THE METAL-INSULATOR TRANSITION IN THE EXPANDED METALS

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THE METAL-INSULATOR TRANSITION IN THE

EXPANDED METALS

T. Ogawa, S. Nara, T. Matsubara

To cite this version:

THE METAL-INSULATOR TRANSITION

IN

T. Ogawa, S. Nara and T. Matsubara

Department of Physics, Faculty o f Science, Kyoto University, Kyoto 606, Japan

Abstract.- The Landau theory of phase transition is extended to include the singular free-energy due to a metal-insulator (MI) transition and to discuss the condensation caused by this. Some relation is pointed out between the behaviours in T > Tc and those in T < Tc. One sees what type of singula- rity the idealized model of the electron system is desired to derive in order to explain the experi- mental facts in the supercritical region.

1 . Introduction.--In 1943, Landau and ZellT~~~icll I 1 1 ~oir,ted out that three cases are theoretically possible in which the M ( metal) -I (insulator) transition 1 ice ends 1) at the critical point of the L(1iguid)-

G(gas) transition, 2) in the L region, and

3) in the G region. Some years ago, we proposed a simple model by which both of the MI- and LG-transitions can be derived

[ 2 - 4 1 . This is essentially a combined model of a lattice gas and the itinerant electrons (the Hubbard model). The sig- nificance of this model is that the origin of the condensation is the energy of the electron system. The equations of states and the phase diagrams of various types, in khich all of the above mentioned three cases are included, were.derived depending on the values of parameters specifying the substances. Most of the results thus ob- tained are qualitatively accep-table except the discontinuous compressibility at the MI-transition in the supercritical region. This singularity seems too strong and is to be ascribed to the adopted ap- proxiriation.

The experimental studies on the expanded

metals have been done mainly on the elec- tric conductivity [ 5 ] . Recently, some in- formations on thermodynamic properties are obtained from the measurements on mercury alocg the isochores by Yao and Endo [ 6 3 .

They have determined the isochores in the

F - T plane from the density measurement using the Archimedean method and deduced the precise values of compressibility. According to this, the isothermal com- pressibility seems to have a weak anom- aly in the density range where the electric conductivity abruptly changes. The slope of the inverse of $he compress- ibility as a function of specific volume is steeper in the small volume side. This fact is consistent with the sound velocity measurement by Suzuki et a1 171.

It is the purpose of this paper to dis- cuss the possible thermodynamic singular- ities caused by the MI-transitions. By extending the Landau theory of phase transitions [ 8 1 , we derive a rather gen- eral relationship between the thermo-, dynamic singularity in the Supercritical

C8-78 JOURNAL DE PHYSIQUE

region on one hand and the relative posi- tion of the end point of the MI-transition line and the critical point of the conden- sation on the other hand. By this rela- tionship, one can know what is desired for the-electron system to have the thermo- dynamic properties observed in experiments.

2. The extension of the L a n d a u theory.- A combined model of a lattice gas and the Hubbard model is used in the previous pa- pers. However, the detail of the model is not so important from the thermodynamic point of view so long as one is interested only in the singularities at the transi- tion. Thermodynamically, the characteris- tic of the MI-transition lies only in the singular contribution of the electronic degree of freedom to the free energy. Therefore one may forget the origin of the singular term in the thermodynamic discus- sions.

It is noted that the concepts of a liquid and a gas are extrinsic or relative since they can be distinguished only through their coexistence. Usually, the volume difference of two phases is taken as the order parameter of the condensation. Let us assume the Helmholtz free energy is regular in volume if the singular contri- bution Fl from the electronic degree of

freedom to the density range of vcvl is absent,

F = F o + F 1 , (1)

X

F1 = A(vl-V) O(vl-V)

,

where the MI-transition takes place at the volume vl and A is a constant. The power

that some electron systems lead to this type [2-4,9-111. The regular part Fo in- cludes all the other contributions and might lead to a condensation without F1. The critical. volume of the system with F= Fo is assumed to be vo. It is of no use to specify Fo in more detail and one may imag- ine any reference system in which both of the repulsions and the attractions are properly taken, for example, such as a van der Waals system and a lattice gas system. The pressure of this system is written as

The volume derivative of the pressure is closely connected with the isothermal com- pressibility K~

,

-

It is noted that the singularity in p is lower than in F by order one and that in cT-l is lower by order one further.

3. The critical point near the MI-tran-

sition.-The isotherms in the p-v plane

may have some maxima and some minima show- ing the existence of some (not necessarily one) condensations. Among them, the pres- ent interest is only in what takes place in the regibn v=vl.

At first, let us investigate the case of A<O. For X=2, the features of the p-v

isotherms are drawn with the mark (-) schematically in Fig.1~. The slope is discontinuous at v=vl. The critical tem-

tally. The critical point beeing at v=vl and p=pO, this case corresponds to the first of three cases by Landau. The boundary of the coexistence region in the p-v plane starts quadratically in the M-

s i d . e ane linearly in the I side (see Fig.2)

The Gutzwiller approximation adopted in the previous papers leads to this case.

For A>2, the critical point lies always in anM state as easily seen from Fig-la. It is noted that this does not exclude the possibility of other condensation in the I- region. The boundary of the coexistence region starts quadratically to both sides of the critical point as in the van der Waals equation of states (see Fig.3). There are some weak singularities i-n both sides of this boundary line at the pres- sure and the temperature of the end point of the MI-transition line. The type of this singularity depends on X and further discussion on this will be given in some other publication.

Next, let us assume A>O. In this case the singular term reduces the tendency for the condensation in the M-states more or less and never brings any new condensation, though it of course remains as a whole. If v1<vO1 the singular term Fl does not affect the condensation near the volume v"v0 and vo is stil a critical volume in

the I-states (see Ffg.4). If vl=vO, vO is

a critical volume (see Fig.5). The detail of this case is rather complicated and is not discussed here. If vl>vo, some other critical point may be in the M-states.

-

-

4 . The s i n g u l a r i t y i n uT '...-I£ X=2, rT jumps at v=vl as schematically shown in Fig.la and the sign of this jump depends on that of A. If 2<X<3, the shape of uT

-

(-1 correspond to the sign of A. If X=3,

K ~ has a kink st v=vl as shown in the - ~

same.convention schematically in Fig.1~. If X*3, KT-' is rather smooth at &=vl a8 shown in Fig.ld but some of its derivatives are discontinuous there.

C8-80 JOURNAL DE PHYSIQUE

gular term F1 is not discussed- The sign _{[lo I Ogawa ,T}

.

_,

.

.

of A and the power X are expected to de- _{Ill1 Ogawa,T. and Yonezawa,F., (in prepara-}

tion). pend on the mechanism of the MI-transition,

for example, on whether it is of Mott-type,

not established that what type of singular- ity these MI-transitions have in the free energy or the ground state energy. The experimental facts cited in section 1 sug- gest that A>O and id3 in Hg in which the type of the MI-transition is believed to be of Wilson-type. The theory of the Wilson transition is desired which leads to A>O since the simplest treatment leads to A<O. As for the Mott transition, it has turned out recently that the simpli- fied version of the Hubbard's approxima- tion, in which only the scattering correc- tion is taken and the resonance broadning correction is dropped, leads to A>O and A=2.5 [lo]. A new variational approxima- tion based on the coherent potential ap- proximation leads to A>O and X=3.5 [ll]. Nobody discussed whether the energy is anomalous or not at the Anderson transi- tion.

R e f e r e n c e s

.-

[l] Landau,L.D. and Zeldovich,J., Acta Phys.-chim.URSS 2(1943)194. [2] Nara,S., Ogawa,T. and Matsubara,T.,

Prog.Theor.Phys. z(1977) 1474. [3] Nara,S., Ogawa,T. and Matsubara,T.,

Prog.Theor.Phys. 61(1979)736. [4] Ogawa ,T., Nara,T. and ~atsubara ,T

..,

Prog.Theor.Phys.Supp1. to be pub- lished.

[5] Hensel,F., 1nst.Phys.Conf.Ser. =(I977 ) 372.

[6] Yao ,M.

.

,

.

.

Proc.6th Int.Conf.Interna1 Friction and Ultrasonic Attenuation in Solids

(Univ. of Tokyo Press, 1977)319. [8] Landau,L.D., JETP 1(1937)19;' z(1937)

627.