A tight-binding calculation of the chemical shift in
trigonal selenium and tellurium
M. Bensoussan, M. Lannoo
To cite this version:
M. Bensoussan, M. Lannoo.
A tight-binding calculation of the chemical shift in
trigonal selenium and tellurium.
Journal de Physique, 1977, 38 (8), pp.921-929.
<10.1051/jphys:01977003808092100>. <jpa-00208660>
HAL Id: jpa-00208660
https://hal.archives-ouvertes.fr/jpa-00208660
Submitted on 1 Jan 1977
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LE JOURNAL DE
TOME
PHYSIQUE
38,
AOUT
1977,
921
Classification
Physics Abstracts
8.138
-
8.520
-
8.662
-
8.810
A TIGHT-BINDING CALCULATION OF THE CHEMICAL SHIFT
IN TRIGONAL SELENIUM AND TELLURIUM
M. BENSOUSSAN
C.N.E.T., 196, rue de Paris, 92220 Bagneux, France
and M. LANNOO
Equipe de Physique des Solides du L.A. au C.N.R.S. N° 253,
I.S.E.N., 3,
rue
F.-Bads, 59046 Lille Cedex, France
(Reçu le 9 fevrier 1977, accepté le 4 mai 1977)
Nous présentons un calcul détaillé du tenseur de déplacement chimique dans le
Résumé.
sélénium et le tellure trigonaux. Ce calcul est fondé sur un modèle en liaisons fortes où les interactions,
entre orbitales moléculaires convenablement définies, sont prises en compte par un traitement de
perturbation au second ordre. La partie diagonale du tenseur peut être complètement interprétée en
utilisant des valeurs raisonnables pour les paramètres. Les résultats sont en accord avec les spectres
optiques qui font ressortir un important caractère liant dans la bande de valence supérieure. Avec ces
valeurs pour les paramètres on ne peut pas rendre compte du signe du terme non diagonal. Les
raisons possibles de ce désaccord sont discutées.
2014
A detailed calculation of the chemical shift tensor in trigonal selenium and tellurium
Abstract.
is presented. It is based on a tight-binding model where interactions between suitably defined molecular orbitals are taken into account by second order perturbation theory. The diagonal part of the
tensor can completely be interpreted using reasonable values for the parameters. The results are in
agreement with optical results which show that there is a substantial amount of bonding character in
the upper valence band. With these values for the parameters the model cannot reproduce the sign of
the non diagonal term. Possible reasons for this discrepancy are discussed.
2014
1. Introduction.
N.M.R. spectra obtained from
trigonal tellurium exhibit three distinct lines [ 1 ] each
showing an anisotropic shift with respect to the orientation of the magnetic field. A similar result is obtained
for trigonal selenium [2]. From this it follows that there
are three distinct sites in the primitive cell of these
crystals. In fact they are equivalent but are differently
oriented relative to the magnetic field. However,
although the structures of the two crystals are the same,
the anisotropic behaviour of the N.M.R. lines, which
is interpreted in terms of a chemical shift tensor a, are
quite different. The N.M.R. properties of a nucleus,
characterized by a, are related to its local electronic
configuration. The applied magnetic field partially
dequenches the orbital momentum of the valence shell
which in turn creates on the nuclei an hyperfine field
related to the local symmetry [3]. Therefore the chemical shift is sensitive to any perturbation of the chemical bonds around a given atom and its study provides information on the local electronic distribution.
A possible theoretical description of the chemical
shift tensor a in trigonal selenium and tellurium has
-
been proposed recently [4]. It was obtained with a
purely molecular model based on the assumption of
perfectly decoupled bonds. The orientation of the
atomic hybrids was deduced from the direction of the
principal axes of a, leading however to orbitals which
did not point towards the nearest neighbours. Such a
result clearly shows that the situation cannot be easily
described by a simple a priori model.
Our aim in this work is then to undertake such a
calculation using a tight-binding approximation. We
want to derive a simple model capable of describing
the results in selenium arid tellurium which are quite
different from each other. For this we start from the
of a molecular model in terms
of a pure s band, a pure p bonding band, a pure p
lone-pair band and then the p antibonding band. As we
shall show this does not predict correct results for 6,
and it is necessary to go further, taking into account
the interactions of these molecular states by second
order perturbation theory.
In a first part we recall the experimental results
concerning the chemical shift tensor. We then detail a
most natural definition
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:01977003808092100
922
theoretical formulation in terms of the p character of
the electron population. This is determined from a
model of the band structure which allows the populations in the different bands to be calculated. Finally
the last part compares the theoretical and experimental results and a discussion of possible refinements is
presented.
If we call u the unit vector along the direction of the
magnetic field Ho, the change in Au along this direction
is given by :
From this it is easy to write down the contributions of
For the sites A, B and C to Au, for rotations of Ho around
Summary of the experimental results.
simplicity we split the chemical shift tensor a into an the Ox, Oy and Oz axes. A fit of these expressions with
isotropic part Qo and an anisotropic part Au, ao being the experimental results allows the parameters occursuch that the trace of Au is equal to zero. Let us ring in eq. (1) to be determined. However only H, J, K
call A, B, C the three sites of the primitive cell (Fig. 1). can be determined unambiguously from experiment.
The magnitude of L can also be determined but its sign
cannot be obtained without knowledge of the crystalline variety as well as the crystalline orientation of the
axes of the sample under study. This has proved
possible for trigonal tellurium [5] from a detailed
analysis of the etchpits (L is found to be negative), but
not for trigonal selenium [2] where only !I Lis known.
The experimental values of H, J, K, L are given in
2.
-
table I.
TABLE I
Experimental values of the elements of u xyz
For the following theoretical study, it is useful to
associate with each atom a local system of axes OX,
OY, OZ defined in figure 1. In this system the tensor
Aa(A) for site A can be written :
Chain structure and definition of the axes in trigonal
selenium and tellurium. a) Unit cell : the Z axis is perpendicular
to the plane XY containing the three atoms A, B, C. b) Right hand
variety. c) Left hand variety.
FIG. 1.
-
From symmetry considerations the tensor Ac
:
the + or - sign are associated with the
and left hand varieties. One can now relate
H, J, K, L to abcd by :
Here
again
right
cor-
responding to site A can be written
Here the axes O.,,y,z are defined on figure 1 and the plus
minus sign correspond respectively to the left hand
or right hand variety.
or
0 is the angle between OZ and the c axis. It is equal to
63027’ for selenium and 62017’ for tellurium.
The results are given in table II. For selenium there
923
are
two sets of values corresponding to the two possible
for L. There is a striking difference between
selenium and tellurium which cannot be explained
from geometrical arguments, the local environment
being the same.
signs
with
TABLE II
Experimental values of the elements of axyz
If one details the matrix elements of L, one then
obtains the expressions given in table III where XYZ
now represent the corresponding p atomic orbitals.
To go further it is now necessary to have a model for
the band structure which allows all populations N to
be calculated.
3. Formulation of the chemical shifl - The chemical shift results from the contribution of several
terms. Usually the most important is due to the
second order interaction of the electron orbital
moment with the nuclear magnetic moment. It can
be written in the form [4] :
are the conduction band states of
energy Ep, 03C8q
the valence band states of energy Eq. The origin for r
is taken at the atomic site.
For a given pair of narrow bands, one can use an
average gap approximation, which allows a to be
reduced to the simpler form :
TABLE III
Exprcssion pf a
in
term of the populations
4/p
avec
We choose the most
all nearest neighbours
interactions as well as some second nearest neighbours
interactions within a chain. We start from a so-called
molecular approximation from which the band structure can be most easily discussed.
We start from the following atomic orbitalss )
and the three p orbitals( pX ),py X !Pz X XYZ
being defined in figure 1. However instead of working
with )px ) and !I py &#x3E; we prefer to build their sum and
their difference which we denotesp1 ) andI P2 )
respectively. These have the advantage of pointing
approximately towards the nearest neighbours in the
chain. The molecular approximation is defined by
including the following matrix elements :
4. A band structure model.
simple description, including
the sum running over the different valence and conduction bands of average energies Ev and Ec; k and k’
are the wave vectors of the Bloch states within each
band. In a tight-binding treatment one writes :
where the I i, a &#x3E; are pure atomic states, i being the
site index, a the orbital index.
In the following we shall only consider intraatomic contributions to (6). This considerably simplifies the calculation and can be considered as a first
step towards a complete treatment. Thus, for atom i,
a(i) reduces to :
-
924
where h is the hamiltonian, ( - fl) the resonance integral between nearest neighbour hybrids of the same
bond; Es and Ep are the free atom s and p energies.
All overlap integrals are neglected. Within this
model, one ends up with four distinct flat bands : a
pure s band, a p bonding band, a pure pz band and
finally a p antibonding band as pictured on figure 2.
This model thus gives a simplified but correct picture
of the electronic structure. The six valence electrons
per atom fill the three lowest bands. This is in agreement with photoemission [6] and pseudopotential
results [7]. The bonding and antibonding orbitals are :
at
band and the antibonding band. This is contrary to the
experimental finding that there is an important peak
in 82(E) [9] which may only be attribuable to this
transition. This can only be understood if there exists
a strong coupling between theZ ) band and the
bonding or s band.
As we shall see later, when one applies perturbation
theory to calculate the populations, there is a first
order term only when one includes an interaction
between second nearest neighbours within a chain. We
shall only retain the most important one, given by
(Fig. 1)
This term, although small, can give a substantial
contribution compared to nearest-neighbour terms
which only contribute to second order.
energies
Atomic
5. Second order perturbation theory.
Nc
Nv
the
valence
and
conduction
and
of
populations
bands, respectively, are calculated by second order
perturbation theory via a Green’s function formalism [10]. The hamiltonian of the molecular model
is h and the total hamiltonian H equals h + V. The
resolvent G = (E - H) -1 can be related to g = (E - h)
and Vby the following power series expansion :
-
In order to derive a more precise band structure,
one must include additional interactions. However,
if the molecular model has some meaning, their main
effect must be to widen the bands as shown on figure 2
while the coupling between different bands must be
weak so that it can be treated by perturbation theory.
In this work we have included all nearest-neighbour
interactions, within a two center approximation [8],
where there are only four independent terms i.e. ss, 66,
s6, nn in the usual notation. Some details are given in
appendix A concerning the interactions in terms of the
molecular orbitals. In this appendix it is also shown
that fl is practically equal to the au term, while the nn
term approximates the width of the p bands.
We shall not derive here a detailed band structure.
Our aim is instead to calculate the effect of these
additional interactions on the chemical shift. Optical
transitions will also be discussed in a qualitative
manner. For, within the molecular model, there is
practically no oscillator strength between theI Z &#x3E;
Any matrix element ( ia )I Gia ’ ) can be expanded
a power series of the following independent terms
of g :
in
Here i is a site index. The origin of the energies is taken
at the p atomic level, L1 being equal to Ep - ES.
iaI Gia’ ) presents simple and multiple poles
at E
0, + fl, - L1 which are the molecular levels
defined above. The multiple poles describe the
widening of these levels into bands. One can calculate
the population Nai,ai in a band derived from the
molecular level E = Ev, by a method analogous to
that used by Deccarpigny and Lannoo [11]. One
=
obtains :
FIG. 2.
-
The formation of bands from the molecular model.
925
This
gives a fairly direct method for evaluating the
populations, with the condition that the power series
expansion is convergent. We shall assume that this is
the case and evaluate eq. (14) to second order. The
complete derivation of all G,. is given in appendix B.
In the same appendix, the populations appearing in
table III are expressed in terms of n, a, 3S6, L1 and y
(here, for simplicity we call n and a the TTTC and uu
interactions, Psa being the s-a one).
The formal results for the populations of interest
are given in table IV in terms of two parameters :
x
=
(n/u)2,
a
=
[0.31(P;a/uL1) - y/2 a]
depend on the ratio Ba/yA., which is in some respects a
measure of the relative importance of the coupling
between second nearest-neighbours within a chain.
TABLE IV
Populations in the
upper and lower signs
different bands. For Nyz
respectively correspond to
the
the
right hand and left hand varieties.
for the dia-
gonal part and N y + 0.184 /x for the non-diagonal part (the - and + signs are related to the right
and left hand variety respectively). Some much
smaller terms have been neglected. The sign of oc will
=
6. Discussion of the chemical shift tensor.
and tables III and IV. To first order in x and a,
-
The chemical shift tensor can be obtained
obtains for the right hand variety :
directly from eq. (8)
one
with
Substracting the isotropic part :
one
obtains :
This can be compared with the experimental values
of table II. Before doing this in detail it is worth
mentioning the following points :
- All terms in x, i.e. (n/u)2 in the diagonal part are
essentially due to the interactions ofI Z &#x3E; orbitals
withB ) andI A &#x3E; orbitals.
The result for the molecular model (with x
0
0 in eq. ( 17)) cannot explain the values of the
and a
diagonal terms and gives zero for the non diagonal
one. It is thus completely unrealistic.
=
-
=
Let us now discuss separately the diagonal terms of
tellurium and selenium. The non-diagonal terms will
be examined afterwards.
a) TELLURIUM. From photoemission spectra [5]
we take fl - 3 eV. We take
-
to take account of the contraction of the wave
functions in the solid compared to the free atom,
as shown for covalent solids [12].
Fitting the diagonal terms thus leads to :
From this
we
deduce
a
ratio
1t/u equal to 0.31. This
926
is satisfactory owing to the fact that usually 7r/a is of
order 0.33 to 0.5. This is in good agreement with the
value 0.27 found elsewhere [7]. From a one can also
deduce y, taking reasonable values for the other
parameters, i.e. :
This set of values leads to y rr 0.22 eV which is quite
coherent with the values obtained for the nearestneighbours’ interactions.
A further confirmation of the coherence of this
model is provided by the isotropic part Qo of the
chemical shift given in (16) and which is found to
equal - 5 190 p.p.m. This is in extremely good
agreement with the result of Willig and Sapoval [13]
deduced from a theoretical calculation of the Te chemical shift in CdTe. Moreover this last value has
allowed an interpretation for all tellurium compounds
to be given.
SELENIUM. - As we do not know the sign of L
shall consider two cases. We take B~ 4 eV [6].
In analogy with tellurium we obtain for the two cases :
both cases x should roughly be greater than 1/3 which
is not coherent with the values obtained from the
diagonal terms. However if this were the case, perturbation theory would no longer be valid. We are
then led to conclude that our model is still too approximate and that it is necessary to introduce further
refinements. For this we believe that the two most
important limitations of this work are the following.
We have assumed that the average gaps were all
given by the molecular model of figure 2 and are thus
fixed by the value of fl. In practice, detailed results
obtained for the band structure [6, 7] show that the
distance between the two upper valence bands is
smaller than the distance between the upper valence
band and the lower conduction band. Preliminary
calculations correcting for this effect show that while
it improves the value for the non-diagonal term, the
agreement is less good for the diagonal one.
Our central approximation has been to retain only
matrix elements corresponding to the site for which 6
is calculated. While this is certainly correct for the term
containing L/r3, there is no clear reason why this
should also be true for the matrix element of L. Even
if one retains only intra-atomic matrix elements a more
general expression is obtained for a :
b)
we
This is
Both cases thus lead to reasonable values.
The essential difference between Se and Te then
comes from the ratio 7c/r (n/a)Te
(7r/a)s.). This
indicates that the bonding character of the upper
valence band is much stronger in Se than in Te. In fact
this turns out to be exactly equal to x which we find to
be of the order of 20 % for Se. This value is in very good
agreement with the result 25 % of a recent numerical
computation [14]. For Te we thus expect that this
bonding character is reduced to about 10 %.
These results are confirmed by optical spectra. In
both materials, the E2(E) spectra exhibit a broad peak
1.5 and 2.0 eV for Te and 2.5
at energies between E
and 4.5 eV for Se. The low energy part of these peaks
may only be attributed to transitions from the upper
valence band to the conduction band. However if one
computes the oscillator strengths associated with
transitions from pureI Z &#x3E; states to the lowest excited
states [15] they are found to be very weak. One then
reaches the conclusion that this peak must be attributed to a non, negligible bonding character in the
upper valence band.
If we now consider the non-diagonal term, the
agreement is not so good. To obtain the correct sign in
=
j =
evidently
an
extension of eq.
(8).
Terms with
i might be important for nearest neighbours. Their
importance is proportional to intersite populations.
They are much more complicated to evaluate and this
needs further work.
7. Conclusion.
In this work we have developed
method of calculating the chemical shift tensor in
trigonal selenium and tellurium, using a tight-binding
approximation. It clearly appears that a molecular
model cannot explain the observed features. For this
it is necessary to take into account the interactions
between molecular states. We have done this to second
order in perturbation theory using an intrasite approximation. We have then shown that it is quite possible to
explain all the results concerning the diagonal terms of
the chemical shift tensor for quite reasonable values of
the interatomic interactions. Furthermore these results
can be related in a coherent manner to the relevant
optical properties. The main conclusion to be drawn
is that there is a non negligible amount of bonding
character in the upper valence band which is of the
order of 20 % for Se and 10 % for Te. For Se, this is in
fair agreement with the result of OPW calculations.
-
a
927
We have also shown that the same values of the
cannot explain the sign of the nondiagonal terms. Therefore, it is necessary to include
additional terms in the expression of the chemical
shift, going beyond the purely intrasite approximation. This is presently being examined.
We believe that this work demonstrates that the
chemical shift is extremely sensitive to any pertur-
parameters
Appendix A
bation of the chemical bonds around a given atom. Any
such perturbation can severely modify the interband
coupling which we have shown to be responsible for
the most important observed properties of the chemical shift tensor. For this reason, it will be interesting
to extend this type of calculation to selenium and
tellurium compounds and try to understand their local
configurations.
: interatomic interactions between nearest neighbours.
-
All terms can be reduced to the follow-
ing matrix elements :
the
labelling
of the sites
being
detailed
FIG. 3.
are
We
can now
of the usual
on
figure
3.
Local environment along a chain. On each atom i there
bonding orbitals Bi, Bi - 1, two antibonding orbitals Ai,
Ai -1 and two free atom orbitals Zi and Si.
-
two
express all these elements in terms of a, 7r, Psa and Pss which denote here the absolute values
and ss two center integrals. From figure 4 one can show that :
ca, nn, s6
928
FIG. 4.
-
Chain structure and detailed definition of local
TABLE V
Nearest-neighbours’ interactions along a chain
axes.
929
It is a simple matter to express all independent terms of (A.1) in terms of our four distinct parameters. The
corresponding results are given in table V. All results are expressed as functions of the angles 2 0 and T approximately equal to 1040 and 100°, respectively, in Se as well as in Te.
The most important term fl of the molecular model can be calculated as a function of J and n. One finds :
Appendix B : intrasite populations. As mentioned in the text one has to calculate the matrix elements
Gxx, Gyy, Gzz and Gyz at a given site, to second order perturbation theory. For this we use :
-
They can be expressed
in terms of (13) and of the second
nearest-neighbour interaction y defined in (11). This
leads to
We can then calculate the populations using (14) and express them as functions of a, n, Bso,
result is given in table IV after neglecting terms which are clearly smaller than the others.
Pss and y. The final
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