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Effect of Cr on electronic and magnetic properties of χ-carbide (Fe,Cr)5C2
Journal of Magnetism and Magnetic Materials 392 (2015) 1–5
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Journal of Magnetism and Magnetic Materials journal homepage: www.elsevier.com/locate/jmmm
Effect of Cr on electronic and magnetic properties of χ-carbide (Fe,Cr)5C2 B. Wang a, Q. Zhang b, Z.F. Zhang b, Z.Q. Lv b,n, W.T. Fu a,nn a b
State Key Laboratory of Metastable Material Science and Technology, Yanshan University, Qinhuangdao 066004, China College of Mechanical Engineering, Yanshan University, Qinhuangdao 066004, China
art ic l e i nf o
a b s t r a c t
Article history: Received 1 November 2014 Received in revised form 8 May 2015 Accepted 8 May 2015 Available online 9 May 2015
From density-function theory calculation, the structural, electronic and magnetic properties of χ-carbides (Fe,Cr)5C2 are investigated. With the increase of Cr content in χ-carbides (Fe,Cr)5C2, the formation energy of χ carbide gradually decrease and energy stability of them increase. The formation energy of Cr5C2 is 0.354 eV/f.u, and the stability of Cr5C2 is higher than other χ carbides (Fe,Cr)5C2, Mn5C2 and Fe5C2. There exists charges transfer from metal cation (Fe/Cr) to C atoms in χ-carbides, and this reveals an ionic contribution to the bonds. The addition of Cr decreases the magnetic moments of χ carbide, and the magnetic moments (Ms) of Cr2Cr2FeC2 and Cr5C2 are 0 μB/f.u., while it expresses opposite magnetic characters of the same atom at different sites in the other χ type (Fe,Cr)5C2 carbides. The 3d states of metal atoms in the majority states (up) move to above the Femi level and some metal atoms (Fe/Cr) in χ type (Fe,Cr)5C2 are undergone the anti-ferromagnetic transformation. & 2015 Elsevier B.V. All rights reserved.
Keywords: χ carbides Phase stability Electronic structure Magnetic behavior
1. Introduction Carbides in steels play a critical role during the process of heat treatment and hot working, which also belong to the most important strengthening phases in carbon steels and white cast. In Fe–C system, these carbides include the well-known θ-Fe3C (cementite), Fe2C (ε and η) and Fe5C2 (Hägg carbide) etc. [1,2]. While some common alloying elements (such as Cr, Mn, Si etc) are often added to steels in order to improve their properties, and then the structure and properties of carbides are also changed due to forming alloying carbides [3,4]. Some researchers [5–8] investigated the structural and electronic properties of iron carbides in Fe–C alloy. Faraon and co-workers [5] calculated the crystalline, electronic, and magnetic structures of θ-Fe3C, χ-Fe5C2 and η-Fe2C using the PLAPW method, and the stability relationship between the three carbides was confirmed. Leineweber et al. [6] studied the structure and elastic properties of χ carbide Fe5C2 using first principles calculations and diffraction experiments. The powderdiffraction data was in agreement with the DFT calculations. Lv and co-workers [7,8] calculated the electronic and magnetic properties of Fe3C (ε and θ) and Fe2C (ε and η) using the pseudopotential plane–wave within the density functional theory, and pointed out the differences in mechanical stability, formation energy and magnetic properties between them. Some researchers n
Corresponding author. Corresponding author. E-mail address: [email protected] (Z.Q. Lv).
nn
http://dx.doi.org/10.1016/j.jmmm.2015.05.023 0304-8853/& 2015 Elsevier B.V. All rights reserved.
[9–12] also studied the phase stability and electronic structure of alloying carbides in steels. Zhou et al. [9] calculated the electronic structure and formation energy of Fe11CrC4 and Fe10Cr2C4 with the Cambridge Serial Total Energy Package (CASTEP) code. Lv et al. [10,11] studied the magnetic properties and phase stability of cementite-type (Fe,M)3C (M¼Cr/Mn/Co/Ni) by means of the pseudopotential-based density functional theory. Recently Gou et al. [12] studied the electronic structures and phase stability of χcarbides (Fe,Mn)5C2 using first-principles technique, and confirmed that Mn enhanced the stability of χ-carbides. Like Mn, element Cr is also often existing in the steels and form alloying carbides such as (Fe,Cr)3C [10], (Fe,Cr)7C3 [13,14], and M23C6 [14,15]. However, the structure and properties of Cr alloying carbides χ-(Fe,Cr)5C2 are not very clear. The present work aims to study the magnetic properties and phase stability of χ-Fe5 xCrxC2 (x¼1–5) using the first-principles calculations and to clarify the phase stability of them in the sight of energy. Cr and Mn doping also affect the magnetic characters of these carbides and alloys [10– 12,14,15], which also play an important role in the design and improvement of materials. In order to make clear the effect of Cr doping on the magnetic properties of χ-carbides, the magnetic moments of χ-Fe5 xCrxC2 (x ¼1–5) and each atom are also calculated from the electronic structure.
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B. Wang et al. / Journal of Magnetism and Magnetic Materials 392 (2015) 1–5
2. Crystal structure and calculation details The crystal structure of the χ-M5C2 carbide (Hägg carbides) is presented in Fig. 1. It is monoclinic with space group C2/c(15) that includes 4 symmetry operations and 14 atoms in the conventional cell [16]. The Wyckoff positions of the atoms are MI 8f (x1, y1, z1), MII 8f (x2, y2, z2), MIII 4e (0, y3, 0.25), and C 8f (x4, y4, z4). When the Fe atoms in different site are replaced by Cr, it form the multicomponent carbides χ-Fe5 xCrxC2 (x ¼1–5). According to the sites of metal atoms in χ type carbides, M5C2 carbides can be expressed as M'2M''2M'''C2 format (see Table 1). All calculations were performed based on the pseudo-potential plane–wave within the density functional theory [17,18]. The exchange-correlation potential was evaluated using the Perdew– Burke–Ernzerhof (PBE) functional [19] within the generalized gradient approximation (GGA) [20]. The interactions between the core and valence electrons were described by ultra-soft pseudopotentials [21], and the Kohn–Sham one-electron states were expanded in a plane wave basis set up to 400 eV. The energy calculations in the first irreducible Brillouin zone were conducted by using 2 6 5 k-point grid of Monkhorst–Pack scheme [22]. The convergence criteria for structure optimization and energy calculation were set to fine quality with the tolerance for the stress concentration factor (SCF), energy, maximum force and maximum displacement of 10 6 eV/atom, 10 5 eV/atom, 0.03 eV/Å and 0.001 Å, respectively. Because of its large effect on magnetic systems, spin polarization was included in the calculations to correctly account for its magnetic properties.
3. Results and discussions 3.1. Structural and energetic stability of the phases The ground state properties of the χ-Fe5 xCrxC2(x¼1–5) are investigated from their total energy, which is calculated as a function of volume. According to the Murnaghan equation of state, the equilibrium lattice constants and total energies of χ-Fe5 xCrxC2(x¼ 1–5) can be obtained (see Table1). The relative stability of Fe5 xCrxC2 can be discussed with energy calculations. The formation energy of a carbide (Fe5 xCrxC)
Table 1 Calculated lattice constants a, b, c (Å), β, v and total cell energy (eV/f.u) of χ-Fe5 xCrxC2 (x¼ 1–5).
Fe5C2 (Ref.12) (Ref.15) Fe2Fe2CrC2 Fe2Cr2FeC2 Cr2Fe2FeC2 Cr2Fe2CrC2 Fe2Cr2 CrC2 Cr2Cr2FeC2 Cr5C2 (Ref.16) Mn5C2 (Ref.12) (Ref.16)
a
b
C
β
v
11.569 (11.588) 11.620 11.453 11.673 11.757 11.519 11.426 11.564 11.804 (11.672)
4.487 (4.579) 4.507 4.527 4.428 4.463 4.535 4.495 4.573 4.528 (4.586)
4.973 (5.059) 4.998 5.039 5.029 5.057 5.080 5.160 5.140 5.100 (5.097)
97.59 (97.75) 97.12 97.68 97.16 97.08 97.73 97.80 97.53 97.49 (97.72)
255.89 (265.99) 259.69 258.91 257.88 263.28 262.96 262.58 269.45 270.24 (270.39)
Total cell energy 4636.48 6239.05 7841.43 7841.49 9443.93 9443.98 11046.49 12649.03 3578.33
can be defined as follows: Ebinding (Fe5 − x Crx C 2) = Etotal (unit) − (5 − x ) Eisolate (Fe ) − xEisolate (Cr) − 2Eisolate (C)
(1)
ΔEf (Fe5 − x Crx C 2 ) = Ebinding (unit) − (5 − x ) Ebinding (Fe ) − xEbinding (Cr) − 2Ebinding (C)
(2)
Here, the binding energy and the formation enthalpy per formula unit are represented by Ebinding (Fe5 xCrxC) and ΔEf (Fe5 xCrxC), respectively. Etotal(unit) is the total energy of the calculated unit. Ebinding(Fe/Cr) is the binding energy of pure element Fe per atom and Eisoalte(Fe/Cr) denotes the total energy of an isolated Fe/Cr atom, finally. The binding energy of pure elements was calculated by using their ground state conventional cell as BCC for Fe, as FCC for Cr and graphite for C. The total energies of isolated atoms were taken from the CASTEP output files directly. It is noted that such estimations from the total energy differences (at T ¼0 K) neglect the vibrations. At T ¼0 K and p ¼0 Pa, the formation enthalpy equals the calculated formation energy, i.e.ΔH (Fe5 − x Crx C2) = ΔE (Fe5 − x Crx C2), when the zero-vibration contribution is ignored [2,8]. The calculated cohesive energy and formation enthalpy of Fe5 xCrxC2 are shown in Table 2. The ΔE of Cr5C2 is 0.354 eV/f.u., which is negative and smaller than the ΔE of Fe5C2 (0.136 eV/f.u) and Mn5C2 ( 0.257 eV/f.u.) [12]. This shows that Cr5C2 is more stable than Fe5C2 and Mn5C2. Fe2Cr2FeC2 and Cr2Fe2FeC2 are with the same contents of Fe/Cr, but the calculated formation energies are different between them. This result is also obtained from the comparison between Cr2Fe2CrC2 and Fe2Cr2CrC2. This indicates that the bonding energies of metal atoms in different sites of χ-M5C2 cell are different. 3.2. Electronic properties Basing on the results from the electronic structure of
Converting
Table 2 Calculated cohesive energy and formation enthalpy of χ-Fe5 xCrxC2 (x ¼ 1–5) (eV/f. u.)
Conventional cell
Primitive cell
Fig. 1. Crystal structure of monoclinic χ-M5C2.
Fe5C2 [12] Mn5C2 [12] Fe2Fe2CrC2 Fe2Cr2FeC2 Cr2Fe2 FeC2 Cr2Fe2CrC2 Fe2Cr2CrC2 Cr2Cr2FeC2 Cr5C2
Binding energy
Formation enthalpy
65.81 64.98 66.84 67.67 67.74 68.64 68.68 69.65 70.64
0.136 0.257 0.066 0.086 0.016 0.044 0.084 0.214 0.354
B. Wang et al. / Journal of Magnetism and Magnetic Materials 392 (2015) 1–5
2 -2 2
CrII
0 -2 2
FeIII
0 -10
-5 0 5 Energy (E-EF) (eV)
10
CrIII
0
CrI
0
-2 -15
2
C
15
DOS (states/eV,atom)
DOS (states/eV,atom)
1 0 -1
3
-2 2
FeII
0 -2 2
FeI
0 -2 1 0 -1 -15
C -10
-5
0
5
10
15
Energy (E-EF) (eV)
Fig. 2. Calculated partial spin-polarized density of states, below 15 eV are not shown (a) Cr2Cr2FeC2 (b) Fe2Fe2CrC2.
Fig. 3. Calculated partial spin-polarized density of states, below 15 eV are not shown (a) Cr2Fe2FeC2 (b) Fe2Cr2FeC2.
χ-Fe5 xCrxC2 (x¼ 1–4) at equilibrium, the plots of density of states (DOSs) were discussed. Spin polarization results from a nearly rigid band shift between the majority spins (up) to lower energy and the minority spins (down) to higher energy due to the gain of energy from exchange. The spin-polarized densities of states for different atoms are shown from Fig. 2 to Fig. 4 at theoretical equilibrium lattice constants. The spin-polarized densities of states of χ-(Fe,Cr)5C2 are similar to that of Fe5C2 and Mn5C2 [12], there are three regions: the lowest valence band, the upper valence band, and the conduction unoccupied states. Comparing the up with down densities, it can be found that the up and down states of C atom is nearly symmetric, but that of Cr and Fe are not symmetric except in Cr2Cr2FeC2 (Fig. 2a). Actually, the lowest valence bands of metal atoms are also almost symmetric, and it is near the Fermi level, the hybridization of C 2p and Fe 3d, that the up and down states are noticeably dissimilar for different χ-(Fe,Cr)5C2 carbides. The lowest valence band ranging from about 14 to 11 eV is consisted of a main C 2s and a small contribution from s, p and d states of metal atoms Fe and Cr. The energy gaps between the lower valence bands and the upper valence bands indicates that the chemical bonds of χ-(Fe,Cr)5C2 take on ionicity. The second part valence band ranges from 7.5 eV to Fermi level, which is constituted by Fe/Cr 3d and the hybridization of C 2p and Fe/Cr 3d. The conduction unoccupied states are mainly formed by 3d of metal atoms.
From Fig. 5 to Fig. 7, the electron density distribution maps of the plane with metal and C atoms of Fe5 xCrxC2 (x ¼1–4) are plotted with the electron density difference map. The electron density difference maps are usually defined as the electron density difference between the isolated atoms and the bonding states with the same atomic configurations, which reflect the loss and gain of electrons in spaces. In this work, the electron density difference was determined as Δρ = (ρcrystal − ∑ ρat ) [7,10], where ρcrystal and
ρat are the valence electron densities for Fe5 xCrxC2 (x¼ 1–4) and corresponding free atoms, respectively. It can be seen that the increment of delocalized electrons is attributed to the metallic bonds in the interstitial regions. The increment of valence electrons is concentrated on the C atoms and the elongated contours correspond to the p-like orbits of C. It can be seen that the electron densities near Fe/Cr atoms decrease and the electron densities of nonmetal C atoms increase. There exists the loss and gain of electrons between Fe/Cr and C atom along the M–C direction. The increment of valence electrons is concentrated on the C atoms. In the interstitial regions, the increment of delocalized electrons is attributed to the metallic bonds. The M–C bonds take on ionicity and the M–M bonds are metallic bonds. It can be seen that χ-Fe5 xCrxC2 (x¼ 1–4), like Fe5C2 and Mn5C2, have mixed characters of metallicity, covalence and ionicity. Combined with the Mulliken population analysis, the loss and
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B. Wang et al. / Journal of Magnetism and Magnetic Materials 392 (2015) 1–5
2
2
CrIII
-2 2
CrII
0 -2 2
FeI
0 -2 1 0 -1 -15
C -10
-5 0 Energy (E-E F) (eV)
5
10
CrIII
0
DOS (states/eV,atom)
DOS (states/eV,atom)
0
-2 2
FeII
0 -2 2
CrI
0 -2 1 0 -1 -15
C
-10
-5 0 5 Energy (E-EF) (eV)
10
15
Fig. 4. Calculated partial spin-polarized density of states, below 15 eV are not shown (a) Fe2Cr2CrC2 (b) Cr2Fe2CrC2.
Cr
Fe C-Cr
Cr
Fe C-Fe
Fe
Fe
C-Cr
Cr
Cr
Fe Cr
Fe
Cr
Fe C
Fe
Fe
C-Cr
Cr
Fe
C-Fe Fe
Fe
Cr
Fe C
C
-0.10
Fe C
Cr
Cr
C
C
C
Fe
Cr
Fe
Fe
C
Fe
0
C
0.10
Cr C
Cr
C
C Fe
C-Fe
Fe
-0.10
0
Fe
0.10
Fig. 5. Electron density differences maps of the plane with metal atom and C atoms plane of: (a) Cr2Cr2FeC2 (b) Fe2Fe2CrC2 the density values are plot from 0.1 electron/Å3 (blue) to þ0.1 electron/Å3 (red). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 6. Electron density differences maps of the plane with metal atom and C atoms plane of: (a) Cr2Fe2FeC2 (b) Fe2Cr2FeC2 the density values are plot from 0.1 electron/Å3 (blue) to þ 0.1 electron/Å3 (red). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
gain electrons of atoms can be clearly exhibited. From the Mulliken population analysis for Cr2Cr2FeC2, the charges of C, CrI, CrII and FeIII atoms distribute to 0.59, þ0.14, þ0.29 and þ0.32, respectively; as for Fe2Fe2CrC2, the charges of C, FeI, FeII and CrIII atoms are 0.66, þ0.27,þ 0.24 and þ0.31, respectively. As for Cr2Fe2FeC2, the charges of C, CrI, FeII and FeIII atoms distribute to 0.63, þ 0.08, þ0.32 and þ0.45, respectively; for Fe2Fe2CrC2, the charges of C, FeI, CrII and FeIII atoms distribute to 0.64, þ0.28, þ0.22 and þ0.28, respectively. As for Cr2Fe2CrC2, the charges of C, CrI, FeII and CrIII atoms distribute to 0.59, þ0.14, þ0.29 and þ0.32, respectively; for Fe2Cr2CrC2, the charges of C, FeI, CrII and CrIII atoms distribute to 0.60, þ0.4, þ0.15 and þ0.12, respectively. The loss and gain of electrons is obvious between Fe/Cr and C atom form the data, which shows the iconicity on the M–C bond. The variations of population values of M–C bonds are from 0.28 to 0.36 in χ-Fe5 xCrxC2 (x ¼1–4), which indicates no repulsion force among these atoms.
3.3. Magnetic properties The spin moments of each atom were calculated from the partial spin-polarized density of states, which were listed in Table 3. It can found that the addition of Cr decreases the magnetic moments of χ carbides, especially Cr in I site. This is mainly due to anti-ferromagnetic transformation of metal atom. Some negative values of M–M overlap population in χ-Fe5 xCrxC2 (x ¼1–4) indicates that there exists repulsion force between these metal atoms and the near neighbors. This may prove the anti-ferromagnetic transformation of metal atoms from the view of mechanics. The magnetic moments (Ms) of Cr2Cr2FeC2 and Cr5C2 are 0 μB/f.u., while other χ carbides (Fe,Cr)5C2 express different magnetic characters. The magnetic moments (Ms) of all χ-Fe5 xCrxC2 (x ¼1–5) are smaller than that of Fe5C2, 8.96 μB/f.u. [12]. Like Fe5C2 and Mn5C2 [5,12], the magnetic moments of metal atoms are also different in the different site of χ-(Fe,Cr)5C2 cell (except from Cr2Cr2FeC2, 0 μB).
B. Wang et al. / Journal of Magnetism and Magnetic Materials 392 (2015) 1–5
4. Conclusions
C
Cr
C-Cr
C-Cr
Fe C-Fe
Fe
Cr
C-Cr
Fe C-Cr
Cr
Fe
C Cr
Cr Cr Fe
Fe
Fe
Fe
Cr
C
C
Cr
C
Cr
C
-0.10
0
5
0.10
Fig. 7. Electron density differences maps of the plane with metal atom and C atoms plane of: (a) Fe2Cr2CrC2 (b) Cr2Fe2CrC2 the density values are plot from 0.1 electron/Å3 (blue) to þ0.1 electron/Å3 (red). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Using first-principles technique, a completely theoretical analysis of the phase stability, electronic and magnetic properties of χ carbides (Fe,Cr)5C2 have been presented. The formation enthalpy of Cr5C2 is 0.354 eV/f.u., and Cr5C2 is more stable than Fe5C2 and Mn5C2. Fe2Cr2FeC2 and Cr2Fe2FeC2 are with the same contents of Fe/Cr, but the calculated formation energies are different between them. This result is also obtained from the comparison between Cr2Fe2CrC2 and Fe2Cr2CrC2. This indicates that the bonding energies of metal atoms in different sites of χ-M5C2 cell are different. There exists the loss and gain of electrons between Fe/Cr and C atom, which reveals an ionic contribution to the bonds in the χ type covalent structure. The addition of Cr decreases the magnetic moments of χ carbides. The magnetic moments (Ms) of Cr2Cr2FeC2 and Cr5C2 are 0 μB/f.u., while it expresses opposite magnetic characters of the same atom in different site in the other χ (Fe, Cr)5C2 carbides. The 3d states of Cr (Fe) in the majority states (up) move to above the Femi level and some metal atoms (Fe or Cr) in χ-(Fe,Cr)5C2 carbides are undergone the anti-ferromagnetic transformation.
Acknowledgments This research was supported by the National Natural Science Foundation of China (No. 51101137 and No. 51171161).
Table 3 Calculated magnetic moments of each atom and χ-Fe5 xCrxC2 (x ¼1–5).
References Species
Fe5C2 [12] Fe5C2 [5] Fe2Fe2CrC2 Fe2Cr2FeC2 Cr2Fe2FeC2 Cr2Fe2CrC2 Fe2Cr2CrC2 Cr2Cr2FeC2 Cr5C2
I (8f site)
II (8f site)
III(4e site)
C(8f site)
Ms
(μB/atom)
(μB/atom)
(μB/atom)
(μB/atom)
(μB/f.u.)
2.18 2.00 1.92(FeI) 1.64(FeI) 1.13(CrI) 0.94(CrI) 1.45(FeI) 0 0
1.72 1.74 1.67(FeII) 0.16(CrII) 1.45(FeII) 1.57(FeII) 0.13(CrII) 0 0
1.16 1.39 1.00(CrIII) 1.29(FeIII) 0.58(FeIII) 0.52(CrIII) 0.56(CrIII) 0 0
0.20 – 0.07 0.10 0.02 0.02 0.01 0 0
8.96 8.81 6.08 4.73 0.09 1.76 2.59 0 0
In the Fe2Fe2CrC2 cell, the magnetic moments of FeI and FeII are positive, while that of CrIII is negative. In the Cr2Fe2FeC2 cell, the magnetic moments of CrI and FeIII are negative, but that of CrIII is negative. The Ms of CrI in Cr2Fe2CrC2 and CrIII in Fe2Cr2CrC2 are negative and opposite to that of metal atom at other sites. The main reason is that the interaction of Fe and Cr in different sites makes the 3d states of metal atom in the majority states (up) move to above the Femi level and some metal atoms (Fe or Cr) are undergone the anti-ferromagnetic transformation. This phenomenon is very interesting, which was also found in χ-Mn5C2 [12] and βMn [23]. Unfortunately, as far as , we know, there are no experimental data available related to these properties in the literature for χ carbides (Fe,Cr)5C2, therefore our calculated values can be considered as prediction of these properties for these compounds. Future experimental work will testify our calculated results.
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Effect of Cr on electronic and magnetic properties of χ-carbide (Fe,Cr)5C2 B. Wang a, Q. Zhang b, Z.F. Zhang b, Z.Q. Lv b,n, W.T. Fu a,nn a b
State Key Laboratory of Metastable Material Science and Technology, Yanshan University, Qinhuangdao 066004, China College of Mechanical Engineering, Yanshan University, Qinhuangdao 066004, China
art ic l e i nf o
a b s t r a c t
Article history: Received 1 November 2014 Received in revised form 8 May 2015 Accepted 8 May 2015 Available online 9 May 2015
From density-function theory calculation, the structural, electronic and magnetic properties of χ-carbides (Fe,Cr)5C2 are investigated. With the increase of Cr content in χ-carbides (Fe,Cr)5C2, the formation energy of χ carbide gradually decrease and energy stability of them increase. The formation energy of Cr5C2 is 0.354 eV/f.u, and the stability of Cr5C2 is higher than other χ carbides (Fe,Cr)5C2, Mn5C2 and Fe5C2. There exists charges transfer from metal cation (Fe/Cr) to C atoms in χ-carbides, and this reveals an ionic contribution to the bonds. The addition of Cr decreases the magnetic moments of χ carbide, and the magnetic moments (Ms) of Cr2Cr2FeC2 and Cr5C2 are 0 μB/f.u., while it expresses opposite magnetic characters of the same atom at different sites in the other χ type (Fe,Cr)5C2 carbides. The 3d states of metal atoms in the majority states (up) move to above the Femi level and some metal atoms (Fe/Cr) in χ type (Fe,Cr)5C2 are undergone the anti-ferromagnetic transformation. & 2015 Elsevier B.V. All rights reserved.
Keywords: χ carbides Phase stability Electronic structure Magnetic behavior
1. Introduction Carbides in steels play a critical role during the process of heat treatment and hot working, which also belong to the most important strengthening phases in carbon steels and white cast. In Fe–C system, these carbides include the well-known θ-Fe3C (cementite), Fe2C (ε and η) and Fe5C2 (Hägg carbide) etc. [1,2]. While some common alloying elements (such as Cr, Mn, Si etc) are often added to steels in order to improve their properties, and then the structure and properties of carbides are also changed due to forming alloying carbides [3,4]. Some researchers [5–8] investigated the structural and electronic properties of iron carbides in Fe–C alloy. Faraon and co-workers [5] calculated the crystalline, electronic, and magnetic structures of θ-Fe3C, χ-Fe5C2 and η-Fe2C using the PLAPW method, and the stability relationship between the three carbides was confirmed. Leineweber et al. [6] studied the structure and elastic properties of χ carbide Fe5C2 using first principles calculations and diffraction experiments. The powderdiffraction data was in agreement with the DFT calculations. Lv and co-workers [7,8] calculated the electronic and magnetic properties of Fe3C (ε and θ) and Fe2C (ε and η) using the pseudopotential plane–wave within the density functional theory, and pointed out the differences in mechanical stability, formation energy and magnetic properties between them. Some researchers n
Corresponding author. Corresponding author. E-mail address: [email protected] (Z.Q. Lv).
nn
http://dx.doi.org/10.1016/j.jmmm.2015.05.023 0304-8853/& 2015 Elsevier B.V. All rights reserved.
[9–12] also studied the phase stability and electronic structure of alloying carbides in steels. Zhou et al. [9] calculated the electronic structure and formation energy of Fe11CrC4 and Fe10Cr2C4 with the Cambridge Serial Total Energy Package (CASTEP) code. Lv et al. [10,11] studied the magnetic properties and phase stability of cementite-type (Fe,M)3C (M¼Cr/Mn/Co/Ni) by means of the pseudopotential-based density functional theory. Recently Gou et al. [12] studied the electronic structures and phase stability of χcarbides (Fe,Mn)5C2 using first-principles technique, and confirmed that Mn enhanced the stability of χ-carbides. Like Mn, element Cr is also often existing in the steels and form alloying carbides such as (Fe,Cr)3C [10], (Fe,Cr)7C3 [13,14], and M23C6 [14,15]. However, the structure and properties of Cr alloying carbides χ-(Fe,Cr)5C2 are not very clear. The present work aims to study the magnetic properties and phase stability of χ-Fe5 xCrxC2 (x¼1–5) using the first-principles calculations and to clarify the phase stability of them in the sight of energy. Cr and Mn doping also affect the magnetic characters of these carbides and alloys [10– 12,14,15], which also play an important role in the design and improvement of materials. In order to make clear the effect of Cr doping on the magnetic properties of χ-carbides, the magnetic moments of χ-Fe5 xCrxC2 (x ¼1–5) and each atom are also calculated from the electronic structure.
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2. Crystal structure and calculation details The crystal structure of the χ-M5C2 carbide (Hägg carbides) is presented in Fig. 1. It is monoclinic with space group C2/c(15) that includes 4 symmetry operations and 14 atoms in the conventional cell [16]. The Wyckoff positions of the atoms are MI 8f (x1, y1, z1), MII 8f (x2, y2, z2), MIII 4e (0, y3, 0.25), and C 8f (x4, y4, z4). When the Fe atoms in different site are replaced by Cr, it form the multicomponent carbides χ-Fe5 xCrxC2 (x ¼1–5). According to the sites of metal atoms in χ type carbides, M5C2 carbides can be expressed as M'2M''2M'''C2 format (see Table 1). All calculations were performed based on the pseudo-potential plane–wave within the density functional theory [17,18]. The exchange-correlation potential was evaluated using the Perdew– Burke–Ernzerhof (PBE) functional [19] within the generalized gradient approximation (GGA) [20]. The interactions between the core and valence electrons were described by ultra-soft pseudopotentials [21], and the Kohn–Sham one-electron states were expanded in a plane wave basis set up to 400 eV. The energy calculations in the first irreducible Brillouin zone were conducted by using 2 6 5 k-point grid of Monkhorst–Pack scheme [22]. The convergence criteria for structure optimization and energy calculation were set to fine quality with the tolerance for the stress concentration factor (SCF), energy, maximum force and maximum displacement of 10 6 eV/atom, 10 5 eV/atom, 0.03 eV/Å and 0.001 Å, respectively. Because of its large effect on magnetic systems, spin polarization was included in the calculations to correctly account for its magnetic properties.
3. Results and discussions 3.1. Structural and energetic stability of the phases The ground state properties of the χ-Fe5 xCrxC2(x¼1–5) are investigated from their total energy, which is calculated as a function of volume. According to the Murnaghan equation of state, the equilibrium lattice constants and total energies of χ-Fe5 xCrxC2(x¼ 1–5) can be obtained (see Table1). The relative stability of Fe5 xCrxC2 can be discussed with energy calculations. The formation energy of a carbide (Fe5 xCrxC)
Table 1 Calculated lattice constants a, b, c (Å), β, v and total cell energy (eV/f.u) of χ-Fe5 xCrxC2 (x¼ 1–5).
Fe5C2 (Ref.12) (Ref.15) Fe2Fe2CrC2 Fe2Cr2FeC2 Cr2Fe2FeC2 Cr2Fe2CrC2 Fe2Cr2 CrC2 Cr2Cr2FeC2 Cr5C2 (Ref.16) Mn5C2 (Ref.12) (Ref.16)
a
b
C
β
v
11.569 (11.588) 11.620 11.453 11.673 11.757 11.519 11.426 11.564 11.804 (11.672)
4.487 (4.579) 4.507 4.527 4.428 4.463 4.535 4.495 4.573 4.528 (4.586)
4.973 (5.059) 4.998 5.039 5.029 5.057 5.080 5.160 5.140 5.100 (5.097)
97.59 (97.75) 97.12 97.68 97.16 97.08 97.73 97.80 97.53 97.49 (97.72)
255.89 (265.99) 259.69 258.91 257.88 263.28 262.96 262.58 269.45 270.24 (270.39)
Total cell energy 4636.48 6239.05 7841.43 7841.49 9443.93 9443.98 11046.49 12649.03 3578.33
can be defined as follows: Ebinding (Fe5 − x Crx C 2) = Etotal (unit) − (5 − x ) Eisolate (Fe ) − xEisolate (Cr) − 2Eisolate (C)
(1)
ΔEf (Fe5 − x Crx C 2 ) = Ebinding (unit) − (5 − x ) Ebinding (Fe ) − xEbinding (Cr) − 2Ebinding (C)
(2)
Here, the binding energy and the formation enthalpy per formula unit are represented by Ebinding (Fe5 xCrxC) and ΔEf (Fe5 xCrxC), respectively. Etotal(unit) is the total energy of the calculated unit. Ebinding(Fe/Cr) is the binding energy of pure element Fe per atom and Eisoalte(Fe/Cr) denotes the total energy of an isolated Fe/Cr atom, finally. The binding energy of pure elements was calculated by using their ground state conventional cell as BCC for Fe, as FCC for Cr and graphite for C. The total energies of isolated atoms were taken from the CASTEP output files directly. It is noted that such estimations from the total energy differences (at T ¼0 K) neglect the vibrations. At T ¼0 K and p ¼0 Pa, the formation enthalpy equals the calculated formation energy, i.e.ΔH (Fe5 − x Crx C2) = ΔE (Fe5 − x Crx C2), when the zero-vibration contribution is ignored [2,8]. The calculated cohesive energy and formation enthalpy of Fe5 xCrxC2 are shown in Table 2. The ΔE of Cr5C2 is 0.354 eV/f.u., which is negative and smaller than the ΔE of Fe5C2 (0.136 eV/f.u) and Mn5C2 ( 0.257 eV/f.u.) [12]. This shows that Cr5C2 is more stable than Fe5C2 and Mn5C2. Fe2Cr2FeC2 and Cr2Fe2FeC2 are with the same contents of Fe/Cr, but the calculated formation energies are different between them. This result is also obtained from the comparison between Cr2Fe2CrC2 and Fe2Cr2CrC2. This indicates that the bonding energies of metal atoms in different sites of χ-M5C2 cell are different. 3.2. Electronic properties Basing on the results from the electronic structure of
Converting
Table 2 Calculated cohesive energy and formation enthalpy of χ-Fe5 xCrxC2 (x ¼ 1–5) (eV/f. u.)
Conventional cell
Primitive cell
Fig. 1. Crystal structure of monoclinic χ-M5C2.
Fe5C2 [12] Mn5C2 [12] Fe2Fe2CrC2 Fe2Cr2FeC2 Cr2Fe2 FeC2 Cr2Fe2CrC2 Fe2Cr2CrC2 Cr2Cr2FeC2 Cr5C2
Binding energy
Formation enthalpy
65.81 64.98 66.84 67.67 67.74 68.64 68.68 69.65 70.64
0.136 0.257 0.066 0.086 0.016 0.044 0.084 0.214 0.354
B. Wang et al. / Journal of Magnetism and Magnetic Materials 392 (2015) 1–5
2 -2 2
CrII
0 -2 2
FeIII
0 -10
-5 0 5 Energy (E-EF) (eV)
10
CrIII
0
CrI
0
-2 -15
2
C
15
DOS (states/eV,atom)
DOS (states/eV,atom)
1 0 -1
3
-2 2
FeII
0 -2 2
FeI
0 -2 1 0 -1 -15
C -10
-5
0
5
10
15
Energy (E-EF) (eV)
Fig. 2. Calculated partial spin-polarized density of states, below 15 eV are not shown (a) Cr2Cr2FeC2 (b) Fe2Fe2CrC2.
Fig. 3. Calculated partial spin-polarized density of states, below 15 eV are not shown (a) Cr2Fe2FeC2 (b) Fe2Cr2FeC2.
χ-Fe5 xCrxC2 (x¼ 1–4) at equilibrium, the plots of density of states (DOSs) were discussed. Spin polarization results from a nearly rigid band shift between the majority spins (up) to lower energy and the minority spins (down) to higher energy due to the gain of energy from exchange. The spin-polarized densities of states for different atoms are shown from Fig. 2 to Fig. 4 at theoretical equilibrium lattice constants. The spin-polarized densities of states of χ-(Fe,Cr)5C2 are similar to that of Fe5C2 and Mn5C2 [12], there are three regions: the lowest valence band, the upper valence band, and the conduction unoccupied states. Comparing the up with down densities, it can be found that the up and down states of C atom is nearly symmetric, but that of Cr and Fe are not symmetric except in Cr2Cr2FeC2 (Fig. 2a). Actually, the lowest valence bands of metal atoms are also almost symmetric, and it is near the Fermi level, the hybridization of C 2p and Fe 3d, that the up and down states are noticeably dissimilar for different χ-(Fe,Cr)5C2 carbides. The lowest valence band ranging from about 14 to 11 eV is consisted of a main C 2s and a small contribution from s, p and d states of metal atoms Fe and Cr. The energy gaps between the lower valence bands and the upper valence bands indicates that the chemical bonds of χ-(Fe,Cr)5C2 take on ionicity. The second part valence band ranges from 7.5 eV to Fermi level, which is constituted by Fe/Cr 3d and the hybridization of C 2p and Fe/Cr 3d. The conduction unoccupied states are mainly formed by 3d of metal atoms.
From Fig. 5 to Fig. 7, the electron density distribution maps of the plane with metal and C atoms of Fe5 xCrxC2 (x ¼1–4) are plotted with the electron density difference map. The electron density difference maps are usually defined as the electron density difference between the isolated atoms and the bonding states with the same atomic configurations, which reflect the loss and gain of electrons in spaces. In this work, the electron density difference was determined as Δρ = (ρcrystal − ∑ ρat ) [7,10], where ρcrystal and
ρat are the valence electron densities for Fe5 xCrxC2 (x¼ 1–4) and corresponding free atoms, respectively. It can be seen that the increment of delocalized electrons is attributed to the metallic bonds in the interstitial regions. The increment of valence electrons is concentrated on the C atoms and the elongated contours correspond to the p-like orbits of C. It can be seen that the electron densities near Fe/Cr atoms decrease and the electron densities of nonmetal C atoms increase. There exists the loss and gain of electrons between Fe/Cr and C atom along the M–C direction. The increment of valence electrons is concentrated on the C atoms. In the interstitial regions, the increment of delocalized electrons is attributed to the metallic bonds. The M–C bonds take on ionicity and the M–M bonds are metallic bonds. It can be seen that χ-Fe5 xCrxC2 (x¼ 1–4), like Fe5C2 and Mn5C2, have mixed characters of metallicity, covalence and ionicity. Combined with the Mulliken population analysis, the loss and
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B. Wang et al. / Journal of Magnetism and Magnetic Materials 392 (2015) 1–5
2
2
CrIII
-2 2
CrII
0 -2 2
FeI
0 -2 1 0 -1 -15
C -10
-5 0 Energy (E-E F) (eV)
5
10
CrIII
0
DOS (states/eV,atom)
DOS (states/eV,atom)
0
-2 2
FeII
0 -2 2
CrI
0 -2 1 0 -1 -15
C
-10
-5 0 5 Energy (E-EF) (eV)
10
15
Fig. 4. Calculated partial spin-polarized density of states, below 15 eV are not shown (a) Fe2Cr2CrC2 (b) Cr2Fe2CrC2.
Cr
Fe C-Cr
Cr
Fe C-Fe
Fe
Fe
C-Cr
Cr
Cr
Fe Cr
Fe
Cr
Fe C
Fe
Fe
C-Cr
Cr
Fe
C-Fe Fe
Fe
Cr
Fe C
C
-0.10
Fe C
Cr
Cr
C
C
C
Fe
Cr
Fe
Fe
C
Fe
0
C
0.10
Cr C
Cr
C
C Fe
C-Fe
Fe
-0.10
0
Fe
0.10
Fig. 5. Electron density differences maps of the plane with metal atom and C atoms plane of: (a) Cr2Cr2FeC2 (b) Fe2Fe2CrC2 the density values are plot from 0.1 electron/Å3 (blue) to þ0.1 electron/Å3 (red). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 6. Electron density differences maps of the plane with metal atom and C atoms plane of: (a) Cr2Fe2FeC2 (b) Fe2Cr2FeC2 the density values are plot from 0.1 electron/Å3 (blue) to þ 0.1 electron/Å3 (red). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
gain electrons of atoms can be clearly exhibited. From the Mulliken population analysis for Cr2Cr2FeC2, the charges of C, CrI, CrII and FeIII atoms distribute to 0.59, þ0.14, þ0.29 and þ0.32, respectively; as for Fe2Fe2CrC2, the charges of C, FeI, FeII and CrIII atoms are 0.66, þ0.27,þ 0.24 and þ0.31, respectively. As for Cr2Fe2FeC2, the charges of C, CrI, FeII and FeIII atoms distribute to 0.63, þ 0.08, þ0.32 and þ0.45, respectively; for Fe2Fe2CrC2, the charges of C, FeI, CrII and FeIII atoms distribute to 0.64, þ0.28, þ0.22 and þ0.28, respectively. As for Cr2Fe2CrC2, the charges of C, CrI, FeII and CrIII atoms distribute to 0.59, þ0.14, þ0.29 and þ0.32, respectively; for Fe2Cr2CrC2, the charges of C, FeI, CrII and CrIII atoms distribute to 0.60, þ0.4, þ0.15 and þ0.12, respectively. The loss and gain of electrons is obvious between Fe/Cr and C atom form the data, which shows the iconicity on the M–C bond. The variations of population values of M–C bonds are from 0.28 to 0.36 in χ-Fe5 xCrxC2 (x ¼1–4), which indicates no repulsion force among these atoms.
3.3. Magnetic properties The spin moments of each atom were calculated from the partial spin-polarized density of states, which were listed in Table 3. It can found that the addition of Cr decreases the magnetic moments of χ carbides, especially Cr in I site. This is mainly due to anti-ferromagnetic transformation of metal atom. Some negative values of M–M overlap population in χ-Fe5 xCrxC2 (x ¼1–4) indicates that there exists repulsion force between these metal atoms and the near neighbors. This may prove the anti-ferromagnetic transformation of metal atoms from the view of mechanics. The magnetic moments (Ms) of Cr2Cr2FeC2 and Cr5C2 are 0 μB/f.u., while other χ carbides (Fe,Cr)5C2 express different magnetic characters. The magnetic moments (Ms) of all χ-Fe5 xCrxC2 (x ¼1–5) are smaller than that of Fe5C2, 8.96 μB/f.u. [12]. Like Fe5C2 and Mn5C2 [5,12], the magnetic moments of metal atoms are also different in the different site of χ-(Fe,Cr)5C2 cell (except from Cr2Cr2FeC2, 0 μB).
B. Wang et al. / Journal of Magnetism and Magnetic Materials 392 (2015) 1–5
4. Conclusions
C
Cr
C-Cr
C-Cr
Fe C-Fe
Fe
Cr
C-Cr
Fe C-Cr
Cr
Fe
C Cr
Cr Cr Fe
Fe
Fe
Fe
Cr
C
C
Cr
C
Cr
C
-0.10
0
5
0.10
Fig. 7. Electron density differences maps of the plane with metal atom and C atoms plane of: (a) Fe2Cr2CrC2 (b) Cr2Fe2CrC2 the density values are plot from 0.1 electron/Å3 (blue) to þ0.1 electron/Å3 (red). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Using first-principles technique, a completely theoretical analysis of the phase stability, electronic and magnetic properties of χ carbides (Fe,Cr)5C2 have been presented. The formation enthalpy of Cr5C2 is 0.354 eV/f.u., and Cr5C2 is more stable than Fe5C2 and Mn5C2. Fe2Cr2FeC2 and Cr2Fe2FeC2 are with the same contents of Fe/Cr, but the calculated formation energies are different between them. This result is also obtained from the comparison between Cr2Fe2CrC2 and Fe2Cr2CrC2. This indicates that the bonding energies of metal atoms in different sites of χ-M5C2 cell are different. There exists the loss and gain of electrons between Fe/Cr and C atom, which reveals an ionic contribution to the bonds in the χ type covalent structure. The addition of Cr decreases the magnetic moments of χ carbides. The magnetic moments (Ms) of Cr2Cr2FeC2 and Cr5C2 are 0 μB/f.u., while it expresses opposite magnetic characters of the same atom in different site in the other χ (Fe, Cr)5C2 carbides. The 3d states of Cr (Fe) in the majority states (up) move to above the Femi level and some metal atoms (Fe or Cr) in χ-(Fe,Cr)5C2 carbides are undergone the anti-ferromagnetic transformation.
Acknowledgments This research was supported by the National Natural Science Foundation of China (No. 51101137 and No. 51171161).
Table 3 Calculated magnetic moments of each atom and χ-Fe5 xCrxC2 (x ¼1–5).
References Species
Fe5C2 [12] Fe5C2 [5] Fe2Fe2CrC2 Fe2Cr2FeC2 Cr2Fe2FeC2 Cr2Fe2CrC2 Fe2Cr2CrC2 Cr2Cr2FeC2 Cr5C2
I (8f site)
II (8f site)
III(4e site)
C(8f site)
Ms
(μB/atom)
(μB/atom)
(μB/atom)
(μB/atom)
(μB/f.u.)
2.18 2.00 1.92(FeI) 1.64(FeI) 1.13(CrI) 0.94(CrI) 1.45(FeI) 0 0
1.72 1.74 1.67(FeII) 0.16(CrII) 1.45(FeII) 1.57(FeII) 0.13(CrII) 0 0
1.16 1.39 1.00(CrIII) 1.29(FeIII) 0.58(FeIII) 0.52(CrIII) 0.56(CrIII) 0 0
0.20 – 0.07 0.10 0.02 0.02 0.01 0 0
8.96 8.81 6.08 4.73 0.09 1.76 2.59 0 0
In the Fe2Fe2CrC2 cell, the magnetic moments of FeI and FeII are positive, while that of CrIII is negative. In the Cr2Fe2FeC2 cell, the magnetic moments of CrI and FeIII are negative, but that of CrIII is negative. The Ms of CrI in Cr2Fe2CrC2 and CrIII in Fe2Cr2CrC2 are negative and opposite to that of metal atom at other sites. The main reason is that the interaction of Fe and Cr in different sites makes the 3d states of metal atom in the majority states (up) move to above the Femi level and some metal atoms (Fe or Cr) are undergone the anti-ferromagnetic transformation. This phenomenon is very interesting, which was also found in χ-Mn5C2 [12] and βMn [23]. Unfortunately, as far as , we know, there are no experimental data available related to these properties in the literature for χ carbides (Fe,Cr)5C2, therefore our calculated values can be considered as prediction of these properties for these compounds. Future experimental work will testify our calculated results.
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