- Email: [email protected]
On the heats of formation of solid germanides and silicides of transition metals
Physica 124B (1984) 247-250 North-Holland, Amsterdam
ON T H E H E A T S OF F O R M A T I O N OF S O L I D G E R M A N I D E S AND S I L I C I D E S OF TRANSITION METALS A. P A S T U R E L and P. H I C T E R Laboratoire de Thermodynamique et Physico Chimie Mdtallurgiques, E.N.S.E.E.G., Domaine Universitaire, B.P. 75, 38402 Saint Martin d'Heres, France
F. C Y R O T - L A C K M A N N G.T.P., C.N.R.S., B.P. 166, 38042 Grenoble Cedex, France
Received 4 July 1983 Silicides and germanides of transition metals are quite similar to other transition metal-p metal compounds, most of them being metallic. We show that the enthalpies of formation of these compounds have two contributions. The first is the energy necessary to convert Si or Ge from the non-metallic into the metallic state and the second is the result of the filling of the d band of the transition metal by the free valence electrons of metallic Si or Ge. This second contribution is calculated according to a band model and finally the calculated enthalpies of formation are compared with available experimental data.
1. Introduction
Pasturel et al. [1] have proposed a simple d band model for the prediction of the heats of formation for binary compounds of a transition metal with a p metal. This model has also been applied successfully to binary liquid systems of transition metals with a p metal or silicon [2]. These approaches have shown that the important negative enthalpies of mixing in these systems are due to the filling of the d band of the transition metal as it is alloyed with a p metal. This assumption of electronic rearrangement when alloying is used to evaluate the change in enthalpy A H = (1 - x) AE2 + x AE1, x being the concentration of the p metal AE1 representing the contribution of the energy due to the change of n u m b e r of free valence electrons, AE2 the contribution of the energy due to the change of n u m b e r of d electrons, In fact, this approach allows to estimate the
change in energy due to the transfer of an electron from an sp orbital to a d orbital. This change in energy may be important in so far as an sp electron has a delocalized character particularly m o r e pronounced than a d electron. T o have a quantitative idea of this change in energy we choose to write that the filling of the d band of transition metal when alloying with a p metal leads the cohesive energy of the alloy to lose its d character. Friedel [6] explained the typical behaviour of the cohesive energy in a transitional metal as due to the d character of their valence states. H e used a tight-binding scheme, whose principles are well known and which gives the essential features of the d band. This procedure permits us to compare directly the energies in the solid and in the gas. The energy in the solid is then given simply by the sum of the one-electron band energies, so that the cohesive energy may be equaled to the d band contribution, rEF E¢ = J ( E 0 - E ) n d ( E ) d E .
0378-4363/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
(1)
A. Pasturel et al. / On the heats of formation of solid germanides and silicides of transition metals
248
E0 is the atomic d level which broadens, as the orbitals overlap, into a band whose density of states is nd(E). This simple formula used by Friedel [6], C y r o t - L a c k m a n n [7] and Ducastelle and C y r o t - L a c k m a n n [8] explains the typical behaviour of the cohesive energy in transitional metals. Any series presents a m a x i m u m of cohesive energy for a nearly half filled band, of the order of several electron volts. By assuming a rectangular d band density of states of width w, Friedel [6] reproduced the parabolic trend of this cohesive energy with band filling,
Ec = d (Eo - w/2) + dw ( 1 0 -
d),
(2)
where d is the n u m b e r of d electrons. Then with the assumption of the filling of the d band of transition metal when it is alloying with a p metal, the contribution AE2 of energy due to the change of n u m b e r of d electrons can be written as the difference between the d band contribution of the alloy and the d metal to the cohesive energy. So AE2 is
which the d band is filled. Similarly, AE1 can be written as EF
AE, = f ( E - B~)n~p(E)dE EFp
- f (E-B~)n~p(E)dE,
where EF and Evp are the Fermi energies of the alloy and the p metal, B~ and nsp(E) are the bottom of the sp band and its density of state, respectively. By use of the free electrons model, AEx = 3/5(EF-- EFp).
a2x .2 ax * AE2 - 2nd(E) ~-~ (d - 5) - B a x * .
where EF and EFa are the Fermi energies of alloy and metal, respectively. W e assume a linear charge transfer and
n~x* = (1 - x*)(10 - d ) .
d + ax* = 10,
(9)
For x > x* the average n u m b e r of free electrons is given by
ax
(d - 5),
(3)
2 = Zpx + (1 - x ) ( Z ° - 10 + d ) . with B = E condition,
(8)
where Zp and Z ° are the n u m b e r of s electrons of the p metal and the d metal, respectively, and ne is the n u m b e r of electrons transferred from the p metal to the d metal, ne is defined at the concentration x* by
- f ( E o - E)na(E) d E ,
a2x 2
(7)
When x < x * , the average n u m b e r of the free electrons by atom in the system is 2 = (1 - x ) Z ° + x(Zp - n~),
EFd
AE2 = - Bax + z,,atc )gs'-7+ -~ n - ~
(6)
These results are valid for 0 < x < x*. For x* < x < 1, the d band of the transition metal is filled and AE2 is given by
EF
AE2 = f ( E o - E)na(E) dE
(5)
w/2 and a is defined from the limit
(4)
where x* is the concentration of a p metal for
(lo)
The purpose of the present communication is to apply the above description of the heats of formation of alloys of d metals with p metals to the alloys of Si or Ge. On the one hand, the pure elements Si and G e are semiconductors with a quite large energy
A. Pasturel et al. / On the heats of formation of solid germanides and silicides of transition metals
gap. On the other hand, silicides and germanides are quite similar to the other transition metal-p metal compounds, most of them being metallic. Since the reference state of Si or Ge is then semiconducting, a metallic state for these elements must be required for calculating the metallic silicides or germanides and a transformation energy is necessary to convert Si or Ge from the non-metallic into the metallic state.
2. Calculation of the enthalpy of formation and discussion To convert Si or Ge into a metallic state, one has thus to take into account the energy needed to delocalize the valence electrons of Si or Ge. (Delocalization due in fact to the interaction with the d electron of the transition metal.) This is similar to what happens at the melting of these elements. Chakraverty [3] has evaluated the electronic contribution AS~ to the entropy of fusion of semiconductors, due to the delocalization of covalent electrons to free electrons. For these elements, the melting is accompanied by the destruction of covalent bands. Considering the different contributions between opposite spins the electronic contribution to the entropy of fusion can be given by
AS[ = 4fR in 2, where [ is a corrective term, given as the ratio of the number of the localized electrons in the solid and the number of free electrons in the liquid. The structure dependent energy is then taken
to be equal to TfAS~ and this is about 9 and 6.8kcalmo1-1, respectively for Si and Ge. So the transition metal silicides and germanides can be treated in the same way as the other compounds with p metals, but, in addition, there will be a positive contribution, i.e. TrASh, to AH which is a constant per gram atom of Si or Ge. In the same way, it is reasonable to assume that the characteristics of the valence electrons of Si or Ge in their metallic state are equal to those of corresponding liquids. The critical composition, x*, is related to the minimum of the experimental entropy or enthalpy of mixing curves of these alloys [1, 2] (see table I). As an example, predictions for heats of formation of (Ge, Co, Ni)-(Si, Ge) compounds have been tabulated and compared with experimental values [5, 6] (see table II). We can see that the agreement is reasonable and that the predicted heats of formation agree with the available information on phase diagrams. Our model shows that the nickel or cobalt silicides compounds are more stable than the iron silicides and are predicted to be stable for a higher concentration in silicon. The comparison between silicides and germanides shows that for a given transition metal, silicides compounds are more stable than germanides compounds. On the other hand, it can be interesting to have some information about the enthalpies of formation of silicides of 4d and 5d transition metals, for which the experimental data are very scarce. By using the same parameters from table I, the calculated enthalpies of formation of these compounds are presented in table III. Stable
Table I Electronic characteristics of metals Metal
Fe Co Ni Si Ge
nd
B
(eV-l)
(eV)
3 3 3
0.3 0.3 0.3
d
7 8.4 9.4
249
Vat
EFp
(cm3)
(eV)
7.1 6.7 6.7 11.2 13.2
12,8 11.6
Xs*
xoe
0.6 0.5 0.35
0.5 0.7
250
A. Pasturel et al. / On the heats o[ [ormation of solid germanides and silicides of transition metals Table II Comparison of calculated and experimental heats of formation Compound
-AH~I¢ -AHcxp (kcal atg -1)
Ref.
Compound
-AHcalc -AHexp (kcal atg -1)
Ref.
Co3Si Co2Si CoSi CoSi2 CoSi3 Fe3Si FeSi
6 7.9 10.7 8.3 5.7 4.4 9
5 5 5 5 5 5 5
Ni2Si Ni2Si NiSi NiGe NiGe CoGe2 CoGe CoGe
10 9 10.5 6.2 7.2 3.9 3.7 3.7
5 5 5 4 4 4 4 4
6.8 9.2 11 8.2 6.4 5 9.4
11 10.7 10.7 5.5 7.6 2.8 3.8 3.6
Table III Predicted heats of formation of solid silicides of 4d and 5d transition metals Compound
-AHf (kcal atg -1)
Compound
-AHf (kcal atg -x)
Pd2Si PdSi PdSi2 Rh2Si RhSi RhSi2 Ru2Si RuSi RuSi2
15.8 16 13.1 9 11.5 8.7 9 10.5 10.4
Pt2Si PtSi PtSi2 Ir2Si IrSi IrSi2 Os2Si OsSi OsSi2
14 16.5 12 3.5 6.5 5 5 4 5
silicide c o m p o u n d s for the three series are predicted and do exist. We can point out a m o r e drastic evolution of the enthalpies of formation in the 4d and 5d series, which can be attributed to a m o r e important difference of volumes between the elements of these two series.
3. Conclusion It was demonstrated that the model can be applied in general to silicides and germanides c o m p o u n d s and it is possible to predict the enthalpies of formation (table II). In comparison with the transition m e t a l - p metal systems, it is necessary to introduce a transformation energy
due to the delocalization of the valence electrons of Si or Ge. References [1] A. Pasturel, P. Hicter and F. Cyrot-Lackmann, J. of Less Common Metals 86 (1982) 181. [2] A. Pasturel, P. Hicter, D. Mayou and F. Cyrot-Lackmann, Scripta Met. 17 (1983) 841. [3] B.K. Chakraverty, J. Phys. Chem. Sol. 1 (1969) 2766. [4] B. Predel and W. Vogelbein, Thermochim. Acta 30 (1979) 201. [5] W. Oelsen and W. Middel, Mitt, KWI Einsenforsch 19 (1937) 1. [6] J. Friedel, in: the Physics of Metals, J.M. Ziman, ed. (Cambridge Univ. Press, London, 1969). [7] F. Cyrot-Laekmann, J. Phys. Chem. Solids 29 (1968) 1235. [8] F. Ducastelle and F. Cyrot-Lackmann, J. Phys. Chem. Solids 32 (1971) 285.
ON T H E H E A T S OF F O R M A T I O N OF S O L I D G E R M A N I D E S AND S I L I C I D E S OF TRANSITION METALS A. P A S T U R E L and P. H I C T E R Laboratoire de Thermodynamique et Physico Chimie Mdtallurgiques, E.N.S.E.E.G., Domaine Universitaire, B.P. 75, 38402 Saint Martin d'Heres, France
F. C Y R O T - L A C K M A N N G.T.P., C.N.R.S., B.P. 166, 38042 Grenoble Cedex, France
Received 4 July 1983 Silicides and germanides of transition metals are quite similar to other transition metal-p metal compounds, most of them being metallic. We show that the enthalpies of formation of these compounds have two contributions. The first is the energy necessary to convert Si or Ge from the non-metallic into the metallic state and the second is the result of the filling of the d band of the transition metal by the free valence electrons of metallic Si or Ge. This second contribution is calculated according to a band model and finally the calculated enthalpies of formation are compared with available experimental data.
1. Introduction
Pasturel et al. [1] have proposed a simple d band model for the prediction of the heats of formation for binary compounds of a transition metal with a p metal. This model has also been applied successfully to binary liquid systems of transition metals with a p metal or silicon [2]. These approaches have shown that the important negative enthalpies of mixing in these systems are due to the filling of the d band of the transition metal as it is alloyed with a p metal. This assumption of electronic rearrangement when alloying is used to evaluate the change in enthalpy A H = (1 - x) AE2 + x AE1, x being the concentration of the p metal AE1 representing the contribution of the energy due to the change of n u m b e r of free valence electrons, AE2 the contribution of the energy due to the change of n u m b e r of d electrons, In fact, this approach allows to estimate the
change in energy due to the transfer of an electron from an sp orbital to a d orbital. This change in energy may be important in so far as an sp electron has a delocalized character particularly m o r e pronounced than a d electron. T o have a quantitative idea of this change in energy we choose to write that the filling of the d band of transition metal when alloying with a p metal leads the cohesive energy of the alloy to lose its d character. Friedel [6] explained the typical behaviour of the cohesive energy in a transitional metal as due to the d character of their valence states. H e used a tight-binding scheme, whose principles are well known and which gives the essential features of the d band. This procedure permits us to compare directly the energies in the solid and in the gas. The energy in the solid is then given simply by the sum of the one-electron band energies, so that the cohesive energy may be equaled to the d band contribution, rEF E¢ = J ( E 0 - E ) n d ( E ) d E .
0378-4363/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
(1)
A. Pasturel et al. / On the heats of formation of solid germanides and silicides of transition metals
248
E0 is the atomic d level which broadens, as the orbitals overlap, into a band whose density of states is nd(E). This simple formula used by Friedel [6], C y r o t - L a c k m a n n [7] and Ducastelle and C y r o t - L a c k m a n n [8] explains the typical behaviour of the cohesive energy in transitional metals. Any series presents a m a x i m u m of cohesive energy for a nearly half filled band, of the order of several electron volts. By assuming a rectangular d band density of states of width w, Friedel [6] reproduced the parabolic trend of this cohesive energy with band filling,
Ec = d (Eo - w/2) + dw ( 1 0 -
d),
(2)
where d is the n u m b e r of d electrons. Then with the assumption of the filling of the d band of transition metal when it is alloying with a p metal, the contribution AE2 of energy due to the change of n u m b e r of d electrons can be written as the difference between the d band contribution of the alloy and the d metal to the cohesive energy. So AE2 is
which the d band is filled. Similarly, AE1 can be written as EF
AE, = f ( E - B~)n~p(E)dE EFp
- f (E-B~)n~p(E)dE,
where EF and Evp are the Fermi energies of the alloy and the p metal, B~ and nsp(E) are the bottom of the sp band and its density of state, respectively. By use of the free electrons model, AEx = 3/5(EF-- EFp).
a2x .2 ax * AE2 - 2nd(E) ~-~ (d - 5) - B a x * .
where EF and EFa are the Fermi energies of alloy and metal, respectively. W e assume a linear charge transfer and
n~x* = (1 - x*)(10 - d ) .
d + ax* = 10,
(9)
For x > x* the average n u m b e r of free electrons is given by
ax
(d - 5),
(3)
2 = Zpx + (1 - x ) ( Z ° - 10 + d ) . with B = E condition,
(8)
where Zp and Z ° are the n u m b e r of s electrons of the p metal and the d metal, respectively, and ne is the n u m b e r of electrons transferred from the p metal to the d metal, ne is defined at the concentration x* by
- f ( E o - E)na(E) d E ,
a2x 2
(7)
When x < x * , the average n u m b e r of the free electrons by atom in the system is 2 = (1 - x ) Z ° + x(Zp - n~),
EFd
AE2 = - Bax + z,,atc )gs'-7+ -~ n - ~
(6)
These results are valid for 0 < x < x*. For x* < x < 1, the d band of the transition metal is filled and AE2 is given by
EF
AE2 = f ( E o - E)na(E) dE
(5)
w/2 and a is defined from the limit
(4)
where x* is the concentration of a p metal for
(lo)
The purpose of the present communication is to apply the above description of the heats of formation of alloys of d metals with p metals to the alloys of Si or Ge. On the one hand, the pure elements Si and G e are semiconductors with a quite large energy
A. Pasturel et al. / On the heats of formation of solid germanides and silicides of transition metals
gap. On the other hand, silicides and germanides are quite similar to the other transition metal-p metal compounds, most of them being metallic. Since the reference state of Si or Ge is then semiconducting, a metallic state for these elements must be required for calculating the metallic silicides or germanides and a transformation energy is necessary to convert Si or Ge from the non-metallic into the metallic state.
2. Calculation of the enthalpy of formation and discussion To convert Si or Ge into a metallic state, one has thus to take into account the energy needed to delocalize the valence electrons of Si or Ge. (Delocalization due in fact to the interaction with the d electron of the transition metal.) This is similar to what happens at the melting of these elements. Chakraverty [3] has evaluated the electronic contribution AS~ to the entropy of fusion of semiconductors, due to the delocalization of covalent electrons to free electrons. For these elements, the melting is accompanied by the destruction of covalent bands. Considering the different contributions between opposite spins the electronic contribution to the entropy of fusion can be given by
AS[ = 4fR in 2, where [ is a corrective term, given as the ratio of the number of the localized electrons in the solid and the number of free electrons in the liquid. The structure dependent energy is then taken
to be equal to TfAS~ and this is about 9 and 6.8kcalmo1-1, respectively for Si and Ge. So the transition metal silicides and germanides can be treated in the same way as the other compounds with p metals, but, in addition, there will be a positive contribution, i.e. TrASh, to AH which is a constant per gram atom of Si or Ge. In the same way, it is reasonable to assume that the characteristics of the valence electrons of Si or Ge in their metallic state are equal to those of corresponding liquids. The critical composition, x*, is related to the minimum of the experimental entropy or enthalpy of mixing curves of these alloys [1, 2] (see table I). As an example, predictions for heats of formation of (Ge, Co, Ni)-(Si, Ge) compounds have been tabulated and compared with experimental values [5, 6] (see table II). We can see that the agreement is reasonable and that the predicted heats of formation agree with the available information on phase diagrams. Our model shows that the nickel or cobalt silicides compounds are more stable than the iron silicides and are predicted to be stable for a higher concentration in silicon. The comparison between silicides and germanides shows that for a given transition metal, silicides compounds are more stable than germanides compounds. On the other hand, it can be interesting to have some information about the enthalpies of formation of silicides of 4d and 5d transition metals, for which the experimental data are very scarce. By using the same parameters from table I, the calculated enthalpies of formation of these compounds are presented in table III. Stable
Table I Electronic characteristics of metals Metal
Fe Co Ni Si Ge
nd
B
(eV-l)
(eV)
3 3 3
0.3 0.3 0.3
d
7 8.4 9.4
249
Vat
EFp
(cm3)
(eV)
7.1 6.7 6.7 11.2 13.2
12,8 11.6
Xs*
xoe
0.6 0.5 0.35
0.5 0.7
250
A. Pasturel et al. / On the heats o[ [ormation of solid germanides and silicides of transition metals Table II Comparison of calculated and experimental heats of formation Compound
-AH~I¢ -AHcxp (kcal atg -1)
Ref.
Compound
-AHcalc -AHexp (kcal atg -1)
Ref.
Co3Si Co2Si CoSi CoSi2 CoSi3 Fe3Si FeSi
6 7.9 10.7 8.3 5.7 4.4 9
5 5 5 5 5 5 5
Ni2Si Ni2Si NiSi NiGe NiGe CoGe2 CoGe CoGe
10 9 10.5 6.2 7.2 3.9 3.7 3.7
5 5 5 4 4 4 4 4
6.8 9.2 11 8.2 6.4 5 9.4
11 10.7 10.7 5.5 7.6 2.8 3.8 3.6
Table III Predicted heats of formation of solid silicides of 4d and 5d transition metals Compound
-AHf (kcal atg -1)
Compound
-AHf (kcal atg -x)
Pd2Si PdSi PdSi2 Rh2Si RhSi RhSi2 Ru2Si RuSi RuSi2
15.8 16 13.1 9 11.5 8.7 9 10.5 10.4
Pt2Si PtSi PtSi2 Ir2Si IrSi IrSi2 Os2Si OsSi OsSi2
14 16.5 12 3.5 6.5 5 5 4 5
silicide c o m p o u n d s for the three series are predicted and do exist. We can point out a m o r e drastic evolution of the enthalpies of formation in the 4d and 5d series, which can be attributed to a m o r e important difference of volumes between the elements of these two series.
3. Conclusion It was demonstrated that the model can be applied in general to silicides and germanides c o m p o u n d s and it is possible to predict the enthalpies of formation (table II). In comparison with the transition m e t a l - p metal systems, it is necessary to introduce a transformation energy
due to the delocalization of the valence electrons of Si or Ge. References [1] A. Pasturel, P. Hicter and F. Cyrot-Lackmann, J. of Less Common Metals 86 (1982) 181. [2] A. Pasturel, P. Hicter, D. Mayou and F. Cyrot-Lackmann, Scripta Met. 17 (1983) 841. [3] B.K. Chakraverty, J. Phys. Chem. Sol. 1 (1969) 2766. [4] B. Predel and W. Vogelbein, Thermochim. Acta 30 (1979) 201. [5] W. Oelsen and W. Middel, Mitt, KWI Einsenforsch 19 (1937) 1. [6] J. Friedel, in: the Physics of Metals, J.M. Ziman, ed. (Cambridge Univ. Press, London, 1969). [7] F. Cyrot-Laekmann, J. Phys. Chem. Solids 29 (1968) 1235. [8] F. Ducastelle and F. Cyrot-Lackmann, J. Phys. Chem. Solids 32 (1971) 285.