3.2. Моделирование сплавов на основе железа при экстремально высоком давлении (до 350 ГПа).
Magnetization density calculated for orthorhombic antiferromagnetic phase of CrN simulated by 2 х 1 х 1 unit cells. The density is shown by iso-surfaces at 0.06, 0.57, 1.14, and 1.71 electrons/Å3. Red (blue) colors correspond to a surplus of majority (minority) spin electrons. It is seen that the magnetization density is well localized at Cr atoms (large bulbs) though some induced spin polarization at N atoms is also seen. Details of calculations are the same as in Ref. [27]. |
The model of the paramagnetic state considered in this work consists of non- collinear and fluctuating local moments, which become disordered above the magnetic order–disorder transition temperature, the Cure TC or Neel TN tempera- ture for ferromagnets and antiferromagnets, respectively. |
Schematic representation of the Disordered Local Moment model. The paramagnetic state consisting of fully disordered non-collinear and fluctuating magnetic moments (Fig. 2) is approximated by fully disordered, collinear and static picture. |
(a) Supercell realization of the disordered local moment model for calculations of the solution energy of an interstial impurity, like C (small yellow atom) in paramagnetic fcc Fe (large red atoms). (b) Using a combination of magnetic special quasirandom structure and magnetic sampling methods, one calculates energies of supercells with different impurity positions in the magnetic SQS representing the host, which are shown here versus the total magnetic moment of Fe atoms located in the first coordination shell of the impurity. Significant variation of the calculated energies depending on the impurity positions in the supercell is clearly seen in the figure. |
Relaxation energies (in eV per f.u.) in the 64-atom CrN supercell as a function of relaxation iteration step. The spurious and relatively large static relaxation energies of the fixed magnetic SQS configuration (black circles) is contrasted with the relaxation energy of the magnetic sampling method (green diamonds) which is nearly vanishing within the accuracy of the method. Here a series of magnetic samples is considered and the forces acting on each nucleus in all the samples are averaged. This average force is then used to guide the nuclei to their optimal static positions. Also the energy of the SQS configuration calculated on the structure obtained with the magnetic sampling method is shown for comparison (red squares). Here we perform one small relaxation step in all the calculations of the magnetic samples. Then we put together the new positions and average them, resulting in the actual relevant new positions. The agreement between the two later methods with magnetic SQS calculations without local lattice relaxations under- lines artificial nature of the conventional static approach. Details of calculations are the same as in Ref. [72] |
Schematic illustration of DLM-MD algorithm. See text for more discussion. |
Histogram of resulting values of local magnetic moments during several pico-seconds run of DLM-MD simulations of fcc-Fe at 1662 K. |
The Hellmann–Feynman forces in ^x, y^, and ^z direction acting on a Cr atom (open symbols) and a N-atom (solid symbols) for each MSM configuration of the supercell. The lines indicate the accumulated average of the forces over MSM configurations in each direction for the Cr-atom (solid lines) and N-atom (dashed line). Details of calculations are the same as in Ref. [72]. |
Mixing enthalpies H (in eV/atom) of paramagnetic bcc Fe–Cr alloys simulated by means of DLM-CPA approach (green dashed line). Excellent agreement with experimental data from [112] (red squares) is obtained in DLM calculations. Ferromagnetic results (FM, black solid line) are shown for comparison. Note that FM bcc Fe was used as the standard state in the calculations of the mixing enthalpies of the ferromagnetic alloys, in contrast to DLM bcc Fe, used as the standard state for calculations of the mixing enthalpies in the paramagnetic alloys. |
Values of solution enthalpy (<Hsol >, in eV) for C, N, V, and Nb impurities, as well as C–C impurity interactions (< E^{int}CC (i) >, in eV) as a function of coordination shell i in paramagnetic fcc iron. |
Valence-band electronic density of states (solid line) of the paramagnetic cubic B1 phase of CrN calculated in Ref. [60] using LDA + U approximation (with U = 3 eV) and a description of magnetic disordered by means of the supercell approach. The experimental ultraviolet photoemission spectroscopy measurement from [138] is shown by a dashed line. |
Example of localized plastic flow autowave generated at the linear work hardening stage in the single crystal of fcc Fe–18% Cr–12% Ni–2% Mo alloy oriented along [001] direction; exx-local elongation; x and y-specimen length and width, respectively; k – nucleus spacing (autowave length); Vaw – autowave propagation rate. The kinetics of macro-localization pattern evolution was investigated using time evolution of local nuclei’s positions [139]. |
Equations of state of orthorhombic antiferromagnetic phase of CrN, calculated with conventional ab inito MD (green squraes), and cubic (B1) param- agnetic phases of CrN, calculated with DLM-MD (blue circles). Experimental data for the orthorhombic (orange triangles up) and cubic (red triangles down) phases are from Ref. [16]. The strong effect on the equation of state due to the neglect of magnetism is demonstrated by non-magnetic calculations shown with long dashed – dotted line. |
Calculated (open circles) and experimentally determined (filled rectangles) Young’s modulus values for fcc Fe–Mn–Cr and Fe–Mn–Co with a Fe = Mn ratio of 2.3. From [128]. Ó IOP Publishing. Reproduced with permission. All rights reserved. |
(a) Mean magnitude of local magnetic moments M ̄ per atom as a function of temperature in paramagnetic (DLM) bcc and fcc Fe. All data points are calculated at the respective experimental volumes. Electronic contributions are considered for the static and the MD calculations by the corresponding Fermi smearing. The shown MD calculations include effects of vibrationally distorted local geometries on the magnetic moments. For comparison, static FM calculations for bcc are shown. (b) Electronic and magnetic free energy contributions to the bcc-fcc competition in Fe as a function of temperature, with and without explicit vibrational effects. |
The calculated total (spin-up + spin-down) electronic DOS of fcc (red shaded) and bcc (black) Fe. (a) Ferromagnetic state in a static lattice, (b) ferromagnetic MD at 1085 K, (c) disordered local moments in static lattice, (d) disordered local moments MD at 1662 K. Calculations are performed at an electronic temperature of T = 0 K in order to be able to clearly distinguish magnetic from vibrational disorder which could otherwise be hidden by electronic smearing. |
Flowchart for Helmholtz free energy calculation algorithm with the TDEP method. The step labeled “Full scale low-accuracy MD simulations” can be done in parallel using different initial supercells. |
Vibrational contribution to Helmholtz Free energy of hcp Fe at 2000 K obtained with eq 5. Red circles show values that were obtained using the upsampling procedure. Initial results obtained with low-precision AIMD are shown as black crosses (in this particular case, the points are on nearly on top of each other). The points are connected with lines as a guide for the eyes. |
Forces Comparison Test between PAW Potentials in VASP and FPLAPW All-Electron Calculation Carried out Using Wien2ka |
ΔU, eq 7, as a function of volume (red circles) The black line represents the linear fit: the deviation is in range from 1 to 5 meV/ atom and is higher at lower volumes. The standard deviation is 4 meV. |
TDEP ground-state energy U0 of hcp Fe at 2000 K obtained with the method described in this paper compared to initial low- precision results. Red circles show values that were obtained using the upsampling scheme. Initial results from low-precision simulation AIMD are shown as gray crosses. The points are connected with lines as a guide for the eyes. |
Unit cell energy Uuc as a function of volume, calculated at electronic temperature 2000 K using high-precision settings and a dense set of volumes. This is the main nonlinear term in eq 7. |
Helmholtz Free energy of hcp Fe at 2000 K obtained with the method described in this paper. Red circles show values that were obtained using high-precision molecular dynamics with enhanced scheme, outlined in this work. Initial results from low-precision simulation AIMD are shown as gray crosses. The points are connected with lines as a guide for the eyes. |
Pressure in hcp Fe as a function of V/V0 at T = 2000 K. Blue line: data from quasiharmonic calculation by Sha and Cohen37 Red data points: this work. |
Data used for pressure estimations. Values with the superscripts are taken from the corresponding experimental studies. The rest are calculated in this study by fitting the equation of state with the Vinet equation. ∗ Bulk modulus and its derivative were calculated. Volume per one atom is in A ̊ 3 ; bulk modulus is in GPa. |
Thomas-Fermi potentials for orbital numbers l = 1 (p electrons) and l = 3 (f electrons). Potentials are plotted for the series of Z from 70 to 80. The color of the line corresponds to Z, as shown in the right palette. |
Left: Band structure of iridium at ambient pressure. Right: Band structure of iridium under a pressure of 80 GPa. |
The relations between С ̃ αβ.. and Cαβ.. . |
Various cases of the cubic crystal deformations. |
The equation of state of bcc tungsten. Dashed line is the result of the regression of calculated P(V) dependency (black squares) with the Birch-Murnaghan equation of state. The solid line is the result of the regression, preformed for combined experimental data [31–34]. |
Pressure dependencies of С44 and С obtained by various methods. Black circles: stress–finite strain relation; red circles: energy–finite strain relation; blue x symbols: stress–infinitesimal strain; green + symbols: energy–infinitesimal strain. The dashed lines are the guide for an eye. |
The TOEC of bcc tungsten and molybdenum given in GPa P = 0 (T = 0 K). |
Pressure dependence of TOEC of bcc tungsten (a) С ̃ 111 ; (b) black squares: С ̃ 112 , blue triangles: С ̃ 155 ; (c) black triangles: С ̃ 123 , blue diamonds: С ̃ 144 , green pentagons: С ̃ 456 . |
Pressure dependence of FOEC for bcc tungsten (T = 0 K ). C ̃ 1111 : triangles, C ̃ 1112 : pentagons, C ̃ 1123 : circles, C ̃ 4444 : diamonds. |
The FOEC of bcc tungsten (T = 0 K); all values С ̃ αβγ δ and P are in GPa. |
Pressure dependence of Gru ̈neisen parameters γj of bcc molybdenum and tungsten. Pentagons: L[110][110] mode of tungsten; diamonds: T[001] [110] mode of tungsten; triangles: T[11 ̄ 0] [110] mode of tungsten; and circles: T[11 ̄ 0] [110] mode of molybdenum. |
The polycrystalline second- and third-order elastic constants W and Mo at P = 0 (GPa). |
Local moment √⟨m2z ⟩ calculated by DFT+DMFT as a function of lattice volume (top) and pressure (bottom). The IMT associated with the lattice volume collapse is shown by vertical solid (top) and dashed lines (bottom). The inset shows the total energy calculated by DFT+DMFT and the Maxwell construction for paramagnetic MnO. Note that for NiO the pressure scale is shrunk by a factor of 5. |
Evolution of the spectral function (a) and orbital occupations (b) calculated by DFT+DMFT for paramagnetic MnO as a function of volume. Spectral function (c) and local spin-spin correlation function χ (τ ) = ⟨mˆ z (τ )mˆ z (0)⟩ (d) of paramagnetic NiO. τ is the imaginary time. At equilibrium volume V0, the 3d electrons are localized to form fluctuating moments [χ (τ ) is seen to be almost constant and close to its maximal value S = 1 for the Ni eg states]. At high compression, the 3d electrons show an itinerant magnetic behavior, implying a localized to itinerant moment crossover under pressure. |
Quasiparticle weight and spectral weight at the Fermi level calculated for the t2g and eg orbitals as a function of lattice volume. Our result for the lattice volume collapse calculated by DFT+DMFT is marked by a red shaded rectangle. The quasiparticle weight Z = [1 − ∂ Im (iω)/∂iω]−1 is evaluated from the slope of the polynomial fit of the imaginary part of the self-energy (iωn) at ω = 0. |
Calculated structural parameters for the paramagnetic B1 phase of transition metal oxides. V0 is ambient pressure volume; Vtr are the lattice volume collapse values; K, bulk modulus for the low/high-pressure phase; K ′ ≡ d K /d P is 4.1 for MnO, FeO, and CoO; K′ = 4.3 for NiO. |