We prove matrix and scalar differential Harnack inequalities for linear parabolic equations on Riemannian and Kahler manifolds.
X-ray magnetic circular dichroism (XMCD) measurements on single-crystal and powder samples of Ba$_{0.6}$K$_{0.4}$Mn$_{2}$As$_{2}$ show that the ferromagnetism below $T_{textrm{C}}approx$ 100 K arises in the As $4p$ conduction band. No XMCD signal is observed at the Mn x-ray absorption edges. Below $T_{textrm{C}}$, however, a clear XMCD signal is found at the As $K$ edge which increases with decreasing temperature. The XMCD signal is absent in data taken with the beam directed parallel to the crystallographic $textrm{c}$ axis indicating that the orbital magnetic moment lies in the basal plane of the tetragonal lattice. These results show that the previously reported itinerant ferromagnetism is associated with the As $4p$ conduction band and that distinct local-moment antiferromagnetism and itinerant ferromagnetism with perpendicular easy axes coexist in this compound at low temperature.
The crystallographic and physical properties of polycrystalline and single crystal samples of BaCo2As2 and K-doped Ba{1-x}K{x}Co2As2 (x = 0.06, 0.22) are investigated by x-ray and neutron powder diffraction, magnetic susceptibility chi, magnetization, heat capacity Cp, {75}As NMR and electrical resistivity rho measurements versus temperature T. The crystals were grown using both Sn flux and CoAs self-flux, where the Sn-grown crystals contain 1.6-2.0 mol% Sn. All samples crystallize in the tetragonal ThCr2Si2-type structure (space group I4/mmm). For BaCo2As2, powder neutron diffraction data show that the c-axis lattice parameter exhibits anomalous negative thermal expansion from 10 to 300 K, whereas the a-axis lattice parameter and the unit cell volume show normal positive thermal expansion over this T range. No transitions in BaCo2As2 were found in this T range from any of the measurements. Below 40-50 K, we find rho ~ T^2 indicating a Fermi liquid ground state. A large density of states at the Fermi energy D(EF) ~ 18 states/(eV f.u.) for both spin directions is found from low-T Cp(T) measurements, whereas the band structure calculations give D(EF) = 8.23 states/(eV f.u.). The {75}As NMR shift data versus T have the same T dependence as the chi(T) data, demonstrating that the derived chi(T) data are intrinsic. The observed {75}As nuclear spin dynamics are consistent with the presence of ferromagnetic and/or stripe-type antiferromagnetic spin fluctuations. The crystals of Ba{0.78}K{0.22}Co2As2 were grown in Sn flux and show properties very similar to those of undoped BaCo2As2. On the other hand, the crystals from two batches of Ba{0.94}K{0.06}Co2As2 grown in CoAs self-flux show evidence of weak ferromagnetism at T < 10 K with small ordered moments at 1.8 K of 0.007 and 0.03 muB per formula unit, respectively.
Using transport measurements, we investigate multicomponent quantum Hall (QH) ferromagnetism in dual-gated rhombohedral trilayer graphene (r-TLG), in which the real spin, orbital pseudospin and layer pseudospins of the lowest Landau level form spontaneous ordering. We observe intermediate quantum Hall plateaus, indicating a complete lifting of the degeneracy of the zeroth Landau level (LL) in the hole-doped regime. In charge neutral r-TLG, the orbital degeneracy is broken first, and the layer degeneracy is broken last and only the in presence of an interlayer potential U. In the phase space of U and filling factor, we observe an intriguing hexagon pattern, which is accounted for by a model based on crossings between symmetry-broken LLs.
Because of its large density-of-states and the 2{pi} Berry phase near its low-energy band-contact points, neutral bilayer graphene (BLG) at zero magnetic field (B) is susceptible to chiral-symmetry breaking, leading to a variety of gapped spontaneous quantum Hall states distinguished by valley and spin-dependent quantized Hall conductivities. Among these, the layer antiferromagnetic state, which has quantum valley Hall (QVH) effects of opposite sign for opposite spins, appears to be the thermodynamic ground state. Though other gapped states have not been observed experimentally at B=0, they can be explored by exploiting their adiabatic connection to quantum Hall states with the same total Hall conductivity {sigma}H. In this paper, by using a magnetic field to select filling factor { u}=2 states with {sigma}H=2e^2/h, we demonstrate the presence of a quantum anomalous Hall (QAH) state for the majority spin, and a Kekule state with spontaneous valley coherence and a quantum valley Hall state for the minority spin in BLG. By providing the first spectroscopic mapping of spontaneous Hall states at { u}=2, our results shed further light on the rich set of competing ordered states in BLG.
We report studies of CaCo{1.86}As2 single crystals. The electronic structure is probed by angle-resolved photoemission spectroscopy (ARPES) measurements of CaCo{1.86}As2 and by full-potential linearized augmented-plane-wave calculations for the supercell Ca8Co15As16 (CaCo{1.88}As2). Our XRD crystal structure refinement is consistent with the previous combined refinement of x-ray and neutron powder diffraction data showing a collapsed-tetragonal ThCr2Si2-type structure with 7(1)% vacancies on the Co sites corresponding to the composition CaCo{1.86}As2 [D. G. Quirinale et al., Phys. Rev. B 88, 174420 (2013)]. The anisotropic magnetic susceptibility chi(T) data are consistent with the magnetic neutron diffraction data of Quirianale et al. that demonstrate the presence of A-type collinear antiferromagnetic order below the Neel temperature TN = 52(1) K with the easy axis being the tetragonal c axis. However, no clear evidence from the resistivity rho(T) and heat capacity Cp(T) data for a magnetic transition at TN is observed. A metallic ground state is demonstrated from band calculations and the rho(T), Cp(T) and ARPES data, and spin-polarized calculations indicate a competition between the A-type AFM and FM ground states. The Cp(T) data exhibit a large Sommerfield electronic coefficient reflecting a large density of states at the Fermi energy D(EF), consistent with the band structure calculations which also indicate a large D(EF) arising from Co 3d bands. At 1.8 K the M(H) data for H|| c exhibit a well-defined first-order spin-flop transition at an applied field of 3.5 T. The small ordered moment of 0.3 muB/Co obtained from the M(H) data at low T, the large exchange enhancement of chi and the lack of a self-consistent interpretation of the chi(T) and M(H,T) data in terms of a local moment Heisenberg model together indicate that the magnetism of CaCo{1.86}As2 is itinerant.
We experimentally investigate the charge (isospin) frustration induced by a geometrical symmetry in a triangular triple quantum dot. We observe the ground-state charge configurations of six-fold degeneracy, the manifestation of the frustration. The frustration results in omnidirectional charge transport, and it is accompanied by nearby nontrivial triple degenerate states in the charge stability diagram. The findings agree with a capacitive interaction model. We also observe unusual transport by the frustration, which might be related to elastic cotunneling and the interference of trajectories through the dot. This work demonstrates a unique way of studying geometrical frustration in a controllable way.
Multi-valued logic gates, which can handle quaternary numbers as inputs, are developed by exploiting the ballistic transport properties of quantum point contacts in series. The principle of a logic gate that finds the minimum of two quaternary number inputs is demonstrated. The device is scalable to allow multiple inputs, which makes it possible to find the minimum of multiple inputs in a single gate operation. Also, the principle of a half-adder for quaternary number inputs is demonstrated. First, an adder that adds up two quaternary numbers and outputs the sum of inputs is demonstrated. Second, a device to express the sum of the adder into two quaternary digits [Carry (first digit) and Sum (second digit)] is demonstrated. All the logic gates presented in this paper can in principle be extended to allow decimal number inputs with high quality QPCs.
Inelastic neutron scattering measurements of paramagnetic SrCo$_{2}$As$_{2}$ at T=5 K reveal antiferromagnetic (AFM) spin fluctuations that are peaked at a wavevector of $textbf{Q}_{mathrm{AFM}}=(1/2,1/2,1)$ and possess a large energy scale. These stripe spin fluctuations are similar to those found in $A$Fe$_{2}$As$_{2}$ compounds, where spin-density wave AFM is driven by Fermi surface nesting between electron and hole pockets separated by $textbf{Q}_{mathrm{AFM}}$. SrCo$_{2}$As$_{2}$ has a more complex Fermi surface and band structure calculations indicate a potential instability towards either a ferromagnetic or stripe AFM ground state. The results suggest that stripe AFM magnetism is a general feature of both iron and cobalt-based arsenides and the search for spin fluctuation-induced unconventional superconductivity should be expanded to include cobalt-based compounds.
Landau level gaps are important parameters for understanding electronic interactions and symmetry-broken processes in bilayer graphene (BLG). Here we present transport spectroscopy measurements of LL gaps in double-gated suspended BLG with high mobilities in the quantum Hall regime. By using bias as a spectroscopic tool, we measure the gap {Delta} for the quantum Hall (QH) state at filling factor { u}={pm}4 and -2. The single-particle gap for { u}=4 scales linearly with magnetic field B and is independent of the out-of-plane electric field E. For the symmetry-broken { u}=-2 state, the measured values of gap are 1.1 meV/T and 0.17 meV/T for singly-gated geometry and dual-gated geometry at E=0, respectively. The difference between the two values arises from the E-dependence of the gap, suggesting that the { u}=-2 state is layer polarized. Our studies provide the first measurements of the gaps of the broken symmetry QH states in BLG with well-controlled E, and establish a robust method that can be implemented for studying similar states in other layered materials.
