Oct 15, 1993 - MC and AN compounds are taken from previous ab initio linear-mufBn-tin-orbitals ... transition-metal carbides and nitrides in the NaC1 crystal.
PHYSICAL REVIEW B
15 OCTOBER 1993-II
VOLUME 48, NUMBER 16
Theory of bonding in transition-metal Department
carbides and nitrides
J. Haglund of Theoretical Physics, Royal Institute of Technology, S-IOO gg Stockholm, Sweden
A. Fernandez Guillermet Consejo Nacional de Investigaciones Cientificas y Tecnicas, Centro Atomico Bariloche, 8/00 San Carlos de Bariloche, Argentina Department
G. Grimvall and M. Korling of Theoretical Physics, Royal Institute of Technology, S IOO-gg Stockholm, Sweden (Received 15 January 1993; revised manuscript received 12 May 1993)
This paper deals with the bonding properties of 3d-, 4d-, and 5d-transition-metal carbides and nitrides. We consider NaCI-structure compounds, MC and MN, as well as carbides and nitrides with more complex crystal structures. The enthalpies of formation at zero temperature A H(0) of the calculations. We MC and AN compounds are taken from previous ab initio linear-mufBn-tin-orbitals describe the structure as formed by a metal fcc lattice in which nonmetal atoms have been inserted at interstitial positions. 4 H(0) is divided into contributions from (i) formation of the metallic fcc lattice, (ii) expansion of this lattice to the lattice spacing of the compound, and (iii) insertion of nonmetal atoms into the metal lattice. E H(0), plotted versus the position of M in the d series, shows a pronounced maximum for the Ti, Zr, and Hf carbides. In agreement with other work, we interpret this as due to the filling of bonding p dhybri-dized states. The maximum in A H(0) is followed 6rst by a decrease and then by an almost constant value. We interpret this as an efFect of a gradual population of antibonding and nonbonding electronic states. Size e8'ects, i.e. , contribution (ii), are small for the 4d- and 5d-series compounds. E H(0) of MN is similar to that of MC, but the largest values occur for M=Sc, Y, La since N has one more valence electron than C. Bonding in the complex carbides and nitrides is given an analogous interpretation.
I. INTRODUCTION The fundamental problem of understanding cohesion and bonding in solids can be approached in many ways, The bonding leading to different kinds of insight. strength is often measured by a single quantity, e.g. , the H or the cohesive energy E, h. enthalpy of formation H or E, h into sepIt may be instructive to divide arate contributions that can be given simple interpretations. The separation is not unique, and its relevance Ab should be tested on a large number of materials. calculations provide a useful initio electronic-structure and flexible tool for such studies, but at present they yield results that contain significant errors. A theoretical discussion of bonding mechanisms should therefore be preceded by a detailed comparison between ab initio results and experimental information in order to get a good understanding of the discrepancies between theory and experiment. ab iniWe have recently carried out extensive calculations and assessments tio electronic-structure information on 3d-, 4d-, and Gdof thermodynamic transition-metal carbides and nitrides in the NaC1 crystal structure, altogether 48 compounds. Further, we have information on the cohesive considered thermodynamic properties of complex 3d-transition-metal (M) carbides M&C2 ) M2C M7C3 M5C2 MBC and M23C6 and on nitrides. An ab inisome complex 3d-transition-metal
tio calculation has also been performed for Fe3C. In the present paper, we rely on these results in an analysis of cohesive properties and bonding energies. Although our previous works contain values both &om theoretical calculations and from assessments of thermodynamic information, the discussion in this paper will be based only on cohesive properties as obtained from ab initio linear-muffin-tin-orbitals (LMTO) band-structure calculations. This is sufhcient for our purposes since we have shown that ab initio calculations, when correctly interpreted, yield cohesive energies and enthalpies of formation in good agreement with experiment, except for an almost constant energy shift. The NaCl structure compounds formed by metals with C and N are often referred to as interstitial compounds. They are then conceived as a metallic lattice in which nonmetal atoms have been inserted at interstitial positions. In order to accommodate the nonmetal atoms, the metallic lattice may have to expand, thus reducing part of the energy associated with metal-metal bonds. Here, we adopt this view in a quantitative study of contributions to the bonding energy. We distinguish between (i) the energy corresponding to the formation of a metal fcc lattice from free metal atoms, (ii) the energy change when expanding this lattice to the lattice spacing of the carbide or nitride, and (iii) the change in chemical bonding energy due to insertion of nonmetal atoms into the metal lattice. The main thermodynamic facts we
0163-1829/93/48{16)/11685{7)/$06.00
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4
&
48
1993
The American Physical Society
J. HAGI. UND
the interpretation of variations in L H. In the following analysis, we will study the formation of a NaCl structure compound MX &om the metallic element in the fcc phase. Then we need the enthalpy difference between M in its stable structure at zero kelvin, M~' ~, and in the fcc structure, M~ "~. Thus, we define
100
E C) C$ 1
O
o
E
SO
et aL
—~
AE„,„,[M] = Et, [M( ")(a
6 7
where ao
ter of
0
M&
E [M('
C4
I
I
SC
Ti
I
V
I
CI
I
I
Fe
Mn
Co
Ni
FIG. 1. Enthalpies of formation b, H(TO) at
To
—298.15
is the theoretical
"~, i.e. ,
want to account for are illustrated in Fig. 1. It shows the enthalpy of formation, b. H(To) at To —298. 15 K, for 3dtransition-metal carbides in the NaCl structure and for several complex 3d-transition-metal carbides. Of particular interest is the maximum in bonding strength for the Ti carbides, the tendency for an almost constant H in the later part of the series, and the difFerence between the NaC1 structure carbides and those having more complex
4
structures. Our paper is organized as follows. In Sec. II we consider the NaCl structure compounds. The relevant thermodynamic quantities are defined, and results from In LMTO band-structure calculations are presented. Sec. III we interpret these ab initio results in terms of a simple model that accounts for hybridization efFects. Section IV contains a discussion of the bonding properties of complex carbides and nitrides. The paper ends with conclusions in Sec. V.
II. (cF8) NaC1-STRUCTURE A. Thermodynamic
~
[M(fcc) (
lattice paramethat minimizes
MA
)]
E
[M(fcc) ( M)]
(2) Using the de6nitions above, the enthalpy of formation per atom in the compound at zero kelvin can be written
A'H(0) [MX'"'] = AE, 1
.„,
[MX'")] 1
+ bE.„,& [M] +— b, E„,„.[M]. —(3) Here we have introduced a term that accounts for the change in chemical bonding energy when carbon or nitrogen atoms are inserted into the metal fcc lattice while keeping the lattice parameter constant,
KEb „q[MX(")] = —(E, [MX(")] —E, , [M( ")(aM )] 2
.
-E. .[x(")] &.
(4)
This quantity has contributions &om M-X interactions, as well as &om changes in M-M bonding energies. AEb „g[MX ' ] will be used in Sec. III for a discussion of hybridization and band-filling efFects in NaCl structure transition-metal carbides and nitrides.
COMPOUNDS
B. A. b
initio results
de6nitions
Experimental information on bonding energies of solid compounds is generally expressed as enthalpies of formation (boH) (cf. Fig. 1). This quantity represents the energy gained in forming a compound (e.g. , MX, with X=C,N) from the constituent elements in their stable (st) modifications M(' ) and X(' ) at some reference temperature. Usually, this corresponds to metals in a certain crystal structure (e.g. , bcc, fcc, or hcp), carbon in the graphite form and nitrogen as a diatomic gas [N2(g)]. In this work, the reference states of carbon and nitrogen are unimportant. In analogy with our previous works, we therefore choose the &ee carbon and nitrogen atoms C~ and N~ as reference states. This choice corresponds to a constant shift in A H and it does not afFect ~
equilibrium
the lattice parameter
)] We also want to separate out the efFect of volume changes in the formation of MX ' . The energy required to expand M~ "& &om the theoretical equilibrium lattice parameter ao to the experimental (or estimated, see beof the compound MX(' ) is low) lattice spacing a
K from thermodynamic
information for carbides of the 3d-transition metals. Curve 1 refers to carbides with the NaCl structure while curves 2 —7 show b. H(To) values for complex carbides with compositions M3CQ M2C M7C3 M5C2 M3C, and M23C6, respectively. Prom Ref. 4.
)] —Et t [M(")],
Following the definitions in Sec. IIA, we have calculated b, E,„&g[M] and bE, t, , [M] using the LMTO method. Our version relies on the semirelativistic and mufBn-tin-potential The present calapproximations. culations were all carried out with the GunnarssonLundqvist density-functional parametrization. Figure 2 shows AE, t, [M) for the 3d-, 4d-, and 5d-transition metals discussed in this work. We note that the largest enthalpy difFerences are found among the elements that are stable in the bcc structure at zero kelvin, i.e. , V, Cr, Fe, Nb, Mo, Ta, and W. Except for Fe, these difFerences are positive, as expected, but they are much larger in magnitude than what is obtained by analyzing thermodynamic information. However, what is relevant for the present study is that bE, t,,„,[M] is much smaller in mag-
„,
„,
THEORY OF BONDING IN TRANSITION-METAL CARBIDES. . .
48 50
I
I
I
I
I
I
— ---
40—
3d 4d
V Cr
10—
0— 10
(f~c)
Sc Tl
20—
g
TABLE I. Ab initio calculated equilibrium lattice parameters ao" of 3d-, 4d- and 5d-transition metals in the fcc structure.
.--- 5d
30— E
I
Mn
I
I
I
Sc Ti Y La
I
V Nb
Zr Hf
Ta
I
I
Cr Mn Fe Mo Tc Ru W Re Os
I
I
Co
Ni
Rh Ir
Pd Pt
11 687
Fe Co Ni
(fee)
(ace)
ao
ao
ao
(a.u. ) 8.55
(a.u. ) 9.40 8.52 8.00 7.61 7.37 7.24 7.24 7.37
(a.u. ) 10.05 8.39 7.98 7.67 7.46 7.36 7.37 7.50
Y
7.68 7.16 6.81 6.59 6.49 6.46 6.53
Zr Nb
Mo
Tc Ru Rh Pd
La Hf
Ta W Re Os Ir
pt
FIG. 2. Calculated structural differences enthalpy AE, &,„,[M] for 3d-, 4d-, and 5d-transition metals [cf. Eq. (1)].
nitude than the variations in A H(0) (Fig. 3). Thus, the fact that the stable structure of the transition metals at zero kelvin changes along the Periodic Table can only account for a small or negligible part of the variation in the total enthalpy of formation of their NaCl structure compounds. The equilibrium lattice parameters ao for all transition metals in the fcc structure are listed in Table I. These values were obtained by Gtting a Murnaghan equation of state to total-energy results from LMTO bandof structure calculations. The lattice parameters a
the carbides and nitrides studied in this paper are listed in Refs. 1—3. All a refer to the stoichiometric composition of MX. The volume expansion energies AE, „pg[M], defined in Eq. (2), are plotted in Fig. 4. We find that AE, „~~[M] has almost the same behavior in all three transition metal series, with a maximum in their later parts. This is understood by noting that AE, „~g[M] approximately equals the product of the bulk modulus and the square of the volume expansion. The calculated variation of these quantities is shown in Fig. 5. The magnitude of AE, „~d[M] is smaller in the 4d and 5d series than in the 3d series. This reQects the larger di8'erence between ao and ao . Still, the variation of b, E,„~g[M] is relatively small, compared with the variations in 4 H(0) (Fig. 3).
150—
400—
100—
300—
50200— I
I
Sc Ti
I
V
I
I
I
I
I
Sc Ti
Cr Mn Fe Co Ni
400—
300—
8
E
200—
N
'K
100—
V
I
Y
Zr
Nb Mo Tc
Ru
Rh
Pd
I
I
Y
400—
30-
300—
20-
200—
I
I
Zr Nb Mo Tc
I
I
Ru Rh
I
Pd
10MN
100— I
La Hf
Cr Mn Fe Co Ni
50403020j0
Ta W
I
Re Os
I
I
Ir
Pt
FIG. 3. Calculated enthalpies of formation at zero kelvin, A H'(0), for 3d-, 4d-, and 5d-transition-metal carbides and nitrides.
La Hf
Ta W
Re Os
Ir
Pt
FIG. 4. Calculated volume expansion energies AE „~s[M] for 3d-, 4d-, and 5d-transition metals [cf. Eq. (2)]. Note the different scales on the energy axis.
J. HAGLUND
11 688
—
500
400 — ~
---
300— Q
3d 4d 5d
~I
200— 100— p
1
I
—
60 5p
40—
3d 4d
302010— p
I
I
I
I
Sc Ti Y Zr
Nb
La
Ta
Hf
V
Cr Mn Fe Mo Tc Ru W Re Os
Co
Ni
Rh
Pd Pt
Ir
FIG. 5. Calculated bulk moduli K and volume
difFerences with respect to NaCl structure carbides and nitrides AV for 3d-, 4d-, and 5d-transition metals in the fcc structure.
III. INTERPRETATIONS AND SIMPLE MODELS
The bonding in NaCl structure transition-metal carbides and nitrides has been studied previously by us For our discussion beand by several other authors. low, we review the general behavior of the electronic energy spectrum in this class of compounds. Figure 6 shows the electronic density of states (DOS) of FeC. It is common practice to consider atom and partial I projections of the total density of states. For systems with more than since one type of atom, this separation is not unique it depends on how the crystal space is divided between the different types of atoms. This is one reason why considerations of projected DOS curves may be dificult or misleading. However, Fig. 6 shows that the lowest-lying Total DOS
I
I
48
et al.
electronic states originate from carbon 8 electrons. No significant electronic DOS is observed at the metal sites in this interval. The energy range that follows the carbon 8 band contains electronic states with both metal and nonmetal character. The essential contributions come from nonmetal p states and from metal d states, which leads to the picture that hybridized p-d states dominate the bonds between metal and nonmetal atoms. For symthere are metal d orbitals with small or metry reasons, negligible overlap with carbon p orbitals. These states contribute to metal-metal bonds in the compound. Our results, based on a quantitative separation of contributions to A H(0) as expressed by Eq. (3), lead to the following picture for the bonding in NaCl structure transition-metal carbides and nitrides. In the beginning of the transition-metal series, the Blling of bonding phybridized states gives rise to maxima in the magnitude of AEb~„g for TiC, ZrC, and HfC (Fig. 7). Since the structural enthalpy diff'erences AE, t, „,[M] and the volume expansion energies b, E,„~g[M] are small, these maxima are present also in b, oH(0) (Fig. 3). The nitrogen atom fills the electronic DOS with one more electron than does the carbon atom. Within a rigid-band model, this explains why the maximum bonding strength occurs already at the first compounds in the series (ScN, YN, and l.aN). This interpretation of the bonding in the NaCl structure compounds formed by carbon or nitrogen with metals in the erst part of the transition-metal series agrees with that of most of the works cited above. The decrease in E H(0) on going &om group V to VII in the Periodic Table for the carbide series, and from group IV to VI for the nitride series, is partly due to an increased volume expansion term AE, „~g[M] in Eq. (3). However, the major reason for the lowered enthalpy of
l
400—
I
I I
MN
50.0—
200— Sc Ti
0.0
V
Cr Mn Fe Co Ni
400O
50.0— C4
04
0.0 10.0— 0.0 5, 0—
0.0 10.0— 0.0 20.0 0.0
-
300—
I
I I I I I I I I
Fe s
I I
200— Y
Zr Nb Mo Tc Ru Rh
Pd
I I
400— C p
300—
~ C s
200—
-0.2 0.0 0.2 0.4 0.6 0.8 1.0 Energy (Ry)
FIG. 6. Calculated electronic density of states for FeC in the NaCl structure. Note the difFerent scales on the DOS axis.
100— La Hf
Ta W
Re Os
FIG. 7. Calculated bonding energies and Sd-transition metals [cf. Eq. (4)].
Ir
Pt
—AE'b „g for
3d-, 4d-,
48
THEORY OF BONDING IN TRANSITION-METAL CARBIDES. . .
formation is a decrease in the magnitude of LEb „g. This is seen by comparing Figs. 3, 4, and 7. We conclude that the electronic energy levels at the Fermi levels of these compounds are best described as antibonding states. For the compounds formed by metals in the last part of the transition-metal series, the AoH(0) values from our ab initio calculations remain almost constant, Fig. 3. This trend is confirmed by our analysis of thermodynamic information. The same slow variation is also present in AEb~„g (Fig. 7) for the 4d- and 5d-series compounds, while LEb „g in the later part of the 3d series shows a slow decrease in magnitude. It is reasonable to describe the electronic states being filled in these compounds as since they do not increase the cohesive ennonbonding, ergy of the compound, relative to that of the pure metal. It is possible that the different trend in the later part of the 3d series is due to magnetic effects in these elements. Figure 8 gives a schematic view of the bonding, antibonding, and nonbonding character of the electronic DOS for the present NaC1 structure compounds. Our model DOS emphasizes the relatively low DOS values in the bonding and antibonding regions compared with the large DOS in the nonbonding energy range. This observation will be used in the following section to qualitatively account for some differences between the NaC1 structure compounds and those with more complex crystal structure. It should be remarked that there is no a priori reason for the existence of a clean separation of the electronic energy spectrum into bonding, antibonding, and nonbonding regions. Such a simplified view can only be adopted if bonding energies, as in this case (Fig. 7), show a clear pattern of variation. In contrast with the transition kom bonding to antibonding states, which often ' is connected with a well-defined minimum in the electronic DOS, the antibonding or nonbonding character of the later part of the MX DOS (Fig. 6) cannot be easThis is partly due to the ambiguous ily distinguished. separation of crystal space in atomic DOS projections which, e.g. , may cause tails of metallic d orbitals to be seen as nonmetal p states. A second complication is that the bonding and antibonding states may be nonsymmetrically distributed on different types of atoms. Schwarz
MX
8
A
N
0A M~X
B
A
N
Energy
FIG. 8. Schematic electronic densities of states illustrating the bonding (H), antibonding (A), and nonbonding (N) character of difFerent energy regions in the energy spectrum for NaC1 structure carbides and nitrides (upper curve) and M3X compounds (lower curve).
11 689
has shown that the antibonding part of the p-d hybridized states has strong metal d character while the nonmetal p character is more pronounced in the bonding electronic states. We suggest that this effect explains why only a small nonmetal p contribution is seen in the antibonding region of the DOS in NaCl structure MX compounds. Gelatt, Williams, and Moruzzi focused on NaCl structure transition-metal monoborides, as an example of interstitial metal-nonmetal compounds, but AoH(0) curves for NaC1 structure carbides and nitrides of the 4d-transition metals were also presented. Prom ab initio calculations of the enthalpy of formation, they concluded that the decrease in the magnitude of 6 H(0) in the middle of a transition-metal series (cf. Fig. 3) is mainly a volume effect, i.e. , it reHects a loss of metalmetal bonding energy as the metal expands to form a metal-nonmetal compound. Moreover, they observed a decrease in the magnitude of A H(0) towards the end of the transition-metal series, which was explained as due to the filling of antibonding p-d hybridized states. Our interpretation of the variation in E H(0) for NaCl structure compounds leads to a picture which is different Rom that of Gelatt, Williams, and Moruzzi We find that the decrease in the magnitude of b, H(0) in the middle of the transition-metal series is mainly the result of reduced chemical bonding, while volume effects account only for a minor part of the variation. Further, we observe an almost constant 4 H(0) in the last part of the transition-metal series. In that region, KoH(0) obtained by Gelatt, Williams, and Moruzzi decreases in magnitude, which explains why they come to a different conclusion regarding the bonding character of the states in the upper part of the electronic energy spectrum.
IV. COMPLEX CARBIDES AND NITRIDES We now turn to the complex carbides M3C2, M2C, M7C3 MSC2) M3C, and M23C6. Many of them have been studied experimentally, when M is a transition metal in the later part of the 3d series. That enabled us to derive4 the enthalpies of formation H(To) which are shown in Fig. 1. We shall now give an interpretation of variations in A H(To), based on the electronic structure. The large number of atoms per unit cell needed to describe the crystal structures of the complex carbides is the reason why an ab initio calculation for alt of them is not yet realistic. We have previously performed bandstructure calculations on Fe3C in its ferromagnetic state. In the present discussion, we will partly invoke bandfilling arguments, and it is more relevant to consider Fe3C in a nonmagnetic electronic configuration. The resulting DOS is shown in Fig. 9. We note that many of the general features in the electronic DOS of NaCl structure compounds (Fig. 6) are found also for FesC. For instance, the lowest lying band is still of carbon s character and well separated f'rom other states. Further, the dominant contributions to the central part of the valence energy spectrum come from carbon p states and metal d states. There, we distinguish a low-energy part which contains
6
J. HAGLUND
11 690
p-d hybridized states and a region which mostly consists of states with metal d character. An important difference between the energy spectrum of Fe3C and that of the NaCl structure compounds is that the DOS for M3C lacks the pronounced. structures that are seen in the DOS of the MC compounds. This is qualitatively explained. by the lower lattice symmetry and the larger number of atoms per crystallographic unit cell in the M3C compounds. In our previous studies, we found that the DOS of all the transition-metal carbides of the NaCl crystal strucTherefore we expect that our ture are very similar. DOS for Fe3C is characteristic also of the other MSC transition-metal compounds. Further, we assume that the general differences found between the DOS of the NaCl (Fig. 6) and the Fes C structure compounds (Fig. 9) hold in a comparison with other complex transition-metal carbide s. We will now argue that many of the features of A H in Fig. 1 can be understood in terms of our simplified model DOS curves as in Fig. 8. Consider the first part of the transition-metal series. The rise in the magnitude of L H in this region for NaCl structure carbides was explained above as due to the filling of bonding, p-d hybridized states. The same increase is seen for all complex carbides. Since we showed that the DOS of Fe3C in this energy range consists of hybridized p-d states, it is reasonable to invoke the same explanation also for complex carbides. We note that the broad minimum which separates bonding and antibonding states in the DOS of the NaC1 structure compounds is not present for Fe3C. This leads to a smaller bandwidth and a larger average DOS for the p-d hybridized region in FesC (cf. Fig. 8) and thus to a smaller energy gain associated with filled bonding p-d hybridized states. A second factor which afFects the increase in the magnitude of L H is the relative carbon (or nitrogen) content of the compound. A lower nonmetal concentration will lead to fewer metalnonmetal bonds and thus to a lower energy associated with these bonds. This agrees well with the sequence of L H curves for complex carbides in Fig. 1. In analogy to what was concluded for the NaCl structure compounds, we suggest that the decrease in the magnitude of H for the complex systems in the middle of the transition-metal series is due to the filling of antibonding states; cf. Fig. 8. This decrease is more pronounced for NaCl structure compounds, since the lower DOS in this energy range implies a population of higher energy states. Finally, the almost constant behavior of A H in the last part of the transition-metal series agrees well with the 611ing of nonbonding states. However, one should note that the L H values in this energy region depend on the position of the nonbonding region in the energy spectrum. Thus, if the nonbonding DOS is displaced towards higher energies for NaCl structure compounds than for complex carbides and nitrides, as illustrated in Fig. 8, the magnitude of the corresponding H will be lower. We suggest that this is the main reason why the A H(TO) curves for the complex carbides and the NaCl structure carbides tend to different values in the later part of the transition-metal series (Fig. 1).
4
4
48
et al. Total DOS
400.0—
I)
200.0-
0.0
I I
Fed
I I
400.0—
I
200.0— CC
0.0
Fe P
400
rA
0A C/0
I
4p. p—
00 n C
1000
I
I
%VX
I
P
I I I I
0.0
I
100.0-
I I I
I I
0.0
I
I
I
g
-0.4 -0.2 0.0 0.2 0.4 0.6 0.8 1.0 Energy (Ry)
FIG. 9. Calculated electronic density of states for nonmagnetic Fe3C. Note the diferent scales on the DOS axis.
4
data on H for some stable comThermodynamic plex 3d-transition-metal nitrides show the same qualitative behavior as for the corresponding carbides. In view of the similarities in the electronic structures of the NaCl structure carbides and nitrides of the 3d series, we expect that the interpretation above of the bonding in complex 3d-transition-metal carbides can be extended to the nitrides. Likewise, we expect complex carbides and nitrides formed by 4d- and 5d-transition metals to follow the same pattern.
V. CONCLUSIONS The purpose of this paper has been to increase the understanding of bonding in transition-metal carbides and nitrides. Our discussion is based on total energies &om ab initio electron-structure calculations. Since ab initio results are known to have significant absolute errors, we have in previous works critically compared them with information. thermodynamic We conclude &om those works that ab initio results on enthalpies of formation H and cohesive energies E, g are adequate for the present analysis of the trends in bonding energies. We have studied in detail the 3d-, 4d-, and 5dtransition-metal carbides and nitrides in the NaCl crystal structure. It is found that L H for the carbides has a characteristic variation along a d series, with an initial increase to a maximum, followed by a d.ecrease and then an almost constant value. These three regions can be given a simple interpretation in terms of the successive Ailing of bonding, antibonding, and nonbonding electron states. The bonding and antibonding states are formed by hybridized carbon p states and metal d states, i.e. , they correspond to metal-nonmetal bonds. The nonbonding part of the electronic energy spectrum is dominated by metal d states, corresponding to metal-metal bonds.
4
THEORY OF BONDING IN TRANSITION-METAL CARBIDES. . .
48
When an interstitial compound is formed &om the constituent elements, one is led to consider the influence on L H &om differences in the atomic size and in the structural enthalpy for the pure elements. We And that these effects are of little importance for the overall variation in 4 H along the transition-metal series. Transition metals M may form a large number of compounds having crystal structures more complex than NaCl, e.g. , with compositions M3C2 M2C, MyC3 M5C2, M3C, and M23C6. There is ample evidence that L H for such systems has a behavior along a transition-metal series similar to that of the NaCl compounds, but with a less pronounced maximum. We suggest that the general pattern of variation in 4 H can be understood within the same picture of bonding, antibonding, and nonbonding
J. Haglund,
G. Grimvall, T. Jarlborg, and A. Fernandez Guillermet, Phys. Rev. B 43, 14400 (1991). A. Fernandez Guillermet, J. Haglund, and G. Grimvall, Phys. Rev. B 45, 11 557 (1992). A. Fernandez Guillermet, J. Haglund, and G. Grimvall, preceding paper, Phys. Rev. B 48, 11 673 (1993). A. Fernandez Guillermet and G. Grimvall, J. Phys. Chem. Solids 53, 105 (1992). A. Fernandez Guillermet and G. Grimvall, Phys. Rev. B 40, 10 582 (1989). J. Haglund, G. Grimvall, and T. Jarlborg, Phys. Rev. B
44, 2914 (1991). G. Hagg, Z. Phys. Chem. B 12, 33 (1931). H. 3. Goldschmidt,
Interstitial Alloys (Butterworths,
Lon-
don, 1967).
L. E. Toth, Transition Metal Carbides and Nitrides (Academic, New York, 1971). O. K. Andersen, Phys. Rev. B 15, 3060 (1975). O. Gunnarsson and B. I. Lundqvist, Phys. Rev. B 13, 4274 (1976). A. Fernandez Guillermet and M. Hillert, Calphad 12, 377 (1988). N. Saunders, A. P. Miodownik, and A. T. Dinsdale, Calphad 12, 351 (1988). G. Grimvall, M. Thiessen, and A. Fernandez Guillermet,
11 691
states. This view is consistent with the results of our ab initio calculation for Fe3C. The fact that the magnitude of 4 H decreases with the carbon content of the complex carbides in the region of p-d hybridized states can be interpreted as a consequence of the reduced number of metal-nonmetal bonds. ACKNOW LEDGMENTS This work was supported by The Swedish National Board for Industrial and Technical Development, The Swedish Research Council for Engineering Sciences, The Swedish Natural Sciences Research Council, and The Goran Gustafsson Foundation.
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