Low- and high-temperature magnetism of Cr and Fe nanoclusters in iron-chromium alloys

Low-energy magnetic states and finite-temperature properties of Cr nanoclusters in bulk bcc Fe and Fe nanoclusters in bulk Cr are investigated using density functional theory (DFT) and the Heisenberg-Landau Hamiltonian based magnetic cluster expansion (MCE). We show, by means of noncollinear magnetic DFT calculations, that magnetic frustration caused by competing ferromagnetic and antiferromagnetic interactions either strongly reduces local magnetic moments while keeping collinearity or generates noncollinear magnetic structures. Small Cr clusters generally exhibit collinear ground states. Noncollinear magnetic configurations form in the vicinity of small Fe clusters if antiferromagnetic Fe-Cr coupling dominates over ferromagnetic Fe-Fe interactions. MCE predictions broadly agree with DFT data on the low-energy magnetic structures, and extend the DFT analysis to larger systems. Nonvanishing cluster magnetization caused by the dominance of Fe-Cr over Cr-Cr antiferromagnetic coupling is found in Cr nanoclusters using both DFT and MCE. Temperature dependence of magnetic properties of Cr clusters is strongly influenced by the surrounding iron atoms. A Cr nanocluster remains magnetic until fairly high temperatures, close to the Curie temperature of pure Fe in the large cluster size limit. Cr-Cr magnetic moment correlations are retained at high temperatures due to the coupling of interfacial Cr atoms with the Fe environment. Variation of magnetization of Fe-Cr alloys as a function of temperature and Cr clusters size predicted by MCE is assessed against the available experimental data.

Figure 1

Schematic representation of atomic positions and atomic magnetic moments (arrows) in small Cr clusters containing between 2 and 5 atoms in a bcc Fe lattice. Gray and yellow spheres denote Cr and Fe atoms, respectively. Cubic unit cells are drawn with solid lines.

Figure 2

Schematic representations of atomic positions and atomic magnetic moments (arrows) of some representative Cr clusters in the lowest-energy collinear state. Some cubic unit cells are drawn for clarity. The orientation of Fe magnetic moments is also shown.

Figure 3

Schematic representations of atomic positions and local magnetic moments (arrows) of Cr clusters containing 15, 24, 22, 41, and 70 atoms in their metastable NCol configurations. The energy difference, per cluster, between these configurations and the respective collinear ground state is shown. Orientation of the lattice Fe moments is also indicated.

Figure 4

Fraction of Cr atoms ($m_i$) with $i$ Cr nearest neighbors ($nn$) versus the size of the Cr cluster ($N_{\text{Cr}}$), for all the octahedral, cubic, and mixed clusters studied.

Figure 5

Absolute value of the angles between local Cr moments and Fe lattice moments in NCol $N_{\text{Cr}}$-cluster configurations ($N_{\text{Cr}}$ = 15, 24, 22, 41, and 70) versus (left) cluster size, and (right) the number of Cr $nn$.

Figure 6

Magnitudes of local moments on atoms in Col and NCol configurations of $n$-Cr clusters versus the number of Cr $nn$.

Figure 7

Schematic representations of atomic positions and magnetic moments (arrows) for the lowest-energy NCol state of a single Fe atom and small Fe clusters in AF bcc Cr. Yellow and gray spheres denote Fe and Cr atoms, respectively. Fe cluster sizes $N_{\text{Fe}}$ are given in parentheses.

Figure 8

Schematic representations of atomic positions and magnetic moments (arrows) for the lowest-energy NCol state of Fe clusters with $N_{\text{Fe}}$ = 6, 9, and 19 in AF bcc Cr. Fe cluster sizes ($N_{\text{Fe}}$) are given in parentheses. Orientations of AF-ordered moments of Cr atoms are shown by the up-down arrow.

Figure 9

Average magnitude of local magnetic moments (in ${\mu }_B$ units), computed for various Col and NCol configurations, shown for all the Fe clusters included in this study. Magnitude of a Fe atom moment in bcc FM lattice is shown for comparison.

Figure 10

Schematic representation of atomic positions and local magnetic moments (arrows) of Fe clusters of 15 and 24 atoms in AF bcc Cr. Two Col (left and right) and an intermediate-energy NCol (middle) configurations are shown together the difference between their energy, per cluster, and that of the ground state. In the Col states, the light yellow atoms have local moments antiparallel to their $nn$ Cr atoms. The Fe cluster size is given in parentheses. Orientation of the lattice Cr atoms is also shown.

Figure 11

15- and 24-Fe clusters in AF bcc Cr: energy per cluster of a magnetic configuration with respect to the magnetic ground state, computed for various Col and NCol states versus the average angle of Fe local magnetic moments. States with angle = ${180}^{\circ }$ and $0^{\circ }$ correspond to Col states shown on the left and right panels of Fig. 10, respectively.

Figure 12

Formation energy per cluster atom [as defined in Eq. (1)] of a Fe or a Cr atom forming a cluster compared with the formation energy of an atom of the same species in either a dilute or a concentrated random solid solution.

Figure 13

Energy difference between a noncollinear ground state and a ground state found using the collinearity constraint for cubic and octahedral chromium clusters, in meV per Cr atom.

Figure 14

Parallel and orthogonal components of Cr magnetic moments (in ${\mu }_B$) computed for the three smallest cubic clusters investigated by MCE.

Figure 15

Angle (in deg) between Fe magnetic moment and moments of Cr atoms in small cubic and octahedral clusters plotted as a function of the number of nearest Cr neighbors (a). The magnitude of magnetic moment of Cr atoms in small cubic and octahedral clusters plotted as a function of the number of nearest Cr neighbors (b).

Figure 16

Schematic representation of atomic positions and local magnetic moments (arrows) of Cr clusters containing 15, 24, 22, 41, and 70 atoms found in MCE simulations. Orientation of the FM-ordered Fe moments is also shown.

Figure 17

Formation energies of Cr and Fe clusters as functions of cluster size (meV per cluster atom). For large clusters, the formation energy decreases in proportion to the relative number of atoms at the interface (which is proportional to $n^{-1/3}$, as shown by solid lines).

Figure 18

Formation energy, per Cr atom, of chromium clusters (1241-atomic cubic cluster and 1469-atomic octahedral cluster) plotted as a function of temperature.

Figure 19

Magnetization (in ${\mu }_B$) of Cr clusters, per atom, plotted vs cluster size for cubic and octahedral clusters.

Figure 20

Temperature dependence of magnetization (in ${\mu }_B$) of cubic (341 atoms) and octahedral (344 atoms) Cr clusters.

Figure 21

Antiferromagnetic correlations in cubic Cr clusters (a) and octahedral Cr clusters (b).

Figure 22

Magnetization, plotted as a function of temperature, for Fe-40 at.$\%$ Cr alloys. Chromium atoms are either distributed randomly in Fe matrix, or form a mixture of cubic nanoclusters of various sizes suspended in Fe-12 at.$\%$ random alloys (see text).

Figure 23

Temperature dependence of magnetization of cubic (341 atoms) and octahedral (344 atoms) Fe clusters, and magnetization of pure iron.