The spectrum and wave function of helical edge modes in Z2 topological insulator are derived on a square lattice using Bernevig-Hughes-Zhang (BHZ) model. The BHZ model is characterized by a "mass" term M (k) = ∆ − Bk 2 . A topological insulator realizes when the parameters ∆ and B fall on the regime, either 0 < ∆/B < 4 or 4 < ∆/B < 8. At ∆/B = 4, which separates the cases of positive and negative (quantized) spin Hall conductivities, the edge modes show a corresponding change that depends on the edge geometry. In the (1, 0)-edge, the spectrum of edge mode remains the same against change of ∆/B, although the main location of the mode moves from the zone center for ∆/B < 4, to the zone boundary for ∆/B > 4 of the 1D Brillouin zone. In the (1, 1)-edge geometry, the group velocity at the zone center changes sign at ∆/B = 4 where the spectrum becomes independent of the momentum, i.e. flat, over the whole 1D Brillouin zone. Furthermore, for ∆/B < 1.354, the edge mode starting from the zone center vanishes in an intermediate region of the 1D Brillouin zone, but reenters near the zone boundary, where the energy of the edge mode is marginally below the lowest bulk excitations. On the other hand, the behavior of reentrant mode in real space is indistinguishable from an ordinary edge mode.
Experimental verification of microwave assistance effect on a perpendicular magnetic CoCrPt based granular film is demonstrated. Significant reduction of coercivity under the assistance of a radio frequency (rf) field is clearly observed. But the coercivity strongly depends on the duration of rf field, indicating that the magnetic moments of constituent grains are severely perturbed by thermal agitation. Analysis based on the Néel-Arrhenius law has revealed that the dynamic coercivity in nanosecond region can be reduced by 50% only by applying a small rf field with the amplitude less than 2% of the anisotropy field.
L10–FePt(001) single-crystal films were grown epitaxially on SrTiO3(001), MgAl2O4(001), and MgO(001) substrates. Their uniaxial magnetic anisotropy Ku and the order structure were examined for the film thickness t range of 2–14 nm. All series of films show large Ku of 4 × 107 erg/cm3 in the thickness range higher than 10 nm, with order parameter S of 0.8 and saturation magnetization Ms of 1120 emu/cm3. Ku decreased gradually as t decreased. The Ku reduction was considerable when t decreased from 4 nm to 2 nm. No marked difference in the thickness dependence of Ku was found in any series of films, although the lattice mismatch between FePt and the substrates was markedly different. Ku reduction showed good agreement with the reduction of S for the films on MgAl2O4 and MgO. The Ku ∼ S2 plot showed an almost linear relation, which is in good agreement with theoretical predictions. Transmission electron microscopy images for a FePt film on MgO substrate revealed that the lattice mismatch between FePt(001) and MgO(001) was relaxed in the initial 1 or 2 layers of FePt(001) lattices, which is likely to be true also for two other series of films.
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