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Multiple ordering and first order transitions: magnetite
Solid State Communications,
Vol. 9, PP. 1041—1043, 1971.
Pergamon Press.
Printed in Great Britain
MULTIPLE ORDERING AND FIRST ORDER TRANSITIONS: MAGNETITE James R. Cullen Naval Ordnance Laboratory, Silver Spring, Maryland 20910 and Earl Callen* American University, Washington, D.C. 20016
(Received 14 April 1971 by R.H. Silsbee)
Using a symmetry argument, it is shown that a three ‘order parameter’ theory, previously introduced to describe the metal—insulator transition in magnetite also explains the first-order nature of this transition.
2 Fe~~ and Fe2~ions on alternate [0011 planes. The relevant Hamiltonian for B-site electrons in the Wannier representation is
IT WAS Landau’ who first showed that not all phase transitions can be second order. Symmetry alone requires some to be first order; for example Landau showed that certain order—disorder transitions in binary alloys must be first order. 1 A
H
=
~
E~ 2C~~C3+ ~ U~N, N~
simplified version of the general argument is as follows. The free energy is expanded in order parameters m, (assumed small). These parameters are basis functions for irreducible representations of the space group of the high symmetry, disordered phase. Below the ordering temperature the minimum free energy is achieved at a set of values of m~(T) not all of which are zero. The cubic ferromagnet is an example of a second order transition. Averages of the x, y and z components of magnetization are the order parameters. Since these change sign under time reversal, only even powers of m~ appear in the expansion. The absence of a cubic term makes the second order transition possible. Now consider the case of magnetite and the Verwey transition. This semi-metal—semi-conductor transition at 119°Kwas considered for years to be characterized by ionic ordering of
~ creates an electron on site i.the (All electrons 3 Because Hamiltonian have the same spin). contains a non-diagonal kinetic energy or hopping term, the number operator Nia does not commute with H. There is then fractional population,3 0 ~ ~ 1 of a site, even at T = 0°K, and there exists the possibility that four tetrahedrally arranged sites a = 1 4, have different charge densities.4 In this way it is possible to understand not only the metal—insulator transition but also the observed Mossbauer spectrum,5 N.M.R.,6 neutron 7 and electron8 diffraction patterns. The N.M.R.6 and neutron diffraction7 experiments, and also resistivity,9 and sound velocity ‘° measurements show the transition to be first order. This paper will show why. We describe the low-temperature phase of Fe 304 in terms of ordering of charge, and restrict
*
Supported in part at American University under
ourselves an unit ordering does not change the size oftothe cell,”which or transfer charge from
NASA Grant NGR—09—003—014.
one tetrahedron to another. Thus, in each
1041
MULTIPLE ORDERING AND FIRST ORDER TRANSITIONS: MAGNETITE
1042
tetrahedron we concentrate upon the two electrons which, in the conventional view character3~and the ize the difference between the two Fe two Fe2~ions. There are then three independent order parameters
2rn,
=
—
+
—
transition; the multidimensional nature of the ordering is essential. There is a very small lattice distortion at the transition, but we believe it is too small to be the origin of the first order
+
—
—
2m3
=
—
—
+
-Under the operations of the space group of magnetite, (Fd3m) these averages transform into one another. It results that rn1, rn2 and rn3 form an irreducible representation of the space group (cubic), the 1 5representation, i.e., like xy, XZ, zy. The free energy expansion takes the form
aXT) ~
No single order parameter theory, unaided by lattice distortion, can account for the first order
2> =
+
necessary to give insulation for reasonable ratio of bandwidth to coulomb energy.1
2m2
=
Vol. 9, No. 13
+
b(T) rn,m2m3 + c(T) ~ rn~
Because the rn’s transform as ~, ~, ~ (rather than as components of a vector as do magnetization components), a cubic term appears in the free energy. At the assumed transition point T0, a = 0, the so sign = ç~ in the disordered state. Whatever of b(T 0), there are 9~/Om~ = 0 and values ~ (m of the m1 for which c 2, rn2, m3) <0, so that the disordered state is unstable at T0. Thus, from symmetry alone the multi-ordering theory leads to a first order transition. [In reference (4) we showed that three orderings were
transition. Thermal expansion and specific heat measurement at T0, now underway at American University, should lay this ghost. Of course group theory cannot assure that
b(T) is non-zero. It is therefore of interest to see the microscopic source of b(T) in the magnetite case. b(T) arises from the non-diagonal coupling ~ A model which relied solely on the Coulomb terms ~ Nia N,c~ could yield neither the fractional population of sites nor a first order transition. 2 treating strains as Anderson and Blount,’ order discussed first energy order transition arisingparameters, from a cubic term in aa free expansion. We think multiple-ordering phenomena more common than is generally supposed, and are presently applying techniques similar to that given here to other many-body systems (Chromium, for example) to determine the nature of the phase transition.
REFERENCES 1.
LANDAU L.D. and LIFSCHITZ E.M., Statistical Physics, Vol. 14, Pergamon Press London,(1958).
2.
VERWEY E.J.W. and HAAYMAN P.W., Physica 8, 979 (1941).
3.
CULLEN J.R. and CALLEN E.R., J. appi. Phys. 41, 879 (1970).
4.
CULLEN J.R. and CALLEN E.R., Phys. Rev. Lett. 26, 236 (1971).
5.
HARGROVE R.S. and KUNDIG W., Solid State Comrnun. 8, 303 (1970).
6.
RUBINSTEIN M., to be published.
7.
SAMUELSON E.J., BLEECKER E.J., DOBREZYNSKI L. and RISTE T., J. appi. Phys. 39, 114 (1968).
8.
YAMADA T., SUZUKI K. and CHIKAZUMI S., Appi. Phys. Lett. 13, 172 (1968).
9.
CALHOUN B.A., Phys. Rev. 94, 1577 (1954).
10.
MORAN T.J. and LUTHI B., Phys. Rev. 187, 710 (1969).
Vol. 9, No. 13
MULTIPLE ORDERING AND FIRST ORDER TRANSITIONS: MAGNETITE
1043
11.
Experiments [see reference (7)11 indicate cell doubling in addition to ordering within the unit cell; such orderings complicate the analysis given here without essentially altering the result.
12.
ANDERSON P.W. and BLOUNT E.I., Phys. Rev. Lett. 14, 217 (1965).
En utilisant la theorie des groupes on a demontré qu’une theorie de trois ‘parametres de l’ordre’ qu’on a presenté auparavant pour expliquer la transition metallique isolateur de Fe304 explique aussi —
pourquoi cette transition est de premiere ordre.
Vol. 9, PP. 1041—1043, 1971.
Pergamon Press.
Printed in Great Britain
MULTIPLE ORDERING AND FIRST ORDER TRANSITIONS: MAGNETITE James R. Cullen Naval Ordnance Laboratory, Silver Spring, Maryland 20910 and Earl Callen* American University, Washington, D.C. 20016
(Received 14 April 1971 by R.H. Silsbee)
Using a symmetry argument, it is shown that a three ‘order parameter’ theory, previously introduced to describe the metal—insulator transition in magnetite also explains the first-order nature of this transition.
2 Fe~~ and Fe2~ions on alternate [0011 planes. The relevant Hamiltonian for B-site electrons in the Wannier representation is
IT WAS Landau’ who first showed that not all phase transitions can be second order. Symmetry alone requires some to be first order; for example Landau showed that certain order—disorder transitions in binary alloys must be first order. 1 A
H
=
~
E~ 2C~~C3+ ~ U~N, N~
simplified version of the general argument is as follows. The free energy is expanded in order parameters m, (assumed small). These parameters are basis functions for irreducible representations of the space group of the high symmetry, disordered phase. Below the ordering temperature the minimum free energy is achieved at a set of values of m~(T) not all of which are zero. The cubic ferromagnet is an example of a second order transition. Averages of the x, y and z components of magnetization are the order parameters. Since these change sign under time reversal, only even powers of m~ appear in the expansion. The absence of a cubic term makes the second order transition possible. Now consider the case of magnetite and the Verwey transition. This semi-metal—semi-conductor transition at 119°Kwas considered for years to be characterized by ionic ordering of
~ creates an electron on site i.the (All electrons 3 Because Hamiltonian have the same spin). contains a non-diagonal kinetic energy or hopping term, the number operator Nia does not commute with H. There is then fractional population,3 0 ~
*
Supported in part at American University under
ourselves an unit ordering does not change the size oftothe cell,”which or transfer charge from
NASA Grant NGR—09—003—014.
one tetrahedron to another. Thus, in each
1041
MULTIPLE ORDERING AND FIRST ORDER TRANSITIONS: MAGNETITE
1042
tetrahedron we concentrate upon the two electrons which, in the conventional view character3~and the ize the difference between the two Fe two Fe2~ions. There are then three independent order parameters
2rn,
=
—
+
—
transition; the multidimensional nature of the ordering is essential. There is a very small lattice distortion at the transition, but we believe it is too small to be the origin of the first order
+
—
—
2m3
=
—
—
+
-Under the operations of the space group of magnetite, (Fd3m) these averages transform into one another. It results that rn1, rn2 and rn3 form an irreducible representation of the space group (cubic), the 1 5representation, i.e., like xy, XZ, zy. The free energy expansion takes the form
aXT) ~
No single order parameter theory, unaided by lattice distortion, can account for the first order
2> =
+
necessary to give insulation for reasonable ratio of bandwidth to coulomb energy.1
2m2
=
Vol. 9, No. 13
+
b(T) rn,m2m3 + c(T) ~ rn~
Because the rn’s transform as ~, ~, ~ (rather than as components of a vector as do magnetization components), a cubic term appears in the free energy. At the assumed transition point T0, a = 0, the so sign = ç~ in the disordered state. Whatever of b(T 0), there are 9~/Om~ = 0 and values ~ (m of the m1 for which c 2, rn2, m3) <0, so that the disordered state is unstable at T0. Thus, from symmetry alone the multi-ordering theory leads to a first order transition. [In reference (4) we showed that three orderings were
transition. Thermal expansion and specific heat measurement at T0, now underway at American University, should lay this ghost. Of course group theory cannot assure that
b(T) is non-zero. It is therefore of interest to see the microscopic source of b(T) in the magnetite case. b(T) arises from the non-diagonal coupling ~ A model which relied solely on the Coulomb terms ~ Nia N,c~ could yield neither the fractional population of sites nor a first order transition. 2 treating strains as Anderson and Blount,’ order discussed first energy order transition arisingparameters, from a cubic term in aa free expansion. We think multiple-ordering phenomena more common than is generally supposed, and are presently applying techniques similar to that given here to other many-body systems (Chromium, for example) to determine the nature of the phase transition.
REFERENCES 1.
LANDAU L.D. and LIFSCHITZ E.M., Statistical Physics, Vol. 14, Pergamon Press London,(1958).
2.
VERWEY E.J.W. and HAAYMAN P.W., Physica 8, 979 (1941).
3.
CULLEN J.R. and CALLEN E.R., J. appi. Phys. 41, 879 (1970).
4.
CULLEN J.R. and CALLEN E.R., Phys. Rev. Lett. 26, 236 (1971).
5.
HARGROVE R.S. and KUNDIG W., Solid State Comrnun. 8, 303 (1970).
6.
RUBINSTEIN M., to be published.
7.
SAMUELSON E.J., BLEECKER E.J., DOBREZYNSKI L. and RISTE T., J. appi. Phys. 39, 114 (1968).
8.
YAMADA T., SUZUKI K. and CHIKAZUMI S., Appi. Phys. Lett. 13, 172 (1968).
9.
CALHOUN B.A., Phys. Rev. 94, 1577 (1954).
10.
MORAN T.J. and LUTHI B., Phys. Rev. 187, 710 (1969).
Vol. 9, No. 13
MULTIPLE ORDERING AND FIRST ORDER TRANSITIONS: MAGNETITE
1043
11.
Experiments [see reference (7)11 indicate cell doubling in addition to ordering within the unit cell; such orderings complicate the analysis given here without essentially altering the result.
12.
ANDERSON P.W. and BLOUNT E.I., Phys. Rev. Lett. 14, 217 (1965).
En utilisant la theorie des groupes on a demontré qu’une theorie de trois ‘parametres de l’ordre’ qu’on a presenté auparavant pour expliquer la transition metallique isolateur de Fe304 explique aussi —
pourquoi cette transition est de premiere ordre.