Local order in liquid tellurium
B. Cabane, J. Friedel
To cite this version:
B. Cabane, J. Friedel. Local order in liquid tellurium. Journal de Physique, 1971, 32 (1),
pp.73-84. <10.1051/jphys:0197100320107300>. <jpa-00207024>
HAL Id: jpa-00207024
https://hal.archives-ouvertes.fr/jpa-00207024
Submitted on 1 Jan 1971
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LE JOURNAL DE
TOME
PHYSIQUE
LOCAL ORDER IN
32,
JANVIER
1971,
1
73
LIQUID TELLURIUM
B. CABANE and J. FRIEDEL
Laboratoire de Physique des Solides
Faculté des Sciences
91-Orsay, France
(Reçu le
17 août
(*)
1970)
Résumé. 2014 Les données structurales et les propriétés électroniques rendent peu vraisemblable
l’existence de chaînes indépendantes dans l’état liquide du tellure. On propose ici que l’ordre local
peut être décrit par des sites à 2 ou 3 premiers voisins, liés par des liaisons présentant un fort caractère covalent. Spatialement, cet ordre local s’étend jusqu’aux seconds voisins. Dans le temps, sa
durée est très faible : les temps de relaxation de ces « reseaux » sont typiques d’un liquide normal,
10-12 s.
Une partie des électrons de valence est localisée dans ces liaisons, les autres formant un gaz
d’électrons typique d’un métal. La valeur correspondante de la densité d’états au niveau de Fermi
est discutée ; sa variation en température est reliée à l’évolution structurale du liquide.
~
Structural data and electronic properties argue against the existence of indepenAbstract.
dent chains in the liquid state of tellurium. It is proposed here that the local order consists of sites
with 2 or 3 first neighbours, joined by bonds with a strong covalent character. In space, that local
order persists up to the second neighbours. In time, it is of very short duration : the relaxation
times for these « networks » are typical of a normal liquid ~ 10-12 s.
Some of the valence electrons are localized in these bonds, with the others forming a metallic
electron gas. The value of the corresponding electronic density of states at the Fermi level is discussed ; its variation with temperature is related to the structural evolution of the liquid.
-
Introduction.
The purpose of this paper is to discuss the existence of chains in pure liquid tellurium.
It has been proposed by many workers (Epstein,
Fritsche and Lark-Horowitz [1], Hodgson [2], Cutler
and Mallon [3], Ioffe and Regel [4]) that liquid
tellurium is a semiconductor, the semiconducting
state being due to retention of the chain structure of
the solid (Buschert [5]). However, more recent experiments measuring the Knight Shift and the Hall
effect show that the density of conduction electron
states at the Fermi level is so large that a metallic
description of these states is more appropriate. This
situation has motivated us to see if a chain structure
model is consistent with such a metallic behaviour.
This problem is considered here with the help of the
new elastic neutron scattering data (Breuil and Tourand [6]), and of other structure related information,
such as the viscosity, the self diffusion coefficient,
and the inelastic scattering of neutrons (Gissler,
Axmann, and Springer [7]). Our approach will be
to ask what types of chain structures, if any, are consistent with these electronic and structural properties,
and then to look whether other types of structures
could provide a better explanation for the experimental data. We will successively consider : (1.1, 1.2)
-
(*) Associé
au
C. N. R. S.
LE JOURNAL DE PHYSIQUE.
-
T.
32,
N°
1,
JANVIER
some chain structures similar to those found in sulfur and selenium, but with a shorter length (Fig. 1),
(I.3) other chain structures with a more pronounced
metallic character, and (II) structures for which no
chains can be defined.
Three basic difficulties occur for the chain models.
First, the configuration of atomic neighbours is not
as observed for liquid tellurium. Second, they imply
the existence of rather long structural relaxation times,
in conflict with the observed low viscosity and high
diffusion coefficient and with the inelastic neutron
scattering spectra. Third, the density of states for the
conduction electrons near the Fermi surface can
hardly be as large as the experimental value.
In contrast to the chain structures, it will be shown
in part II that the observed local order can be described by a three-dimensional « network » of covalent
bonds. One possible structure for the high temperature situation (around 900 °C) is discussed in more
detail. In that description, the « network » of bonds
extends over the whole liquid (i. e. the liquid is continuous : there are no clusters nor molecules) ; on the
other hand, the corresponding local order persists
in space only up to the second neighbours, and relaxes
very fast in time : the life time of these (partially)
covalent bonds should be of the order of 10- 12 s.
This provides a consistent explanation for all of
1971.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:0197100320107300
74
the observed properties mentioned above. Furthermore, their temperature variations between the boiling
point (990°C) and the melting point (450oC can be
interpreted as reflecting an evolution between the
high temperature three-dimensional network and the
usual « chain structure » which applies properly only
to undercooled liquid Te.
I. Chain
CHAINS ».
structures
-
structures.
-
Two elements
« SEMICONDUCTING
known to have chain
state : sulfur and sele-
1.1
are
(Fig. 1) in the liquid
bound to these 2 nearest neighbours by (1 bonds, i. e.
overlap of atomic p orbitals in the direction of the
bonds. The amount of s mixing for these orbitals is
determined by the deviation of the bond angle 0
from 900 : for selenium, 6
106° gives 22 % s-hybridization. Two électrons per atom participate to these
bonds, two others lie on the deeper s levels, and the
two last ones occupy the remaining p levels : they
fill up completely the n band (which includes together
bonding and antibonding states). The antibonding
u states are empty (each atom has only 6 valence
electrons), and the gap between these and the n band
is responsible for the semiconducting properties of the
long chains (Fig. 2).
For chains of finite length, there is one unpaired
electron at each end, left over from the broken bond
(Chen and Das [14]). If that electron is not localized
on the end group, one
gets 2 conduction electrons for
each chain (and 2 localized acceptor states, or eventually 2 holes).
=
FIG. 1.
Helical chains of Te or Se in the solid state : d1 is
the length of a covalent bond inside a chain (2 first neighbours).
d2 is the distance between an atom and its 4 nearest neighbours
on the next chains. d3 is the distance between the axes of neighbouring chains ; in the hexagonal lattice of Se and Te, it corresponds to 6 neighbours. d4 is the distance of the second (covalent)
neighbours inside the chain. d3 and d4 are the same for tellurium.
-
nium. The liquid state in these substances is achieved
by rapid relative motions of chains, rotation around
covalent bonds, and dissociation of a few covalent
bonds, giving chains of finite length [8], [9], [10]. A
simple LCAO scheme for the electronic structure in
the solid state has been used by Reitz [11], Hulin [12],
and Olechna and Knox [13]. In the approximation
where the only interactions considered are between
one atom and its 2 nearest neighbours inside the chain,
this picture of the electronic structure should still
be valid for the liquid state (1). Each atom is then
(1) This nearest neighbour approximation neglects the interactions with atoms on other chains. It may be reasonable
for sulfur and selenium where the bonds inside the chains
(covalent) are much stronger than the bonds between the chains
(Van der Waals type for sulfur). This is substantiated by the
existence of independent chains in the liquid state of these elements. That approximation is however much worse for Te where
the interactions between chains are only 2 or 3 times smaller
than those inside the chains [12], [7].
Relative positions of energy bands in solid Te, in
FIG. 2.
the simplest LCAO description. The deepest band corresponds
to the 5 s atomic levels filled with 2 electrons per atom (« unshared electron pair »). The next band is made of bonding levels
(a bond), using essentialy 5 p atomic levels hybridized with
N 15 % 5 s levels. The last full band corresponds to the n electrons, which have pure 5 p character. The (empty) conduction
band is build with the antibonding states corresponding
to the a bond.
-
How would such a « long chains » structure fit
the experimental data for liquid tellurium ? First
consider the number and the distance of the nearest
neighbours to a tellurium atom, as seen by the elastic
neutron scattering measurements of the radial distribution function g(r) (Breuil and Tourand [6]). These
results are shown schematically in figure 3a. The
75
Indeed, following the same kind of arguments which
allowed us to assert the persistence of the covalent bonds in the liquid state, we should assume that
the bonds between the chains are also preserved. At
this point, it is no longer clear whether it is possible
to define any chains in the liquid state of tellurium.
Now let us compare the dynamic properties of
liquid tellurium with those of known chain systems
like selenium. Inelastic neutron scattering has been
done on both liquids by Gissler, Axmann and Sprin-
[7]. Optical phonons corresponding to stretching
twisting vibrations in the chains are observed in
liquid selenium, where they give well defined peaks
in the time of flight spectra. But for tellurium they
all completely disappear on melting, including those
corresponding to vibrations of the bond lengths.
Since neutron scattering allows the exploration of the
whole phonon spectrum down to wavelengths on the
order of a few interatomic distances, it would be expected to show such vibrations in liquid Te if strong
covalent bonds were preserved there. That on the
ger
and
contrary the spectrum of these vibrations seems to
be completely smeared out argues against the persistence of chain structures with a lifetime much longer
than typical liquid relaxation times.
FIG. 3.
a) experimental g(r) from Tourand and Breuil
The distances dl, d2, d3 and d4 correspond to figure 2. At 930 °C
the areas under the peaks correspond to : 3 ± 0.15 neighbours
at di ; 3.7 ::l: 0.20 neighbours at d2 ; 7.1 ::l: 0.20 neighbours at
d3 + d4 ; b) expected density distribution for independant chains.
The neighbouring chains being allowed free translation and
rotation, the atomic repartition seen by an atom on the reference chain is uniformly distributed on a cylinder. The projection of that density distribution on the r axis gives figure 3b.
c) Same as figure 3b, but taking into account the broadenings
seen on the first peak of the experimental g(r).
-
initial peak of g(r) occurs for a distance corresponding
to the 2 covalent nearest neighbours which exist for
the chains in the solid state (Fig. 1). Its position and
width do not change with temperature. This suggests
that the covalent bonds are present in the liquid state
even at high temperature (930 °C). However, the area
of that peak corresponds to 3 ± 0.15 atoms at 900 OC,
and somewhat less at lower temperatures, as shown
in figure 4. The fact that one observes 3 instead of
2 nearest neighbours is unexplained in any chain
picture.
peak would correspond to the nearest
neighbouring chains in the solid (Fig. 1),
The second
atoms
on
FIG. 4.
Number of nearest neighbours versus temperature
deduced from the radial distribution function of Tourand
and Breuil.
-
as
and the third to the distance between the axes of the
chains (6 atoms) and also to the 2 second neighbours
inside the reference chain. These latter two peaks are
as sharp as the first one, and well separated, even at
the highest temperature.
This requires very strong interactions between the
chains : for example, no free translations or rotations of
one chain with respect to the next ones are allowed,
as remarked by Breuil and Tourand (Fig. 3b, c).
Additional information is
provided by the measureselenium, these
properties are dominated by the average chain length,
which decreases exponentially with temperature (as
shown by the measurement of the concentration of
broken bonds [10]). The value of the viscosity should
also give the chain length as a function of temperature,
and the viscosity activation energy is expected to
correspond to the energy required to break a covalent
ments of the
viscosity.
In sulfur and
76
bond. The
resulting viscosity for selenium is one order
magnitude larger than the one for metals at comparable reduced temperatures (q 25 poises near
the melting point 220 °C, 13 centipoises at 500 oC,
and 2,7 cp at 800 °C) [15] (2) while the one for Te
has a value much like that of a metal (il
2 cp at
melting point 450 °C, and 0,8 cp at 900 °C, compared
to AI : 1
1,13 cp and Sb : fi
1,30 cp near their
melting points). It is nevertheless difficult to draw
quantitative conclusions from these values. The
of
=
=
=
=
we
get :
=
variation of the apparent activation energy with
temperature is more significant. For Se it keeps a
high value over the whole range of temperatures
(4 kcal/mole at 500 °C, 2 kcal/mole at 800°C). For
Te it is comparable at the melting point (2 kcal/mole),
but falls to a very low value at high temperature
( 0,1 kcal/mole around 800 °C). This fact and the
low value of the viscosity indicate that the concentration of « strong » bonds (i. e. bonds with a lifetime
significantly longer than typical « jump » times in
liquids, - 10-12 s) is already small after melting, and
that they progressively disappear at high temperature.
It contradicts the persistence of the chain structure
which would be inferred from the great stability of
the radial distribution curve (Breuil, Tourand).
Finally consider the value of the electronic density
of states at the Fermi level, as obtained from measurements of the Knight shift (Cabane and Froidevaux [16]) and the Hall effect (Tieche and Zareba [17],
Enderby and Walsh [18], Perron [19]). (A qualitative
discussion of the Knight shift is presented here ; a
more detailed interpretation is given in the Appen-
dix.)
The Knight shift is proportional to the product of
the Pauli electron spin susceptibility by the average
probability density at the nucleus for electrons at the
Fermi level. The first term gives the electronic density
of states n(EF) if the exchange enhancement of the
electron spin susceptibility is known. We assume here
that the enhancement of the susceptibility xs relative
to the value Xo for independent free electrons is comparable to what is found in other substances, i. e. that
xs/x° is between 1 and 2. The second term in the Knight
shift is the amount of s-character of the wave functions at the Fermi level. In the simplest LCAO description [11 ], the upper valence states (the rc band)
are completely p-type, while the antibonding u states
(the conduction band) have 15 % s-hybridization
(Fig. 2). Clearly, these antibonding states can be mixed
with a larbe number of excited states, in particular
with 6 s2 levels. This would increase their s-character, but it is still unlikely that it will exceed 30 %.
(2) The addition of iodine reduces the chain length by
forming I-Se-Se----Se-I molecules. The average chain length
then be deduced from the iodine concentration. The viscosity of liquid Se with a few percent iodine has been measured
by Krebs and Morsch [15]. For an average length of 15 atoms,
the viscosity at 300 °C is 20 cp.
can
With these assumptions we try to get an estimate of
the number n of « free electrons » which, in a free
electron model, would correspond to the observed
density of states as deduced from the Knight shift.
0.49 %),
From the Knight shift value at 750°C (K
where ç is the
amount of s character of the wavefunctions of the electrons at the Fermi level. We assumed
that ç is probably of the order of 0.15, while XSIXOS is
between 1 and 2. The measured value of the Knight
shift then shows that the actual number of « free electrons » per atom at 750 °C is not smaller than 1.
Now consider n(EF) obtained from the Hall effect.
Although earlier measurements yielded erroneous
data [1]] [4], agreement on the value of the Hall
constant has now been found in the three latest experiments [17] [18] [19]. These measurements indicate
that, in a free electron picture, n(EF) corresponds to
2.6 free electrons per atom at 750 °C.
Both the Knight shift and Hall effect are inexplicable
with long chains. For example, an average length of
10 atoms would give but 0.2 « free electrons » per
atom, while experimental data indicate that there
are at least 1, and probably 2 or 3 free electrons per
atom.
SHORT «MOLECULES». - From the preit is clear that a metallic density of
section
ceeding
states and the observed g(r) can not be obtained from
real chains in liquid Te. We now consider whether
it is possible to overcome these difficulties with very
short molecules. For example, molecules of 3 to 4 atoms
would begin to give an adequate number of conduction electrons (3) and a more reasonable value for
the viscosity.
The radial distribution function nevertheless still
requires very strong interactions between these molecules. These interactions will be especially strong at
the end groups, where the vacant orbitals are likely
to make bonds with other molecules. A method for
fitting the observed g(r) with this picture is a follows.
An average of 3 neighbours at the covalent distance
d, can be accounted for with 2 neighbours bound
at the distance di for atoms inside the molecules,
and 3 or 4 neighbours bound at the same distance
dl for the atoms at the ends of the molecules. But the
fact that the bond length is in both cases the covalent
distance d1 (the first peak in g(r) is sharp and unsplit)
I .2 VERY
(3) In the preceeding scheme, 4 atoms molecules would give
0.5 free electron per atom. Furthermore, it is likely that this
disruption of the periodicity of the lattice will cause .the gap
to disappear, so that the electronic density of states at the Fermi
level might be substantial [20] [21].
77
that these new bonds between an end group
and other molecules are as good covalent bonds as
the others inside the molecules. At this point we no
longer have individual molecules, but « bridged
chains » or rather a three dimensional network of
covalent bonds. The discussion of this structure will
be continued in part II.
means
I.3 SEMIMETALLIC CHAINS.
It is not essential to
expect that the conduction electrons come from the
end groups only : even in the solid state, tellurium
is almost semimetallic (the gap width is only 0.34 eV).
Thus it is reasonable to consider the possibility that
the disorder of the liquid state will weaken that small
gap to a « pseudogap » [20] [21]. In other words, the
memory of the solid state gap in the liquid could be
only a minimum of the electron density of states n(E),
whose importance should be related to the amount
of local order. It would then be possible to fit the
observed values of the density of states at the Fermi
level n(EF) with the pseudogap height as an adjustable
parameter. But it is not clear whether a liquid of
strongly correlated chains (in the sense of the previous
discussion of g(r)) can allow the necessary amount of
disorder for this model (4).
-
Such a model still has all the problems associated
with any chain picture :
-
The number of nearest neighbours would still be
2 instead of 3.
No explanation is provided for the very strong
correlations between chains.
-
-
The
-
No
neutron
would still be too
observed.
viscosity
to what is
high compared
explanation is provided for the inelastic
scattering data.
Again, these difficulties could be overcome by a
drastic reduction of the chain length (around 2 atoms).
It has been shown in the preceeding section that in this
case the bonds between these groups of a few atoms
should produce a 3 dimensional covalent network,
with a distribution of sites with 2, 3, or more nearest
neighbours. This point is discussed further in the
next section.
II. Three dimensional covalent networks. It has
argued in section 1 that chain structures are not
good picture for liquid tellurium. Now we turn to
been
a
alternative structures in order to improve the agreeexperimental data. In a first step, the radial
distribution function g(r) suggests the existence of
many bridges between the chains, increasing somewhat
this way the correlations between chains and the
average coordination number. The existence of these
bridges in liquid Te is not quite a surprise. Consider
for example Hulin’s calculations of the electron band
structure in the solid state [12]. They have shown that
the interactions of a given Te atom with the 4 nearest
atoms on different chains are only 2 or 3 times smaller than the ones with the first 2 neighbours inside the
chain. Moreover, the elastic coupling constants between chains in the solid state can be deduced from the
frequencies of the optical vibrations [7]. Compared to
the ones inside the chains, they are found to be 3 times
smaller in Se, and only 2 times in Te.
But in fact, the initial peak of the radial distribution
function g(r) at 900 °C corresponds to 3 nearest neighbours at the same covalent distance. This suggests
that the transformation of the short range order
(with regard to the chain structure) is even more substantial, and that it is worth trying to understand what
structure can be constructed with 3 covalent bonds
per tellurium atom.
Of course, one could consider some sites with more
than 3 covalent bonds. However, this would imply
a large change in the type of bonding. It will be shown
indeed in section II.4 that with 3 covalent bonds it is
still possible to use the same s - p hybrid orbitals as
in the case of only 2 bonds. But each atom has only
3 p orbitals in the valence shell, allowing only 3 such
bonds with the same amount of hybridization. Formation of 4 bonds would require complete sp3 hybridization, which is unlikely because the 5 s levels are much
deeper (10 eV) than the 5 p levels. Formation of
6 bonds in a more or less cubic arrangement would
correspond to pure p bonding orbitals. This is possible
at higher temperatures and pressures, where an evolution towards a metallic dense packing similar to that
of Polonium is expected. But the bond lengths and
angles would then be substantially changed with
respect to the sp hybrid covalent bonds ; furthermore,
g(r) should have only 2 peaks instead of 3. This is not
observed in the present range of the experiments i. e.
up to 930 °C.
We discuss now in detail one possible structure with
3 covalent bonds per atom.
ment with
II. 1 HIGH
A
(4)
large
explain the
disappearance of the energy gap : if the local order is not affected, the electron band structure should not change very much.
Inside the chains, the disorder can be achieved by almost free
rotation around the bonds. That kind of disorder affects only
the 2 second neighbours in the chain for the reference atom.
It is also necessary to consider the disorder in the positions of
the 4 nearest atôms on the next chains (Fig. 1). Such a disorder
is inconsistent with the strong correlations between chains shown
amount of disorder is necessary to
by g(r) (Fig. 3).
TEMPERATURE STRUCTURE.
-
The struc-
ture which to us appears most plausible for liquid
Te at 900 °C is shown in figure 5. It is the arsenic (A7)
type of structure. Let us consider what characteristics
it would give to g(r).
peak of the radial distribution function
g(r) corresponds to 3 first neighbours, joined by covalent bonds, at the distance di . We assume that the
bond length and bond angle are almost the same as in
-
The first
78
the solid state. The exact value of the bond angle is
calculated in the next step.
by bond lengths and angles may explain why the 3d
peak is so narrow, even at high temperature. Incidently,
The covalent neighbours of these first neighbours give 6 atoms at the distance d4 which all come
into the third peak of g(r) if the bond angle is comparable to the one in the solid state. Reciprocally, from
their exact distance, i. e. from the position of the third
peak, we get the value of the bond angle, wich is 1000,
compared to 1030 in the solid state. Rotation around
the covalent bonds does not change the distance of
these atoms to the central one (Fig. 5).
that amount of rotation around the bonds may be
quite large : the corresponding energy barrier, for
group VI B elements, is given by Pauling [22] as
~ 2 or 3 kcal/mole, while kT is 1.4 kcal/mole at
4500 C and 2.3 kcal/mole at 900 °C. This may provide
also an explanation for the absence of any peak in
g(r) after the 3d one. Finally, the second peak corresponds to 3 or 4 atoms in this model, their number
and positions being fixed mainly by excluded volume
effects (depending on the positions of the 6 preceeding ones-see Fig. 5). Their actual number, as given
by g(r), is 3.7 (solid state 4).
It is stressed again that this model does not involve
any molecules nor clusters : the network of bonds
extends over thé whole liquid (5). Or course, this
does not imply that long range order is retained :
rotation around the covalent bonds, disorder in the
positions and numbers of the second and third neighbours insure a large amount of randomness after 2
or 3 interatomic distances.
-
There is still some place for 3 or 4 more atoms,
somewhat further than the first neighbours, and in
opposite directions. They go into the second peak
-
(Fig. 5).
=
II.2 EVOLUTION
FIG. 5.
High temperature structure. Covalent bonds (full
lines) join each atom and its 3 nearest neighbours. Their angle
is calculated from the distance of the 6 covalent next neighbours
(dots) of the reference atom. It should be remarked that even
free rotation around the covalent bonds does not change the
distance of these covalent next neighbours. The 3 second neighbours (crosses) of the reference atom are located nearly in opposite directions to the first neighbours. Théir bonds to the refe-
atom are of the same nature as the bonds between chains
in Solid Te. Their number is mainly determined by excluded
volume effects. For example, some rotation around the covalent
bonds might afford some place for more than 3 of them (see
rence
The comparison with the experimental g(r) gives,
by construction, a good agreement on the position
and the area of the first peak. The third peak is also
found at the right place, the number of corresponding
neighbours being 6 in this model, compared with the
experimental value = 7 and the solid state value 8.
The fact that their distance is unaffected by rotation
around the covalent bonds, and determined only
=
OF
THE STRUCTURE
WITH TEMPE-
When the icmpérature is lowered from
930 to 490 °C, the average number of first neighbours
falls from 3 to 2.5 (Fig. 4). This suggests that near
the melting point, many atoms have only 2 nearest
neighbours. These sites with 2 covalent bonds will be
called binary sites, or chain sites, while the others
with 3 bonds will be called ternary sites in the following discussion. The liquid is no longer stable (with
respect to the solid) at temperatures where that
average coordination number would approach 2,
so that no real chains can be observed in the usual
liquid range. Formation of small series of sites with
2 nearest neighbours cannot be expected before the
concentration of these « chain sites » is substantially
larger than one half (both species are assumed to be
miscible, so that there is no tendency for association
of binary sites. This is clearly the case down to the
RATURE.
-
melting point).
The structure at intermediate temperatures is then
likely to be a mixture of binary and ternary sites, the
proportions being given by the average coordination
number. For example, at 490 °C, the average coordination number is 2.5 ± 0.2, corresponding to about
equal concentrations of binary and ternary sites. A
two dimensional representation of such a possible
network is shown in figure 6. This provides a single
structural parameter (i. e. the concentration of ternary sites) for the interpretation of structural and
electronic properties.
(5) For example, in the case of a stalking of clusters involving
central atom, his 3 first neighbours and his 6 second covalent
neighbours (Fig. 5) the 6 atoms on the frontiers of the cluster
would not have an adequate number of covalent bonds. The
initial peak in g(r) would then be smaller and broader.
a
79
associated with the local shears and rotations).
Furthermore, it would not explain the observed tendancy towards higher coordination numbers, which
is in fact quite general among liquid semiconductors.
Finally, one could also consider the entropy of
the electron gas associated with the electron states
close to the Fermi level : this term favours metallic
situations, with a high density of states at the Fermi
level. But in the liquid state, that contribution to the
total entropy is also expected to be small compared
with the configurational and vibrational entropies.
-
2022
FIG. 6.
Structure at intermediate temperatures. This is an
example of how the structure at intermediate temperatures
might appear in 2 dimensions. Near the melting point, the
concentrations of binary and ternary sites are about equal in
liquid Te. This is inferred from the average coordination number,
which is given by the area under the first peak in the radial distribution curve 4 03C0r2 g(r).
-
That evolution in temperature shows that the
enthalpy for the liquid would be
made of binary sites only (this is the hypothetical
situation at sufficiently low temperatures). In other
words, the ternary bonding system corresponds to a
state of higher energy than the binary one (6) ; it
will be seen from the discussion of the electronic
structure (11.4) that this instability with respect to
the chain sites corresponds to the promotion of one
electron per atom into an antibonding state. The
increasing concentration of ternary sites at higher
temperatures is due to the entropic terms in the free
energy :
First, the vibrational entropy of the site can be
estimated from its Debye temperature, i. e. from the
corresponding elastic constants. It will clearly be
larger for ternary sites, because the potential well
is not nearly as deep as for binary sites.
state of minimum
-
Second, the configuration entropy of the liquid
likely to be dominated by local shears and rotations.
-
is
Qualitatively, that entropy increases when the local
environment becomes more isotropic (for example,
when the distances of the first and second neighbours
become equal). Thus, this term also favours sites with
a higher coordination number.
There is also a configuration entropy which is
associated with the mixing of binary and ternary sites.
However, the entropy associated with this alloy disorder is expected to be a small part of the entropy of the
liquid (in particular, much smaller than the entropy
-
So far, we have been commenting on the implications of the changes in temperature on the first peak
in g(r). It is observed that the 2nd and 3d peak are
quite unsensitive to temperature variations (’). This
is consistent with the present description. Indeed, the
positions and numbers of the neighbours corresponding to the 2nd and 3d peak in g(r) are almost the same
for binary and ternary sites. In particular, this occurs
for the 3d peak because in the solid state (and in the
hypothetical chain liquid) the distance between the
axes of the chains is just the same as the distance of
the 2 covalent second neighbours inside the chains
(see Fig. 1). This way, binary sites have 2 covalent
+ 6 non covalent neighbours at that distance, while
ternary sites have 6 covalent + 2 non covalent neighbours at the same distance (Fig. 5).
In this section we comment
II. 3 DYNAMICS.
qualitatively on the relation-ship between the observed dynamic properties of liquid Te (viscosity, selfdiffusion ; inelastic neutron scattering) and our model
for the high temperature atomic structure.
It has been shown in the previous section that the
binary configuration corresponds to a lower energy
for a Te site than the ternary one. This suggests that
the potential barrier for the jump of a tellurium
atom from a ternary site to a different position is
weak compared to that for an atom in a chain like
configuration. This corresponds to a relatively high
jump frequency, a short lifetime of all the bonds, and
a substantial self diffusion coefficient. The microscopic value of the self diffusion coefficient has been
measured with radioactive tracers by Potard [23] :
2.9 x 10- 5 cm2 js at 490°C. This agrees well with
D
the microscopic value which can be estimated from
the width of the elastic peak in the inelastic neutron
scattering experiment [7] : D 2.6 x 10- 5 cm2/s at
460°C. Using the common diffusion equation :
-
=
=
(6) The bonds of a ternary tellurium atom are nevertheless
stronger than inusual liquid metals : this is indicated by the
small width of the peaks in g(r), and is described by what we
call their « covalent character ». At first sight, this would suggest
a viscosity somewhat higher than in liquid metals. But the number of bonds to break for each jump is smaller than in the liquids
with a dense atomic packing, and the excluded volume effects
are less important (there is more « free volume » available
for the diffusion process, because of the small coordination
number).
(7) In fact, there is a small decrease in the height of the 2"d
peak of g(r) as the temperature goes up. This is consistent with
the idea that, when an extra covalent bond is formed on a binary
site with one of the next-nearest neighbours, that distance moves
from the 2nd peak to the lst peak in g(r).
80
get a crude extimation of the correlation time for
self diffusion ’te ’" 2.5 x 10-12 s.
That short lifetime for the covalent bonds also
explains the moderate value of the viscosity : all
the structural relaxation times in liquid Te will be
on the order of 10-12 s, hence typical of a good
we
liquid (6).
Finally, the
neutron inelastic scattering results are
consistent with that liquid behaviour. Indeed, the
vibration modes which are present in the chains of
liquid selenium should be completely damped in liquid
Te, first by the short lifetime of the bonds (10-12 s
compared to the optical vibration frequencies
lO-13 s), and also by the lack of periodicity in the
liquid (increased by the presence of 2 types of sites).
This damping will smear out the dispersion curves
for these vibrations, and broaden the corresponding
peaks in the scattering time-of-flight spectra. That
they actually completely disappear on melting for
Te means that this broadening is rather severe.
-
FiG. 7.
Hybrid atomic orbitals for a ternary tellurium site A.
The amount of 5 s hybridization (or the s-character) of the bonding orbitals fIIl, fil 2, f113 is determined by their angle. The last
of the 6 electrons of atom A, located on antibonding orbitals,
is not represented.
-
II .4 ELECTRON STATES :: HIGH TEMPERATURE LIMIT.
- In this section an elementary discussion of the
electronic structure for a liquid of ternary tellurium
sites is presented. It is suggested that the existence of
3 strong bonds with bond angle 1000 for each tellurium atom determines the nature of the valence
band and its filling by the electrons. The conduction states are then considered.
In the condensed state, the existence of strong bonds
allows the description of the bonding orbitals as
linear combinations of atomic hybrid orbitals [24].
In that scheme, the hybridization of the atomic orbitals can be obtained from the knowledge of the site
structure [22] [14] : the symmetry axis of the ternary
site being taken as z axis, we write the 3 bonding
orbitals of atom A (see Fig. 7) :
where SA is the 5 s orbital of the atom, and XA, YA,
ZA, its three 5 p orbitals. rx2 is the amount of s character of these 3 hybrid orbitals, and fixes their angle,
i. e. the bond angle : 1090 would give complete hybridization (a2
0.25), 900 pure p character of the bonds
(a2 0). The actual value for liquid Te is 1000, and
yields OE2 = 0.15. The bonding orbitals are thus primarily 5 p orbitals, with 15 % 5 s mixing.
The last orbital is :
=
=
For small mixing (a « 1) it would give the 5 s2 pair
of electrons. The actual value of a shows that this
pair has 45 % p character.
In the free Te atom, the 5 s levels are 10 eV lower
than the 5 p levels [25]. For a ternary site, the hybridization reduces the distance in energy between the
electrons of the unshared pair and the ones on bonding states to 40 % of the energy difference between
5 s and 5 p levels. These 4 eV are nevertheless sufficient to keep the pair deeper in energy compared to
bonding levels.
The energy bands
are
then filled in the
following
way :
deeper « unshared pair »,
3 electrons on bonding levels,
1 in the conduction band, build with antibonding
states in our description. Clearly, these last states will
-
2 electrons in the
-
-
be in fact a full mixing of antibonding and excited
states, and would be much better described in a nearly
free electrons picture, their wave functions being
described as plane waves orthogonalized to the valence
states.
of bands is shown in
that
such
a picture the conducin
figure
tion is definitely metallic and « n type » ; the conduction band is filled with one electron per atom, so that
there is no « holes » conduction in the high temperature limit. This is in qualitative agreement with the
Hall constant value (which is negative, but 3 times too
small for 1 « free electron » per atom), and in conflict
with the thermoelectric power, which is positive.
This problem is general for many disordered systems
like amorphous or liquid semiconductors.
The Knight shift does not give any indication of
the sign of the carriers ; its value can be interpreted
in this model with 1 free electron per atom if one assumes an exchange enhancement of the electron spin
The
resulting arrangement
8a. It is
seen
81
in temperature of the structure and of the electronic
density of states at the Fermi level n(EF). The simplest
approximation is to assume that the binary sites give
no contribution to n(EF). This may be reasonable in
a liquid composed essentially of such binary sites,
i. e. of chains (it is the « semiconducting chains »
hypothesis), but it is surely not valid when the concentration of these binary sites becomes one half or
less : their electronic structure is then likely to be
widely changed by the presence of the ternary sites.
The concentration in ternary sites and the results of
the electronic density measurements at different temperatures are compared in figure 9.
FIG. 8.
-
a) Qualitative representation of the density of
energy for
states
network of ternary sites. The « unshared
electron pair » corresponds to the 2 electrons on deeper 5 s levels,
and the « bonding states » to 3 electrons (one for each bond).
There is one more electron which goes into the conduction band.
b) Same as a), but with a small overlap between the valence
and conduction bands, as it should probably be the case for
substances with the A7 structure. The possible effect of structural disorder on the shape of the density of states is tentatively shown by the dashed line.
versus
a
a factor of 2, and an s-character of
functions at the Fermi level of about 15 %
(see the Appendix for more details). In the LCAO
scheme, this implies that the upper part of the conduction band is very strongly hybridized with 6 s levels.
In fact, it seems that this number of 1 « free electron » per atom deduced from the previous description of the electron bands leads to some difficulties
for a quantitative interpretation of Hall effect. A more
realistic point of view would be to assume that the
electron density of states at the Fermi level is higher
than that deduced from the filling of a free electron
band by 1 electron per atom. This could be achieved
through some overlap of the valence and conduction
bands (Fig. 8b). This is actually the case in many
materials with the A7 structure, like arsenic [26].
The best fit for the Hall effect is obtained with 2.6
« free electrons » per atom at 750 OC. This number
can provide an adequate interpretation of the Knight
shift too : for example, with the same s-character for
the wave functions, the necessary exchange enhancement of the electron spin susceptibility would be
susceptibility by
the
wave
II.5 TEMPERATURE
The next
DENSITY.
-
VARIATIONS OF THE ELECTRONIC
point is
to
relate the variations
FIG. 9.
Evolution with temperature of the structure and of
the electronic density. Dots : concentration of ternary sites,
as deduced from g(r) ; squares : number of « free electrons »
from Hall effect [17] [18] [19] ; crosses : number of « free electrons » from Knight Shift. Knight Shift data are plotted assuming an exchange enhancement of the spin susceptibility by a
factor of 2, and an s-character for the wave functions at the
Fermi level of 22 %. This choice, which gives a good agreement
of the absolute values of the numbers of free electrons deduced
from the Knight Shift and Hall constant, is rather arbitrary.
The number of free electrons deduced from the Knight Shift
could be smaller or larger by a factor - of 2.
-
a proper description of the electronic structure
intermediate temperatures, one has to consider a
singe band structure for all electrons, whether they
come from binary or ternary sites, instead of separating the contributions of « semiconducting » and
« metallic » sites. One could, for instance, try to calculate the density of states from the knowledge of the
In
at
82
short range order ; for this, one has to take into
account 2 kinds of disorder : the « solid solution
type » disorder in the repartition of binary and ternary sites, and the «liquid type » disorder. The
« liquid type » disorder affects only the atomic configuration at distances larger than 2 or 3 interatomic
distances, but the « solid solution type » disorder
disturbs also the nature of the nearest neighbours.
The actual band structure might then be very different from a simple average over the band structures
of the high temperature and low temperature liquid.
Conclusion.
Various temperatures give access
wide range of situations in liquid Te. At high
temperature (900 °C), the short range order has been
described by a structure where most atoms have 3
first neighbours, joined by strong bonds similar to
the covalent bonds of the chains in solid Te. That
structure has been used to explain the metallic behaviour of liquid Te in the high temperature limit. In
fact, a further increase of the average coordination
number can be speculated for still higher temperatures
and pressures : the appearance of sites with simple
cubic coordination (6 atoms in the first coordination
shell, instead of 3 + 3 at 9000 without pressure) would
not be surprising, in view of the cubic structure of
polonium [27]. At lower temperatures (near the melting point 450 °C), an increasing number of Te atoms
tends to form only 2 covalent bonds. Evolution towards
the semiconducting situation expected for a liquid
of chains can be achieved by strong supercooling or/
and addition of selenium. Perron’s resistivity data
on TeSe alloys [28] suggest that this reduces the
density of states at the Fermi level to a point where
the corresponding states become localized. The conduction should then be due to carriers excited across
that mobility gap.
On the whole, the dominant feature of liquid Te
appears to be the existence of a network of strong
covalent bonds. But covalent networks, where each
atom makes a small number of strong bonds with
its nearest neighbours, are commonly associated with
semiconducting properties. On the other hand,
metallic behaviour is usually produced by a metallic
type of binding, associated with a dense packing.
Nevertheless, liquid Te shows an interesting situation
where a network of covalent bonds (probably stronger than in the semimetals Sb, As..., because they are
not destroyed on melting) coexists with a metallic
like electron gas : the concentration of « free electrons » (2 or 3 « free electrons » per atom) is much
larger than for those seminetals, and comparable
to what is found in most metals.
-
to a
Appendix : Knight shift. - In a NMR experion a metal, the conduction electrons are responsible for a relative enhancement of the local magnetic
field at the nucleus compared to the field which would
ment
exist in
is given
a
nonmetallic substance. This contribution
by [29] :
where xs is the electron spin susceptibility per atom,
and
1 t/1F(O) ,2 &#x3E; is the average over all the electronic states at the Fermi level of the electronic probability
density at the nucleus. Only s-electrons, whose wave
functions have a finite amplitude at the nucleus,
contribute to this contact interaction. In most usual
metals (i. e. when both n(EF) and
1 t/1F(O) 12 &#x3E;
are not small) it gives the dominant contribution to
the Knight shift.
There are also some noncontact contributions to
the Knight shift :
The dipolar interaction of the nucleus with the
electron spin gives rise to anisotropic shifts in solid
metals of low symmetry. In liquid metals it vanishes,
because of the rapid thermal motion.
-
For electrons with a large p character, there is
orbital contribution to the susceptibility and to
the NMR line shift : the orbital magnetic moment of
these electrons, although quenched in first order by
the crystal field, gives a contribution through the
second order in perturbation, In solid Te, these terms
are responsible for the large (-0.1
%) chemical
shifts
These
shifts observed by Bensoussan [30].
than
smaller
are however one order of magnitude
their
isotropic
the observed Knight shift (0.5 %), and
part is small. In metals, these orbital effects are responsible for the Kubo-Obata contribution to the
line shift [31]. In contrast to the direct contact interaction, this term arises from all electrons and not
just those at the Fermi surface. In some cases where
the density of states at the Fermi level is small, like
in superconductors [32] or in semimetals [33] [34],
the orbital interaction can then be comparable to the
direct contact interaction expressed in equation (1).
Such a contribution exists in liquid Te, for the wave
functions of the electrons have a large p character.
But we think that the density of states n(EF) and the
amount of s character of the wave functions at the
Fermi level (at least 15 %) are large enough to make
the direct contact interaction dominant. This is however a naïve assertion, and tedious calculations would
be necessary to get any sound idea about the relative
orders of magnitude of the direct contact and orbital
contributions to the line shift.
The orbital moment of the conduction electrons
can also interfere through the spin orbit coupling,
leading approximately to an enhancement of the
electron spin susceptibility by a factor or 1 + 03BB/0394,
where  is the spin orbit coupling constant, and A
the mean width of the conduction band [31]. A crude
estimate from the band structure in the solid state
-
an
83
- a few eV (i. e. 1 or 2 eV), while À is about
[11] [12]. This suggests that the spin orbit
coupling increases the line shift at most by 50 %.
Finally, there is a contribution to the line shift
from the polarization of the ionic core states by the
conduction electrons. For nontransition metals, this
contribution is one order of magnitude smaller than
the contact interaction [35].
yields 4
0.5 eV
We will therefore assume that the dominant interaction between the nucleus and electrons is the direct
contact interaction given by expression (1). Spin orbit
coupling and other orbital contributions can induce
our results in error by about 50 %.
Expression (1) can be rewritten in the following
way
[29] :
where c
1 fi F( 0) 12 &#x3E; .
the
ives approximately thé
1
I gives
iili
proportion of S character of the
=
wave
functions at
the Fermi level, a(s) is the hyperfine coupling constant, and g1
Ill/l, 1 being the nuclear spin, III its
magnetic moment, and y, the Bohr magneton. From
the knowledge of the shift, and a theoretical estimate
of ç, one can get the value of the electron spin susceptibility xs. It is then necessary to make assumptions
on the effects of electron correlations,
exchange enhancement, and non-free electron behaviour of the
conduction electrons in order to relate it to the density of states n(EF) :
=
with
a
describing essentially the exchange enhance-
ment of X..
As an example, let us try to make a reasonable choice
for the parameters a and ç, in order to get a good
agreement on n(EF) with the Hall effect :
The value of the Knight shift at 750°C is 0.49 %.
- The exchange effects usually enhance the electron spin susceptibility by a factor which lies in the
2. We choose a
1.5.
range 1
- We assume that the wave functions at the Fermi
level in liquid Te are made primarily of antibonding
states. This would give ç of the order of 0.15, but the
mixing 6 s levels can increase somewhat this value.
In fact, the best fit , with the Hall effect at 750°C is
0.15. In this case, the «équivalent
obtained for ç
number of free electrons » which, in a free electron
band, would give the same n(EF) as what is observed,
is 2.6 (Fig. 9).
-
-
=
=
What is the
hand, ç
can
possible range for that value ? On one
hardly be smaller than 0.15, and oc is
at least 1, so that the upper limit for the number of
free electrons is of the order of 6. This result is not
very useful, the number of valence electrons being
also 6. On the other hand, in nontransition metals
a is not likely to be larger than 2 ; ç could perhaps
increase up to about 50 %, if the conduction electrons
are nearly free. This shows that the lower limit for
the equivalent number of free electrons at 750°C
is about 1 free electron per atom.
M. Breuil and G. Tourand
have made essential contributions to this work. We
are also indebted to G. Cabane, W. G. Clark, F. Cyrot,
M. Hulin, M. Papoular and C. Potard for innumerable discussions.
Acknowledgments.
-
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