidirectional charge density waves (UCDW) or ``striped'' states at low temperatures and at Landau level filling fractions of the form nu = M + x with 4 < M < 10 an integer and 0.4 < x < 0.6. Following previous work, we model the striped state using a Hartree-Fock approach. We construct the low energy excitations of the system by making smooth deformations of the stripe edges analogous to the construction of edge state excitations of quantum Hall droplets. These low energy excitations are described as a coupled Luttinger liquid theory, as discussed previously by MacDonald and Fisher (PRB61 5724 2000). Here, we extend that work and explicitly derive all of the parameters of this low energy theory using a Hartree Fock approach. We also make contact with the equivalent low energy hydrodynamic approach of Fogler and Vinokur (PRL84 5828 2000) and similarly derive the parameters of this theory. As examples of the use of these results, we explicitly calculate the low-energy excitation spectrum and study tunneling into the striped state.

ANISOTROPY IN THE COMPRESSIBLE QUANTUM HALL STATE

Using a mean field theory on the von Neumann lattice, we study compressible anisotropic states around ν=l+1/2 in the quantum Hall system. The Hartree-Fock energy of the unidirectional charge density wave (UCDW) are calculated self-consistently. In these states the UCDW seems to be the most plausible state. We show that the UCDW is regarded as a collection of the one-dimensional lattice fermion systems which extend to the uniform direction. The kinetic energy of this one-dimensional system is induced from the Coulomb interaction term and the self-consistent Fermi surface is obtained.

New results at half-fillings in the second and third Landau level

This paper presents some recent results on the anisotropic electronic transport at half-filling factors in the second and third Landau levels. In the third Landau level at ν=92and112, the electronic transport is already strongly anisotropic in perpendicular magnetic field. This anisotropic phase is believed to be due to the formation of a unidirectional charge density wave. Applying an in-plane field overcomes the initial pinning and reorients the direction of this anisotropy. In the second Landau level, the even-denominator fractional quantum Hall state at ν=52and72 is destroyed and transformed into an anisotropic phase by an in-plane magnetic field. This result provides additional evidence that the ground state at ν=52and72 is spin-polarized.

Paired Hall states versus unidirectional CDW in a tilted field forν=52

We formulate the composite fermions in the presence of an in-plane magnetic field. As the in-plane field increases, if we assume the state at $\nu=5/2$ turns into the mixed state between the unidirectional charge density wave domains and paired Hall state, we can phenomenologically fit the theoretically defined gap to the experimental measured results. We explain the destruction of the paired Hall states and then a phase transition from the paired Hall state to the unidirectional charge density wave from a symmetry point of view.

Stripes in quantum Hall double-layer systems

We present results of a study of double layer quantum Hall systems in which each layer has a high-index Landau level that is half-filled. Hartree-Fock calculations indicate that, above a critical layer separation, the system becomes unstable to the formation of a unidirectional coherent charge density wave (UCCDW), which is related to stripe states in single layer systems. The UCCDW state supports a quantized Hall effect when there is tunneling between layers, and is {\it always} stable against formation of an isotropic Wigner crystal for Landau indices $N \ge 1$. The state does become unstable to the formation of modulations within the stripes at large enough layer separation. The UCCDW state supports low-energy modes associated with interlayer coherence. The coherence allows the formation of charged soliton excitations, which become gapless in the limit of vanishing tunneling. We argue that this may result in a novel {\it ``critical Hall state''}, characterized by a power law $I-V$ in tunneling experiments.

Reorientation of anisotropy in a square well quantum hall sample.

We have measured magnetotransport at half-filled high Landau levels in a quantum well with two occupied electric subbands. We find resistivities that are isotropic in perpendicular magnetic field but become strongly anisotropic at nu = 9/2 and 11/2 on tilting the field. The anisotropy appears at an in-plane field, B(ip) approximately 2.5 T, with the easy-current direction parallel to B(ip) but rotates by 90 degrees at B(ip) approximately 10 T and points now in the same direction as in single-subband samples. This complex behavior is in quantitative agreement with theoretical calculations based on a unidirectional charge density wave state model.

Compressible anisotropic states around the half-filled Landau levels

Using the von Neumann lattice formalism, we study compressible anisotropic states around the half-filled Landau levels in the quantum Hall system. In these states the unidirectional charge-density wave (UCDW) state seems to be the most plausible state. The charge-density profile and Hartree-Fock energy of the UCDW are calculated self-consistently. The wavelength dependence of the energy for the UCDW is also obtained numerically. We show that the UCDW is regarded as a collection of the one-dimensional lattice Fermi-gas systems which extend to the uniform direction. The kinetic energy of the gas system is generated dynamically from the Coulomb interaction.

In-plane magnetic field-induced anisotropy and orientation energy of stripe phases at half-filled high Landau levels

We report on detailed Hartree–Fock calculations of the unidirectional charge density wave orientation energy induced by a tilted magnetic field. We find that for current experimental samples stripes are oriented perpendicular to the in-plane field, consistent with experiment. For wider two-dimensional electron systems we predict tilt-induced stripe states with variable anisotropy energy sign.

Field-tilt anisotropy energy in quantum Hall stripe states

Recently reported giant anisotropy in the longitudinal resistivity of a 2D electron system with valence Landau level index $N \ge 2$ has been interpreted as a signal of unidirectional charge density wave (UCDW) ground states. We report on detailed Hartree-Fock calculations of the UCDW orientation energy induced by a tilted magnetic field. We find that for current experimental samples stripes are oriented perpendicular to the in-plane field, consistent with experiment. For wider 2D electron systems we predict tilt-induced stripe states with variable anisotropy energy sign.

Exact results for interacting electrons in high Landau levels.

We study a two-dimensional electron system in a magnetic field with a fermion hardcore interaction and without disorder. Projecting the Hamiltonian onto the n-th Landau level, we show that the Hartree-Fock theory is exact in the limit n \rightarrow \infty, for the high temperature, uniform density phase of an infinite system; for a finite-size system, it is exact at all temperatures. In addition, we show that a charge-density wave arises below a transition temperature T_t. Using Landau theory, we construct a phase diagram which contains both unidirectional and triangular charge-density wave phases. We discuss the unidirectional charge-density wave at zero temperature and argue that quantum fluctuations are unimportant in the large-n limit. Finally, we discuss the accuracy of the Hartree-Fock approximation for potentials with a nonzero range such as the Coulomb interaction.

Wigner transition temperature of a two-dimensional electron gas in a strong magnetic field

Within the Hartree-Fock approximation the transition temperature ${T}_{c}$ of a two-dimensional electron gas of density $n$ in a strong magnetic field $B$ is studied in the nondegenerate limit $\frac{n}{B}\ensuremath{\rightarrow}0$. By means of an asymptotically correct calculation it is shown that the ansatz of a strongly anharmonic, unidirectional charge-density wave (CDW) for the condensed state leads, apart from weak logarithmic corrections, to the limiting form ${T}_{c}\ensuremath{\sim}{n}^{\frac{1}{2}}$. The ansatz of an essentially harmonic, though triangular symmetric CDW leads to much smaller limiting values ${T}_{c}\ensuremath{\sim}n{B}^{\ensuremath{-}\frac{1}{2}}$. This demonstrates the importance of the anharmonicity of the CDW or, equivalently, of the sharp localization of electronic wave functions in the nondegenerate limit.

Magnetic field induced phase transition of a three-dimensional electron gas

It is shown within the Hartree-Fock approximation that the electron gas in a strong magnetic field undergoes a phase transition to a charge density wave (CDW) state. The transition temperature T c is calculated as a function of the magnetic field strength B. With increasing B, the nature of the electron system at T c changes from degenerate to non-degenerate. For moderate values of B a second order transition to a unidirectional CDW is obtained, and for large B a first order transition to a CDW with the symmetry of a hexagonal lattice. If material parameters are adapted to Hg .8Cd .2Te, T c is of the order of 1K for B values near 5T.

Classical and Quantum Two-Dimensional Wigner Crystal in Strong Magnetic Fields

The ground state of a two-dimensional electron system in a strong magnetic field is investigated under the condition that only the ground Landau level is partially filled. Within the unrestricted Hartree-Fock approximation the ground state proves to be a charge-density-wave (CDW) state. When the density is low, wave functions of nearby lattice sites are well separated so that the CDW is regarded as a classical Wigner crystal. It is shown that the amplitude of the CDW is very large at arbitrary densities. Density dependence of the ground state energy is discussed using (1) a unidirectional CDW structure which allows one to take full account of the exchange effect, and (2) the square lattice for which self-consistent solution is obtained in the case of the half-filled Landau level as well as in the low density classical regime.