Conduction through localized states in a single crystal of the TlGa<Subscript>0.5</Subscript>Fe<Subscript>0.5</Subscript>Se<Subscript>2</Subscript> alloy

Layered single crystals of the TlGa_0.5Fe_0.5Se₂ alloy in a dc electric field at temperatures ranging from 128 to 178 K are found to possess variable-range-hopping conduction along natural crystal layers through states localized in the vicinity of the Fermi level. The parameters characterizing the electrical conduction in the TlGa_0.5Fe_0.5Se₂ crystals are estimated as follows: the density of states near the Fermi level N_F = 2.8 × 10¹⁷ eV^?1 cm^?3, the spread in energy of these states ΔE = 0.13 eV, the average hopping length R_av = 233 Å, and the concentration of deep-lying traps N _t = 3.6 × 10¹⁶ cm^?3.     

Studies on structural and electrical properties of Mg<Subscript>0. 5+y</Subscript> (Zr<Subscript>2-y</Subscript>Fe<Subscript>y</Subscript>) <Subscript>2</Subscript> (PO<Subscript>4</Subscript>) <Subscript>3</Subscript> ceramic electrolytes

The sample of Mg_{0. 5+y} (Zr_1-y Fe_y) ₂ (PO₄) ₃ (0.0 ≤y ≤0.5) was synthesized using the sol-gel method. The structures of the samples were investigated using X-ray diffraction and Fourier transform infrared spectroscopy measurement. XRD studies showed that samples had a monoclinic structure which was iso-structured with the parent compound, Mg_0.5Zr (PO₄) ₃. The complex impedance spectroscopy was carried out in the frequency range 1–6 MHz and temperature range 303 to 773 K to study the electrical properties of the electrolytes. The substitutions of Fe³⁺ with Zr⁴⁺ in the Mg_0.5Zr (PO₄) ₃ structure was introduced as an extrainterstitial Mg²⁺ ion in the modified structured. The compound of Mg_0.5+y (Zr_1-y Fe_y)₂(PO₄)₃ with y?=?0.4 gives a maximum conductivity value of 1.25?×?10^?5 S cm^?1 at room temperature and 7.18?×?10^?5 S cm^?1 at 773 K. Charge carrier concentration, mobile ion concentration, and ion hopping rate are calculated by fitting the conductance spectra to power law variation, σ _ac (ω)?=?σ _o ? +?Aω ^α . The charge carrier concentration and mobile ion concentration increases with increase of Fe³⁺ inclusion. This implies the increase in conductivity of the compounds was due to extra interstitial Mg²⁺ ions.     

Long-time relaxation phenomena in Pb<Subscript>0.94</Subscript>Ba<Subscript>0.06</Subscript>Sc<Subscript>0.5</Subscript>Nb<Subscript>0.5</Subscript>O<Subscript>3</Subscript> (PBSN-6) single crystals

Long-time polarization relaxation in the temperature range where PBSN-6 single crystals reside in the relaxor state was studied. An analysis of the time dependence of the permittivity ε′(t) performed at measuring frequencies from 1 Hz to 1 kHz in weak electric fields E₀ showed that the relaxation (or freezing) times derived by extrapolating relations of the type ε′(t) ～ log(t/t₀) and ε′(t) ～ exp{?[ln(t/t₀)]_β} range from 10⁸ to 10¹¹ min and depend substantially on the bias voltage applied to the sample. A study of the pattern of the dielectric response in moderate and strong infralow-frequency fields revealed that, after a sample was maintained under a bias lower than the coercive force, it no longer exhibited the additional anomalies in the amplitude dependences of the effective loss tangent tan_eff(E₀) than were observed in a thermally recuperated sample.     

Relaxation phenomena in TlGa<Subscript>0.99</Subscript>Fe<Subscript>0.01</Subscript>Se<Subscript>2</Subscript> single crystals

The relaxation electronic phenomena occurring in TlGa_0.99Fe_0.01Se₂ single crystals in an external dc electric field are investigated. It is established that these phenomena are caused by electric charges accumulated in the single crystals. The charge relaxation at different electric field strengths and temperatures, the hysteresis of the current-voltage characteristic, and the electric charge accumulated in the TlGa_0.99Fe_0.01Se₂ single crystals are consistent with the relay-race mechanism of transfer of a charge generated at deep-lying energy levels in the band gap due to the injection of charge carriers from the electric contact into the crystal. The parameters characterizing the electronic phenomena observed in the TlGa_0.99Fe_0.01Se₂ single crystals are determined to be as follows: the effective mobility of charge carriers transferred by deep-lying centers μ_f=5.6×10^?2 cm²/(V s) at 300 K and the activation energy of charge transfer ΔE=0.54 eV, the contact capacitance of the sample C _c =5×10^?8 F, the localization length of charge carriers in the crystal d _c =1.17×10^?6 cm, the electric charge time constant of the contact τ=15 s, the time a charge carrier takes to travel through the sample t _t =1.8×10^?3 s, and the activation energy of traps responsible for charge relaxation ΔE _σ = ΔE _Q = 0.58 eV.     

Optical properties and mechanisms of current flow in Cu<Subscript>2</Subscript>ZnSnS<Subscript>4</Subscript> films prepared by spray pyrolysis

Thin films Cu₂ZnSnS₄ (up to 0.9 μm thick) with p-type conductivity and band gap E_g = 1.54 eV have been prepared by the spray pyrolysis of 0.1 M aqueous solutions of the salts CuCl₂ · 2H₂O, ZnCl₂ · 2H₂O, SnCl₄ · 5H₂O, and (NH₂)₂CS at a temperature T_S = 290°C. The electrophysical properties of the films have been analyzed using the model for polycrystalline materials with electrically active grain boundaries. The energy and geometric parameters of the grain boundaries have been determined as follows: the height of the barriers is E_b ≈ 0.045–0.048 eV, and the thickness of the depletion region is δ ≈ 3.25 nm. The effective concentrations of charge carriers p₀ = 3.16 × 10¹⁸ cm^–3 and their mobilities in crystallites μ_p = 85 cm²/(V s) have been found using the technique for determining the kinetic parameters from the absorption spectra of thin films at a photon energy hν ≈ E_g. The density of states at grain boundaries N_t = 9.57 × 10¹¹ cm^–2 has been estimated.     

Spectroscopic study of double sodium-yttrium fluoride crystals doped with erbium Na<Subscript>0.4</Subscript>Y<Subscript>0.6</Subscript>F<Subscript>2.2</Subscript>:Er<Superscript>3+</Superscript>: I. Intensities of luminescence spectra and kinetics of luminescence

Na_0.4Y_0.6F_2.2:Er³⁺ (NYF:Er) crystals with an erbium concentration as high as 100 at. % (Na_0.4Er_0.6F_2.2) were grown by the Bridgman-Stockbarger method. The optical spectra were investigated at low (6 K) and room temperatures. It is shown that the absorption spectrum of NYF:Er crystals contains wide bands (790–801 and 965–980 nm) corresponding to the emission range of laser diodes. The peak absorption cross section σ_a for the band peaked at λ=970.4 nm is 0.15×10^?20 cm². On the basis of the analysis of the absorption and luminescence spectra at low (6 and 12 K) temperatures, the structure of the Stark splitting of erbium levels was determined as a structure of quasi-centers for which Stark components are inhomogeneously broadened. The oscillator strengths of the transitions from the ground state of erbium to excited multiplets were calculated from the absorption spectra measured at T=300 K, and the intensity parameters Ω_t were determined by the Judd-Ofelt method: Ω₂=1.65×10^?20 cm², Ω₄=0.56× 10^?20 cm², and Ω₆=1.01×10^?20 cm². These values of the intensity parameters were used to calculate the probabilities of radiative transitions and the branching ratios. The rates of multiphonon nonradiative transitions in NYF: Er were estimated. The luminescence decay kinetics for radiative levels of erbium ions upon their selective excitation by nanosecond laser pulses was studied. The intracenter lifetimes of radiative levels of erbium ions were determined from the luminescence kinetics upon selective ion excitation by low-intensity light in a sample with a low erbium concentration (0.5%). It is demonstrated that, with an increase in temperature from 6 to 300 K, luminescence from the ⁴ G _11/2, ² G(H)_9/2, and ⁴ F _9/2 levels is quenched as a result of multiphonon nonradiative transitions. Luminescence from the ⁴ I _9/2 level is quenched only insignificantly with increasing temperature, and no quenching of luminescence from the ⁴ I _11/2 and ⁴ I _13/2 levels is observed. The spectra of steady-state luminescence of NYF:Er(0.5–15%) crystals were investigated upon broadband excitation by UV and UV-visible lamp light and selective time-resolved laser excitation. It is shown that low-lying levels of erbium ions separated by an energy gap smaller than 2500 cm^?1 are populated via cascade mechanisms. On the basis of the results obtained, it is concluded that NYF:Er ³⁺ crystals are promising candidates for active media of tunable diode-pumped lasers.     

Optical and x-ray diffraction studies of the symmetry of distorted phases of the (NH<Subscript>4</Subscript>)<Subscript>3</Subscript>ZrF<Subscript>7</Subscript> crystal

(NH₄)₃ZrF₇ single crystals were grown, and polarization-optical and x-ray diffraction studies were performed on powders and crystalline plates of various cuts over a wide temperature range. Phase transitions are revealed at temperatures T _1↑ = 280 K, T _2↑ = 279.6 K, T _3↑ = 260–265 K, and T _4↑ = 238 K on heating and at T _1↓ = 280 K, T _2↓ = 269–270 K, T _3↓ = 246 K, and T _4↓ = 235 K on cooling. The sequence of changes in symmetry is established to be as follows: O _h ⁵ (Z = 4) ? D _2h ²⁵ (Z = 2) ? C _2h ³ (Z = 2) ? C _i ¹ (Z = 108) ? monoclinic²(Z = 216).     

A tilted rotatory frame method for the measurement of nuclear spin diffusion coefficients in solids doped with paramagnetic centers: Mn-Doped CaF<Subscript>2</Subscript>

Nuclear magnetic resonance (NMR) experiments recording the recovery of the magnetization of the nuclei in one phase, following the excitation of the nuclei in the other phase, is a classical way of studying blends inhomogeneous at the nanometer scale. Interpretation of the time recovery in terms of the spatial dimension requires knowledge of the two-phase spin diffusion coefficientsD ₀. A new method of measurement of_{D 0} is proposed on the basis of variable angle-tilted rotatory frame relaxation in homogeneous samples doped with paramagnetic centers. The choice of the tilt angle allows one to finely balance the direct relaxation by the paramagnetic center and the spin diffusion. The shape of the relaxation is analyzed with the solution for the diffusion-limited regimeM(t)/M(0)=exp[?(r ₂ t)^1/2?r ₁ t] andD ₀ then calculated fromr ₁,r ₂ and the concentration of paramagnetic centers. Conditions where reliable results can be obtained both theoretically and numerically are explored. The method has been implemented and applied to polycrystalline Mn-doped CaF₂ leading toD ₀=540±60 nm²/s, in agreement with existing values on this model compound.     

Growth and optical properties of Pr<Superscript>3+</Superscript>:KLu(WO<Subscript>4</Subscript>)<Subscript>2</Subscript> laser crystal: a candidate for red emission laser

A Pr³⁺:KLu(WO₄)₂ crystal with dimension of 30 × 30 × 15 mm³ was grown in the K₂W₂O₇ flux. A slice was cut from the crystal, and the polarized absorption and fluorescence spectra were measured at room temperature. Based on the J-O theory, the oscillator intensity parameters Ω _t (t = 2, 4, 6), spontaneous emission probabilities and branch ratios were estimated and good results had been obtained. Furthermore, the crystal has a relatively large emission cross-section in the region of 615–630 nm with the highest value of 14.5 × 10^?20 cm², which indicates that the crystal is good for the application in red emission laser. The emission decay time for ¹D₂ and ³P₀ multiplets was discussed. By adapting the I-H model to fit the emission decay curves, the lifetime for ¹D₂ at 607 nm and ³P₀ at 615 nm are 19.72 μs and 8.95 μs, respectively. Then the corresponding fluorescence quantum efficiencies of the two multiplets reach 83.7 % and 87.9 %, respectively. All the studies illustrate that this crystal is potential in red emission laser application.     

Photoionization of the Xe atom and Xe@C<Subscript>60</Subscript> molecule

Photoionization of the Xe atom and Xe@C₆₀ molecule have been studied usingthe random phase approximation with exchange (RPAE) method. The Xe atom was described byrelaxed orbitals including overlap integrals. The C₆₀ fullerene has beenrepresented by an attractive short range spherical well with potentialV(r), given byV(r) =  ?V ₀ forr _i  < r < r _o ,otherwise V(r) = 0 wherer _i andr _o are respectively, the inner and outerradii of the spherical shell. The time independent Schrödinger equation was solved usingboth regular and irregular solutions and the continuous boundary conditions atr _i andr _o . The results demonstrate improvementto previous calculations for both the Xe atom and Xe@C₆₀ molecule and comparevery well with the recent experimental data.     

Emergent nesting of the Fermi surface from local-moment description of iron-pnictide high-T<Subscript>c</Subscript> superconductors

We uncover the low-energy spectrum of a t-J model for electrons on a square lattice of spin-1 iron atoms with 3d _xz and 3d _yz orbital character by applying Schwinger-boson-slave-fermion mean-field theory and by exact diagonalization of one hole roaming over a 4 × 4 × 2 lattice. Hopping matrix elements are set to produce hole bands centered at zero two-dimensional (2D) momentum in the free-electron limit. Holes can propagate coherently in the t-J model below a threshold Hund coupling when long-range antiferromagnetic order across the d + = 3d _{(x + iy)z} and d ? = 3d _{(x ? iy)z} orbitals is established by magnetic frustration that is off-diagonal in the orbital indices. This leads to two hole-pocket Fermi surfaces centered at zero 2D momentum. Proximity to a commensurate spin-density wave (cSDW) that exists above the threshold Hund coupling results in emergent Fermi surface pockets about cSDW momenta at a quantum critical point (QCP). This motivates the introduction of a new Gutzwiller wavefunction for a cSDW metal state. Study of the spin-fluctuation spectrum at cSDW momenta indicates that the dispersion of the nested band of one-particle states that emerges is electron-type. Increasing Hund coupling past the QCP can push the hole-pocket Fermi surfaces centered at zero 2D momentum below the Fermi energy level, in agreement with recent determinations of the electronic structure of mono-layer iron-selenide superconductors.     

Linear and nonlinear optical spectroscopy of gadolinium iron borate GdFe<Subscript>3</Subscript>(BO<Subscript>3</Subscript>)<Subscript>4</Subscript>

The optical spectra and the second-harmonic generation (SHG) are studied in a noncentrosymmetric GdFe₃(BO₃)₄ magnet. In the region of weak absorption (α～20–400 cm^?1) below ～3 eV, three absorption bands are distinguished, which can be unambiguously assigned to forbidden electronic transitions from the ground ⁶A₁ state of the Fe³⁺ ion to its excited states ⁴T₁(～1.4 eV), ⁴T₂(～2 eV), and ⁴A₁, ⁴E(～2.8 eV). Intense absorption begins in the region above 3 eV (α～2–4×10⁵ cm^?1), where two bands at ～4.0 and 4.8 eV are observed, which are caused by allowed electric dipole charge-transfer transitions. The spectral features of SHG in the 1.2–3.0-eV region are explained by a change in the SHG efficiency caused by a change in the phase mismatch. It is shown that in the weak absorption region, phase matching can be achieved for SHG.     

Nonergodic state of relaxation ferroelectric Cd<Subscript>2</Subscript>Nb<Subscript>2</Subscript>O<Subscript>7</Subscript> in a constant electric field

The temperature dependence of the residual polarization of the nonergodic relaxation state (NERS) obtained from the measurements of pyroelectric current during zero-field heating (ZFH) in the temperature interval from 10 to 295 K is investigated for the Cd₂Nb₂O₇ relaxation system in two cases: (1) after sample cooling in a constant electric field E (FC) from T = 295 K to a preset temperature, which is much lower than the “freezing” temperature of the relaxation state (T _f ≈ 182 K), field removal, and subsequent cooling in zero field (ZFC) to T = 10 K and (2) after ZFC from T = 295 K to the same temperature below T _f , application of the same field, and FC to T = 10 K. The behavior of the P _r ^FC (T) and P _r ^ZFC (T) dependences is analyzed. In the field E < 2 kV/cm, the P _r ^ZFC curves as functions of 1/T have a broad low-intensity peak in the region T ≈ T _f , which vanishes in stronger fields, when the P _r ^FC (1/T) curves intersect at T ≈ T _f and have no anomalies. The difference in the behavior of P _r ^ZFC (T) and P _r ^FC (T) indicates the difference in the nature of NERS formed during ZFC and FC of the system upon a transition through T _f . In the ZFC mode, NERS exhibits glasslike behavior; in the FC regime, features of the ferroelectric behavior even in the weak field. Analogous variations of P _r ^ZFC (T) and P _r ^FC (T) in a classical ferroelectric KDP are also given for comparison.     

Comparative study of solid-state reaction and sol-gel process for synthesis of Zr-doped Li<Subscript>0.5</Subscript>La<Subscript>0.5</Subscript>TiO<Subscript>3</Subscript> solid electrolytes

We investigated the synergistic influences of synthesis methods (solid-state reaction vs. sol-gel process) and Zr⁴⁺ doping on the structure and ionic conductivity of perovskite-structured Li_0.5La_0.5TiO₃ (LLTO) solid electrolytes. The lithium-ion conductivity of Li_0.5La_0.5Ti_1???x Zr _x O₃ ceramic specimens was evaluated as a function of x value and compared carefully between those two synthesis methods. Regarding the conductivity, sol-gel process is better for the synthesis of LLTO than solid-state reaction. As a result, the highest grain conductivity is obtained in the sol-gel-derived pure LLTO sample with x?=?0, reaching 1.10?×?10^?3 S?·?cm^?1. Partial substitution of Zr⁴⁺ enlarges the LLTO’s grain aggregate size and increases the total superficial area of aggregates. Consequently, Zr⁴⁺ substitution not only affects the grain (bulk) conductivity, but more importantly, also improves the grain boundary conductivity and the total conductivity. The highest total conductivity is 5.84?×?10^?5 S?·?cm^?1 with x?=?0.04 by sol-gel process.     

Electronic excitation energy transfer and nonstationary processes in KH<Subscript>2</Subscript>PO<Subscript>4</Subscript>:Tl crystals

We report the results of our experimental study and numerical simulation of the electronic excitation energy transfer to impurity centers under conditions where nonstationary processes take place in the hydrogen sublattice of potassium dihydrogen phosphate (KH₂PO₄) single crystals doped with mercury-like Tl⁺ ions (KDP:Tl). We present the experimental results of our investigation of the decay kinetics of the transient optical absorption (100 ns–50 s) of intrinsic defects in the hydrogen sublattice of KDP:Tl obtained by pulsed absorption spectroscopy and the results of our study of the dynamics of the change in steady-state luminescence intensity with irradiation time (1–5000 s). To explain the transfer of the energy being released during electron recombination involving intrinsic KDP:Tl lattice defects, we formulate a mathematical model for the transfer of this energy to impurity Tl⁺ luminescence centers. Within the model being developed, we present the systems of differential balance equations describing the nonstationary processes in the electron subsystem and the hydrogen sublattice; provide a technique for calculating the pair correlation functions Y(r, t) of dissimilar defects based on the solution of the Smoluchowski equation for the system of mobile hydrogen sublattice defects; calculate the time-dependent reaction rate constants K(t) for various experimental conditions; and outline the peculiarities and results of the model parametrization based on our experimental data. Based on our investigation, the dramatic and significant effect of a gradual inertial increase by a factor of 50–100 in steady-state luminescence intensity in the 4.5-eV band in KDP:Tl crystals due to the luminescence of mercury-like Tl⁺ ions has been explained qualitatively and quantitatively.     

<Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">e</Emphasis>3</Subscript> Decay Studies in the OKA Experiment

Recent results from OKA setup concerning form factor studies in K_e3 decay are presented. About 5.25 M events obtained for decays of 17.7 GeV/cK⁺ are selected for the analysis. The linear and quadratic slopes for the decay form factor f₊(t) are measured: λ'₊ = 2.95 ± 0.022 ± 0.018 × 10 ^-2 for the linear slope fit and λ₊ = 2.611 ± 0.035 ± 0.028 × 10 ^-2, λ"₊ = 1.91 ± 0.19 ± 0.14 × 10 ^-3 for the quadratic one. The scalar and tensor contributions are compatible with zero. Several alternative parametrizations are tried: the Pole fit parameter is found to be M _V = 891 ± 3 MeV; the parameter of the dispersive parametrization is measured to be Λ₊ = 2.458 ± 0.018 × 10^-2.     

Mechanisms of low-temperature high-frequency conductivity in systems with a dense array of Ge<Subscript>0.7</Subscript>Si<Subscript>0.3</Subscript> quantum dots in silicon

High-frequency (HF) conductivity in systems with a dense (with a density of n = 3 × 10¹¹ cm^?2) array of self-organized Ge_0.7Si_0.3 quantum dots in silicon with different boron concentrations n_B is determined by acoustic methods. The measurements of the absorption coefficient and the velocity of surface acoustic waves (SAWs) with frequencies of 30–300 MHz that interact with holes localized in quantum dots are carried out in magnetic fields of up to 18 T in the temperature interval from 1 to 20 K. Using one of the samples (n_B = 8.2 × 10¹¹ cm^?2), it is shown that, at temperatures T ≤ 4 K, the HF conductivity is realized by the hopping of holes between the states localized in different quantum dots and can be explained within a two-site model in the case of

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, where ω is the SAW frequency and τ₀ is the relaxation time of the populations of the sites (quantum dots). For T > 7 K, the HF conductivity has an activation character associated with the diffusion over the states at the mobility threshold. In the interval 4 K < T < 7 K, the HF conductivity is determined by a combination of the hopping and activation mechanisms. The contributions of these mechanisms are distinguished; it is found that the temperature dependence of the hopping HF conductivity approaches saturation at T* ≈ 4.5 K, which points to a τ₀ ≤ 1. A value of τ₀(T*) ≈ 5 × 10^?9 s is determined from the condition ωτ₀(T*) ≈ 1.

     

Study of anharmonic effects in the IR spectrum of a solution of CF<Subscript>3</Subscript>Br in liquid argon

The vibrational spectrum of CF₃Br with the natural abundance ratio of isotopologues is studied in solutions in liquid Ar at T = 90 K in the frequency range 4000–400 cm^?1 with a resolution of 0.1 cm^?1 for the concentration range 3.1 × 10^?7?6.3 × 10^?3 mol %. The parameters of the vibrational spectrum of the molecule are determined: the frequencies are measured accurate to ± 0.1 cm^?1, and the transition probabilities are found up to the fourth order inclusive. One hundred fifty absorption bands of CF₃Br are interpreted, including the bands belonging to all the isotope modifications of this compound; the halfwidths of these bands are determined. For all the fundamental frequencies, the isotope shifts are obtained. Vibrational ?-resonance is studied. Using the experimental data obtained and taking into account this resonance, a complete set of parameters describing the experimental frequencies with the error δ ≈ 0.3 cm^?1 was found. This set consists of 6 harmonic frequencies and 30 anharmonicity constants, including the constants r _ik related to vibrational ?-resonances.     

Thermophysical study of structural phase transitions in Na<Subscript>0.95</Subscript>Li<Subscript>0.05</Subscript>NbO<Subscript>3</Subscript> solid solution

Temperature dependences of specific heat C_p(T) and coefficient of thermal expansion ;(T) for Na_0.95Li_0.05NbO₃ sodium-lithium niobate ceramic samples are investigated in the temperature range of 100–800 K. The C_p(T) and α(T) anomalies at T₃ = 310 ± 3 K, T₂ = 630 ± 8 K, and T₁ = 710 ± 10 K are observed, which correspond to the sequence of phase transitions N ? Q ? S(R) ? T2(S). The effect of heat treatment of the samples on the sequence of structural distortions was established. It is demonstrated that annealing of the samples at 603 K leads to splitting of the anomaly corresponding to the phase transition Q → R/S in two anomalies. After sample heating to 800 K, the only anomaly is observed in both the C_p(T) and ;(T) dependence. Possible mechanisms of the observed phenomena are discussed.     

Long-range spin correlations in a honeycomb spin model with a magnetic field

We consider the spin-1/2 model on the honeycomb lattice [A. Kitaev, Ann. Phys. 321, 2 (2006)] in the presence of a weak magnetic field h _α ? J. Such a perturbation treated in the lowest nonvanishing order over h _α leads [K.S. Tikhonov, M.V. Feigel’man, and A.Yu. Kitaev, Phys. Rev. Lett. 106, 067203 (2011)] to a powerlaw decay of irreducible spin correlations 《s ^z (t, r)s ^z (0, 0)》 ∝ h _z ² f(t, r), where f(t, r) ∝ [max(t, Jr)]^–4. We have studied the effects of the next order of perturbation in h _z and found an additional term of the order h _z ⁴ in the correlation function 《s ^z (t, r)s ^z (0, 0)》 which scales as h _z ⁴ cosγ/r ³ at Jt? r, where γ is the polar angle in the 2D plane. We demonstrate that such a contribution can be understood as a result of a perturbation of the effective Majorana Hamiltonian by the weak imaginary vector potential A _x ∝ i h _z ² .