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Electronically induced lattice distortion in WO3 and stability of ReO3 lattice
Solid State Communications, Vol. 34, pp. 433—435. Pergamon Press Ltd. 1980. PrInted In Great Britain. ELECTRONICALLY INDUCED LATTICE DISTORTION IN W03 AND STABILITY OF Re03 LATTICE A. Fujnnorl and N. Tsuda National Institute for Researches in Inorganic Materials, Sakura-mura, Nlihari-gun, Ibaraki 305, Japan (Received 9 January 1980 by W. Sasald)
Structural phase transitions in W03 and the stability of cubic Re03 and M~WO3lattices are attributed to the screening effect involving W or Re 5d and 0 2p states on the lattice vibrations. The presence of the conduction electrons in Re03 or M~WO3is shown to suppress this effect and to stabilize the cubic lattice. THE CUBIC LATTICE of Re03 is stable down to liquidhelium temperatures, while closely related W03 shows several phase transitions and remains tetragonal up to the melting point 1770K: WO3 has distorted Re03 structures changing from tetragonal—orthorhombic— monoclinic—tridlinic—monoclinic during cooling [11. On the other hand, when metal ions enter the wo3 lattice forming bronzes M~WO3,the cubic lattice becomes stable for large x (x S 0.4 forM = Na). Electronic energy bands for ReO3 [2], cubic W03, and NaWO3 [3] have been calculated and it has been shown that theyare similar except that conduction bandsareemptyforWO3.InNaWO3,theNaatom donates its electron to the WO3 conductionbands, making NaWO3 isoelectronic to Re03. Thus the stability of the cubic lattice in Re03 would be attributed to the occupancy of the conduction band. In this comniunication, electron screening effect on the lattice vibrations is shown to be responsible for the instability of W03 and to be suppressed by the presence of conduction electrons in Re03 or M~WO3.The screening in the present case is not a free-electron like one as in the simple metals but an atomic character, metal d or oxygen p, plays an important role. Such a type of screening has been suggested to cause ferroelectric or other phase transitions in some dielectrics [4—6]. This mechanism is formulated in a simple secondorder perturbation expression as follows. The Hamiltonian of the system is given by H
=
H1
=
H,
=
H0 +H, + H1,
(la)
~
(ib)
2
E (P~,P~ +~
(ic)
~,,
H,~= ~
~ I
(id)
where Q)L~,and P~,are the normal coordinate and its conjugate momentum of the vth vibration with the wave vector k, and E is the summation over the electrons. H, consists of interactions among atomic cores and of firstorder energy changes due to changes in the orbital overlaps resulting from the atomic displacement. Orbital mixing arisingfrom the displacement is represented by H11, which polarizes electrons and screens the “bare” interaction H,: H1, lowers the energy of the electron— lattice system by ~Q~v)
=
Ep n:~ ~~
P+~I0kIP~>IQ~Q~
0CC
for the Icy mode, where e~is a one-electron energy of 1k,,). If ~~ ~, Q~,+ ~ <0, (3) the lattice becomes unstable with respect to this mode at least at low temperatures. Equation (2) indicates that if occupied and unoccupied states connected by H0, are close in energy the screening is expected to be strong. In Re03 or M~WO3,the lowest conduction-band states (around l’—~—X)are populated [2] and Interband screening involving the conduction and valence bands is largely suppressed, while H, can be regarded to be approximately the same for these compounds and W03. In the tetragonal phase adjacent W atoms in one layer are displaced to the opposite directions as shown in Fig. 1 [7]. This displacement has the M~symmetry [k = (u/a) (1, 1,0)]. For( n)=1’~5,~,X3 in equation (2), matrix elements non-zero by symmetry are listed in Table 1. Corresponding energy denominators are given after Mattheiss’ values for Re03 (adjusted bands) [2]. Notice that the notation of the irreducible representation is defined with the W atom at the origin following the convention of electronic states, not that of lattice dynamics. From Table 1 one can see that
433
434
LATTICE DISTORTION iN Wa3 AND STABILITY OF Re03 LATTICE
Vol. 34, No.6
1H
Table 1. Representative non-zero matrix elements oj 0, and corresponding energy denominators in ReO3 that states with the symmetry in the third column are out of the energy range of the band calculation [2]
—
means
Energy separation between
Normal mode
The lowest conduction band
Valence or higher conductionband
the lowest conduction ban~ and valence/higher conduction band (Ry)
M~
l’~
M~ M~ R~ R~2 X~ X~
—I—
X3
R~5
R2 R12 R~5 R’15 M2 M5 T5 X1
l~5
I;
X~ X~
r15
0.13/—
-II-
0.12/— 0.15/— 0.38/0.30 —~0.06, 0.14/— 0.25 /—
0.50/— 0.38/0.30 0.05/— 0.51/— 0.26/0.22 0.5/— 0.1,03/0.2 —/--
-I--
0.12/— 0.16/— 0.06, 0.14/—
0.06,0.17/—
“25
0.10/
M~ M~
-/-
M~
-/-
-I-
L’12
0.06/0.16 0.10/
x3
-I0.06, 0.14/— 1~I~
—I—
L~i5
—0.06/—
mixings X~’ X~,X~’ X~,Mi” P~etc, where v and c stand for the valence and conduction bands respectively, are responsible for the screening of the M~mode. In ReO3 or M~WO3,this screening vanishes whereas that involving the 1’~—A~—X3 and higher conduction bands appears.i However, the latter screening is weak due to generally large energy separation or due to small matrix elements as will be shown. The tetragonal phase transforms into the orthorhombic one at 1010K. This transition consists of the —
—
—
displacement of the W atoms [k = (ir/a)(l, 0, 0), X~,J and the rotation of the W06 octahedra [k = (u/a) (1, 1, 1), R ‘15x] [1] in addition to the above tetragonal distortion (Fig. 1). Non-zero matrix elements of H,, and energy denominators in this case are also given in Table 1, which also favours the screening effect. The orthorhombic-to-monoclinic transition at 580 K is the rotation of the octahedra with respect to another axis (R’15~)[1,8]. To evaluate the magnitude of (2), the matrix
Vol. 34, No.6
LATTICE DISTORTION IN W01 AND STABILITY OF ReO3 LATTICE
X~Y ~e~(1.O.0)
M~ ~~(1,1~O)
435
Ri’s~~=g(1.1,1)
Fig. 1. Normal modes of the ReO3 lattice corresponding to the phase transitions in W03. The notation for the irreducible representation is defmed with the transition-metal atom (filled circles) at the origin. Open circles represent the oxygen atoms. elementshave to be estimated. The M~mode is chosen an example. For (pm) = X~energy lowering due to the X~’ X~ mixIng is given by
above results satisfy the condition (3), theretbre, the instability of the M~mode in W03 and the stability in ReO3 or M~WO3may well be attributed to the screening effect. As for the R~5modes, ~AE is smaller because
Q(M~)2, 2 e(1?) 2 e(X1)
elements. However, as the bare terms potential itself is they do not the matrix supposed to include be smalla~du)/ar (zero within theinsimple force-
—
(4a)
—
where the factor 2 arises from the spin degeneracy, with
~-
a
constant model of [11] and [12]), theR’ 15 mode could be unstable in WO3.
Matrix elements between the lowest and highest q conduction bands are small, because they consist mainly (4b) of changes in the W 5d—W Sd overlaps.
+~
In Table 1 the I’~~ mode which leads to a ferro-
in the LCAO model with the two-center approximation [9]. In equation (4) (pdir) is the transfer integral between nearest-neighbour W Sd and 0 2p orbitals, q is the displacement of W along the [00 1] direction, and the displacement of 0 is not assumed. Mattheiss’ value (pdur) = 0.120 Ry for Re03 [2] is used. The distance dependence of (pdur) is estimated by assuming it to be proportional to corresponding overlap integral S~.For a crude estimation of s;’ aS,r/arw..o the value between 1,isused [10], Ti3dandO2pin SrTiO3,—0.85 a.u althoughS~may critically depend on the form of the 2
wave functions. Thus (4a) is estimated to be 0.6(7 Ry, where q is given in atomic units. Comparable contribution comes from the 12’— X~mixing. For (pm) = 1’~the energy lowering is 1.4q2 Ry. When the summation with respect to pin equation (2)is done, we roughly get t~E(Q(M~)) —, Nq2 Ry, where N is the number of unit cells in the system. On the other hand, the bare potential could be evaluated using the force constants of ReO 3 derived from the elastic constants [11, 12]: 2QQW~)2 4N(4Hq2 + 2Kq2) 0.6Nq2 Ry, —
electric transition is also given, indicating that Re03 is stable with respect to this mode, too. For some other modes that do not become soft in W03, the selection rules for H0, were found not to favour the screening. — The authors are grateful to Drs T. Chiba and T. Akahane for valuable discussions and information.
Acknowledgements
1.
2.
3. 4.
—
—
~(M~)
where H and K are force constants for the O—Re-O bending and the Re—O stretching, respectively. The
5. 6. 7. 8.
REFERENCES E. Salje,Acta Cryst. B33, 574 (1977). L.F.Mattheiss,Thy& Rev. 181, 987 (1969). L. Kopp, B.N. Hermon & S.H. Uu, Solid State comm~.22,677(1977). N. Kristoffel & P. Konsin, Ferroelectrics 6,3 (1973). T. Hidaka,Phys. Rev. B17, 4363 (1978). T. Hidaka, Phy& Rev. B20, 2769 (1979). W.L. Kehl, R.G. Hay & D~Wahl,J. AppL PhyL 23, 212 (1952). S. Tanisaki,J. Phys. Soc. Japan 15, 573 (1960).
9. L.F. Mattheiss, Phys. Rev. B6, 4740 (1972). 10. J.D. Zonk & T.N. Casselman,Phys. Rev. Lett. 31, 11. N. 960(1960). Tsuda, Y. Sumino, I. Ohno & T. Akahane, J. Phys. Soc. Japan 41, 1153 (1976). 12. M. Ishli, T. Tanaka, T. Akahane & N. Tsuda, J. Phys. Soc. Japan 41,908(1976).
Structural phase transitions in W03 and the stability of cubic Re03 and M~WO3lattices are attributed to the screening effect involving W or Re 5d and 0 2p states on the lattice vibrations. The presence of the conduction electrons in Re03 or M~WO3is shown to suppress this effect and to stabilize the cubic lattice. THE CUBIC LATTICE of Re03 is stable down to liquidhelium temperatures, while closely related W03 shows several phase transitions and remains tetragonal up to the melting point 1770K: WO3 has distorted Re03 structures changing from tetragonal—orthorhombic— monoclinic—tridlinic—monoclinic during cooling [11. On the other hand, when metal ions enter the wo3 lattice forming bronzes M~WO3,the cubic lattice becomes stable for large x (x S 0.4 forM = Na). Electronic energy bands for ReO3 [2], cubic W03, and NaWO3 [3] have been calculated and it has been shown that theyare similar except that conduction bandsareemptyforWO3.InNaWO3,theNaatom donates its electron to the WO3 conductionbands, making NaWO3 isoelectronic to Re03. Thus the stability of the cubic lattice in Re03 would be attributed to the occupancy of the conduction band. In this comniunication, electron screening effect on the lattice vibrations is shown to be responsible for the instability of W03 and to be suppressed by the presence of conduction electrons in Re03 or M~WO3.The screening in the present case is not a free-electron like one as in the simple metals but an atomic character, metal d or oxygen p, plays an important role. Such a type of screening has been suggested to cause ferroelectric or other phase transitions in some dielectrics [4—6]. This mechanism is formulated in a simple secondorder perturbation expression as follows. The Hamiltonian of the system is given by H
=
H1
=
H,
=
H0 +H, + H1,
(la)
~
(ib)
2
E (P~,P~ +~
(ic)
~,,
H,~= ~
~ I
(id)
where Q)L~,and P~,are the normal coordinate and its conjugate momentum of the vth vibration with the wave vector k, and E is the summation over the electrons. H, consists of interactions among atomic cores and of firstorder energy changes due to changes in the orbital overlaps resulting from the atomic displacement. Orbital mixing arisingfrom the displacement is represented by H11, which polarizes electrons and screens the “bare” interaction H,: H1, lowers the energy of the electron— lattice system by ~Q~v)
=
Ep n:~ ~~
P+~I0kIP~>IQ~Q~
0CC
for the Icy mode, where e~is a one-electron energy of 1k,,). If ~~ ~, Q~,+ ~ <0, (3) the lattice becomes unstable with respect to this mode at least at low temperatures. Equation (2) indicates that if occupied and unoccupied states connected by H0, are close in energy the screening is expected to be strong. In Re03 or M~WO3,the lowest conduction-band states (around l’—~—X)are populated [2] and Interband screening involving the conduction and valence bands is largely suppressed, while H, can be regarded to be approximately the same for these compounds and W03. In the tetragonal phase adjacent W atoms in one layer are displaced to the opposite directions as shown in Fig. 1 [7]. This displacement has the M~symmetry [k = (u/a) (1, 1,0)]. For( n)=1’~5,~,X3 in equation (2), matrix elements non-zero by symmetry are listed in Table 1. Corresponding energy denominators are given after Mattheiss’ values for Re03 (adjusted bands) [2]. Notice that the notation of the irreducible representation is defined with the W atom at the origin following the convention of electronic states, not that of lattice dynamics. From Table 1 one can see that
433
434
LATTICE DISTORTION iN Wa3 AND STABILITY OF Re03 LATTICE
Vol. 34, No.6
1H
Table 1. Representative non-zero matrix elements oj 0, and corresponding energy denominators in ReO3 that states with the symmetry in the third column are out of the energy range of the band calculation [2]
—
means
Energy separation between
Normal mode
The lowest conduction band
Valence or higher conductionband
the lowest conduction ban~ and valence/higher conduction band (Ry)
M~
l’~
M~ M~ R~ R~2 X~ X~
—I—
X3
R~5
R2 R12 R~5 R’15 M2 M5 T5 X1
l~5
I;
X~ X~
r15
0.13/—
-II-
0.12/— 0.15/— 0.38/0.30 —~0.06, 0.14/— 0.25 /—
0.50/— 0.38/0.30 0.05/— 0.51/— 0.26/0.22 0.5/— 0.1,03/0.2 —/--
-I--
0.12/— 0.16/— 0.06, 0.14/—
0.06,0.17/—
“25
0.10/
M~ M~
-/-
M~
-/-
-I-
L’12
0.06/0.16 0.10/
x3
-I0.06, 0.14/— 1~I~
—I—
L~i5
—0.06/—
mixings X~’ X~,X~’ X~,Mi” P~etc, where v and c stand for the valence and conduction bands respectively, are responsible for the screening of the M~mode. In ReO3 or M~WO3,this screening vanishes whereas that involving the 1’~—A~—X3 and higher conduction bands appears.i However, the latter screening is weak due to generally large energy separation or due to small matrix elements as will be shown. The tetragonal phase transforms into the orthorhombic one at 1010K. This transition consists of the —
—
—
displacement of the W atoms [k = (ir/a)(l, 0, 0), X~,J and the rotation of the W06 octahedra [k = (u/a) (1, 1, 1), R ‘15x] [1] in addition to the above tetragonal distortion (Fig. 1). Non-zero matrix elements of H,, and energy denominators in this case are also given in Table 1, which also favours the screening effect. The orthorhombic-to-monoclinic transition at 580 K is the rotation of the octahedra with respect to another axis (R’15~)[1,8]. To evaluate the magnitude of (2), the matrix
Vol. 34, No.6
LATTICE DISTORTION IN W01 AND STABILITY OF ReO3 LATTICE
X~Y ~e~(1.O.0)
M~ ~~(1,1~O)
435
Ri’s~~=g(1.1,1)
Fig. 1. Normal modes of the ReO3 lattice corresponding to the phase transitions in W03. The notation for the irreducible representation is defmed with the transition-metal atom (filled circles) at the origin. Open circles represent the oxygen atoms. elementshave to be estimated. The M~mode is chosen an example. For (pm) = X~energy lowering due to the X~’ X~ mixIng is given by
above results satisfy the condition (3), theretbre, the instability of the M~mode in W03 and the stability in ReO3 or M~WO3may well be attributed to the screening effect. As for the R~5modes, ~AE is smaller because
Q(M~)2, 2 e(1?) 2 e(X1)
elements. However, as the bare terms potential itself is they do not the matrix supposed to include be smalla~du)/ar (zero within theinsimple force-
—
(4a)
—
where the factor 2 arises from the spin degeneracy, with
~-
a
constant model of [11] and [12]), theR’ 15 mode could be unstable in WO3.
Matrix elements between the lowest and highest q conduction bands are small, because they consist mainly (4b) of changes in the W 5d—W Sd overlaps.
+~
In Table 1 the I’~~ mode which leads to a ferro-
in the LCAO model with the two-center approximation [9]. In equation (4) (pdir) is the transfer integral between nearest-neighbour W Sd and 0 2p orbitals, q is the displacement of W along the [00 1] direction, and the displacement of 0 is not assumed. Mattheiss’ value (pdur) = 0.120 Ry for Re03 [2] is used. The distance dependence of (pdur) is estimated by assuming it to be proportional to corresponding overlap integral S~.For a crude estimation of s;’ aS,r/arw..o the value between 1,isused [10], Ti3dandO2pin SrTiO3,—0.85 a.u althoughS~may critically depend on the form of the 2
wave functions. Thus (4a) is estimated to be 0.6(7 Ry, where q is given in atomic units. Comparable contribution comes from the 12’— X~mixing. For (pm) = 1’~the energy lowering is 1.4q2 Ry. When the summation with respect to pin equation (2)is done, we roughly get t~E(Q(M~)) —, Nq2 Ry, where N is the number of unit cells in the system. On the other hand, the bare potential could be evaluated using the force constants of ReO 3 derived from the elastic constants [11, 12]: 2QQW~)2 4N(4Hq2 + 2Kq2) 0.6Nq2 Ry, —
electric transition is also given, indicating that Re03 is stable with respect to this mode, too. For some other modes that do not become soft in W03, the selection rules for H0, were found not to favour the screening. — The authors are grateful to Drs T. Chiba and T. Akahane for valuable discussions and information.
Acknowledgements
1.
2.
3. 4.
—
—
~(M~)
where H and K are force constants for the O—Re-O bending and the Re—O stretching, respectively. The
5. 6. 7. 8.
REFERENCES E. Salje,Acta Cryst. B33, 574 (1977). L.F.Mattheiss,Thy& Rev. 181, 987 (1969). L. Kopp, B.N. Hermon & S.H. Uu, Solid State comm~.22,677(1977). N. Kristoffel & P. Konsin, Ferroelectrics 6,3 (1973). T. Hidaka,Phys. Rev. B17, 4363 (1978). T. Hidaka, Phy& Rev. B20, 2769 (1979). W.L. Kehl, R.G. Hay & D~Wahl,J. AppL PhyL 23, 212 (1952). S. Tanisaki,J. Phys. Soc. Japan 15, 573 (1960).
9. L.F. Mattheiss, Phys. Rev. B6, 4740 (1972). 10. J.D. Zonk & T.N. Casselman,Phys. Rev. Lett. 31, 11. N. 960(1960). Tsuda, Y. Sumino, I. Ohno & T. Akahane, J. Phys. Soc. Japan 41, 1153 (1976). 12. M. Ishli, T. Tanaka, T. Akahane & N. Tsuda, J. Phys. Soc. Japan 41,908(1976).