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II. PHASE TRANSITIONTHE CONDUCTION PROCESSES IN MAGNETIC AND IN
INTERMEDIATE VALENCY COMPOUNDS
N. Mott
To cite this version:
N. Mott. II. PHASE TRANSITIONTHE CONDUCTION PROCESSES IN MAGNETIC AND IN
INTERMEDIATE VALENCY COMPOUNDS. Journal de Physique Colloques, 1980, 41 (C5), pp.C5-
51-C5-58. �10.1051/jphyscol:1980510�. �jpa-00219945�
JOURNAL
DE
PHYSIQUEColloque C5, suppldinent au n o 6, Tome 41, juin 1980, page C5-51
I PHASE TRANSITION.
THE CONDUCTION PROCESSES IN MAGNETIC AND IN INTERMEDIATE VALENCY COMPOUNDS
N.F.
MottCavendish Laboratory.
Abstract.- A comparison is made between two classes of naterial in which d or f orbitals show intermediate valencies
;in one group is Fe 0 and Ti 0 , in the
other SmS (above 6 kbar), SmB and TmSe. It is pointed ou2 that the4e?ectrons do not order in the former if thg ratio of electrons to sites differs from 0.5 due to alloying by more than a few per cent. The electron gas in the latter case, and above the Verwey temperature in the former, is described as a "Wigner glass". Some speculations are added about Eu S and Sm S . In SmS and SmB the Coqblin-Blandin model is developed further, and3t&e naturs $f hybridisation getween an f-band and a conduction band is defined. In contrast to Fe304 the two charge states do not order, and the reason for this is discussed.
4f 5 (magnetic) and 4f6 (non-magnetic) in the ratio 3:7, through on each site the configuration is a mixture of the two. The conductivity above
%100 K depends little on temperature and has a value /3,4/charac- teristic of a poor metal (about 3 x 10 3 Q - ~ cm-'1. SmS, a semiconductor at low pressures, makes a first order transition to a "metallic" state of this kind with increasing pressure at 6 kbar. These materials are strongly paramagnetic, and
1. Introduction.- In this lecture I shall there is no ordering (superlattice) of the describe two quite different classes of sm2+ and sm3+ ions.
material, namely magnetite and other mate-
Fig. 1
:Conductivity of Fe 0 F for the values of x shown below (~half-&~al ./l3/)
1. x=0.025, 2. x=O.O5, 3. x=O.l, 4. x=O.O15.
It will be seen that the Verwey transition disappears somewhere between x=0.025 and 0.05.
rials undergoing a Verwey transition, and
2
the intermediate-valency rare earthcompourds - -
SmB6, SmS at high pressures (above 6 kbar), - 'E
I -and TmS. In magnetite (Fe304) the ions on
athe B-sites of the inverse spinel structure 2 -
O -have the configurations ~ e and ~ ~ + e in ~ +
m- 1
equal concentrations
;at low temperatures
their positions are ordered, the exact
-2structure being still a matter for investi-
gation (see for instance refs./l/ and
/2/, -3while at the Verwey temperature Tv (%I19 K) a first-order transition occurs to a state
-4in which long-range order has disappeared,
- 5accompanied by an increase in the conducti-
vity by two orders to about 10~52-~cm-~
- 6(see Fig.1). Ferrimagnetic order persists
up to
%850 K. The material can thus be
- 7The purpose of this paper is to
-
-
-
- - - -
-
I 1 :I
10 20
contrast the behaviour of these two subs- tances, and also that of Eu3S4 or Sm3S4.
considered of intermediate valency.
30In SmB6 intermediate valency is also
10' K / Testablished, the Sm ions being in the states
There is no lack of theoretical work on both substances /5,6/ but we hope to be able to bring forward a few new points.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1980510
JOURNAL DE PHYSIQUE
2. The Verwey transition in magnetite.- Characteristic of magnetite (Fe 0
)and
3 4
other materials (e.g./7/ Ti407) which show a Verwey transition is that the number of mobile electrons (e.g. the extra electron on Fe2+ contrasted with Fe3+) is half the number of available sites. The ordering is thought to be a consequence of the Coulomb repulsion between the electrons (with energy Uv
%e / ~ a ) , 2 so that the ordered state is rather similar to a F7igner crystal. Cullen and Callen /8/ investigated the condition for ordering at T
=0
;this depends on the crystal structure but is of the form
UV/B :, 3
1(1
where B is the width of the d-band. Thus unless the d-band is fairly narrow, with large 6f fective mass for the particles, the material should be metallic. The present author /5/ has suggested that a sufficient effective mass can be obtained only if the particles are polarons, and there is strong evidence that this is so for the similar material Ti407 (in this case bipolarons /7/?
and probably in Fe304 where the Verwey tem- perature increases /9/ by %lo% on substitu- ting 40% of the heavy oxygen isotope 0 18 .
In Fe304 there is little doubt in principle about the mechanism for conducti- vity below Tv. An electron is excited from an FE!~+ site onto an ~ e site, and can ~ + thus move from one to the other. If the energy to excite it is
E ,the conductivity will vary as exp (-&E/~T) and the thermo- power as (k/e) (~/2kT + const). In most experimental work this variation holds only a little below the Verwey transition, and hqping conduction /lo/ in an impurity band due to.non-stoichiometry dominates the thermopower and conductivity at low T.
p.uch theoretical work, notably that of Ihle and Lorenz /11/, starts from the
"electrostatic" model, or more exactly the assumption that the Cullen-Callen ratio (1) is large compared with unity. In such calcu- lations, the demonstration by Anderson /12/
is important that, for the particular form of order postulated by<Verwey, long range
order can be destroyed without changing nearest neighbour interactions, so that the Verwey temperature Tv is much less than Uv/k -
From the point of view of this lecture, perhaps the most interesting property of Fe304 is that in the alloys Fe3 (04-xFx) with x as low as 0.03 the fluorine suppres- ses the ordered phase /13,14/. The plot of conductivity against 1/T is shown in figure 1. Very similar behaviour is shown /15/ by
(Til-xVx)407 with x in the neighbourhood of 0.01. It appears that, if the number of electrons deviates by a few per cent from one half of the number of sites, ordering does not occur. The state of the material is then what we describe as a "Fermi glass"
or "Wigner glass". I would next like to explain what I mean by these terms.
As Anderson /16/ first showed in 1954 the one-electron Schrbdinger equation
2 2m
V $+ - (E-V)
$= 0 (2 fi2
has only localized solutions if the poten- tial energy function V(xyz) for an electron in a solid has a sufficiently large randon element. If vO2 is the mean square deviation of the potential from the periodic form, and B is the original band width, the condition that all states within the band are localized is
where the constant depends on various condi- tions but is of order 2. If one considers a degenerate electron gas under these condi- tions, electron states being occupied up to a limiting Fermi energy EF, Pauli spin paramagnetism and electronic specific heat proportional to N(E) are predicted, but all states are localized, and the conductivity a should be by variable range hopping, depending on T in the limit of low tempera-, tures (1) as /7/ .
1
o
=A exp (-B/T~) (4) In any "Fermi glassl','the potential energy must be partly due, in the sense of Hartree-
~ o c k , to the other randomly localized
electrons
;in the materials discussed here it must be mainly so
;in analogy with VJigner crystallization we use the term Wigner glass. The low temperature conducti- vity is observed to conform approximately to equation (4) .
The present author /lo/ has proposed that in "pure" Fe304 above the Verwey temperature the material is a Wigner glass, the field V(x, y, z ) being wholely due to the other electrons. Of course such a material could not be stable at zero tempe- rature. It is stabilized by the entropy which is proportional to N(E), and not
directly due to the disorder, as was supposed in earlier theoretical work. The latent heat at the first order transition is thus proportional to N (EF) k 2 ~ 2 . Conduc- tion is then by the hopping of electrons near the Fermi energy. To explain the reason for this assumption, we must look more closely into the mechanism of hopping conduction.
The many derivations of eq, (4) neglect the interaction between electrons on diffe- rent sites. As pointed out by Knotek and Pollak /18/, Pollak /19/, Matt /20/ and Efros and Sklovskii /21/, when this is taken into account, at any occupied site the neighbours have a higher probability than average of being unoccupied. This lowers the energy of the electron by an amount of order e / ~ a , 2 where 5 is the dis- tance between sites. The so-called "Coulomb gap" is produced
;its magnitude (Pollak, priv. corn) is e / ~ a 2 if the disorder energy Vo is not much greater than EG, otherwise
2 2
it is (e / ~ a : (e /rasG)'.
Eq. (4) is then only va1i.d if kT > E G' If kT << EG, an electron can only move to a site of nearly equal energy, so that the activation energy tends to zero with T as in eq.(4), if many electron hops take place.
What is envisaged is shown in figure 2a.
Accordinq to hnott /10,20/, the resultin?
(1) This is perhaps not certain as will be seen below.
behaviour of a is as shown in figure 3 in the low temperature limit, the same value of B is predicted but a much smaller value of A.
Fig. 2
:(a) Showing the path of an electron hopping from A to B, with subsidiary short hops. (b) Many-electron hopping as envisa- ged by Knotek and Pollak.
Fig. 3
:Sugges ed behaviour of Rn a as a
h
function of 1/T , as explained in the text.
(a) single-electron hops, (b) intermediate region, (c) many-electron hops.
Knotek and Pollak /19/ give argument5 which we think probably correct, to suggest that many-electron hops of the type shown in figure 3b may occur, resulting in a beha- viour of the type a = A exp (-B/T') , s < 4;
and its exact value uncertain. The conclu-
sion of Efros and Sklovskii that the index
s is 4 has been criticised (2) by the
present author /21/.
C5-54
JOURNAL DE PHYSIQUEOf particular interest is the inter- mediate region shown in figure 3. The thermo- power for hopping is given by (ref./17/,
p.65) s S - i e (T
QT) -gg- (5) ,
00_ / ^ — « \ .
and probably some very similar or even / / ^ \ \ .
identical behaviour should occur for many- " // ^ \ S . electron hopping. Here T is the auantity // / \ ^ ^ ^ - - ^
determining a in exp (-T /T) " * . But the - // / ^ \ ^ ^ 5
present author /10/ has suggested that the 50 - /
3intermediate regime the electron may move
r/
out of its "Coulomb well", and execute
1^ ^ ^ ^o _L_
several hops before it creates a second
T/
Kwell. In this case, the thermopower should
be given by S = (k/e) (e/2kT + Const). Fig. 5 : Thermopower S in yV/oK of Fe
30 F
„. „ i. • • i „ i,_4. _ „. j- for low and moderate temperatures. 1. x =
x xFigure 4 shows in principle what we expect,
Q^
2 > x = 0 > 0 ?^ ^
x*
QQ5 ( G r a e n e r e tand figure 5 what is observed /14/ for a l ) . Fe (O F )
3 1-x x 4* It also explains the results of Schlegel and Wachter / 2 4 / see also Schlegel et al, this volume, on the free-carrier con- b tribution to the optical constants above T ;
/ \ 20
vS / N.
c^__^ they find a free carrier density of 10 / J^*^
c mmuch smaller than would be the case if / ^^"^ the material were a metal, as in sometimes /yS^ assumed. We think that these authors are /^ measuring the number of electrons excited y out of the Coulomb gap, and that this is / 20 -3
/ 10 cm at room temperature.
/ 2+
It 3. Eu, S. and SnuS .- In Eu 0 the ions Eu and E u
3 +occupy sites in the crystal which Fig. 4 : Showing schematically the thermo-
a r e n o tequivalent. The electrical proper- power S to be expected for variable-range ties are equivalent to those of other rare
^ r s ' ^ g l i l l e c ^ r o n ' h S ^ i f w h i c T t h ^ e i e c - ^ t h insulators /24/. On the other hand tron is excited out of the Coulomb wells,
E u3
S4
a n d S m3
S4 ^
a v e astructure (ThuP.) so that S * (k/e) (e/2kT.) (c) kT greater
r e l a t t i c e s i t e s a r eor comparable with Coulomb gap, single elec-
tron hops. equivalent. There seems to be no ordering of . _ .. , , . „ „_ the two charge states (Wachter, priv. comm) Since therefore above T the behaviour of
3'
e. . -. ^ £ 4 . 1 , n ., „ „j„^ but in the MSssbauer spectrum of Eu,S. the the pure material and of the alloy are simi-
r2+ 3+
, .. „... T „
u t v„ „ • „ f4-„ two lines corresponding to Eu and Eu lar, and, the "Wigner glass hypothesis fits ^ ^
, , <• J.U i 4-j- -a. , „ „ * ,
a r eclearly distinguishable /25/ at low T.
the data for the latter, it seems a reason- •*
3' '
, , , , .« J.U
cA possible hypothesis is that two ions form
able model for the former.
r JCa "Wigner glass", in the sense discussed in , „, , . . JTT^I. L . , , /,,/ the last section, stabilised possibly by a (2) The conclusion of Kobayashi et al./2 3/ *
2 Jthat s = 4- without correlation contradicts slight non-stoichiometry.
all other theoretical work, and is, we
T h e c o n d u c t i v i t o f s s s i l ebelieve, wrong.
J3 4
3crystals is about /26/
2 -AE/kT ,-1 cm-l
10 e
Iwith AE changing from 0.142 eV to 0.132 eV when l/kT
%8 x The mechanism is not clear -- perhaps hopping in the intermediate range of figure 3, if our hypothesis about a Wigner glass is correct.
A model in which the sulphur ions are closely coupled to the respective valence states of the rare earth has been given by Mulak and Stevens /35/
;the sulphur ion is thought to be displaced towards the trebly-charged ion. This would doubtless decrease the rate of transfer of the charge in the sense described by Hewson and Newns /34/. One way of describing such a system is as a degenerate gas of polarons, the polaron being treated simply as a
heavy particle. With such a model, for which the Cullen-Callen condition (1) must be well satisfied, it is however difficult to understand why a superlattice is not set UP
4. Intermediate valency compounds.- The materials SmB6, SmS and TmSe under high pressure have been investigated for many years, and several theories produced. In this lecture I want to go back to a nodel due to Coqblin and Blandin /28/, which I discussed /29/ in 1974 and which is develo-- ped further by Allen in this volume /30/.
In this I draw in figure 6 a conduction band and a 4f band overlapping and hybridi- zing with each other. In drawing this picture, I make the following points
:a) In considering an f band, I suppose that the 4f6 state can be considered as moving through the 4f5 matrix (or vice versa) , without of course spin-orbit couplincj breaking down. The "pseudoparticle" can have a wave function before hybridization of the form
( 5 )
1 exp (ika,) n $s 5 (q) qn 6 (q)
n s = 1-n-1, n+l-N
where $s is the wave function for the 4f 5 5 state on atom s, yln 6 for 4f6 on atom n. We suppose these pseudoparticles are heavy
fermions and a Fermi distribution up to a limiting energy can occur.
Fig. 6
:Energy of pseudo-particle in SmB6 as a function of wavenumber,k. H denotes the hybridized band for a single pseudo- particle, as explained in the text. XY represents the gap.
b) The ratio (7: 3) in which 4f5 and 4f are mixed must be determined by minimi- 6 sing the energy of the conduction (5d+6s) electrons
;the f band acts as a sort of sink.
C) Hybridization takes place between the f and conduction bands, the "pseudo- particles" thus taking the form
Here Yn represents the wave function of 6 all the f electrons, all in the state 4f 5
-
except that on atom n which is in the state 4f6
;y refers to the situation where all 5 f electrons are in the state 4f5
;Qn is a Wannier function for the conduction elec-
tron on site n. As pointed out by Jefferson and Stevens, /31/ @ must be chosen so that the second term in (6) has the same parity and j- value as the first. X,~X; are functions of the nuclei round site n, to take account of any expansion /3/round the larger ion (4 f 1 6 .
d) Hybridization may lead to a gap, if the point X in figure 6 lies below Y for
(3) On this point see the discussion by
Hewson and Newns /34/.
JOURNAL DE PHYSIQUE
all directions in k-space. For the materials under consideration there appears to be a small gap, AE, of the order 100 K. A dif- ferent model for the gap is proposed by Haen et a1./37/, based on the proposal of Jullien et a1./33/ that in a "Kondo latticd' a gap should open if there is one electron per localized spin. This however does not seem tobe the case here.
e) If kT > A, the conductivity shows metallic behaviour. Since the conductivity should be given by
where f is the Fermi distribution function and o (E) the conductivity at energy E, then if kT > Aa gapwill make little diffe- rence. Since s+ f transitions are exceedin- gly probable on account of the high density of states in the f bind, a very small amount of phonon, impurity or electron- electron scattering should give a mean free path of order L a a (the minimum value), so that the resistivity is almost indepen- dent of T and of order e /3fia 2 a 3000 Ctql cm -1 as observed /3,4/ (Figure 7) .
Fig. 7
:Behaviour of resistivity of SmB or TmS (schemat'c). (a) pure material (bf impurity band. a
non-metallic (T
)impurity band (c) metallic f) Below a 100 K the resistivity
rises, on account of the gap. Nickerson et a1./3/ and also Allen /4/ find a flattening off at low T. This in our view is due to an impurity band. For a material with so small a gap the dielectric constant must be high, so that the Mott condition for metallic conduction
should be satisfied for a small value of N (the concentration of impurities) even though meff is not particularly small.
Allen's measurements of the Hall effect in this region give a value of N, from which if one calculates the minimum metallic conductivity
gin
=0 .26 e2/lia, a = N - ~ / ~ ,
one obtains a value an order greater than that observed. In our view, however, at low T themeasurements may still be in the T-' hopping region
;in this region a will
be less than omin, and the Hall coefficient will certainly overestimate the number of electrons, perhaps by a large factor /4/.
Haen et a1 /32/ for the analogous case of TmxSe give evidence that, when x = 0.05,
p goes to infinity as T
+0, which seems to support our hypothesis for this material.
--The next point that I want to make is to ask, why the 4f5 and 4f6 ions do not order, as the 3d7 and 3d8 ions do in Fe304.
Here the conduction electrons do of course screen the Coulomb interaction, but if the field falls off as exp (-r/r
)with ro a
0
0.5 the Coulomb interaction should still be much greater than the 4f band-width, as estimated from the gap (a 50 K) . If the
4f5 and 4f states were p,resent in equal concentrations, it seems highly probable that the Cullen-Callen condition (1) would give ordering. But what our discussion of Fe3(04-xFx) shows is that, if the two valency states are not gresent in equal concentrations, a state with long-range order does not have lower energy than
- - -
( 4 ) The Hall coefficient for variable-range
hopping has not been observed with certainty
and is probably much smaller than l/nec.
the Wigner glass. We think that, by analogy, if the Cullen-Callen calculation were carried out for full and empty sites present in unequal concentrations, one would not find an ordered state, but a metal except for very small B. And of course a Wigner glass is impossible, since the lowest state cannot have entropy of disorder ; the observed behaviour seems the only possible one.
Finally, we may compare our treat- ment with that of Jefferson and Stevens
/ 3 1 / , who describe the system by a product
of the terms in (6) in the square brackets (but without the Xis). We suspect that these treatments may, if refined, turn out to be equivalent : but theirs does not give so simple an explanation of a gap.
Acknowledgments.- I am grateful to Prof.
K.W.H. Stevens for a useful discussion about this problem.
Note on some recent development. - D . J accard ,
J . Sierro and E. Bucher (Solid State communJ 2 (1979) 713 have measured the thermopower S of TmSe down to 5~10-~IZ.
Though S clearly tends to zero with T , it shows approximately a T ' law rather than T , suggesting variable range hopping.
T. Kasuya, K. Takegahara, T . Fujita, T .
T anaka and E . Bannai (J . Physique Colloq .
~ 5 , 40 (1979), 308) find deviation from
the 3 law below 1 K, but a view of the
consideration of Knotek and Pollak this
is perhaps not surprising.
JOURNAL D E PHYSIQUE
