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Cation disordering in Ag2HgI4 and Cu2HgI4
Solid State Communications, Vol. 23, pp. 629—631,1977.
Pergamon Press.
Printed in Great Britain.
CATION DISORDERING IN Ag
2HgI AND Cu2HgI4* S.N. Girvin and G.D. ~ahan Physics Department Indiana University Bloomington, Indiana 47401
(Received 13 June 1977 by A.A. Maradudin)
The cation order—disorder transitions in Ag2HgI4 and Cu211g14 are first order. This is unusual, since in other superionic conductors the cation disordering is gradual with temperature if there is no structural phase transition. These two materials are also unique in that they have two disordering cations rather than one. A study of a two species lattice gas model shows that this extra degree of freedom is responsible for the first order nature of this transition on the fcc lattice.
1.
Ag I,~doped with Hg. The divalent mercury ions in~r8duceholes into the acation 5 As resultlattice, of this, as Ag noted by Ketelaar. 211g14 satisfies the necessary (but not sufficient) condition for good ionic conduction that there be incomplete occupation of the available ca— tion sites. This condition is satisfied in the “undoped” crystal, cz—AgI only above 144°Cwhere
Introduction
Many superionic conductors exhibit one or more phase transitions at temperatures below the melting point. These phase transitions generally fall into two separate categories.~ Class I transitions are first order and have a discontinuity in the ionic conductivity resulting from a symme— try change in the lattice. Class II transitions are second order and the ionic conductivity is continuous through the transition although there may be a change in the activation energy. Since they are second order, class II transitions involve no lattice symmetry change other than an order—disorder transition on the sublattice containing the conducting ions. The lattice gas Hamiltonian has been successfully used to ex— plain the thermodynamics of class II transitions and the characteristic change in the conduction activation energy ~l,2 There are two prominent exceptions to the general scheme outlined above. They are the compounds Ag2HgI4 and Cu2ligI4. These substances have a class I transition with no associated syumetry change in the anion lattice. That is, the cation order—disorder transition is first order. The thermodynamics of the AD lattice gas Ramiltonian has been investigated for var3’4 to It have is ious lattices and has always been found the thephase present paper to show that onlypurpose second of order transitions, the unique nature of the phase transitions for Ag 2HgI4 and Cu2HgI4 is a result of the fact that these compounds have two disordering ion ape— des rather than just one. When proper account is taken of this additional degree of freedom, the lattice gas Ramiltonian correctly predicts a first order phase transition.
[1 EJ -
I
=
®
U
=
9
~. The structure of the fcc catian sublat— tice in 8—Ag2HgI4 and B—Cu2HgI4.
AgI undergoes a class I transition in which the anion sublattice changes synmetry from fcc to bcc. Since we are concerned only with the fcc cation sublattice, it is convenient to subtract 2. Structure from the charge of each particle an amount equal to the average charge per site on the cation sub— The compound B—AgI has the zincblende lattice. This means that a mercury ion is repre— structure which separately distributes the ca— sented as a particle of charge +1, a silver atom tions and anions on two interpenetrating fcc is represented as a neutral vacancy, and an actual sublattices. In smalogy with the doping of a vacancy is represented as a particle of charge —1. semiconductor, one may think of Ag2Hg14 as Using this scheme the cation sublattice structure * Research supported by Air Force Grant No. AFOSR 76—3106. 629
630
CATION DISORDERING IN Ag
2 HgI~AND Cu2 Hg114
Vol. 23, No. 9.
for the low temperature phase of Ag2HgI4 is shown in Figure 1. Note that within this scheme, pure AgI would. appear as an empty lattice and pure RgI2 would have the so—called copper—gold struc— ture shown in Figure 2. The compound Ag2HgI4 forms because it is slightly favored in Ce’ulomb energy relative to the separated phases of AgI and HgI2. The Madelung constant (referred to the short unit cell side) is 2.28(5) for the Ag2HgI4 structure and 2.25(5) for the competing copper— gold structure.
silver is approximately forty times that of the mercury) but ought not to adversely affect the thermodynamics. The usual single species lattice gas Hamil— tonian is equivalent to a spin one—half Ising model with a magnetic field. The present two species problem is equivalent to a spin one sys— tem. We define a spin variable, S~on each site by: A B s~ = n~— n1 . (2)
•
Since any given site can be at most singly occu— pied, represents the charge on the iu~site, and the expression:
N
(3)
0
~
ii gives the occupation of the site. (2) and (3) into (1) yields: 2 I I —~.ES j I K = —J E S i S j —liES
I I
U
Substituting ,
(4)
where the “magnetic field’1, h Is given by:
I
~
h
(P~_~~)/2.
(5)
and the “crystal field splitting”, ~ is given by: •
0
=
2. The structure tice in 8—HgI 2.
=
9
Overall charge neutrality requires
—JE S~S. J —~ES~ i .
=
(7)
Note that the coupling is antiferromagnetic so that J is negative. From the partition function: —~K Z5Tre , (8)
Model Hamiltonian
In order to make the thermodynamic calcu— lations feasible, we will treat the cations as a lattice gas with nearest neighbor interactions. The Ramiltonian is given by: K
(6)
of the fcc cation sublat—
At 52°C(67°Cfor Cu2HgI4), the cation sub— lattice in Ag HgI undergoes an order—disorder phase transition ~n which both species of cation 6 in the anion sublattice volume disorder. Since thereandisessentially no symmetry nochange change during the transition, we need be con— cerned only with the Ramiltonian and the thermo— dynamics of the fixed fcc cation sublattice. 3.
(~~+~i~)/2.
—E J(i4—n~) (n~_n~)_E(pA4+pBn~), (1) ii> ~ ~ i
<
where the sum is over pairs of nearest neighbor lattice sites, A and B refer respectivelyAtO the p~sitive and negative particle species, n~and n are nuither operators, and and ~B are chem— i~al potentials. This Bamiltonian ignores the contribution of particle hopping to the thermo— dynamics and treats the mercury ions as if they were indistinguishable from the silver except for thei~charge. Th~latterseems reasonable since Ag , 9u+ and Hg all have the same electronegativity. Although Pauling’s table assigns a somewhat larger radius to silver than mercury, they are abo?t the sane size according to Waddington’s table. The difference in mass and the be expected to contribute the mobility dif— small difference in bondingto characteristics can ference of the two species (the mobility of the
one may obtain the lattice gas concentration, c = CA + CB by: c
ialn Z P
,
(9)
where P is the inverse temperature. This Hamiltonlan is the antiferromagnetic version of the Blume—Capel model~-°~5Because the fcc lattice is not alternant’6, the nature of the antiferromagnetic system is rather dif— ferent from the ferromagnetic cases studied pre— viously. The phase diagrams for the f cc anti— ferromagnet are shown in Figures 3 and 4. They were calculated using a renormalization group me— thod which is similar to those in references (4~ and (14) and which will be described elsewhere. The Ag Hg1 4 system has a lattice gas concen— tration of ~O%. We see from the phase diagrams that the transition is first order at that con— centration (in agreement with experiment). The detailed nature of the two mixed phase regimes can not be expected to agree with the 18 phase Our near— dia— eat the system. same ground state gramneighbor for the model actual gives AgI—Hg12 energy for the Ag 2HgI4 structure (which has
Vol. 23, No. 9.
CATION DISORDERING IN Ag
2 HgIt~ AND Cu2 Ugh,
631
0.10
/
004
\
0.08•
I
-
fcc —
0.06
0.03
\\ fcc
~0.O4~
ANTIFERROMAGNETIC
ANTIFERROMAGNETIC
~O.O2~ I
\\
\
0.02
________
0.01
_______________________
MIXED PHASE
.
0 I
0
-
.04
I
.08
~/12
.12
.16
.20
j
3. Phase diagram for the two species f cc lattice gas showing critical temperature vs • the chemical potential, i~. Note that ~ is negative. cl.O).and We need to Include next nearest c.5) thewould copper—gold structure (which has neighbor Interactions in order to have the Ag~Hg— 14 structure be preferred at low temperatures, but this does not appear to be numerically fea— sible.
4.
Conclusions
0.2
I
0.4
I
0.6
I
0.8
4. Phase diagram for the two species fcc lattice gas showing critical temperature vs. concentration. 1has~) transitions of other Ag HgI (and Cu UgI is first order unlikesuper— the ionic conductors. The lattice gas Hamiltonian, no~—st~uctured p which correctly predicts a second order transi— tion for the single species problem, has been investigated for the case of two disordering species. This unique extra degree of freedom found in Ag Hg1 4 and Cu2HgI appears to account for the unusual thermodynamic characteristics of these systems.
The Cation order—disorder transition in
REFERENCES
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
17. 18.
1.0
CONCENTR~ITION
PARDEE, W. J. and MAHAN, G. 0., J. Solid State Chem. 15, 310 (1975). MA}IAN, G. D., Phys. Rev. Bl4, 780 (1976). SUBBASWAHY, K. R. and MAHAN, G. D., Phys. Rev. Lett. 37, 642 (1976). CLARO, P. and MAHAN, G. D., J. Phys. ClO, L73 (1977); and to be published. KETELA.AR, J. A. A., Trans. Faraday Soc. 34, 874 (1938). BROWALL, K. W., KASPER, J. S., WIEDEMEIER, H., J. Solid State Chem. 10, 20 (1974); 15, 54 (1975). PAULING, LINUS, The Nature of the Chemical Bond, (3rd. ed., Cornell University Press, 1960), p 93. PAULING, LINUS, op. cit., p 514. WADDINGTON, T. C., Trans. Faraday Soc. 62, 1482 (1966). BLUME, M., Phys. Rev. 141, 517 (1966). CAPEL, H. W., Physlca 32, 966 (1966); 33, 295 (1966); 37, 423 (1967). BLUME, H., EMERY, V.J. and GRIFFITHS, R.B., Phys. Rev. A4, 1071 (1971). MUKA}IEL, D. and BLUME, N., Phys. Rev. AlO, 610 (1974). BERKER, A.N. and WORTIS, MICHAEL, Phys. Rev. B14, 4946 (1976). BURKHARDT, T. W. and KNOPS, H.J.F., Phys. Rev. BlS, 1602 (1977). An alternant lattice is defined to be a lattice which can be divided Into two sublattices such that all the nearest neighbors of a point in one sublattice belong to the other sub— lattice. MAJIAN, G. D. and GIRVIN, S.N., to be published. Gmelin Handbuch der anorganishche Chemie, 8. Auflage, System 61, Tell B4, (Springer—Verlag, 1974), p 163.
Pergamon Press.
Printed in Great Britain.
CATION DISORDERING IN Ag
2HgI AND Cu2HgI4* S.N. Girvin and G.D. ~ahan Physics Department Indiana University Bloomington, Indiana 47401
(Received 13 June 1977 by A.A. Maradudin)
The cation order—disorder transitions in Ag2HgI4 and Cu211g14 are first order. This is unusual, since in other superionic conductors the cation disordering is gradual with temperature if there is no structural phase transition. These two materials are also unique in that they have two disordering cations rather than one. A study of a two species lattice gas model shows that this extra degree of freedom is responsible for the first order nature of this transition on the fcc lattice.
1.
Ag I,~doped with Hg. The divalent mercury ions in~r8duceholes into the acation 5 As resultlattice, of this, as Ag noted by Ketelaar. 211g14 satisfies the necessary (but not sufficient) condition for good ionic conduction that there be incomplete occupation of the available ca— tion sites. This condition is satisfied in the “undoped” crystal, cz—AgI only above 144°Cwhere
Introduction
Many superionic conductors exhibit one or more phase transitions at temperatures below the melting point. These phase transitions generally fall into two separate categories.~ Class I transitions are first order and have a discontinuity in the ionic conductivity resulting from a symme— try change in the lattice. Class II transitions are second order and the ionic conductivity is continuous through the transition although there may be a change in the activation energy. Since they are second order, class II transitions involve no lattice symmetry change other than an order—disorder transition on the sublattice containing the conducting ions. The lattice gas Hamiltonian has been successfully used to ex— plain the thermodynamics of class II transitions and the characteristic change in the conduction activation energy ~l,2 There are two prominent exceptions to the general scheme outlined above. They are the compounds Ag2HgI4 and Cu2ligI4. These substances have a class I transition with no associated syumetry change in the anion lattice. That is, the cation order—disorder transition is first order. The thermodynamics of the AD lattice gas Ramiltonian has been investigated for var3’4 to It have is ious lattices and has always been found the thephase present paper to show that onlypurpose second of order transitions, the unique nature of the phase transitions for Ag 2HgI4 and Cu2HgI4 is a result of the fact that these compounds have two disordering ion ape— des rather than just one. When proper account is taken of this additional degree of freedom, the lattice gas Ramiltonian correctly predicts a first order phase transition.
[1 EJ -
I
=
®
U
=
9
~. The structure of the fcc catian sublat— tice in 8—Ag2HgI4 and B—Cu2HgI4.
AgI undergoes a class I transition in which the anion sublattice changes synmetry from fcc to bcc. Since we are concerned only with the fcc cation sublattice, it is convenient to subtract 2. Structure from the charge of each particle an amount equal to the average charge per site on the cation sub— The compound B—AgI has the zincblende lattice. This means that a mercury ion is repre— structure which separately distributes the ca— sented as a particle of charge +1, a silver atom tions and anions on two interpenetrating fcc is represented as a neutral vacancy, and an actual sublattices. In smalogy with the doping of a vacancy is represented as a particle of charge —1. semiconductor, one may think of Ag2Hg14 as Using this scheme the cation sublattice structure * Research supported by Air Force Grant No. AFOSR 76—3106. 629
630
CATION DISORDERING IN Ag
2 HgI~AND Cu2 Hg114
Vol. 23, No. 9.
for the low temperature phase of Ag2HgI4 is shown in Figure 1. Note that within this scheme, pure AgI would. appear as an empty lattice and pure RgI2 would have the so—called copper—gold struc— ture shown in Figure 2. The compound Ag2HgI4 forms because it is slightly favored in Ce’ulomb energy relative to the separated phases of AgI and HgI2. The Madelung constant (referred to the short unit cell side) is 2.28(5) for the Ag2HgI4 structure and 2.25(5) for the competing copper— gold structure.
silver is approximately forty times that of the mercury) but ought not to adversely affect the thermodynamics. The usual single species lattice gas Hamil— tonian is equivalent to a spin one—half Ising model with a magnetic field. The present two species problem is equivalent to a spin one sys— tem. We define a spin variable, S~on each site by: A B s~ = n~— n1 . (2)
•
Since any given site can be at most singly occu— pied, represents the charge on the iu~site, and the expression:
N
(3)
0
~
ii gives the occupation of the site. (2) and (3) into (1) yields: 2 I I —~.ES j I K = —J E S i S j —liES
I I
U
Substituting ,
(4)
where the “magnetic field’1, h Is given by:
I
~
h
(P~_~~)/2.
(5)
and the “crystal field splitting”, ~ is given by: •
0
=
2. The structure tice in 8—HgI 2.
=
9
Overall charge neutrality requires
—JE S~S. J —~ES~ i .
=
(7)
Note that the coupling is antiferromagnetic so that J is negative. From the partition function: —~K Z5Tre , (8)
Model Hamiltonian
In order to make the thermodynamic calcu— lations feasible, we will treat the cations as a lattice gas with nearest neighbor interactions. The Ramiltonian is given by: K
(6)
of the fcc cation sublat—
At 52°C(67°Cfor Cu2HgI4), the cation sub— lattice in Ag HgI undergoes an order—disorder phase transition ~n which both species of cation 6 in the anion sublattice volume disorder. Since thereandisessentially no symmetry nochange change during the transition, we need be con— cerned only with the Ramiltonian and the thermo— dynamics of the fixed fcc cation sublattice. 3.
(~~+~i~)/2.
—E J(i4—n~) (n~_n~)_E(pA4+pBn~), (1) ii> ~ ~ i
<
where the sum is over pairs of nearest neighbor lattice sites, A and B refer respectivelyAtO the p~sitive and negative particle species, n~and n are nuither operators, and and ~B are chem— i~al potentials. This Bamiltonian ignores the contribution of particle hopping to the thermo— dynamics and treats the mercury ions as if they were indistinguishable from the silver except for thei~charge. Th~latterseems reasonable since Ag , 9u+ and Hg all have the same electronegativity. Although Pauling’s table assigns a somewhat larger radius to silver than mercury, they are abo?t the sane size according to Waddington’s table. The difference in mass and the be expected to contribute the mobility dif— small difference in bondingto characteristics can ference of the two species (the mobility of the
one may obtain the lattice gas concentration, c = CA + CB by: c
ialn Z P
,
(9)
where P is the inverse temperature. This Hamiltonlan is the antiferromagnetic version of the Blume—Capel model~-°~5Because the fcc lattice is not alternant’6, the nature of the antiferromagnetic system is rather dif— ferent from the ferromagnetic cases studied pre— viously. The phase diagrams for the f cc anti— ferromagnet are shown in Figures 3 and 4. They were calculated using a renormalization group me— thod which is similar to those in references (4~ and (14) and which will be described elsewhere. The Ag Hg1 4 system has a lattice gas concen— tration of ~O%. We see from the phase diagrams that the transition is first order at that con— centration (in agreement with experiment). The detailed nature of the two mixed phase regimes can not be expected to agree with the 18 phase Our near— dia— eat the system. same ground state gramneighbor for the model actual gives AgI—Hg12 energy for the Ag 2HgI4 structure (which has
Vol. 23, No. 9.
CATION DISORDERING IN Ag
2 HgIt~ AND Cu2 Ugh,
631
0.10
/
004
\
0.08•
I
-
fcc —
0.06
0.03
\\ fcc
~0.O4~
ANTIFERROMAGNETIC
ANTIFERROMAGNETIC
~O.O2~ I
\\
\
0.02
________
0.01
_______________________
MIXED PHASE
.
0 I
0
-
.04
I
.08
~/12
.12
.16
.20
j
3. Phase diagram for the two species f cc lattice gas showing critical temperature vs • the chemical potential, i~. Note that ~ is negative. cl.O).and We need to Include next nearest c.5) thewould copper—gold structure (which has neighbor Interactions in order to have the Ag~Hg— 14 structure be preferred at low temperatures, but this does not appear to be numerically fea— sible.
4.
Conclusions
0.2
I
0.4
I
0.6
I
0.8
4. Phase diagram for the two species fcc lattice gas showing critical temperature vs. concentration. 1has~) transitions of other Ag HgI (and Cu UgI is first order unlikesuper— the ionic conductors. The lattice gas Hamiltonian, no~—st~uctured p which correctly predicts a second order transi— tion for the single species problem, has been investigated for the case of two disordering species. This unique extra degree of freedom found in Ag Hg1 4 and Cu2HgI appears to account for the unusual thermodynamic characteristics of these systems.
The Cation order—disorder transition in
REFERENCES
1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16.
17. 18.
1.0
CONCENTR~ITION
PARDEE, W. J. and MAHAN, G. 0., J. Solid State Chem. 15, 310 (1975). MA}IAN, G. D., Phys. Rev. Bl4, 780 (1976). SUBBASWAHY, K. R. and MAHAN, G. D., Phys. Rev. Lett. 37, 642 (1976). CLARO, P. and MAHAN, G. D., J. Phys. ClO, L73 (1977); and to be published. KETELA.AR, J. A. A., Trans. Faraday Soc. 34, 874 (1938). BROWALL, K. W., KASPER, J. S., WIEDEMEIER, H., J. Solid State Chem. 10, 20 (1974); 15, 54 (1975). PAULING, LINUS, The Nature of the Chemical Bond, (3rd. ed., Cornell University Press, 1960), p 93. PAULING, LINUS, op. cit., p 514. WADDINGTON, T. C., Trans. Faraday Soc. 62, 1482 (1966). BLUME, M., Phys. Rev. 141, 517 (1966). CAPEL, H. W., Physlca 32, 966 (1966); 33, 295 (1966); 37, 423 (1967). BLUME, H., EMERY, V.J. and GRIFFITHS, R.B., Phys. Rev. A4, 1071 (1971). MUKA}IEL, D. and BLUME, N., Phys. Rev. AlO, 610 (1974). BERKER, A.N. and WORTIS, MICHAEL, Phys. Rev. B14, 4946 (1976). BURKHARDT, T. W. and KNOPS, H.J.F., Phys. Rev. BlS, 1602 (1977). An alternant lattice is defined to be a lattice which can be divided Into two sublattices such that all the nearest neighbors of a point in one sublattice belong to the other sub— lattice. MAJIAN, G. D. and GIRVIN, S.N., to be published. Gmelin Handbuch der anorganishche Chemie, 8. Auflage, System 61, Tell B4, (Springer—Verlag, 1974), p 163.