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An LCAO band structure and density of states of lead
Solid State Communications,
Vol. 14, pp. 395—397, 1974.
Pergamon Press.
Printed in Great Britain
AN LCAO BAND STRUCTURE AND DENSITY OF STATES OF LEAD A. Breeze Dept. of Pure and Applied Chemistry, University of Strathclyde, Glasgow Gil XL, Scotland (Received 25 October 1973; in revised form 12 November 1973 by L. Hedin)
An LCAO investigation of the electronic structure of lead metal shows that crystal field effects are responsible for the splittings observed, by XPS, in the 6p states.
VERY little theoretical work has been attempted on the face-centred cubic metal lead. Loucks1 has calculated the band structure using the Relativistic APW method, which gave excellent agreement with a calculation by Anderson & Gold2 paranietrised to fit the Fermi surface measurements.
N(E)
Recently, Mathewson et al.3 have obtained the optical spectrum of lead and have interpreted it in detail by using Anderson and Gold’s results to calculate the joint density of states. The XPS valence band spectrum of lead has been reported by Ley et al.,4 and important features in this spectrum have been interpreted by them as stemming from spin—orbit splitting.
-no
FIG. 1. Density of states of lead. We have calculated the non-relativistic band structure and density of states of lead, both in order to test the applicability of the LCAO approach, described earlier,5 to metals and to test the importance of relativistic effects on the density of states of heavy metals.
Trial calculations on several metal systems lead us to d lopt a value for the proportionality constant, appea •iig in the two-centre integral approximation5 of 2.50. The Orbital lonisation Potentials were obtained as described in our earlier work and took the values I~= 15.59 eV,J~= 7.42 eV. The remaining parameters, the Slater orbital exponents (&s), were treated as variables. Upon parametrisation against the XPS results, the values; = 3.0 and cs~= 2.7 were chosen.
Loucks1 clearly, showed that relativistic effects were an essential part of the correct description of the Fermi surface. We therefore chose to parametrise our calculations against the observed peak positioning in the )(PS spectrum. This process was made practicable because of the relatively small amount of computing time required to obtain a density of states using an s and p atomic orbital basis.
The density-of-states histogram shown in Fig. 1, plotted at 0.2 eV intervals, was obtained by calculation at 152 points in 1/48th of the Brilouin zone. 395
396
AN LCAO BAND STRUCTURE W
Vol. 14, No.5
r
I.
0
e.V.
r,,
~
~
-10
/
~
—12
A,
K,_
-
\r/
FIG. 2. Band structure of lead.
Table 1. Fermi Surface Dimensions (a.uT’)
3—4 5—6 7—9 8—9 3—11 12—13
N(E)
This work
Loucks’
0.161 0.242 0.309 0.202 0.148 0.242
0.158 0.259 0.338 0.184 0.146 0.238
For numbering see Fig. 2. of 6s states and the rest of the 6p states.
___________________________________________ 0
2
4
6
eV
FIG. 3. Joint density of states,
followed by linear interpolation to yield 101,329 points in the whole zone. Ef occurs at 7.6 eV. The low-energy portion is composed almost entirely —
The agreement with the XPS spectrum, both in the relative heights of the two filled 6p peaks and in the asymmetry of the 6s peak, is very good. From this single result, it is clear that the observed splitting of the 6p states is primarily a crystal field and not a spin—orbit effect. Where comparison can be made, the band structure, Fig. 2, is closely similar to that of Loucks. The main discrepancies occur, as expected, where
Vol. 14, No.5
AN LCAO BAND STRUCTURE
Loucks found the largest relativistic splittings. In Table I, using Loucks’ numbering, we compare our calculated Fermi surface dimensions with his and note that the greatest differences can be associated with the symmetry point K. The calculated joint density of states is shown in Fig. 3. The general features are in agreement with those calculated and observed by Mathewson et a!. The narrowness of the main peak we suspect to be due to the inadequacy of the simple LCAO method in reproducing the greater free-electron-band character of the top band.
397
The general agreement we have obtained with the variety of experimental and calculational results discussed above shows the LCAO approach to be of value even in the study ofheavy metals. In conclusion, we have shown that the interpretation of spectra reflecting the density of states of lead, and presumably similar heavy metals, can be made with little reference to the relativistic effects so important to the adequate description of the Fermi surface.
REFERENCES 1. 2.
LOUCKST.L.,Phys.Rev.Lett. 14,1072(1965). ANDERSON J.R. and GOLD A.V.,Phys. Rev. 139, A1459 (1965).
3. 4.
MATHEWSON A.G., MYERS H.P. and NILSSON P.O.,Phys. Status Solidi(b) 57, K31 (1973). LEY L., POLLACK R., KOWALCZYK S. and SHIRLEY D.A.,Phys. Lett. 41A, 429 (1972).
5.
BREEZE A. and PERKINS P.G., Solid State Commun. (in press).
Une étude par la mèthode LCAO, de la structure electronique de plombe metallique montre que les effets du champs crystale sont responsible pour les rendants XPS observes, dans les états 6p.
Vol. 14, pp. 395—397, 1974.
Pergamon Press.
Printed in Great Britain
AN LCAO BAND STRUCTURE AND DENSITY OF STATES OF LEAD A. Breeze Dept. of Pure and Applied Chemistry, University of Strathclyde, Glasgow Gil XL, Scotland (Received 25 October 1973; in revised form 12 November 1973 by L. Hedin)
An LCAO investigation of the electronic structure of lead metal shows that crystal field effects are responsible for the splittings observed, by XPS, in the 6p states.
VERY little theoretical work has been attempted on the face-centred cubic metal lead. Loucks1 has calculated the band structure using the Relativistic APW method, which gave excellent agreement with a calculation by Anderson & Gold2 paranietrised to fit the Fermi surface measurements.
N(E)
Recently, Mathewson et al.3 have obtained the optical spectrum of lead and have interpreted it in detail by using Anderson and Gold’s results to calculate the joint density of states. The XPS valence band spectrum of lead has been reported by Ley et al.,4 and important features in this spectrum have been interpreted by them as stemming from spin—orbit splitting.
-no
FIG. 1. Density of states of lead. We have calculated the non-relativistic band structure and density of states of lead, both in order to test the applicability of the LCAO approach, described earlier,5 to metals and to test the importance of relativistic effects on the density of states of heavy metals.
Trial calculations on several metal systems lead us to d lopt a value for the proportionality constant, appea •iig in the two-centre integral approximation5 of 2.50. The Orbital lonisation Potentials were obtained as described in our earlier work and took the values I~= 15.59 eV,J~= 7.42 eV. The remaining parameters, the Slater orbital exponents (&s), were treated as variables. Upon parametrisation against the XPS results, the values; = 3.0 and cs~= 2.7 were chosen.
Loucks1 clearly, showed that relativistic effects were an essential part of the correct description of the Fermi surface. We therefore chose to parametrise our calculations against the observed peak positioning in the )(PS spectrum. This process was made practicable because of the relatively small amount of computing time required to obtain a density of states using an s and p atomic orbital basis.
The density-of-states histogram shown in Fig. 1, plotted at 0.2 eV intervals, was obtained by calculation at 152 points in 1/48th of the Brilouin zone. 395
396
AN LCAO BAND STRUCTURE W
Vol. 14, No.5
r
I.
0
e.V.
r,,
~
~
-10
/
~
—12
A,
K,_
-
\r/
FIG. 2. Band structure of lead.
Table 1. Fermi Surface Dimensions (a.uT’)
3—4 5—6 7—9 8—9 3—11 12—13
N(E)
This work
Loucks’
0.161 0.242 0.309 0.202 0.148 0.242
0.158 0.259 0.338 0.184 0.146 0.238
For numbering see Fig. 2. of 6s states and the rest of the 6p states.
___________________________________________ 0
2
4
6
eV
FIG. 3. Joint density of states,
followed by linear interpolation to yield 101,329 points in the whole zone. Ef occurs at 7.6 eV. The low-energy portion is composed almost entirely —
The agreement with the XPS spectrum, both in the relative heights of the two filled 6p peaks and in the asymmetry of the 6s peak, is very good. From this single result, it is clear that the observed splitting of the 6p states is primarily a crystal field and not a spin—orbit effect. Where comparison can be made, the band structure, Fig. 2, is closely similar to that of Loucks. The main discrepancies occur, as expected, where
Vol. 14, No.5
AN LCAO BAND STRUCTURE
Loucks found the largest relativistic splittings. In Table I, using Loucks’ numbering, we compare our calculated Fermi surface dimensions with his and note that the greatest differences can be associated with the symmetry point K. The calculated joint density of states is shown in Fig. 3. The general features are in agreement with those calculated and observed by Mathewson et a!. The narrowness of the main peak we suspect to be due to the inadequacy of the simple LCAO method in reproducing the greater free-electron-band character of the top band.
397
The general agreement we have obtained with the variety of experimental and calculational results discussed above shows the LCAO approach to be of value even in the study ofheavy metals. In conclusion, we have shown that the interpretation of spectra reflecting the density of states of lead, and presumably similar heavy metals, can be made with little reference to the relativistic effects so important to the adequate description of the Fermi surface.
REFERENCES 1. 2.
LOUCKST.L.,Phys.Rev.Lett. 14,1072(1965). ANDERSON J.R. and GOLD A.V.,Phys. Rev. 139, A1459 (1965).
3. 4.
MATHEWSON A.G., MYERS H.P. and NILSSON P.O.,Phys. Status Solidi(b) 57, K31 (1973). LEY L., POLLACK R., KOWALCZYK S. and SHIRLEY D.A.,Phys. Lett. 41A, 429 (1972).
5.
BREEZE A. and PERKINS P.G., Solid State Commun. (in press).
Une étude par la mèthode LCAO, de la structure electronique de plombe metallique montre que les effets du champs crystale sont responsible pour les rendants XPS observes, dans les états 6p.