Electronic structural and magnetic properties of Mn5Ge3 clusters H. K. Yuan, H. Chen, A. L. Kuang, C. L. Tian, and J. Z. Wang Citation: The Journal of Chemical Physics 139, 204307 (2013); doi: 10.1063/1.4832741 View online: http://dx.doi.org/10.1063/1.4832741 View Table of Contents: http://scitation.aip.org/content/aip/journal/jcp/139/20?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Structures and magnetic properties of Si n Mn ( n = 1 – 15 ) clusters J. Chem. Phys. 130, 164514 (2009); 10.1063/1.3123805 The geometric, electronic, and magnetic properties of Ag 5 X + ( X = Sc , Ti, V, Cr, Mn, Fe, Co, and Ni) clusters J. Chem. Phys. 124, 184319 (2006); 10.1063/1.2191495 Electronic structure and magnetic anisotropy for nickel-based molecular magnets J. Appl. Phys. 97, 10M522 (2005); 10.1063/1.1859972 Magnetic properties of MnN: Influence of strain and crystal structure Appl. Phys. Lett. 86, 164105 (2005); 10.1063/1.1905787 Electronic structure and magnetic properties of Al 1 − x Mn x N alloys J. Appl. Phys. 96, 6565 (2004); 10.1063/1.1818351
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THE JOURNAL OF CHEMICAL PHYSICS 139, 204307 (2013)
Electronic structural and magnetic properties of Mn5 Ge3 clusters H. K. Yuan, H. Chen,a) A. L. Kuang, C. L. Tian, and J. Z. Wang School of Physical Science and Technology, Southwest University, Chongqing 400715, People’s Republic of China
(Received 29 June 2013; accepted 1 November 2013; published online 25 November 2013) Theoretical understanding of the stability, ferromagnetism, and spin polarization of Mn5 Ge3 clusters has been performed by using the density functional theory with generalized gradient approximation for exchange and correlation. The magnetic moments and magnetic anisotropy energy (MAE) have been calculated for both bulk and clusters, and the enhanced magnetic moment as well as the enlarged MAE have been identified in clusters. The most attractive achievement is that Mn5 Ge3 clusters show a fine half-metallic character with large energy scales. The present results may have important implications for potential applications of small Mn5 Ge3 clusters as both emerging spintronics and next-generation data-storage technologies. © 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4832741] I. INTRODUCTION
The emerging field of spintronics, regarded as next generation electronics, would be dramatically boosted if room temperature ferromagnetism could be added to semiconductor devices and integrated circuits that are compatible with silicon technology.1–3 The magnetic Mn5 Ge3 compound, being full compatibility to the silicon-based microelectronics, has attracted considerable attention mainly motivated by its substantial spin polarization up to about 42% and persistent ferromagnetism up to about 296 K.4–7 In addition, it has a good lattice matching with the Ge (or Si) semiconductor and thus can grow directly onto Ge (or Si) substrates forming an epitaxial high quality Mn5 Ge3 layer, which would not only allow a direct injection of spin polarized current into the normal semiconductors but also avoid an insulating layer between the semiconductor and the ferromagnet in realization of the gatetunable spin devices.8 Up to now, various fabrication techniques have been employed to prepare magnetic Mn5 Ge3 films or dilute magnetic semiconductor (DMS). Mnx Ge1 − x films that are epitaxially grown on Ge substrates,7–20 including single-crystal growth, solid phase epitaxy, and molecular beam epitaxy. However, irrespective of the fabrication techniques, the formations of thermodynamically favorable ferromagnetic precipitates of small Mn5 Ge3 nanoclusters often appear when the growth temperature is too high and/or the Mn content exceeds the solubility limit due to a low solubility of Mn in Ge.2, 20 The Mn5 Ge3 precipitates have been proved to play an important role in determining the electromagnetic properties of these Ge-based systems. To gain further insight into the formations of Mn5 Ge3 clusters as well as the complex magnetism of Mn5 Ge3 /Ge heterostructures, free Mn5 Ge3 clusters can be used as prototype models because, similar to internal boundaries and surfaces, free clusters have a large fraction of atoms exposed to surface where strain can be released a) Author to whom correspondence should be addressed. Electronic mail:
[email protected] 0021-9606/2013/139(20)/204307/6/$30.00
by a larger dopant atom. Moreover, although bulk and films of Mn5 Ge3 compounds have been actively studied for several years,4–7, 21, 22 there has an alternative attempt of reducing their dimensionality in nanoscale. It will be ideal for spintronic applications if the advanced electronic and magnetic characterizations are enhanced in Mn5 Ge3 clusters, because this would open an entirely new and promising dimension. In the recent works, magnetic Mn5 Ge3 clusters have been successfully grown,23, 24 where the spherical structures are randomly distributed on Ge substrate and exhibit a broad size distribution. The experiments also indicated that the easy axial of Mn5 Ge3 nanoclusters is aligned along the hexagonal [0001] direction of the nanocrystal. Theoretical investigation of the structural and magnetic properties of Mn5 Ge3 materials is still at the early stages, and almost nothing is known about Mn5 Ge3 clusters. Using modern computational techniques, we present a first study relating the nanostructures to their electronic and magnetic properties. Even if, in general, no effort will be made to explore the lowest energy atomic structure of all our considered clusters, i.e., we mainly report the structure calculations on Mn5 Ge3 clusters at the corresponding bulk arrangement, we believe that our results could be important since, in real systems, it is possible that during the growth sequence clusters might rearrange themselves around these atomic arrays. Three fundamental points have been addressed (i.e., how about the structural stabilities of spherical clusters; how the electronic and magnetic properties evolve with cluster size; how they depend on the details of local structural environment). In what follows, we will first describe the computational methodology in Sec. II, and then present our results and discussions in Sec. III. Finally, a summary is given in Sec. IV.
II. METHODOLOGY
To model Mn5 Ge3 nanoclusters, we build a large supercell of the hexagonal crystal Mn5 Ge3 structure (P63 /mcm; a = 7.18 Å and c = 5.05 Å)21 and then cut out spherical
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FIG. 1. The optimized structures of Mn5 Ge3 bulk and spherical clusters (Mn5 Ge3 )n (n = 1, 6, 17). The Mn30 Ge18 and Mn84 Ge54 clusters whose chemical formula can be approximately written as (Mn5 Ge3 )6 and (Mn5 Ge3 )17 are stoichiometrically similar to bulk compound. The dark balls (red) represent Ge atoms, while the gray and taupe balls (blue and green) represent Mn1 and Mn2 atoms, respectively.
skeletons with different specific radius. In this study, three cluster sizes consisting of 8, 48, and 138 atoms, whose chemical compositions exactly/or closely match the stoichiometry of bulk Mn5 Ge3 , have been considered. They are the most stoichiometric and spherical clusters from all the range that we looked at, starting from 8 atoms and going up to 150 atoms. In Fig. 1, we show the top and side views of the crystal and cluster structures. The Mn5 Ge3 hexagonal cell contains 10 Mn atoms and 6 Ge atoms, and Mn atoms can be distinguished into two different sublattices (Mn1 and Mn2 types) due to their different coordination (Mn1 /Mn2 type is in a sixfold/fourfold position). For the constructed Mn5 Ge3 clusters, there are two different atomic planes perpendicular to [0001] direction as illustrated by Mn84 Ge54 structure: 2, 4, 6 layers contain only Mn1 type forming a hexagonal two-dimensional lattice; 1, 3, 5 layers contain Mn2 type and Ge atoms. After geometrical optimization of our largest cluster Mn84 Ge54 , its overall size is 1.5 nm as obtained from the distance between the most distant parts of the ionic radius surfaces at the both Mn atoms. Note that no passivation of the surface is made in contrast to the quantum dots of semiconductor materials. All calculations are performed based on the generalized gradient approximation (GGA) to the density functional theory (DFT). For the geometrical optimizations and magnetic moment calculations, the exchange-correlation functional of Perdew, Burke, and Ernzerhof (PBE) as well as the effective core potential (ECP) are adopted by using the DMOL3 package.25, 26 The convergence criteria are set to 1 × 10−6 eV
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for total energies and 5 × 10−3 eV/Å for the forces. Initial structures are first set in the highest possible symmetries and then carried out optimizations within symmetry constraint. If initial high symmetry does not change after the optimization, a low symmetry is imposed to ensure that the result is not affected by initial symmetry. The spin multiplicity is automatically determined during the optimization of small clusters. We also crosscheck the magnetic results in such way that the optimizations are done by initially imposing moment of 5 μB on each Mn atom, and we interestingly find that they often result in the same magnetic moment. Thus, initial moment of 5 μB is imposed on each Mn atom for large cluster calculations. For the calculation of magnetic anisotropy energy (MAE), the PBE functionals as well as the projected augmented wave (PAW) method within a plane wave cut-off energy of 351 eV are adopted by using the VASP package.27–29 If spin-orbit coupling (SOC) is not included, there is no need to define the spin quantization axis, because the energy does not depend on the direction of the spin magnetic moment, i.e., rotating overall spin magnetic moments by different angles results in principle exactly in the same energy. As the SOC is included, the SOC couples the magnetization to the lattice and determines the direction of magnetization, called the easy- (or hard-) axis. Hence, the MAE Eδγ = Eδ −Eγ can be calculated in terms of energy differences between two independent magnetization orientations δ and γ under the SOC mechanism. The optimal structures obtained from the DMOL3 optimizations are treated as the starting structures for three self-consistent field calculations, where three magnetization orientations are orderly aligned parallel to [001], [010], and [100] directions. Three clusters consisting of 8, 48, and 138 atoms (corresponding overall sizes are about 5, 10, and 15 Å) are placed into a large simple cubic box with edges of 15, 20, 25 Å, respectively. This ensures that the cluster and its imaging clusters are separated by a sufficiently high amount of vacuum to exclude the inter-cluster interactions. The convergence criterion is 1 × 10−6 eV for the self-consistent electronic loop. In contrast to the bulk calculations of MAE and density of state (DOS) where we used 8 × 8 × 10 MonkhorstPack grid k points, we perform cluster calculations using the Gamma point only. For the test calculations, Mn30 Ge18 cluster has been performed with different k points varying between the Gamma point and 4 × 4 × 4 points. However, it always leads to the similar shapes of DOSs and the close MAEs, which indicates that the Gamma point is a reasonable compromise between the accuracy and computational effort for further large 138-atom cluster. The benchmark calculation is done on bulk Mn5 Ge3 . We find that the ferromagnetic state with lattice parameters of a = 7.22 Å, c = 5.03 Å and magnetic moment of 30 μB per computational unit cell is 0.81 eV more stable than the ferrimagnetic state (Mn1 and Mn2 type atoms are in antiferromagnetic arrangement) with lattice parameters of a = 7.17 Å, c = 5.06 Å and magnetic moment of 12 μB per computational unit cell. In the ferromagnetic state, the local magnetic moment of Mn1 -type atoms is 2.72 μB , which is smaller than 3.30 μB of Mn2 -type atoms. Beside the best fitting of our lattice parameters with experimental values (a = 7.18 Å, c = 5.05 Å),21 our total moment of the bulk as well as the
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local moments of different atomic species are in agreement with corresponding experimental measurements (Mn1 = 1.96 μB ; Mn2 = 3.23 μB ; μcell = 26 μB )6, 21 and previous theoretical reports (Mn1 = 2.12 μB ; Mn2 = 3.16 μB ; μcell = 26 μB ).5 In addition, Tawara and Sato have experimentally indicated that the easy axis of bulk Mn5 Ge3 is along [0001] direction and it possesses MAE about 0.49 meV.22 Our calculations reproduce their easy axis and obtain MAE of 0.31 meV.
III. RESULTS AND DISCUSSION
After the relaxations, the atomic orders of the constructed clusters remain as they are initially made, in accordance with the structure of Mn5 Ge3 compound as shown in Fig. 1. The structure of the smallest cluster (Mn5 Ge3 )1 is a distorted hexagonal bipyramid with C2v symmetry, which can be regarded as the basic unit in constructing large clusters. The average bond lengths of Mn1 –Ge and Mn1 –Mn1 (Mn2 –Ge and Mn2 –Mn1 ) are 2.71 and 2.68 Å (2.55 and 3.02 Å), which are slightly elongated (contracted) with respect to corresponding lengths of 2.53 and 2.53 Å (2.56 and 3.08 Å) in bulk. For (Mn5 Ge3 )6 and (Mn5 Ge3 )17 clusters, surface relaxation is noticeable and appears to affect bonding in the atomic layers that are close to the surface. Generally, the clusters have their peripheral bonds exhibiting about 5% elongations along the c axis (e.g., Mn1 –Mn1 and Mn2 –Mn2 between two layers) as the compensation of contraction of the peripheral bonds in each layer (e.g., Mn2 –Mn2 and Mn2 –Ge in one layer). To have a clear view of bonding characters, the deformation density that is defined as the total charge density with the density of the isolated atoms subtracted has been calculated for Mn54 Ge84 cluster. We depicted the regions with positive deformation in Fig. 2, where the accumulation strongly encircles Ge atoms and partially localizes on Mn2 atoms (panel (a)). The former accumulation is built up mainly from the increased Ge-2p delocalization electrons while the latter accumulation is from the increased Mn-3d localization electrons, both of which are mainly transferred from the atomic Mn-4s orbitals. It reveals the ionic bonding characteristic in Mn54 Ge84 cluster (Qc from Mn atoms to Ge atoms, Table I). In addition, the bonding features of the d-d interaction can be seen in the following part of the electronic DOS. Although there is negligible density overlap between neighboring Ge atoms of the adjacent layers, the density feature represents a typical hybridization for three Ge atoms in each hexagonal ring illustrated by layer 5 in Fig. 2 (panel (b)). It reflects the non-negligible degree of covalency bond between
FIG. 2. (a) and (b) show the isosurfaces for the excess charge of Mn54 Ge84 cluster at 0.025 e/a.u3 . The map plane of (b) only contains layer 5. (c) and (d) show the orbitals for HOMO and LUMO of Mn54 Ge84 cluster at 0.01 e/a.u3 .
Ge atoms. In addition, Figure 2 shows that the HOMO orbitals are strongly localized on the peripheral Mn–Mn bonds and partially on the inside Mn atoms (panel (c)), while the LUMO orbitals spread out over the entire structure (panel (d)). From the orbital shapes around each Mn atom, it indicates that the Mn-3d electrons contribute to these states. In Fig. 3, theoretical results are given for the magnetization profile of Mn54 Ge84 cluster. From the up panel for different layers, Mn magnetic moments increase slightly as one goes from the center layer to the surface layer (Mn1 : 2.99, 3.17, 4.15 μB for j = 6, 4, 2 layers; Mn2 : 3.88, 4.02, 3.99 μB for j = 5, 3, 1 layers), either of which are much larger than the corresponding calculated bulk moments (Mn1 : 2.72 μB ; Mn2 : 3.30 μB ). From the down panel for a particular layer exemplified by layer 5, Mn magnetic moment increase gradually from the innermost atoms (equivalent atomic sites on circle 1) to the outmost atoms (equivalent atomic sites on circle 4). However, the charge transfers on Mn atoms demonstrate an opposite picture, where small (large) charge transfer coincides with large (small) magnetic moment. This is due to the
TABLE I. The normalized atomic compositions, binding energies Eb and magnetic exchange interaction energy E (eV/Mn5 Ge3 unit), averaged total magnetic moment μaver (μB /Mn5 Ge3 unit), and the averaged local magnetic moments (μB /atom) with charge transfers Q (e). Cluster Mn5 Ge3 Mn30 Ge18 Mn84 Ge54 Bulk
Per formula unit
Eb
E
μaver
μMn1 (Q)
μMn2 (Q)
μGe (Q)
(Mn5 Ge3 )1 (Mn5 Ge3 )6 (Mn5 Ge3 )17 Mn5 Ge3
15.55 21.75 24.37 28.28
0.38 0.17 0.31 0.41
23.00 18.33 17.53 15.00
4.54 (0.49) 4.11 (0.09) 3.53 (1.05) 2.67 (1.04)
5.05 (0.12) 3.70 (1.10) 3.96 (0.46) 3.52 (1.18)
−0.41 (−0.44) −0.47 (−0.82) −0.40 (−1.08) −0.41 (−1.85)
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FIG. 3. Local magnetic moment and charge transfer of Mn atoms for 1.3 nm Mn54 Ge84 cluster. The j labels the nonequivalent atomic sites ordered by increasing distance to the central atom.
evidence that the more charge transfer from Mn to Ge comes into being, the more back-donation of the Ge-4s electrons to the minor Mn-4d orbitals does occur. These variational trends are common to all studied Mn5 Ge3 clusters in different sizes. Although large magnetic moments 0.4–0.5 μB are induced at Ge atoms (Table I), which are antiferromagnetically coupled with Mn atoms, the significant increasing in Mn moments cannot be fully offset by the negative Ge moments. Consequently, the calculated moment μ¯ = 17.53 μB per Mn5 Ge3 formula remains larger than 15 μB of bulk. However, it is demonstrated in Table I that the averaged magnetic moments can be decreased by increasing cluster size. We also calculated the magnetic exchange interaction between Mn1 and Mn2 atoms in different systems, where the parallel and antiparallel orientations of magnetic moments are imposed on Mn1 and Mn2 atomic positions. The energy differences E between ferrimagnetic state (antiparallel arrangement) and ferromagnetic (parallel arrangement) state for different systems have been listed in Table I. The stability of high-spin state of Mn5 Ge3 clusters points to the possibility that for such nanoparticles dispersed in/or deposited on, for example, Ge matrix/or Ge surface, the magnetic moment of the entire particle will align to an applied magnetic field, analogous to the phenomenon of ferromagnetic. By using the bulk magnetic anisotropy constant K2 = 4.2 × 105 J/m3 ,22 our estimated MAE = K2 V = 0.49 meV is close to our DFT value of 0.31 meV for bulk Mn5 Ge3 . Since bulk Mn5 Ge3 exhibits large MAE with uniaxial anisotropy along the [0001] direction, we expect these characteristics can be held in clusters. According to the transmission electron microscopy (TEM) analysis, an easy magnetization corresponding to the hexagonal [0001] direction has been determined for the majority of deposited Mn5 Ge3 clusters.23, 24 If we adopt the magnetic anisotropy constant K2 = 3.3 × 105 J/m3 as experimentally estimated by Jain et al.,24 we can estimate
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MAE = K2 V = 0.12 and 0.11 meV (per Mn5 Ge3 formula) for Mn30 Ge18 and Mn84 Ge54 clusters, respectively, which are in the same order of magnitude to our precise DFT results of 0.42 and 0.31 meV (per Mn5 Ge3 formula). The quantitative divergences of MAE in clusters can be attributed to the anisotropy constant K2 that were derived from the deposited cluster model in large particle sizes. Indeed, Jain et al.24 have pointed out that the magnetic anisotropy in Mn5 Ge3 clusters is very sensitive to the slight structural distortions. Recently, it has been experimentally proved that the isolated Fe4 singlemolecule magnets with large MAE of 2 meV can be used for storing information at a very low temperature 0.5 K.31 Longterm magnetic data storage requires that spontaneous magnetization reversals should occur significantly less often than once in 10 years, which requires that the total MAE of each magnetic particle should exceed 40kB T.30 Here, total MAE of 5.27 meV of Mn84 Ge54 cluster could guarantee 10-year stability of an information bit at the temperature of liquid nitrogen, T = 1.53 K. On the other hand, if room temperature 300 K is used to evaluate the thermal stability of one domain cluster, the smallest size should be larger than 4.9 nm. Under the bulk condition, Mn5 Ge3 compound exhibits a metallic behavior with about 42% spin polarization.4 To exploit the electronic properties of Mn5 Ge3 clusters for device applications, the total and partial DOS have been displayed in Fig. 4. The plots of free Mn and Ge atoms as well as Mn5 Ge3 bulk are also shown for comparison. It can be seen that the DOS of Mn30 Ge18 and Mn84 Ge54 clusters are similar in shape. The valence band in the DOS spectra consists of mainly Mn-3d-orbitals mixed with small contributions from Ge-sp-orbitals, presenting the strong sp-d hybridizations between Mn and Ge atoms. Despite the widely spread d states owning to the strong hybridization between the Mn atoms, the bandwidths of both majority and minority Mn-3d channels in clusters decrease significantly as compared with that in bulk. Importantly, the peak of majority (minority) spin states is shifted to the lower energies by about 3.0 (2.2) eV as compared to the bulk case, resulting in 100% minority spin polarization of the conductance electrons (for more rigorous calculations of the spin polarization, see Ref. 32). The DOS shifting can be probably explained by the following reasons: (1) the size effect decreases the delocalization of Mn-3d electrons thereby narrowing the width of Mn-3d band, especially for the majority band; (2) the bond length elongations along [0001] direction weaken the interactions between Mn atoms in adjacent layers, further narrowing the width of majority Mn-3d band and localizing the DOS peak close to the position of free Mn atom; (3) the centroids of the spin-down orbitals of free Mn and Ge atoms are energetically close together, so that a relatively large hybridization between Mn and Ge atoms can occur easily. In clusters, the bond length contractions between Mn2 and Ge atoms in each layer can enhance this interaction, which would push down the spindown band of both Mn and Ge. The moving of the majority and minority bands with respect to the Ef would finally affect both metallic and magnetic properties of clusters. Thus, small Mn5 Ge3 clusters are well half-metallic ferromagnets. If these results can be preserved for deposited Mn5 Ge3 clusters, then it is very promising for integrating Mn5 Ge3 clusters
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FIG. 4. The PDOS of different clusters within Fermi level at zero energy.
into ferromagnetic-semiconductor heterostructures, the ultimate goal being the realization of spintronic devices. IV. CONCLUSION
In summary, we have investigated the electronic and magnetic properties of different (Mn5 Ge3 )n clusters (n = 1, 6, 17; compositions exactly/or closely match the stoichiometry of bulk Mn5 Ge3 ) on the basis of DFT calculations. We found that unlike the bulk, small Mn5 Ge3 clusters are characterized as ferromagnetic state within 100% minority spin polarized half-metals. The calculations show that both bulk and clusters can exhibit large magnetic anisotropy energy in the order of magnitude of 0.4 meV per Mn5 Ge3 formula. These results bring us to an ideal that small Mn5 Ge3 clusters are expected to be good candidates for magnetic and spintronics applications. It may be that small Mn5 Ge3 clusters offer a new and, heretofore, unexplored type of spintronics materials. ACKNOWLEDGMENTS
This work was supported by the National Natural Science Foundation of China (Project Nos. 10904125 and 91121013), the Chongqing’s Natural Science Foundation of China (Project Nos. CSTC-2008BB4253 and CSTC2011BA6004), and the Fundamental Research Funds for the Central Universities (Project No. XDJK2012B008). 1 S.
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THE JOURNAL OF CHEMICAL PHYSICS 139, 204307 (2013)
Electronic structural and magnetic properties of Mn5 Ge3 clusters H. K. Yuan, H. Chen,a) A. L. Kuang, C. L. Tian, and J. Z. Wang School of Physical Science and Technology, Southwest University, Chongqing 400715, People’s Republic of China
(Received 29 June 2013; accepted 1 November 2013; published online 25 November 2013) Theoretical understanding of the stability, ferromagnetism, and spin polarization of Mn5 Ge3 clusters has been performed by using the density functional theory with generalized gradient approximation for exchange and correlation. The magnetic moments and magnetic anisotropy energy (MAE) have been calculated for both bulk and clusters, and the enhanced magnetic moment as well as the enlarged MAE have been identified in clusters. The most attractive achievement is that Mn5 Ge3 clusters show a fine half-metallic character with large energy scales. The present results may have important implications for potential applications of small Mn5 Ge3 clusters as both emerging spintronics and next-generation data-storage technologies. © 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4832741] I. INTRODUCTION
The emerging field of spintronics, regarded as next generation electronics, would be dramatically boosted if room temperature ferromagnetism could be added to semiconductor devices and integrated circuits that are compatible with silicon technology.1–3 The magnetic Mn5 Ge3 compound, being full compatibility to the silicon-based microelectronics, has attracted considerable attention mainly motivated by its substantial spin polarization up to about 42% and persistent ferromagnetism up to about 296 K.4–7 In addition, it has a good lattice matching with the Ge (or Si) semiconductor and thus can grow directly onto Ge (or Si) substrates forming an epitaxial high quality Mn5 Ge3 layer, which would not only allow a direct injection of spin polarized current into the normal semiconductors but also avoid an insulating layer between the semiconductor and the ferromagnet in realization of the gatetunable spin devices.8 Up to now, various fabrication techniques have been employed to prepare magnetic Mn5 Ge3 films or dilute magnetic semiconductor (DMS). Mnx Ge1 − x films that are epitaxially grown on Ge substrates,7–20 including single-crystal growth, solid phase epitaxy, and molecular beam epitaxy. However, irrespective of the fabrication techniques, the formations of thermodynamically favorable ferromagnetic precipitates of small Mn5 Ge3 nanoclusters often appear when the growth temperature is too high and/or the Mn content exceeds the solubility limit due to a low solubility of Mn in Ge.2, 20 The Mn5 Ge3 precipitates have been proved to play an important role in determining the electromagnetic properties of these Ge-based systems. To gain further insight into the formations of Mn5 Ge3 clusters as well as the complex magnetism of Mn5 Ge3 /Ge heterostructures, free Mn5 Ge3 clusters can be used as prototype models because, similar to internal boundaries and surfaces, free clusters have a large fraction of atoms exposed to surface where strain can be released a) Author to whom correspondence should be addressed. Electronic mail:
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by a larger dopant atom. Moreover, although bulk and films of Mn5 Ge3 compounds have been actively studied for several years,4–7, 21, 22 there has an alternative attempt of reducing their dimensionality in nanoscale. It will be ideal for spintronic applications if the advanced electronic and magnetic characterizations are enhanced in Mn5 Ge3 clusters, because this would open an entirely new and promising dimension. In the recent works, magnetic Mn5 Ge3 clusters have been successfully grown,23, 24 where the spherical structures are randomly distributed on Ge substrate and exhibit a broad size distribution. The experiments also indicated that the easy axial of Mn5 Ge3 nanoclusters is aligned along the hexagonal [0001] direction of the nanocrystal. Theoretical investigation of the structural and magnetic properties of Mn5 Ge3 materials is still at the early stages, and almost nothing is known about Mn5 Ge3 clusters. Using modern computational techniques, we present a first study relating the nanostructures to their electronic and magnetic properties. Even if, in general, no effort will be made to explore the lowest energy atomic structure of all our considered clusters, i.e., we mainly report the structure calculations on Mn5 Ge3 clusters at the corresponding bulk arrangement, we believe that our results could be important since, in real systems, it is possible that during the growth sequence clusters might rearrange themselves around these atomic arrays. Three fundamental points have been addressed (i.e., how about the structural stabilities of spherical clusters; how the electronic and magnetic properties evolve with cluster size; how they depend on the details of local structural environment). In what follows, we will first describe the computational methodology in Sec. II, and then present our results and discussions in Sec. III. Finally, a summary is given in Sec. IV.
II. METHODOLOGY
To model Mn5 Ge3 nanoclusters, we build a large supercell of the hexagonal crystal Mn5 Ge3 structure (P63 /mcm; a = 7.18 Å and c = 5.05 Å)21 and then cut out spherical
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FIG. 1. The optimized structures of Mn5 Ge3 bulk and spherical clusters (Mn5 Ge3 )n (n = 1, 6, 17). The Mn30 Ge18 and Mn84 Ge54 clusters whose chemical formula can be approximately written as (Mn5 Ge3 )6 and (Mn5 Ge3 )17 are stoichiometrically similar to bulk compound. The dark balls (red) represent Ge atoms, while the gray and taupe balls (blue and green) represent Mn1 and Mn2 atoms, respectively.
skeletons with different specific radius. In this study, three cluster sizes consisting of 8, 48, and 138 atoms, whose chemical compositions exactly/or closely match the stoichiometry of bulk Mn5 Ge3 , have been considered. They are the most stoichiometric and spherical clusters from all the range that we looked at, starting from 8 atoms and going up to 150 atoms. In Fig. 1, we show the top and side views of the crystal and cluster structures. The Mn5 Ge3 hexagonal cell contains 10 Mn atoms and 6 Ge atoms, and Mn atoms can be distinguished into two different sublattices (Mn1 and Mn2 types) due to their different coordination (Mn1 /Mn2 type is in a sixfold/fourfold position). For the constructed Mn5 Ge3 clusters, there are two different atomic planes perpendicular to [0001] direction as illustrated by Mn84 Ge54 structure: 2, 4, 6 layers contain only Mn1 type forming a hexagonal two-dimensional lattice; 1, 3, 5 layers contain Mn2 type and Ge atoms. After geometrical optimization of our largest cluster Mn84 Ge54 , its overall size is 1.5 nm as obtained from the distance between the most distant parts of the ionic radius surfaces at the both Mn atoms. Note that no passivation of the surface is made in contrast to the quantum dots of semiconductor materials. All calculations are performed based on the generalized gradient approximation (GGA) to the density functional theory (DFT). For the geometrical optimizations and magnetic moment calculations, the exchange-correlation functional of Perdew, Burke, and Ernzerhof (PBE) as well as the effective core potential (ECP) are adopted by using the DMOL3 package.25, 26 The convergence criteria are set to 1 × 10−6 eV
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for total energies and 5 × 10−3 eV/Å for the forces. Initial structures are first set in the highest possible symmetries and then carried out optimizations within symmetry constraint. If initial high symmetry does not change after the optimization, a low symmetry is imposed to ensure that the result is not affected by initial symmetry. The spin multiplicity is automatically determined during the optimization of small clusters. We also crosscheck the magnetic results in such way that the optimizations are done by initially imposing moment of 5 μB on each Mn atom, and we interestingly find that they often result in the same magnetic moment. Thus, initial moment of 5 μB is imposed on each Mn atom for large cluster calculations. For the calculation of magnetic anisotropy energy (MAE), the PBE functionals as well as the projected augmented wave (PAW) method within a plane wave cut-off energy of 351 eV are adopted by using the VASP package.27–29 If spin-orbit coupling (SOC) is not included, there is no need to define the spin quantization axis, because the energy does not depend on the direction of the spin magnetic moment, i.e., rotating overall spin magnetic moments by different angles results in principle exactly in the same energy. As the SOC is included, the SOC couples the magnetization to the lattice and determines the direction of magnetization, called the easy- (or hard-) axis. Hence, the MAE Eδγ = Eδ −Eγ can be calculated in terms of energy differences between two independent magnetization orientations δ and γ under the SOC mechanism. The optimal structures obtained from the DMOL3 optimizations are treated as the starting structures for three self-consistent field calculations, where three magnetization orientations are orderly aligned parallel to [001], [010], and [100] directions. Three clusters consisting of 8, 48, and 138 atoms (corresponding overall sizes are about 5, 10, and 15 Å) are placed into a large simple cubic box with edges of 15, 20, 25 Å, respectively. This ensures that the cluster and its imaging clusters are separated by a sufficiently high amount of vacuum to exclude the inter-cluster interactions. The convergence criterion is 1 × 10−6 eV for the self-consistent electronic loop. In contrast to the bulk calculations of MAE and density of state (DOS) where we used 8 × 8 × 10 MonkhorstPack grid k points, we perform cluster calculations using the Gamma point only. For the test calculations, Mn30 Ge18 cluster has been performed with different k points varying between the Gamma point and 4 × 4 × 4 points. However, it always leads to the similar shapes of DOSs and the close MAEs, which indicates that the Gamma point is a reasonable compromise between the accuracy and computational effort for further large 138-atom cluster. The benchmark calculation is done on bulk Mn5 Ge3 . We find that the ferromagnetic state with lattice parameters of a = 7.22 Å, c = 5.03 Å and magnetic moment of 30 μB per computational unit cell is 0.81 eV more stable than the ferrimagnetic state (Mn1 and Mn2 type atoms are in antiferromagnetic arrangement) with lattice parameters of a = 7.17 Å, c = 5.06 Å and magnetic moment of 12 μB per computational unit cell. In the ferromagnetic state, the local magnetic moment of Mn1 -type atoms is 2.72 μB , which is smaller than 3.30 μB of Mn2 -type atoms. Beside the best fitting of our lattice parameters with experimental values (a = 7.18 Å, c = 5.05 Å),21 our total moment of the bulk as well as the
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local moments of different atomic species are in agreement with corresponding experimental measurements (Mn1 = 1.96 μB ; Mn2 = 3.23 μB ; μcell = 26 μB )6, 21 and previous theoretical reports (Mn1 = 2.12 μB ; Mn2 = 3.16 μB ; μcell = 26 μB ).5 In addition, Tawara and Sato have experimentally indicated that the easy axis of bulk Mn5 Ge3 is along [0001] direction and it possesses MAE about 0.49 meV.22 Our calculations reproduce their easy axis and obtain MAE of 0.31 meV.
III. RESULTS AND DISCUSSION
After the relaxations, the atomic orders of the constructed clusters remain as they are initially made, in accordance with the structure of Mn5 Ge3 compound as shown in Fig. 1. The structure of the smallest cluster (Mn5 Ge3 )1 is a distorted hexagonal bipyramid with C2v symmetry, which can be regarded as the basic unit in constructing large clusters. The average bond lengths of Mn1 –Ge and Mn1 –Mn1 (Mn2 –Ge and Mn2 –Mn1 ) are 2.71 and 2.68 Å (2.55 and 3.02 Å), which are slightly elongated (contracted) with respect to corresponding lengths of 2.53 and 2.53 Å (2.56 and 3.08 Å) in bulk. For (Mn5 Ge3 )6 and (Mn5 Ge3 )17 clusters, surface relaxation is noticeable and appears to affect bonding in the atomic layers that are close to the surface. Generally, the clusters have their peripheral bonds exhibiting about 5% elongations along the c axis (e.g., Mn1 –Mn1 and Mn2 –Mn2 between two layers) as the compensation of contraction of the peripheral bonds in each layer (e.g., Mn2 –Mn2 and Mn2 –Ge in one layer). To have a clear view of bonding characters, the deformation density that is defined as the total charge density with the density of the isolated atoms subtracted has been calculated for Mn54 Ge84 cluster. We depicted the regions with positive deformation in Fig. 2, where the accumulation strongly encircles Ge atoms and partially localizes on Mn2 atoms (panel (a)). The former accumulation is built up mainly from the increased Ge-2p delocalization electrons while the latter accumulation is from the increased Mn-3d localization electrons, both of which are mainly transferred from the atomic Mn-4s orbitals. It reveals the ionic bonding characteristic in Mn54 Ge84 cluster (Qc from Mn atoms to Ge atoms, Table I). In addition, the bonding features of the d-d interaction can be seen in the following part of the electronic DOS. Although there is negligible density overlap between neighboring Ge atoms of the adjacent layers, the density feature represents a typical hybridization for three Ge atoms in each hexagonal ring illustrated by layer 5 in Fig. 2 (panel (b)). It reflects the non-negligible degree of covalency bond between
FIG. 2. (a) and (b) show the isosurfaces for the excess charge of Mn54 Ge84 cluster at 0.025 e/a.u3 . The map plane of (b) only contains layer 5. (c) and (d) show the orbitals for HOMO and LUMO of Mn54 Ge84 cluster at 0.01 e/a.u3 .
Ge atoms. In addition, Figure 2 shows that the HOMO orbitals are strongly localized on the peripheral Mn–Mn bonds and partially on the inside Mn atoms (panel (c)), while the LUMO orbitals spread out over the entire structure (panel (d)). From the orbital shapes around each Mn atom, it indicates that the Mn-3d electrons contribute to these states. In Fig. 3, theoretical results are given for the magnetization profile of Mn54 Ge84 cluster. From the up panel for different layers, Mn magnetic moments increase slightly as one goes from the center layer to the surface layer (Mn1 : 2.99, 3.17, 4.15 μB for j = 6, 4, 2 layers; Mn2 : 3.88, 4.02, 3.99 μB for j = 5, 3, 1 layers), either of which are much larger than the corresponding calculated bulk moments (Mn1 : 2.72 μB ; Mn2 : 3.30 μB ). From the down panel for a particular layer exemplified by layer 5, Mn magnetic moment increase gradually from the innermost atoms (equivalent atomic sites on circle 1) to the outmost atoms (equivalent atomic sites on circle 4). However, the charge transfers on Mn atoms demonstrate an opposite picture, where small (large) charge transfer coincides with large (small) magnetic moment. This is due to the
TABLE I. The normalized atomic compositions, binding energies Eb and magnetic exchange interaction energy E (eV/Mn5 Ge3 unit), averaged total magnetic moment μaver (μB /Mn5 Ge3 unit), and the averaged local magnetic moments (μB /atom) with charge transfers Q (e). Cluster Mn5 Ge3 Mn30 Ge18 Mn84 Ge54 Bulk
Per formula unit
Eb
E
μaver
μMn1 (Q)
μMn2 (Q)
μGe (Q)
(Mn5 Ge3 )1 (Mn5 Ge3 )6 (Mn5 Ge3 )17 Mn5 Ge3
15.55 21.75 24.37 28.28
0.38 0.17 0.31 0.41
23.00 18.33 17.53 15.00
4.54 (0.49) 4.11 (0.09) 3.53 (1.05) 2.67 (1.04)
5.05 (0.12) 3.70 (1.10) 3.96 (0.46) 3.52 (1.18)
−0.41 (−0.44) −0.47 (−0.82) −0.40 (−1.08) −0.41 (−1.85)
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FIG. 3. Local magnetic moment and charge transfer of Mn atoms for 1.3 nm Mn54 Ge84 cluster. The j labels the nonequivalent atomic sites ordered by increasing distance to the central atom.
evidence that the more charge transfer from Mn to Ge comes into being, the more back-donation of the Ge-4s electrons to the minor Mn-4d orbitals does occur. These variational trends are common to all studied Mn5 Ge3 clusters in different sizes. Although large magnetic moments 0.4–0.5 μB are induced at Ge atoms (Table I), which are antiferromagnetically coupled with Mn atoms, the significant increasing in Mn moments cannot be fully offset by the negative Ge moments. Consequently, the calculated moment μ¯ = 17.53 μB per Mn5 Ge3 formula remains larger than 15 μB of bulk. However, it is demonstrated in Table I that the averaged magnetic moments can be decreased by increasing cluster size. We also calculated the magnetic exchange interaction between Mn1 and Mn2 atoms in different systems, where the parallel and antiparallel orientations of magnetic moments are imposed on Mn1 and Mn2 atomic positions. The energy differences E between ferrimagnetic state (antiparallel arrangement) and ferromagnetic (parallel arrangement) state for different systems have been listed in Table I. The stability of high-spin state of Mn5 Ge3 clusters points to the possibility that for such nanoparticles dispersed in/or deposited on, for example, Ge matrix/or Ge surface, the magnetic moment of the entire particle will align to an applied magnetic field, analogous to the phenomenon of ferromagnetic. By using the bulk magnetic anisotropy constant K2 = 4.2 × 105 J/m3 ,22 our estimated MAE = K2 V = 0.49 meV is close to our DFT value of 0.31 meV for bulk Mn5 Ge3 . Since bulk Mn5 Ge3 exhibits large MAE with uniaxial anisotropy along the [0001] direction, we expect these characteristics can be held in clusters. According to the transmission electron microscopy (TEM) analysis, an easy magnetization corresponding to the hexagonal [0001] direction has been determined for the majority of deposited Mn5 Ge3 clusters.23, 24 If we adopt the magnetic anisotropy constant K2 = 3.3 × 105 J/m3 as experimentally estimated by Jain et al.,24 we can estimate
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MAE = K2 V = 0.12 and 0.11 meV (per Mn5 Ge3 formula) for Mn30 Ge18 and Mn84 Ge54 clusters, respectively, which are in the same order of magnitude to our precise DFT results of 0.42 and 0.31 meV (per Mn5 Ge3 formula). The quantitative divergences of MAE in clusters can be attributed to the anisotropy constant K2 that were derived from the deposited cluster model in large particle sizes. Indeed, Jain et al.24 have pointed out that the magnetic anisotropy in Mn5 Ge3 clusters is very sensitive to the slight structural distortions. Recently, it has been experimentally proved that the isolated Fe4 singlemolecule magnets with large MAE of 2 meV can be used for storing information at a very low temperature 0.5 K.31 Longterm magnetic data storage requires that spontaneous magnetization reversals should occur significantly less often than once in 10 years, which requires that the total MAE of each magnetic particle should exceed 40kB T.30 Here, total MAE of 5.27 meV of Mn84 Ge54 cluster could guarantee 10-year stability of an information bit at the temperature of liquid nitrogen, T = 1.53 K. On the other hand, if room temperature 300 K is used to evaluate the thermal stability of one domain cluster, the smallest size should be larger than 4.9 nm. Under the bulk condition, Mn5 Ge3 compound exhibits a metallic behavior with about 42% spin polarization.4 To exploit the electronic properties of Mn5 Ge3 clusters for device applications, the total and partial DOS have been displayed in Fig. 4. The plots of free Mn and Ge atoms as well as Mn5 Ge3 bulk are also shown for comparison. It can be seen that the DOS of Mn30 Ge18 and Mn84 Ge54 clusters are similar in shape. The valence band in the DOS spectra consists of mainly Mn-3d-orbitals mixed with small contributions from Ge-sp-orbitals, presenting the strong sp-d hybridizations between Mn and Ge atoms. Despite the widely spread d states owning to the strong hybridization between the Mn atoms, the bandwidths of both majority and minority Mn-3d channels in clusters decrease significantly as compared with that in bulk. Importantly, the peak of majority (minority) spin states is shifted to the lower energies by about 3.0 (2.2) eV as compared to the bulk case, resulting in 100% minority spin polarization of the conductance electrons (for more rigorous calculations of the spin polarization, see Ref. 32). The DOS shifting can be probably explained by the following reasons: (1) the size effect decreases the delocalization of Mn-3d electrons thereby narrowing the width of Mn-3d band, especially for the majority band; (2) the bond length elongations along [0001] direction weaken the interactions between Mn atoms in adjacent layers, further narrowing the width of majority Mn-3d band and localizing the DOS peak close to the position of free Mn atom; (3) the centroids of the spin-down orbitals of free Mn and Ge atoms are energetically close together, so that a relatively large hybridization between Mn and Ge atoms can occur easily. In clusters, the bond length contractions between Mn2 and Ge atoms in each layer can enhance this interaction, which would push down the spindown band of both Mn and Ge. The moving of the majority and minority bands with respect to the Ef would finally affect both metallic and magnetic properties of clusters. Thus, small Mn5 Ge3 clusters are well half-metallic ferromagnets. If these results can be preserved for deposited Mn5 Ge3 clusters, then it is very promising for integrating Mn5 Ge3 clusters
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FIG. 4. The PDOS of different clusters within Fermi level at zero energy.
into ferromagnetic-semiconductor heterostructures, the ultimate goal being the realization of spintronic devices. IV. CONCLUSION
In summary, we have investigated the electronic and magnetic properties of different (Mn5 Ge3 )n clusters (n = 1, 6, 17; compositions exactly/or closely match the stoichiometry of bulk Mn5 Ge3 ) on the basis of DFT calculations. We found that unlike the bulk, small Mn5 Ge3 clusters are characterized as ferromagnetic state within 100% minority spin polarized half-metals. The calculations show that both bulk and clusters can exhibit large magnetic anisotropy energy in the order of magnitude of 0.4 meV per Mn5 Ge3 formula. These results bring us to an ideal that small Mn5 Ge3 clusters are expected to be good candidates for magnetic and spintronics applications. It may be that small Mn5 Ge3 clusters offer a new and, heretofore, unexplored type of spintronics materials. ACKNOWLEDGMENTS
This work was supported by the National Natural Science Foundation of China (Project Nos. 10904125 and 91121013), the Chongqing’s Natural Science Foundation of China (Project Nos. CSTC-2008BB4253 and CSTC2011BA6004), and the Fundamental Research Funds for the Central Universities (Project No. XDJK2012B008). 1 S.
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