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The electronic band structure and density of states of palladium
Volume 39A,
22 May 1972
PHYSICS LETTERS
number 4
THE ELECTRONIC BAND STRUCTURE AND DENSITY
OF STATES OF PALLADIUM D.L. ROGERS and C.Y. FONG Department of Physics, University of California, Davis, California 95616, USA Received 14 March 1972
The electronic energy band structure and density of states are calculated for palladium using the Empirical Results are compared with ones obtalned by the APW method.
pseudopotential method.
This letter reports a first application of the empirical pseudopotential method [11 to the paramagnetic transition element palladium. We report here only the electronic band structure and density of states.
The pseudo-hamiltonian is of the form (1) H=p2/2m + VL(r)+ VNL (r) where VL (r) is the local pseudopotential and VNL (r) is the non-local pseudopotential. The expansion of the local potential in reciprocal lattice vectors G was truncated for G 12 ~ 11 in units of(2ir/a)2 where a equals 3.89 A, the lattice constant of Pd: VL(r)=
~
V(IGI)exp(iG.r)
(2)
GI2~11 Since Pd has the FCC structure this leaves 4 parametersforlGl23,4,8and 11.
The non-local d-like potential is of the form
2
(5)
where a and sc along with A2 andR5 are treated as parameters for the non-local potential. Local pseudopotential parameters were used to fit mainly the positions of the s and the s-d hybridization while the non-local pseudopotential parameters were used to fit the position and width of the d-bands relative to the s and p bands. We determine the parameters by using the experimental results of Yu and Spicer [2] and the energy distribution curves as calculated by Janak et al.[3]. The wave functions and eigenvalues were calculated by truncating the expansion of2the wave(2ir/a)2. func~50.l tion over plane wave states with 1G1 The plane waves with 26.1 ~IGI2 ~50.l were treated using the Läwden-Brust perturbation scheme [4] and the resulting size of the matrix is of the order of
l5OX 150.
VNL(r)=EP~ U(Ir—R 11) ‘~‘2
I’
.X(kIVNLIk’>exp _a(~(~”))
(3)
E(R~) 08
where the R,’s are the lattice vectors and P~, P2 are projection operators that project out 12 states. U(r) is a square well potential of radius R5 and depth A2
06 05 04
U(r)A2 forr~R5
03
0forr>R5.
(4)
To insure convergence an exponential damping factor is added to the matrix elements: (kIV~LIk’)= exp( a((I
07
ic))2~
~
02
0I
ii.
Fig. 1. The band structure of Pd along various symmetry lines.
345
Volume 39A, number 4
PHYSICS LETTERS
22 May 1972
Table 1 Numerical values of the parameters. Parameters for the non-local pseudo-
Parameters for the local pseudopotential
potential 2
V(IGJ
V(4) =
3(2,r/a)2) =
=
—
0.05 Ry.
R 5=
—
0.04
A2
V(8) = 0.0073
=
V(11) = 0.002
K
=
=
0.926 A —8.4288 Ry.
0.433 A’ 1.872 (2w/a)
a (lattice constant) = 3.89 A
Table
2
Comparison between the APW [5J calculations and the EPM
[Rogersand Fong] calculations. Symmetry
APW (Ry)
EPM (Ry)
0.232 0.335
0.227 0.338
X1 -+ L3 X1 -+ X4’ X3 —~X4’ L1~+L3hJ*
0.37 1 0.704 0.673 0.399
0.383 0.685 0.665 0.354
L3u_*L2~
0.133 0.38 1
0.134
r1 r1
-.
r~’
-÷
EF —~X1 Superscript u
and that the width of the d-bands, as represented by EF X1 is about 5% or 0.2 eV wider. -~
0.402
means the upper L3 states.
The electronic band structure, along the symmetry lines in the 1/48th of Brillouin zone is plotted in fig. 1. It is seen that from r to X and from I’ to L the usuals s-d hybredization occurs. The Fermi energy lies at about 0.46 Ry above r1 making L~the top of the d bands. This agrees well with the corresponding results of 0.43 Ry obtained by Janak et al. [3]. A comparison of our results with those of an APW calculation by Mueller et al. [5] are presented in table 2. It is seen that all the energy gaps agree to within 8%
346
Fig. 2. The electronic density ofstates for Pd.
We would like to thank Professor Marvin L. Cohen for providing certain computer time for this calculation. References [1] C.Y. Fong and M.L. Cohen,
Phys. Rev. Lett. 24
(1970)
306.
Yu and W.E. Spicer, Phys. Rev. 169 (1968) 497. [31J.F. Janak, D.E. Eastman and AR. Williams, Solid State Comm. 8(1970)271. [4] D. Brust, Phys. Rev. 134 (1964) A1337. [5] F.M. Mueller, AJ. Freeman, J.O. Dimmock and AM. [2] A.Y-C.
Furdyna, Phys. Rev. Bi (1970) 4617.
22 May 1972
PHYSICS LETTERS
number 4
THE ELECTRONIC BAND STRUCTURE AND DENSITY
OF STATES OF PALLADIUM D.L. ROGERS and C.Y. FONG Department of Physics, University of California, Davis, California 95616, USA Received 14 March 1972
The electronic energy band structure and density of states are calculated for palladium using the Empirical Results are compared with ones obtalned by the APW method.
pseudopotential method.
This letter reports a first application of the empirical pseudopotential method [11 to the paramagnetic transition element palladium. We report here only the electronic band structure and density of states.
The pseudo-hamiltonian is of the form (1) H=p2/2m + VL(r)+ VNL (r) where VL (r) is the local pseudopotential and VNL (r) is the non-local pseudopotential. The expansion of the local potential in reciprocal lattice vectors G was truncated for G 12 ~ 11 in units of(2ir/a)2 where a equals 3.89 A, the lattice constant of Pd: VL(r)=
~
V(IGI)exp(iG.r)
(2)
GI2~11 Since Pd has the FCC structure this leaves 4 parametersforlGl23,4,8and 11.
The non-local d-like potential is of the form
2
(5)
where a and sc along with A2 andR5 are treated as parameters for the non-local potential. Local pseudopotential parameters were used to fit mainly the positions of the s and the s-d hybridization while the non-local pseudopotential parameters were used to fit the position and width of the d-bands relative to the s and p bands. We determine the parameters by using the experimental results of Yu and Spicer [2] and the energy distribution curves as calculated by Janak et al.[3]. The wave functions and eigenvalues were calculated by truncating the expansion of2the wave(2ir/a)2. func~50.l tion over plane wave states with 1G1 The plane waves with 26.1 ~IGI2 ~50.l were treated using the Läwden-Brust perturbation scheme [4] and the resulting size of the matrix is of the order of
l5OX 150.
VNL(r)=EP~ U(Ir—R 11) ‘~‘2
I’
.X(kIVNLIk’>exp _a(~(~”))
(3)
E(R~) 08
where the R,’s are the lattice vectors and P~, P2 are projection operators that project out 12 states. U(r) is a square well potential of radius R5 and depth A2
06 05 04
U(r)A2 forr~R5
03
0forr>R5.
(4)
To insure convergence an exponential damping factor is added to the matrix elements: (kIV~LIk’)= exp( a((I
07
ic))2~
~
02
0I
ii.
Fig. 1. The band structure of Pd along various symmetry lines.
345
Volume 39A, number 4
PHYSICS LETTERS
22 May 1972
Table 1 Numerical values of the parameters. Parameters for the non-local pseudo-
Parameters for the local pseudopotential
potential 2
V(IGJ
V(4) =
3(2,r/a)2) =
=
—
0.05 Ry.
R 5=
—
0.04
A2
V(8) = 0.0073
=
V(11) = 0.002
K
=
=
0.926 A —8.4288 Ry.
0.433 A’ 1.872 (2w/a)
a (lattice constant) = 3.89 A
Table
2
Comparison between the APW [5J calculations and the EPM
[Rogersand Fong] calculations. Symmetry
APW (Ry)
EPM (Ry)
0.232 0.335
0.227 0.338
X1 -+ L3 X1 -+ X4’ X3 —~X4’ L1~+L3hJ*
0.37 1 0.704 0.673 0.399
0.383 0.685 0.665 0.354
L3u_*L2~
0.133 0.38 1
0.134
r1 r1
-.
r~’
-÷
EF —~X1 Superscript u
and that the width of the d-bands, as represented by EF X1 is about 5% or 0.2 eV wider. -~
0.402
means the upper L3 states.
The electronic band structure, along the symmetry lines in the 1/48th of Brillouin zone is plotted in fig. 1. It is seen that from r to X and from I’ to L the usuals s-d hybredization occurs. The Fermi energy lies at about 0.46 Ry above r1 making L~the top of the d bands. This agrees well with the corresponding results of 0.43 Ry obtained by Janak et al. [3]. A comparison of our results with those of an APW calculation by Mueller et al. [5] are presented in table 2. It is seen that all the energy gaps agree to within 8%
346
Fig. 2. The electronic density ofstates for Pd.
We would like to thank Professor Marvin L. Cohen for providing certain computer time for this calculation. References [1] C.Y. Fong and M.L. Cohen,
Phys. Rev. Lett. 24
(1970)
306.
Yu and W.E. Spicer, Phys. Rev. 169 (1968) 497. [31J.F. Janak, D.E. Eastman and AR. Williams, Solid State Comm. 8(1970)271. [4] D. Brust, Phys. Rev. 134 (1964) A1337. [5] F.M. Mueller, AJ. Freeman, J.O. Dimmock and AM. [2] A.Y-C.
Furdyna, Phys. Rev. Bi (1970) 4617.