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Anomalous physical properties in disordered alloys with strong chemical interactions
Solid State Communications, Vol. 53, No. 1, pp. 9 1 - 9 3 , 1985. Printed in Great ~ritain.
0038-1098/85 $3.00 + .00 Pergamon Press Ltd.
ANOMALOUS PHYSICAL PROPERTIES IN DISORDERED ALLOYS WITH STRONG CHEMICAL INTERACTIONS D. Mayou, D. Nguyen Manh, A. Pasturel*, F. Cyrot-Lackmann Laboratoire d'Etudes des Propri6t~s Electroniques des Solides, C.N.R.S., B.P. 166, 38042 Grenoble Cedex, France
(Received 17 September 1984 by E.F. Bertaut) Some alloys are characterized by an abnormal behaviour of their physical properties around a given composition; this composition can be estimated from a simple band-scheme with the assumption of the occurence of a pseudogap; this approach will be more particularly applied to semiconductor liquid alloys. great with respect to bandwidths; then the overlap of bands is small. On the other hand hybridization can mix the states of the initial bands and lead to a splitting into bonding and antibonding states. In a first part, we show that under reasonable assumption the states number on both sides of the pseudogap obeys a simple sum rule. Then this sum rule will be applied to some more characteristic alloys.
1. INTRODUCTION IN THE LAST FEW years, a number of liquid alloys have been discovered which, although formed by two metallic components, exhibit a non-metallic behaviour in a given composition range. The liquid Gold-Caesium alloy is one of the most typical examples and has received considerable attention, both experimental and theoretical [1, 2]. As a rule, this non metallic behaviour is characterized by large charge transfer and local ordering occurring throughout this same composition range. Examples of physical behaviour of these liquid alloys are extrema in the resistivity, in the density and magnetic susceptibility and important effects in Neutron diffraction experiments at a given composition [3, 4]. Highly negative thermodynamic data accompanied these phenomena and allowed some authors to analyse the behaviour of these alloys from the existence of chemical complexes or associates [5]. The associationmodel has certainly led some success in rationalizing the many available experimental data, in so far as it is a representative model but it does not estimate the critical composition for which all experimental data showed anomalies. Moreover, there is no information on the mechanism stabilizing the associates and their use is full of objections. Evidently a more fundamental description should start from the electronic structure of these alloys and could allow a better understanding of their behaviour; indeed following the Mott's intuitive arguments [6], the abnormal behaviour of numerous physical properties at a given composition in these alloys may be interpreted as being due to the occurence of a pseudogap in the density of states. Two mechanisms are generally proposed for the creation of a pseudogap in disordered systems. One is of ionic origin and appears when the difference in energy of the atomic levels is
2. THE SUM RULE The starting point of our approach is the description of the electronic properties of the alloys A 1-xBx from a Hamiltonian characterized by the different parameters which are atomic sites energies including charge transfer and hopping integrals between each site and its neighbours. For simplicity, we discuss first these alloys for which there is one orbital on each lattice site the values of atomic sites energies in the alloy being eA* and eB* (we choose e.A* < eB*). Let us consider the fictitious situation obtained for a large separation of atomic site energies i.e.e.A M and e/fit (e.AM < eBM); the density of states spectrum of the alloy would then be composed of two bands separated by a gap and constituted of A-type and B-type states respectively. Then, we assume that the gap is conserved when the atomic energies varies from cA M to e.,4* and e / ~ to eB* respectively. The eigenenergies varying continuously, they cannot go through the gap from one band to the other and then the states number in each subband is conserved, equal to that one of initial atomic states although their eigenstates are mixed by hybridization. In a general way, we call S÷ and S_ the atomic states, including charge transfer, which are respectively above and below the pseudogap. Then we assume the existence of a continuous transformation which conserves the pseudogap and leads to a separation of the spectrum in two subbands of pure S÷ and S_ states. Therefore, the number of states of the real system below and above the pseudogap must be equal to the number of S_ and S÷
*L.T.P.C.M.-E.N.S.E.E.G.B.P. 75, 38042 Saint Martin d'H6res Cedex, FRANCE. 91
92
DISORDERED ALLOYS WITH STRONG CHEMICAL INTERACTIONS
states respectively. Our argument is that of the wellknown localization theorem [7]. We want to turn now to the question of the depth of the pseudogap; indeed our approach is strictly valid if a gap occurs; in the case of pseudogap, as long as it remains well-defined, we assume that the states conservation can also be used; typically experimental values of conductivity can give information on the depth of the pseudogap [6] ; using this criterion, we can say that our approach remains valid for alloys which are either in the localized electron range (o < 300 fZ-1 cm -1 ) or in the diffusive motion range (300 g2-1 cm -I < o < 3000 g2-1 cm- 1). 3. DISCUSSION In the following discussion, the use of the sum rule will allow us to calculate the critical composition x* which corresponds to the filling of states below the pseudogap by the valence electrons. The average number of states below the pseudogap is given by: D a v = xDB + (1--X)DA,
Zav = x Z z + (1 - - X ) Z A ,
(2)
where ZB and ZA are the valence numbers of B and A. The critical composition x* corresponds to the equality of equation (1) and (2): X* =
Table 1. Comparison between experimental an~l calculated critical composition.
system
x* cal
x* exp.
cr ~2-a cm -1
Au-Rb Au-Cs Au-K Mg-Bi Li-Bi Li-Pb Cs-Sb Na-Pb Na-Bi Na-Sb
0.5 0.5 0.5 0.4 0.25 0.2 0.25 0.2 0.25 0.25
0.5 0.5 0.5 0.4 0.25 0.2 0.25 0.2 0.25 0.25
5.5 3.0 1100 45 500 2100 2.0 2400 528 5.0
Table 2. Comparison between experimental and calculated compos#ion for tellurides.
system
(1)
where DB andDA are the number of atomic levels of B and A below the pseudogap and the average number of valence electrons is given by:
Vol. 53, No. 1
Cu-Te Ag-Te Sn-Te Pb-Te Zn-Te Cd-Te Ga-Te In-Ye TI-Te
x* cal.
x* exp.
o
0.33 0.33 0.5 0.5 0.5 0.5 0.33 0.33 0.33
0.33 0.33 0.5 0.5 0.5 0.5 0.6 0.6 0.33
500 100 1400 1100 50 50 13 60 70
~'~-1 cm-I
ZA -- DA DB -- DA + ZA -- ZB "
(3)
For the alkali metals-gold or polyvatent metals alloys A a-xBx (A = Cs, Li, Na and B = Pb, Sn, Bi, Sb, Au), the atomic levels of the more electronegative metal, i.e. B, are located well below the atomic levels of the more electropositive metal [8]. In Table 1, we compare the so calculated critical composition with the experimental critical composition for the more characteristic systems; we have shown the experimental values of conductivities which are available information of the depth of the pseudogap [9, 10]. It may be asked whether such an approach can be applied to other important systems like tellurides. We must quote that in the case of polyvalent metal-Te alloys, the atomic level ns 2 of the polyvalent metal is located at a lower energy than the atomic level 5p ¢ of Te [8] ; as a rule, the p states of the two elements are mainly concerned by the alloying effect and the formation o f the pseudogap may occur between the two p subbands. Table 2 gives comparison between experimental and calculated critical composition; we can see that the agreement is satisfying expected for Ga,
In-Te alloys. In this case, as it has been already suggested by Cutler [9], we assume that the charge transfer is sufficiently important to inverse the ns 2 atomic level of the polyvalent metal and the atomic level 5p 4 of the Tellure; then, all the states of Te are located below the gap and we obtain a calculated critical composition equal to 0.6, in good agreement with the experimental one. 4. CONCLUSION Generally, the critical composition is calculated from the valence rule which however requires a total charge transfer. The proposed sum rule generalizes this valence rule even in the cases where the states on both sides of the pseudogap are mixed and charged transfer is not complete. REFERENCES 1.
H. Hoshino, R.W. Schmutzler & F. Hensel, Phys.
2.
Lett. AS1, 7 (1975). J.R. Franz, F. Brouers & C. Holzhey,J. Phys. F10,
235 (1980).
Vol. 53, No. 1 3. 4. 5. 6.
DISORDERED ALLOYS WITH STRONG CHEMICAL INTERACTIONS
A. Boss & S. Steeb,Phys. Lett. 63A, 333 (1977). N. Nicoloso, R.W. Schmutzler & F. Hensel, Ber. Buns, Phys. Chem. 82, 62l (1978). B. Predel & G. Oehme, Z. Metallkunde 70,450 (1979). N.F. Mott and E.A. Davis Electronic Processes in Non-Crystalline Materials (Oxford: Clarendon) 1971.
7. 8. 9. 10.
93
S. Kirkpatrick, B. Velicky & H. Ehrenreich, Phys. Rev. BI, 3250 (1970). F. Herman & S. Skillman, Atomic Structure Calculations (Prentice-Hall, Englewood Cliffs, New Jersey) 1963. M. Cutler, Liquid Semiconductors (Ed. by Academic Press, New York) 1977. W.F. Callaway & M.L. Saboungi, J. Phys. F. 13, 1213(1983).
0038-1098/85 $3.00 + .00 Pergamon Press Ltd.
ANOMALOUS PHYSICAL PROPERTIES IN DISORDERED ALLOYS WITH STRONG CHEMICAL INTERACTIONS D. Mayou, D. Nguyen Manh, A. Pasturel*, F. Cyrot-Lackmann Laboratoire d'Etudes des Propri6t~s Electroniques des Solides, C.N.R.S., B.P. 166, 38042 Grenoble Cedex, France
(Received 17 September 1984 by E.F. Bertaut) Some alloys are characterized by an abnormal behaviour of their physical properties around a given composition; this composition can be estimated from a simple band-scheme with the assumption of the occurence of a pseudogap; this approach will be more particularly applied to semiconductor liquid alloys. great with respect to bandwidths; then the overlap of bands is small. On the other hand hybridization can mix the states of the initial bands and lead to a splitting into bonding and antibonding states. In a first part, we show that under reasonable assumption the states number on both sides of the pseudogap obeys a simple sum rule. Then this sum rule will be applied to some more characteristic alloys.
1. INTRODUCTION IN THE LAST FEW years, a number of liquid alloys have been discovered which, although formed by two metallic components, exhibit a non-metallic behaviour in a given composition range. The liquid Gold-Caesium alloy is one of the most typical examples and has received considerable attention, both experimental and theoretical [1, 2]. As a rule, this non metallic behaviour is characterized by large charge transfer and local ordering occurring throughout this same composition range. Examples of physical behaviour of these liquid alloys are extrema in the resistivity, in the density and magnetic susceptibility and important effects in Neutron diffraction experiments at a given composition [3, 4]. Highly negative thermodynamic data accompanied these phenomena and allowed some authors to analyse the behaviour of these alloys from the existence of chemical complexes or associates [5]. The associationmodel has certainly led some success in rationalizing the many available experimental data, in so far as it is a representative model but it does not estimate the critical composition for which all experimental data showed anomalies. Moreover, there is no information on the mechanism stabilizing the associates and their use is full of objections. Evidently a more fundamental description should start from the electronic structure of these alloys and could allow a better understanding of their behaviour; indeed following the Mott's intuitive arguments [6], the abnormal behaviour of numerous physical properties at a given composition in these alloys may be interpreted as being due to the occurence of a pseudogap in the density of states. Two mechanisms are generally proposed for the creation of a pseudogap in disordered systems. One is of ionic origin and appears when the difference in energy of the atomic levels is
2. THE SUM RULE The starting point of our approach is the description of the electronic properties of the alloys A 1-xBx from a Hamiltonian characterized by the different parameters which are atomic sites energies including charge transfer and hopping integrals between each site and its neighbours. For simplicity, we discuss first these alloys for which there is one orbital on each lattice site the values of atomic sites energies in the alloy being eA* and eB* (we choose e.A* < eB*). Let us consider the fictitious situation obtained for a large separation of atomic site energies i.e.e.A M and e/fit (e.AM < eBM); the density of states spectrum of the alloy would then be composed of two bands separated by a gap and constituted of A-type and B-type states respectively. Then, we assume that the gap is conserved when the atomic energies varies from cA M to e.,4* and e / ~ to eB* respectively. The eigenenergies varying continuously, they cannot go through the gap from one band to the other and then the states number in each subband is conserved, equal to that one of initial atomic states although their eigenstates are mixed by hybridization. In a general way, we call S÷ and S_ the atomic states, including charge transfer, which are respectively above and below the pseudogap. Then we assume the existence of a continuous transformation which conserves the pseudogap and leads to a separation of the spectrum in two subbands of pure S÷ and S_ states. Therefore, the number of states of the real system below and above the pseudogap must be equal to the number of S_ and S÷
*L.T.P.C.M.-E.N.S.E.E.G.B.P. 75, 38042 Saint Martin d'H6res Cedex, FRANCE. 91
92
DISORDERED ALLOYS WITH STRONG CHEMICAL INTERACTIONS
states respectively. Our argument is that of the wellknown localization theorem [7]. We want to turn now to the question of the depth of the pseudogap; indeed our approach is strictly valid if a gap occurs; in the case of pseudogap, as long as it remains well-defined, we assume that the states conservation can also be used; typically experimental values of conductivity can give information on the depth of the pseudogap [6] ; using this criterion, we can say that our approach remains valid for alloys which are either in the localized electron range (o < 300 fZ-1 cm -1 ) or in the diffusive motion range (300 g2-1 cm -I < o < 3000 g2-1 cm- 1). 3. DISCUSSION In the following discussion, the use of the sum rule will allow us to calculate the critical composition x* which corresponds to the filling of states below the pseudogap by the valence electrons. The average number of states below the pseudogap is given by: D a v = xDB + (1--X)DA,
Zav = x Z z + (1 - - X ) Z A ,
(2)
where ZB and ZA are the valence numbers of B and A. The critical composition x* corresponds to the equality of equation (1) and (2): X* =
Table 1. Comparison between experimental an~l calculated critical composition.
system
x* cal
x* exp.
cr ~2-a cm -1
Au-Rb Au-Cs Au-K Mg-Bi Li-Bi Li-Pb Cs-Sb Na-Pb Na-Bi Na-Sb
0.5 0.5 0.5 0.4 0.25 0.2 0.25 0.2 0.25 0.25
0.5 0.5 0.5 0.4 0.25 0.2 0.25 0.2 0.25 0.25
5.5 3.0 1100 45 500 2100 2.0 2400 528 5.0
Table 2. Comparison between experimental and calculated compos#ion for tellurides.
system
(1)
where DB andDA are the number of atomic levels of B and A below the pseudogap and the average number of valence electrons is given by:
Vol. 53, No. 1
Cu-Te Ag-Te Sn-Te Pb-Te Zn-Te Cd-Te Ga-Te In-Ye TI-Te
x* cal.
x* exp.
o
0.33 0.33 0.5 0.5 0.5 0.5 0.33 0.33 0.33
0.33 0.33 0.5 0.5 0.5 0.5 0.6 0.6 0.33
500 100 1400 1100 50 50 13 60 70
~'~-1 cm-I
ZA -- DA DB -- DA + ZA -- ZB "
(3)
For the alkali metals-gold or polyvatent metals alloys A a-xBx (A = Cs, Li, Na and B = Pb, Sn, Bi, Sb, Au), the atomic levels of the more electronegative metal, i.e. B, are located well below the atomic levels of the more electropositive metal [8]. In Table 1, we compare the so calculated critical composition with the experimental critical composition for the more characteristic systems; we have shown the experimental values of conductivities which are available information of the depth of the pseudogap [9, 10]. It may be asked whether such an approach can be applied to other important systems like tellurides. We must quote that in the case of polyvalent metal-Te alloys, the atomic level ns 2 of the polyvalent metal is located at a lower energy than the atomic level 5p ¢ of Te [8] ; as a rule, the p states of the two elements are mainly concerned by the alloying effect and the formation o f the pseudogap may occur between the two p subbands. Table 2 gives comparison between experimental and calculated critical composition; we can see that the agreement is satisfying expected for Ga,
In-Te alloys. In this case, as it has been already suggested by Cutler [9], we assume that the charge transfer is sufficiently important to inverse the ns 2 atomic level of the polyvalent metal and the atomic level 5p 4 of the Tellure; then, all the states of Te are located below the gap and we obtain a calculated critical composition equal to 0.6, in good agreement with the experimental one. 4. CONCLUSION Generally, the critical composition is calculated from the valence rule which however requires a total charge transfer. The proposed sum rule generalizes this valence rule even in the cases where the states on both sides of the pseudogap are mixed and charged transfer is not complete. REFERENCES 1.
H. Hoshino, R.W. Schmutzler & F. Hensel, Phys.
2.
Lett. AS1, 7 (1975). J.R. Franz, F. Brouers & C. Holzhey,J. Phys. F10,
235 (1980).
Vol. 53, No. 1 3. 4. 5. 6.
DISORDERED ALLOYS WITH STRONG CHEMICAL INTERACTIONS
A. Boss & S. Steeb,Phys. Lett. 63A, 333 (1977). N. Nicoloso, R.W. Schmutzler & F. Hensel, Ber. Buns, Phys. Chem. 82, 62l (1978). B. Predel & G. Oehme, Z. Metallkunde 70,450 (1979). N.F. Mott and E.A. Davis Electronic Processes in Non-Crystalline Materials (Oxford: Clarendon) 1971.
7. 8. 9. 10.
93
S. Kirkpatrick, B. Velicky & H. Ehrenreich, Phys. Rev. BI, 3250 (1970). F. Herman & S. Skillman, Atomic Structure Calculations (Prentice-Hall, Englewood Cliffs, New Jersey) 1963. M. Cutler, Liquid Semiconductors (Ed. by Academic Press, New York) 1977. W.F. Callaway & M.L. Saboungi, J. Phys. F. 13, 1213(1983).