^{1,a)}, James E. Saal

^{1}, Zhi-Gang Mei

^{1}, Yi Wang

^{1}and Zi-Kui Liu

^{1}

### Abstract

Exploration of longstanding issues in magnetic materials, for example the nature of Curie/Néel temperature and the Schottky anomaly of heat capacity, appeals to reliable models at finite temperatures. Based on first-principles calculations and partition function approach with the microstates being the collinear magnetic configurations, the magnetic thermodynamics of fcc Ni, including the heat capacity and the pressure-dependent Curie temperature, is predicted well and compared with the results from experiments and mean-field approach. As demonstrated in fcc Ni, it is found that the magnetic thermodynamics containing anomalies stems from the magnetic configurational entropy caused by the competition of various magnetic states.

This work is funded by the Office of Naval Research (ONR) under the Contract No. N0014-07-1-0638 and the National Science Foundation (NSF) through Grant No. DMR-1006557. First-principles calculations were carried out partially on the LION clusters supported by the Materials Simulation Center and the Research Computing and Cyber infrastructure unit at the Pennsylvania State University, and partially on the resources of NERSC supported by the Office of Science of the U.S. DOE under Contract No. DE-AC02-05CH11231.

I. INTRODUCTION

II. THEORY AND METHODOLOGY

A. Magnetic thermodynamics from partition function approach

B. Canonical magnetic ensemble

C. First-principles methodology and the estimation of quasiharmonic approach

III. RESULTS AND DISCUSSIONS

IV. SUMMARY

### Key Topics

- Nickel
- 25.0
- Curie point
- 19.0
- Entropy
- 17.0
- Heat capacity
- 17.0
- Ab initio calculations
- 8.0

## Figures

supercell of fcc Ni (left), where the fcc cell is represented by the distorted bcc with . The arrows illustrate the collinear spin alignments of the eight-layer Ni atoms with spin up and spin down. str1 is the FMC, str2 is the nonmagnetic configuration (not shown), others are the spin-flipping configurations (SFCs: str3–str7).

supercell of fcc Ni (left), where the fcc cell is represented by the distorted bcc with . The arrows illustrate the collinear spin alignments of the eight-layer Ni atoms with spin up and spin down. str1 is the FMC, str2 is the nonmagnetic configuration (not shown), others are the spin-flipping configurations (SFCs: str3–str7).

First-principles calculated total energies and the projected magnetic moments as a function of volume for str6 (see Fig. 1), the EOS [see Eq. (7)] fitted energy vs volume curve is also shown.

First-principles calculated total energies and the projected magnetic moments as a function of volume for str6 (see Fig. 1), the EOS [see Eq. (7)] fitted energy vs volume curve is also shown.

Predicted energies for all possible spin-up and spin-down configurations within the supercell of fcc Ni (see Fig. 1) by CEM (Ref. 14) (circles) with inputs from first-principles calculated relative energies (plus symbols, see Table I). Note that (i) the effective cluster interactions up to the second nearest pair are employed in CEM; (ii) the CEM predicted energies of high concentrations are not shown due to the symmetric energies, and (iii) the “1–7” structure [1 spin up (down) and 7 spin down (up) Ni atoms] is unstable according to first-principles calculations.

Predicted energies for all possible spin-up and spin-down configurations within the supercell of fcc Ni (see Fig. 1) by CEM (Ref. 14) (circles) with inputs from first-principles calculated relative energies (plus symbols, see Table I). Note that (i) the effective cluster interactions up to the second nearest pair are employed in CEM; (ii) the CEM predicted energies of high concentrations are not shown due to the symmetric energies, and (iii) the “1–7” structure [1 spin up (down) and 7 spin down (up) Ni atoms] is unstable according to first-principles calculations.

Experimental (Ref. 25) (open circles) and predicted (curves) heat capacities at constant pressure (pressure ) by PFA and MFA for fcc Ni. The magnetic configurational entropy contribution is obtained by partition function due to the competition among magnetic states. A protrusion is clearly predicted with the maximum of 594 K (from ), pertaining to the Curie temperature and closing to the measured 625 K (Ref. 25).

Experimental (Ref. 25) (open circles) and predicted (curves) heat capacities at constant pressure (pressure ) by PFA and MFA for fcc Ni. The magnetic configurational entropy contribution is obtained by partition function due to the competition among magnetic states. A protrusion is clearly predicted with the maximum of 594 K (from ), pertaining to the Curie temperature and closing to the measured 625 K (Ref. 25).

Predicted thermal populations of FMC and SFCs for fcc Ni as a function of temperature and under pressure . The cross point of 706 K corresponds to another definition of Curie temperature and is comparable with the measured 625 K (Ref. 25). The structures (str1–str7) are illustrated in Fig. 1.

Predicted thermal populations of FMC and SFCs for fcc Ni as a function of temperature and under pressure . The cross point of 706 K corresponds to another definition of Curie temperature and is comparable with the measured 625 K (Ref. 25). The structures (str1–str7) are illustrated in Fig. 1.

Experimental (Ref. 30) (symbols) and calculated Curie temperatures ’s of fcc Ni as a function of pressure. The predictions are based on the maximum of (see Fig. 4) and the equal of thermal populations between FMC and the sum of SFCs (, see Fig. 5). The predicted forms a band to separate the FM and PM regions.

Experimental (Ref. 30) (symbols) and calculated Curie temperatures ’s of fcc Ni as a function of pressure. The predictions are based on the maximum of (see Fig. 4) and the equal of thermal populations between FMC and the sum of SFCs (, see Fig. 5). The predicted forms a band to separate the FM and PM regions.

## Tables

First-principles predicted equilibrium properties of the supercell for fcc Ni (see Fig. 1) using the four-parameter Birch–Murnaghan EOS [see Eq. (7)], including the equilibrium volume ( per eight atoms), bulk modulus (GPa), and its pressure derivative , the relative energy (eV per eight atoms) with respect to str1 (the FMC structure), and the multiplicity (degeneracy factor) for each structure.

First-principles predicted equilibrium properties of the supercell for fcc Ni (see Fig. 1) using the four-parameter Birch–Murnaghan EOS [see Eq. (7)], including the equilibrium volume ( per eight atoms), bulk modulus (GPa), and its pressure derivative , the relative energy (eV per eight atoms) with respect to str1 (the FMC structure), and the multiplicity (degeneracy factor) for each structure.

Predicted Curie temperatures ’s of fcc Ni in terms of the MFA and the present PFA, where is magnetic exchange energy used in MFA, see Eq. (8).

Predicted Curie temperatures ’s of fcc Ni in terms of the MFA and the present PFA, where is magnetic exchange energy used in MFA, see Eq. (8).

Article metrics loading...

Full text loading...

Commenting has been disabled for this content