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Attempt to determine the band parameters of graphite by a theoretical calculation
F. Dujardin, J.P. Decruppe
To cite this version:
F. Dujardin, J.P. Decruppe. Attempt to determine the band parameters of graphite by a theoretical calculation. Journal de Physique, 1981, 42 (8), pp.1167-1174. �10.1051/jphys:019810042080116700�.
�jpa-00209307�
Attempt to determine the band parameters of
graphite
by a theoretical calculationF. Dujardin and J. P. Decruppe
Laboratoire de Physique du Solide, Ile du Saulcy, 57000 Metz, France (Reçu le 5 décembre 1980, révisé le 23 mars 1981, accepté le 23 avril 1981)
Résumé. 2014 Les auteurs ont essayé de déterminer les paramètres 03B3i du modèle de Slonczewski-Weiss du graphite
par un calcul théorique utilisant des fonctions d’onde atomique type Slater et le potentiel atomique tabulé par Herman et Skillman. Le potentiel cristallin est relié à la densité de charge par l’équation de Poisson et développé
en série de Fourier pour utiliser la méthode décrite par Lafon et Lin. L’énergie des bandes 03C0 est calculée sur la
ligne KFMK de la zone de Brillouin afin de comparer aux résultats d’autres auteurs. Le lissage des courbes d’énergie
par les expressions littérales de Slonczewski et Weiss est un autre moyen d’accéder à ces paramètres.
Abstract. 2014 In this work, the authors tried to determine the parameters 03B3i of the Slonczewski and Weiss model of graphite by a theoretical calculation using Slater’s type atomic wave functions and the Herman-Skillman atomic
potential. The crystalline potential is related to the charge density by Poisson’s equation and written as a Fourier series to use the method described by Lafon and Lin. The energy of the 03C0 bands is calculated on the line K0393MK of the Brillouin zone in order to compare our results with those of different authors. The fitting of the energy
curves by means of the Slonczewski and Weiss energy functions is another way to estimate these parameters.
Classification
Physics Abstracts
31.20E - 71.25R
1. Introduction. - In his study of the band struc- ture of graphite in the tight binding approximation,
Wallace [1] introduced only a few parameters to
account for the different atomic interactions and
neglected the overlap of atomic orbitals.
The further works of Slonczewski and Weiss [2]
established a model widely used for the study of the
electronic properties of graphite. This model, cons-
tructed from the results of a group theoretical ana-
lysis [3] deals particularly with the energy at the
corner of the Brillouin zone and seven parameters determinable by experiments are introduced.
Several attempts have been made to calculate the band structure of graphite and numerous values have been proposed for the parameters y.
We can mention the variational method of Painter and Ellis [4] ; the self consistent LCAO calculation
of Zunger [5], and thé methods using the tight binding approximation [6] or a combined OPW-tight binding approximation [7-8] and fitted with experimental
results.
Our purpose is to estimate the band parameters by a simple method using the tight binding approxi-
mation without any intermediate fitting.
On the whole, the order of magnitude of the results
is correct.
2. Slonczewski-Weiss band parameters. - In their attempt to describe the band structure of graphite,
Slonczewski and Weiss [2] introduced seven para- meters in their model, two of these having already
been defined by Wallace [1].
These parameters show the interactions between atoms of the same plane (yo), of adjacent planes (Y1, Y3, 74) and of second neighbour planes (y2, ys)
while A shows the difference in the crystallographic
nature of atoms A and B (Fig. 1).
Fig. 1. - Slonczewski-Weiss parameters.
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphys:019810042080116700
1168
2 .1 DEFINITION OF THE PARAMETERS. - The Hamil-
tonian, H, of an electron 2p,, in the crystal can be
written as : .,«
where Ho represents the Hamiltonian of an electron in the isolated atom. Since
it follows that H’ = V - U ; V, U represent respec-
tively the potential energy operators in the lattice and in the isolated atom.
The parameters are thus defined by :
x(r), the wave function of the 2p electron, will be of
the Slater type [9] r cos 0 where
A general formulation of the y’s will be used in the
following
the indices 1 and 2 representing two different atomic sites.
2.2 DETERMINATION OF U, , . - In order to per- form the calculation of the term U12, we shall use
the Herman-Skillman atomic potential [10].
In Rydberg units :
The different values of the functions u(r), equal to
one for r = 0 and 1/Z for r > R, are obtained from Herman-Skillman’s Tables [10].
In the vicinity of the nucleus, the electron moves in the field of a charge Z, while, after the limit R, it is placed in the field of a unit charge. In the case of carbon, Z = 6 and R = 1.21 Á, a distance smaller than the smallest carbon-carbon distance.
Therefore the function u(r) takes the form :
and
The first integral in the right hand side is easily performed by means of the spheroidal coordinates while the second integration is restricted to a sphere
of radius R and centred on O1 (Fig. 2).
Fig. 2. - Coordinates of two-centre integrals.
The different values of U12 appear in table I.
We note that for distances of the same range, the interaction is stronger between two atoms on the C axis than between atoms of the same plane. This is probably the consequence of the shape of the 2p., wave
functions.
When the two sites are identical we get
Table 1. - Values of U12 (eV) for different distances carbon-carbon. d and h are expressed in Á.
2.3 DETERMINATION OF Y12. - When we consider
the crystal potential as the superposition of atomic potentials the calculation of V12 requires the eva-
luation of three centre integrals.
A method, worked out by Lafon and Lin [11]
removes these difficulties.
If
then
2. 3 .1 Calculation of VK. - The potential energy consists of two parts : the Coulomb interaction and the exchange factor. The Coulomb potential energy
Vc(r) is related to the charge density p(r) by Poisson’s equation [12] :
The charge density has also the periodicity of the
lattice and can therefore be written as a Fourier series
Using Poisson’s equation we relate V CK (Fourier
coefficient of the Coulomb interaction) and’
The term Vco(K = 0) will be calculated separately.
The charge density is :
g(r - Rij) represents the different nuclei at points Rij,
and n(r - Rij) the electrons in the different orbitals
xnlm(r - Rij). The index j denotes the elementary
cell while i refers to the atom in it (i = 1,4)
The coefficient anlm represents the occupation of the
state n, 1, m.
It turns out that :
The sum over all atomic sites becomes
where Tj is a lattice translation and Ri marks the
atom i in the elementary cell.
The different wave functions used in the deter- mination of n(r - Rij) are
The Fourier coefficients of the charge density are
then :
For the case K = 0 we consider K as a continuous variable and develop elK.r near zero.
The integral containing etK.r becomes :
1170
and
The exchange terms VXK are calculated with Slater’s approximate formulation of VX(r) [13]
where
The determination is greatly simplified if we use the approximate form proposed by Woodruff [14]
The atomic charge densities overlap in the crystal but
it is assumed that the difference introduced by using the approximate form of VX(r) is small and that VXK VCK-
The Fourier coefficients VXK are :
These integrals are evaluated by numerical methods for each K.
The approximate formula for VX(r) is not suitable
for the case K = 0 since the integral in VxK is very sensitive to the behaviour of n(r) at long distance.
Woodruff [14] again proposes to change n(r) for
where Q. = Q/4 and X represents the number of valence electrons (X = 4 for carbon) and
and
Finally the expressions of V o and V K are
and
2.3.2 Calculation of 1 K’ - Since ri = r - Ri, we get for K # 0
And with the notation used in figure 2
Â, y, v represent the three components of K in an ortho- normal basis.
We note that
with
The integral I is calculated according to the method
described by Lafon and Lin [11]. We obtain after differentiation :
with :
When we perform the product V K 1 x the term SK e - iK.R1 takes different expressions according as R 1
denotes the position of an atom A or B. But in both cases, the general form will be Co + i Si, Co and Si representing respectively a sum of cosines and sines.
Table Il. - Values of V 12 (eV) when one atom, at least, is of type A or C.
Table III. - Values of V 12 (eV) when both atoms are of type B or D.
Finally we get for V 12
where S 12 = I0 is the overlap integral between sites 1 and 2
and the summation over K is extended to half the space.
The different results for V 12 appear in tables II and III.
From these results, we note that the difference of
crystallographic nature between the atoms A and B does not appear clearly in these calculations.
(VAA = - 52.20 # VBB = - 52.17 eV). But for two
atoms of second neighbour planes, the difference is
quite appreciable since VAA = 4.26 x 10-4 eV, VBB = 3.83 x 10-4 eV when d = 0 and h = 6.708 Á.
This can be justified by the fact that a third atom C lies between two atoms A which is not the case for the two atoms B.
From the terms Vij and Vij we can evaluate the different band parameters (Table IV). The theoretical value found for yo by this method is beyond the upper
boundary of the interval of recent experimental values
2.26 [15]-3.16 [16] and the absolute value of L1 seems to be too high though its sign is correct.
Table IV. - Calculated and experimental band para- meters.
The calculated values are too small for the other parameters. Boyle and Nozières [17] proposed 0.14 eV
for yi, but this value has been disputed later [18]
and the actual value is about 0.4 eV.
Concerning y4, a value smaller than ours has
recently been proposed [19] but the values the most
widely found in call publications are higher than the
one we propose.
It may be noticed that our values for y3 and y4 are
of the same magnitude : this follows from the fact that our calculations do not show the crystallogra- phic difference between the atoms A and B.
1172
The results are founded on the definitions given in part 2.1 and these imply that the basis be built either with Wannier functions or, which is the same, with
non overlapping atomic orbitals (orthonormal basis).
As the overlap of the Slater type functions is not negligible, we shall calculate the energy values on the line KTMK of the Brillouin zone taking into account long range interactions. We hope to reach the para- meters by fitting the calculated energies on the edge
and near it with the S.W. dispersion laws.
3. Band structure of graphite. - 3.1 ENERGY CAL-
CULATED ON THE FKM LINE. - The different Bloch functions are expressed as linear combinations of Slater type atomic orbital :
and the secular determinant is calculated on the line KTMK of the Brillouin zone (Fig. 3) and in table V
we compare our results with several authors. (Figure 4
shows the definitions of the quantities compared in
table V.)
Fig. 3. - Brillouin zone and symmetry points (from Koster).
Table V. - Comparison between several models and
experimental values.
The numerical values presented in the different columns are obtained by the following authors.
Columns 1 and 2 show the results we obtained by including only the first neighbours and those obtained
by taking into account all the calculated parameters.
Column 3 : Painter and Ellis [4] model : variational method using LCAO and the Herman-Skillman
potential.
Column 4 : Two dimensional model of Bassani and Pastori [6] : semi empirical method with adjustable
parameters.
Fig. 4. - n bands along the line KFMK.
Column 5 : Tsukada’s model. Combination of the OPW and tight binding methods [20]. The screened Coulombian potential introduced is
Column 6 : Nagayoshi [7] same model as the preced- ing but the value chosen for A and R are different.
Column 7 : Self-consistent calculation of Zunger [5].
Finally, in the last column we give two experimental
values obtained by Greenaway [21] from the study of
the reflectivity of graphite and by Bianconi [22] from
the emission spectra of graphite.
We note that our results are relatively similar to
those obtained by Bassani [6] and that the method
including only first neighbour seems to give better
results.
On the edge HKH of the Brillouin zone, the eigen-
values E1, E2, E3 associated respectively to the eigen-
Fig. 5. - Energies on the edge KH.
functions
-V v
Y 3 = YD and Y4 = YD are :
If we label the different energies as in the S.W. model
we note an inversion of the position of E1 and E2
which leads to a negative value for y1 (y1 = - 0.08 eV).
We must emphasize that this inversion of E1 and E2
occurs when the overlap of the different functions are
introduced. At point H of the edge, the difference
Ei - E3 gives 4 = - 0.034 eV (Fig. 5).
3.2 DETERMINATION OF THE OTHER PARAMETERS. -
In the corner of the Brillouin zone, near the edge HKH, the energies near E1 and E2 are [23] :
We introduce then P1 and P2
with The vector PN is defined in
figure 6.
Fig. 6. - In the vicinity of the corner.
And the functions F+(a) and F-(a)
The exact resolution of the secular determinant for small Q ( 6 0.08) leads to E1 - and E2 + and then
we can construct f, (a) and f-(a).
Fig. 7. - Determination of yo using the slope of f+(u).
The variation of f + (a) with a (Fig. 7) is linear although the results are slightly different when the calculations are performed on the line KF or KM.
From the slopes of these lines we deduce yo and its value is 2 eV when only the first neighbours are
included. Within the same hypotheses, the slope of f - (a) gives for 74 an average value of - 0.045 eV
(Fig. 8).
Fig. 8. - Determination of y4 using the slope of f-(a).
In the vicinity of E3, the solutions are
where
with
Combining the expressions of the energy for a = 0 and oc = n/3 we obtain y3, but the value is not constant
as the calculation of E1- and E2 + along KF and KM gives slightly different results with variations of the
same magnitude as those of E3 ± calculated for
x = 0 or x = n/3.
1174
4. Conclusion. - In this work, we tried to deter- mine the band parameters of graphite from a theoreti-
cal method, but we realized that with every step of the calculations we had to choose between several
possibilities thus bringing our method nearer to the
other ones.
The wave functions the analytical form of which should bring the analytic calculation as far as possible,
the different forms of atomic and crystalline potentials,
the exchange potential modulated by a variable
factor a, Woodruff’s approximation of the exchange potential form are, in fact, a set of adjustable entities
as the different parameters introduced by other
authors do. A study of the influence of all these factors
on the final results would be interesting but unfor- tunately the numerical calculations are so long that
we worked only with one set of wave functions and
potentials.
Despite these choices and approximations, we must
admit that the general band shape along the line KMr of the Brillouin zone may be compared to the results of other authors and especially to the model of Bassani [6]. It would be interesting to see if this latter leads to a set of accurate yi’s because the fitting method, comparing the calculated energy values with those
predicted by the approximate S.W. formulae is very sensitive to the behaviour of the bands around the
point K. Moreover we can point out that the two ver-
sions of Nagayoshi’s works [7] and [8] using different
sets of adjustable parameters give similar band models which lead to quite poor yi’s in the first case and to excellent results in the second. (Paradoxically, the
agreement with the optical transition between H bands at point M is better in the first case.)
In our case the fitting gives too small values for every parameter. However this method gives 2 eV
for yo.
The first method gives 3.64 eV for yo and 0.14 eV
for y1, while the second leads to y0 ~ 2 eV ; these values are outside of the experimental interval (yo - 2.6-3 eV) and the difference between our two values of yo results from the importance of the over- laps.
Boyle and Nozières [17] did propose, for yi, a value similar to ours but actually, the most widely accepted
value is about 0.4 eV.
Finally, the tight binding method may not be the
best method for the study of the band structure of
graphite where the overlap of 2p,- wave functions are important and an OpW method should be better
adapted to describe the behaviour of the conduction electrons. Therefore the work of Tsukada-Nagayoshi mixing the OPW and tight binding methods to
account for a weak interaction between planes seems interesting.
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