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(1.School of Physics Mechatronic Engineering, GuiZhou Minzu University, Guiyang 550025, China; 2.School of Materials Science and Engineering, Guizhou Minzu University, Guiyang 550025, China; 3.Engineering Training Center, Guizhou Minzu University, Guiyang 550025, China)

Abstract:The electronic structures and magnetic properties of the bulk Fe2Ge and its (001) surfaces were calculated by the pseudo-potential plane wave method based on the first-principle of density functional theory. Here, two types of the terminated (001) surfaces were considered: Ge(Ⅰ)-(001) surface and Ge(Ⅱ)-(001) surface. For the electronic structures, the different types of the Fe2Ge (001) surfaces all show metallic characteristics, which are in agreement with the bulk counterpart. For the magnetism, the Ge atoms are ferromagnetic spin ordering in the bulk and Ge(Ⅱ)-(001) surface, while the Ge atoms are ferrimagnetic spin ordering in the first layer of the Ge(Ⅰ)-(001) surface. Moreover, the spin magnetic moment of the Ge atoms in the Ge(Ⅱ)-(001) surface are better than those of the bulk and Ge(Ⅰ)-(001) surface. These results are related to the hybridization between the Fe d and Ge p states, which were discussed by analyzing their density of states.

Key words:Fe2Ge (001) surface; electronic structure; magnetic property; first-principle; density functional theory; spin polarizability

0 Introduction

As a magnetic material, the binary alloy Fe2Ge has attracted growing attention[1-3]for its excellent magneto-transport characteristics and potential applications in magnetic storage devices[4]. Fe2Ge is hexagonal structure with space group ofP63/mmc(No.194). There is only one symmetry type of Fe atoms, which has a trigonal bipyramid environment surrounded by four nearest neighbor Ge atoms.

Since the 1960s, it has been proved by experiments[5-7]that the intermetallic compound Fe2Ge has hexagonal structure of B82type asβphase[5]. Neutron and X-ray diffraction adopted by Adelson et al[6]. indicated that Fe2Ge is a ferromagnetic metal phase. Magnetization measurements adopted by Hall et al[7]. indicated that Fe2Ge exhibits large anisotropy in the hyperfine magnetic field due to the domination of the anisotropy of crystal field. With further experimental researches and theoretical calculations, it is found that there are a great deal number of discrepancies[8-16]among previously experimental magnetic moments for the hexagonal-βphase of the intermetallic compound Fe2Ge. Recently, the investigation on magnetic properties of Fe2Ge thin film, which is obtained by MBE code position on Ge (111), has provided a new method to restudy the previously controversial results on the magnetic property of the hexagonal-βphase Fe2Ge[17]. In addition, stable hexagonal phase Fe2Ge was obtained by adding proper amount of intermetallic compound Fe2Ge to MnNiSi-based alloys. The FM/PM magnetic structure transition (MST) was obtained by adjusting the Fe2Ge content at a wide range of Curie temperature[18].

The compound Fe2Ge has high magnetization, low dielectric constant, excellent electromagnetic properties and high Curie temperature up to 400 K predicted by theory. Meanwhile, the Fe element in the compound Fe2Ge has +2, +3 and other valence status, which is a good oxidation reduction catalyst with preferable surface activity. In addition, the compound Fe2Ge is also a new environment-friendly electromagnetic material with a close-packed hexagonal structure, which exists obvious anisotropy inc-axis direction and perpendicular toc-axis direction[2-4,17], i.e., the Fe positive ion and adjacent Ge negative ions comprise two-dimensional regular hexagonal structure centered on Fe positive ion. The charge centers of these positive and negative ions coincide with each other without any external forces. But if there are any external forces, the charge centers of the cation and anion will shift to each other, generating a magnetic moment. The superposition of the couple magnetic moments produced by all the units in the crystal results in a macroscopic distribution of the electrical potential along the surface. At the same time, the 3d states electrons of Fe atoms in the Fe2Ge are not filled, forming the outer electronic configuration of 3d64s2, while the outer electronic configuration of Ge atoms is 3d104s24p2. Therefore, the complex Fe-Ge chemical bonds formed at the Fe2Ge surface lead to the changes of electronic structure of the surface. However, the magnetic mechanism of the surface is not clear.

These researches motivated us to study the electronic structures and magnetic properties of Fe2Ge (001) surface, as well as that of the bulk through the Ultrasoft pseudo-potential (USPP) plane wave method based on the first-principle of density functional theory using a slab model.

1 Computational details

The binary compound Fe2Ge is hexagonal system with space group ofP63/mmc(No.194) and lattice parameters ofa=0.500 27 nm andc=0.405 48 nm. Each crystal cell contains four Fe atoms and two Ge atoms. In the unit cell of Fe2Ge, Fe atoms occupy the (0.333, 0.667, 0.25) and (0.333, 0.667, 0.75) sites and Ge atom occupies the (0, 0, 0) site. The crystal structure of Fe2Ge is shown in Fig.1.

The electronic structure and magnetic properties were calculated using the first-principle pseudopotential plane wave method based on functional density theory. All calculations were completed by Cambridge sequential total energy package (CASTEP). Initially, the surface model of Fe2Ge was established and the Broyden Fletcher Goldfarb and Shannon (BFGS) algorithm was used to optimize the geometry of the model. It was found that Fe2Ge (001) surface is stable. Subsequently, the electronic structure and magnetic property of Fe2Ge (001) surface were calculated on the basis of geometry optimization. Ultrasoft pseudo-potentials (USPP) was used to deal with the interaction between ions and electrons. The exchange correlation potential is selected as the generalized gradient approximation (GGA). The wavefunctions of valence electrons were expanded by the plane wave sets. The cutoff energy of plane wave was 350 eV and the accuracy of iterative convergence was 5.0×10-6eV. The total energy was calculated in reciprocal space and the Brillouin zone integral was calculated by Monkhorst-Pack method. Considering the strong exchange correlation potential between the d state electrons of the transitional metal Fe element, the GGA with Hubbard U (GGA+U) method was used to calculate the electronic properties and magnetic properties of the bulk and surface. Here, the U value was set to 2.5 eV, which was referred to the Reference [19]. The calculated results are in agreement with the theorical data given in Reference [20]. Besides, during the calculation, the convergence tests were conducted by increasing gradually vacuum width and the number of atomic layers. The slab thickness of 6 atomic layers were adopted to carry out the convergence tests of the vacuum width. After the vacuum width increases to 0.8 nm, the total energy converges to a constant as the vacuum width tends to infinity. For the convergence test of the atomic layers, after the slab thickness increases to 8 atomic layers using a vacuum width of 1 nm, the surface energy tends to be stable as the atomic layers tend to infinity. In order to ensure the enough accuracy of calculated results, 10 atomic layers and 1.2 nm vacuum width were used to simulate the Fe2Ge (001) surface.

Fig.1 Crystal structure of hexagonal Fe2Ge

2 Results and discussion

2.1 Stability: chemical structural and mechanical

The chemical, structural and mechanical stabilities of the compound Fe2Ge were identified by calculating its formation energy, cohesive energy and elastic constants, respectively. The formation energy formula for this alloy is given as follows:

(1)

(2)

Finally, the elastic constants were examined to determinate the mechanical stability. As presented in Table 2, the elastic constants,C11=253.265 GPa,C33=323.686 65 GPa,C44=4.323 3 GPa,C12=188.580 2 GPa andC13=62.462 58 GPa, satisfy the mechanical stability criterion for the hexagonal structure listed as follows[21]:

C44>0

(3)

C11>|C12|

(4)

C33(C11+|C12|)>2C13

(5)

It is suggested that this alloy of hexagonal structure has the mechanical stability at the ground state.

Table 2 Elastic constants C11, C33, C44, C12 and C13

2.2 Band structures

The band structures of the bulk Fe2Ge and its (001) surfaces were investigated due to the differences of their electronic structures. Their band structures are presented respectively in Fig.2 and Fig.3. As it can be seen in Fig.2, the band structure expands along the G-A-H-K-G-M-L-H high symmetry direction of the Brillouin zone. The different spin energy bands all overlap withEF. This indicates that the Fe2Ge exhibits metallic behavior. For the (001) surface, there are two types of the (001) terminated surfaces. The Ge(Ⅰ)-(001) terminated surface is a Ge-terminated (001) surface in which the Fe atom layers and Ge atom layers are alternatively arranged, whereas the Ge(Ⅱ)-(001) terminated surface is a Ge-terminated (001) surface where the second and third layers all are Fe atom layers (see Fig.4). Fig.3 shows the spin-up and spin-down band structures of both the Ge(Ⅰ)-(001) and Ge(Ⅱ)-(001) terminated surfaces, respectively. Their band structures are more organized than that of the bulk. However, their spin bands in different directions all intersect the Fermi level line at many points, indicating that the thin film Fe2Ge remains the metallic characteristics of the bulk.

Fig.2 Band structure of bulk Fe2Ge

Fig.3 Surface band structures with Ge(Ⅰ)-(001) and Ge(Ⅱ)-(001) terminations

Fig.4 Surfaces with Ge(Ⅰ)-(001) and Ge(Ⅱ)-(001) terminations

Fig.5 Total, Fe atoms and Ge atoms density of states

2.3 Densities of states

In order to further understand the electromagnetic mechanism of these metallic behaviors, the characteristics of their electronic structures were analyzed. The total DOS and atom-projected DOS are shown in Fig.5. The total DOS shows the metallic nature in both spin-up channel and spin-down channel. The spin-polarized rate of the bulk Fe2Ge is about 34.23%, which is calculated by the spin-polarized formula[22]:

(6)

whereN↑(E)is the number of the electrons in spin-up channel;N↓(E)is the number of the electrons in spin-down channel. The DOS of Ge atoms is nearly symmetry with respect to energy axis, implying that magnetic moments carried by Ge atoms are close to zero. However, the DOS of Fe atoms is not symmetry with respect to energy axis, indicating that magnetic moments carried by Fe atoms are ferromagnetic spin ordering. These results reveal that the Fe2Ge is a metallic ferromagnet.

Fig.6 and Fig.7 represent the partial DOS of Fe atoms and Ge atoms, respectively. In the partial DOS of Fe atoms, the states in the energy region from 10 eV to 18 eV mainly come from p states, the states in the energy region from 4 eV to 10 eV are dominated by contributions from p states and s states, while the states in the surroundingEFare mainly contributed by d states. In the partial DOS of Ge atoms, the states in the energy from -6.5 eV to -4 eV are dominated by contributions from p states, the states in the energy from -12 eV to -6.5 eV are mainly contributed by s states, while the states in the surroundingEFconsist mainly of both p and s states.

Fig.6 Fe atom-projected density of states

Fig.7 Ge atom-projected density of states

Fig.8 shows the total DOS of both the Ge(Ⅰ)-(001) and Ge(Ⅱ)-(001) terminated surfaces. Their total DOS atEFare nonzero indicating that the Ge(Ⅰ)-(001) and Ge(Ⅱ)-(001) slabs exhibit the metallic behavior. For the Ge(Ⅰ)-(001) terminated slab, the spin-polarized rate is about 43.84%, which is higher than that of the bulk. For the Ge(Ⅱ)-(001) terminated slab, the spin-polarized rate is about 18.23%, which is lower than that of the bulk.

Fig.8 Surface density of states with Ge(Ⅰ)-(001) and Ge(Ⅱ)-(001) terminations

The atom-projected density of states of the Ge(Ⅰ)-(001) and Ge(Ⅱ)-(001) terminated surfaces are given in Fig. 9 and Fig. 10, respectively. Obviously, their atom-projected density of states all are similar to that of the bulk. However, for the Ge(Ⅰ)-(001) terminated surface, the peaks formed by the Fe d and Ge p states are not sharp like that of the bulk, which is caused by the decreased hybridization between the Fe d and Ge p states. For the Ge(Ⅱ)-(001) terminated surface, the peaks formed by the Fe d and Ge p states are sharper than that of the bulk, which result from the increased hybridization between the Fe d and Ge p states.

Fig.9 Atom-projected density of states of Fe and Ge in Ge(Ⅰ)-(001) terminated surface

Fig.10 Atom-projected density of states of Fe and Ge in Ge(Ⅱ)-(001) terminated surface

2.4 Magnetic properties

The atom-projected and total magnetic moments of the unit cell for Fe2Ge are presented in Table 3. Here, the magnetic moments for Fe atoms in the unit cell are positive indicating the existence of ferromagnetic coupling between Fe atoms. However, the negatively magnetic moment for Ge atom in the unit cell implies its antiferromagnetic coupling with Fe atoms. In the unit cell, the proportion of the magnetic moment for Fe atoms to total magnetic moments is highly 99.796%. This indicates that the magnetic property of bulk is dominated by contributions from the atoms projected magnetic moments of Fe atoms. As shown in Table 3, the value of magnetic moment per Fe atom is close to Fe (2.25 μB) in Fe2Ge in Reference [20].

Table 3 Total and atom-projected magnetic moments

Fig.11 Atom-projected magnetic moments of different terminated surfaces

The atom-projected magnetic moments of the Ge(Ⅰ)-(001) and Ge(Ⅱ)-(001) terminated surface in the first, second, third and fourth layers (S1, S2, S3, S4) are shown in Fig. 11. For the Ge(Ⅰ)-(001) terminated slab, the magnetic moment of the Fe atom in S2 get the maximum value. The magnetic moment of the Fe atom in S4 is about 2.47 μB, which is very close to that of the bulk (see Table 3). The value of the Ge atom magnetic moment in S1 is equal to that in S3 but they have different signs. The Ge (S1) with positively magnetic moment indicates the ferromagnetic spin ordering whereas the Ge (S3) with negatively magnetic moment shows the ferrimagnetic spin ordering. For the Ge(Ⅱ)-(001) terminated slab, the magnetic moment for Fe atom increases as the layer varies from S2 to S3 and the Fe atom in S3 gets the highest magnetic moment. The magnetic moments for the Ge atoms in S1 and S4 are negative, exhibiting the ferrimagnetic spin ordering. These results imply that the Ge spin-polarized magnetism of the Ge(Ⅱ)-(001) terminated surface is stronger compared with the Ge(Ⅰ)-(001) terminated surface. In terms of the changes of the geometric structure, in the Ge(Ⅱ)-(001) terminated surface, the distance between the Ge atom in S1 and the Ge atom in S4 is about 0.37 nm, which is larger than the values of other Ge-Ge bond lengths in the bulk Fe2Ge and Ge(Ⅰ)-(001) terminated surface. It is suggested that the interaction between the Ge atoms in the Ge(Ⅱ)-(001) terminated surface is weak. Consequently, it promotes the magnetic property of Ge atoms. On the other hand, the DOS of Ge atoms of the Ge(Ⅱ)-(001) terminated surface is similar to that of the bulk and Ge(Ⅰ)-(001) terminated surface except that owing to the increased hybridization between the Fe d states and Ge p states, it has a very sharp peak of spin-up states at -4.5 eV and a very sharp peak of spin-down states at -3.5 eV (see Fig.7, Fig.9 and Fig.10). Therefore, in the Ge(Ⅱ)-(001) terminated surface, the p states of Ge atoms has strong localization and large exchange splitting (see Fig.10), which results in large enhancement of the spin magnetic moments of the Ge atoms.

3 Conclusions

The electronic structure and magnetic properties of the Fe2Ge bulk and (001) surface were calculated by the first-principle pseudopotential plane wave method based on functional density theory. For the bulk, the Fe2Ge shows the metallic characteristic with the 34.23% spin polarization. Near theEFlevel, the DOS of Fe atoms is mainly come from the contributions of the d states, and the DOS of Ge atom is contributed by the s states and p states. However, the total DOS is dominated by the contributions from the d states. Similarly, the total magnetic moment of the bulk almost comes from the contributions (99.796%) from the Fe atoms magnetic moments. For the surface, two different types of the Fe2Ge (001) surface were taken into consideration: Ge(Ⅰ)-(001) terminated surface and Ge(Ⅱ)-(001) terminated surface. They all exhibit metallic nature, which remain the bulk corresponding behavior. Their spin polarizations are 43.84% and 18.23%, respectively. In the magnetism, the Fe atoms magnetic moments in all surface are agree well with the corresponding magnetic moments of the bulk. The Ge atoms in S1 for the Ge(Ⅰ)-(001) terminated surface show ferromagnetic spin ordering, which is opposite to the magnetic spin ordering of the Ge atoms of the bulk. This is caused by the decreased hybridization between the Fe d and Ge p states in the Ge(Ⅰ)-(001) terminated surface. In addition, the spin magnetic moment of Ge atoms for Ge(Ⅱ)-(001) terminated surface are stronger than those of bulk and Ge(Ⅰ)-(001) terminated surface.