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ELECTRON CORRELATION IN METALS EVIDENCED BY COMPTON SCATTERING
A. Issolah, J . Chomilier, Y. Garreau, G. Loupias
To cite this version:
A. Issolah, J . Chomilier, Y. Garreau, G. Loupias. ELECTRON CORRELATION IN METALS
EVIDENCED BY COMPTON SCATTERING. Journal de Physique Colloques, 1987, 48 (C9), pp.C9-
851-C9-854. �10.1051/jphyscol:19879152�. �jpa-00227264�
ELECTRON CORRELATION I N METALS EVIDENCED BY COMPTON SCATTERING
A. ISSOLRH, J. CHOMILIER, Y. GARREAU and G. LOUPIAS*
Laboratoire d e Mineralogie-Cristallographie, UPM et Universite Paris V I I , CNRS, T 16, 4, Place Jussieu, F-75252 Paris Cedex 05, France
and " L U R E , CNRS-CEA-MEN, Bat. 2090. Universit6 Paris-Sud, F-91405 Orsay Cedex, France
FEsUvlE
La difference entre profils Compton theorique -pseudopotentiel- et experimental permet d'evaluer I'importance de la correlation aectronique dans les metaux. Pour le beryllium elle est quasi independante de la direction etudiee, alors que pour un compose lamellaire comme le graphite, elle est anisotrope. Dans le premier cas, un traitement de correlations electroniques pour un gaz d'6lectrons suppose homogene, reduit de moitie la difference. Cette mbthode ne peut s'appliquer au graphite pour ameliorer le calcul mono electronique et il est necessaire d'utiliser des fonctions d'onde multi6lectroniques.
The difference between theoretical -pseudopotential- and experimental Compton profiles allows to estimate the electron correlations in metals. For beryllium it is almost directionally independent, while for a lamellar compound such as graphite, also of hexagonal symmetry, the difference is anisotropic. In the first case the correlation effects evaluated for a homogenous electron gas, reduced the difference almost by afactor 2. This method cannot be used to improve the one-electron calculation in graphite and more advanced theories using multi electron wave functions are required.
Directionnal Compton profile (DCP) is related to ~ ( p ) , the Fourier transform of the ground state wave function of the electron, such that:
J(q)=
I
x(P) x*(P) d2pwhere p i s the initial momentum of the electron and q its projection on the scattering vector. As the wave function extension behave opposite ways from one space to the other, valence wave functions are particularly localized in p space. So Compton profile is more appropriate than diffraction studies as long as bonding electrons are concerned.lt is easy in an experiment to obtain the contribution of the valence electrons only, by subtracting the large and flat core electron profile from the total experimental one.
Comparing experimental and calculated DCP furnishes a check of the quality of the wave function used in the computation. We are dealing in this paper with hexagonal symmetry structure crystals, namely beryllium and graphite. Inelastic scattering experiments have been performed at LURE-DCI on such compounds, and DCP have been derived for the valence electrons, after subtraction of an isotropic calculated atomic core profile.
For both materials, theoretical DCP have been derived from pseudopotential wave functions, i.e. with a single electron treatment, and the difference between experiment and calculation is a measure of the importance of the electron correlation in metallic systems.
It is interesting to study beryllium electron density because half the whole electrons are involved in the conduction band. A particular aspect of this metal is a large covalent character, due to a 2p wave function overlap, that leads to a reduction of the c/a ratio (1.568), compared to the ideal case of the closed packet hexagonal (1.633). P. J. Brown (1) admitted that the p,
-
orbital contribute around 113 to the density.
DCP have been computed by Rennert et al. (2) in a pseudo potential scheme, which has
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19879152
C9-852 JOURNAL DE PHYSIQUE been improved in a second calculation by Rennert (3), taking into account:
-the orthogonalization with the core wave functions. This contribution is 10% of the total correction term and is anisotropic.
-
the electron correlation term, i.e. the high momentum tail of the Bloch functions which produces a deviation of the DCP from the nearly free electron profile. This treatment uses the method developed by Daniel and Vosko (4), in the RPA approximation. The Coulomb interaction is introduced as a perturbation in the treatment of the momentum distribution; this leads to a simple correlated momentum density correction which is isotropic. Fig. 1 shows the difference between the experimental and calculated profile of <001> Be (2); this difference is compared to the amplitude of the correlation correction and to the amplitude of both correlation and orthogonalization corrections (3). It is obvious that these corrections explain most of the difference between experiment and theory.Fig. 1 :
...
<001> DCP difference of Be (ref 3) Fig. 2: <001> DCP difference of Be (ref. 5) Correction calculations:-
without correlation correction---correlation alone
---
with correlation correction in calculated DCP ---plus orthogonalizationAn ab-initio pseudopotential calculation of the DCP of beryllium has been performed by Chou et al. (5) in which the wave function are othogonalized to the core. Seven directions have been investigated, experimentally and theoreticaly. Although the measured anisotropies (difference between two DCP's) present a good agreement with calculated ones, the comparison between calculation from one elctron wave function and measurement for each direction show the same discrepancy, almost direction independent. These calculations systematically overestimate the profile for small momentum (i.e. for long distances) and underestimate it for large momentum (short distances) using either Rennert (2) or Chou et al. (5) method.
Besides, the same oscillations in a yRay experiment (6) occur at low values of q when performing the difference with the same calculation, so one can assess that these structures are not an artefact due to the experiment. An isotropic correction accounting the electron correlation effects, has been added, calculated as Rennert. Fig. 2 shows the <001> difference profile, and it is found that electron correlation improves by a factor 2 the discrepancy with the calculation. It is interesting, at the light of these pseudopotential models, i. e. one might guess not the best to explain the outer electron behaviour, that they are both improved by a simple treatment of the electron correlations. The remaining discrepancy might be improved by performing a more sophisticated correction taking into account the electron correlation of anisotropic inhomogeneous systems.
Graphite is a lamelar compound highly anisotropic, a prototype dimensional materials; for instance the electric conductivity ratio odoa is around
interlayer electrons are delocalized in r space, they are localized in p space, and these free electrons lead to a shrinked DCP for the scattering vector perpendicular to the six fold axis because the contribution at low q is very high.
Fig 3: Difference CP for beryllium(-) and graphite(--), scattering vetor perpendicular to c DCP's of graphite have been computed from wave functions provided by the ab-initio SCF pseudopotential calculation of their energy band structures (8). The electron correaltion has been included with use of the local density approximation of Hedin Lundqvist (9) for the exchange correlation potential, i.e. non homogeneous approximation. To evidence the effects of electronic correlations in graphite we compare on Fig. 3 the difference between theory and experiment of the DCP's of graphite and beryllium, both for the scattering vector perpendicular to the six fold axis; actually the experiment on graphite is performed on a sample (HOPG) with this axis well determined, and as a powder in the basal plane; subsequently the DCP of beryllium presented is an average of DCP in the basal plane. This last one is presented in units of q / p ~ ? where p~ is the Fermi momentum, and the ordonate axis is scaled to the percentage of the value of the valence profile for q=O.O, for reasons o: easy comparison. For the graphite, a scaling parameter ps is used to obtain the reduced momentum; ps is chosen in order to match the zero crossing of both curves; it is worth 1.58 a.u. and is larger than the value of pF=l ,256 a.u.
(10). Then the profile differences show astonishing similarities both for the relative amplitude and the position of these structures. One must mention that the present experimental data slightly differ from previous ones (1 1) because the profiles have been symetrized between -1.0 and +1.0 a.u. in order to increase the statistics. This is justified by the symetry of the core profile calculated in our experimental conditions beyond the impulse approximation (12).
The Fig. 4 presents the difference DCP's of graphite for the scattering vector parallel and perpendicular to c. The curves for these two directions are much more different than in the case of beryllium (continuous lines of Fig. 2 and 3), the zeo crossings are not at the same places and near the Fermi momentum the shape is "softer". So if this discrepancy with the one electron correlations is a measurement of the electron correlation, we observe a larger anisotropic correlation in graphite that in beryllium, following the electronic properties. This anisotropy of the electron correlation in graphite is actually due to the valence, because we have shown (1 2) that the carbon 1 s electron leads to an isotropic core CP. To improve the agreement between calculated and observed CP, we need more advanced theories using more realistic wave functions than mono electronic ones.
C9-854 JOURNAL DE PHYSIQUE
Fig. 4: Difference DCP of graphite. Scattering vector parallel to c(-) and perpendicular (--)
lB€mcEs
1. Brown P. J., Phil. Mag. (1972) 1377.
2. Rennert P., Dbrre Th. and Glasser U., Phys. stat. sol.
M
(1978) 221.3. Rennert P., Phys. stat. sol.
b105
(1981) 567.4. Daniel E. and Vosko S. H., Phys. Rev.
120
(1960) 2041.5. Chou M. Y., Lam P. K., Cohen M. L., Loupias G., Chomilier J. and Petiau J., Phys. Rev. Lett.
(1982) 1452.
6. Hansen N. K., Pattison P. and Schneider J. R., Z. Phys.
W
(1979) 215.7. Dresselhauss M. and Dresselhauss ,Advances in Physics, (1981).
8. Holzwarth N. A. W., Louie S. and Rabii S., Phys. Rev.
B26
(1 982) 5382.9. Hedin L. and Lundqvist B. I., J. Phys.
U
(1 971) 2064.10. Felsteiner J. and Gertner I., J. Appl. Phys. !j2 (1981) 1592.
11. Chou M. Y., Cohen M. L. and Louie S., Phys. Rev. (1986) 6619.
12. lssolah A. , Chomilier J., Loupias G., Levy B. and Beswick A., "Calculated Compton profile:
experimental check of the deviation from the impulse approximation in graphite", this conference.
