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4f energy levels in band theory and the Fermi surface structure of La and Ce compounds
PHYSICA
Physica B 186-188 (1993) 179-181 North-Holland
4f energy levels in band theory and the Fermi surface structure of La and Ce compounds Osamu Sakai and Yasunori Kaneta Department of Physics, Tohoku University, Sendai 980, Japan We calculate 4f energy levels by a non-empirical band theory including the self-interaction correction and the unoccupied states potential correction. The 4f levels of La compounds appear at about 6 eV above the Fermi energy consistently with Bremsstrahlung isochromat spectroscopy experiments, but further improvements are necessary for the Fermi surface structure calculation.
In this paper, we attempt to calculate the 4f energy levels by a non-empirical method in the band calculation. The standard band theory based on the local spin density (LSD) approximation often fails to predict the 4f energy levels in rare earth compounds [1]. The unoccupied 4f energy levels usually appear to be lower compared with those obtained by Bremsstrahlung isochromat spectroscopy (BIS), while the occupied levels appear to be shallower compared with photoemission spectroscopy (PES). Previous works on La and Ce compounds show that the Fermi surface structures (FSS) are reproduced fairly well by calculations, shifting the 4f levels to be consistent with BIS and PES from those of the LSD band calculation [2,3]. Recently, importance of the self-interaction correction ( S I C ) [ 4 , 5 ] for the occupied states has been extensively discussed for 3d compounds [6]. It drastically improves the LSD results for the magnitude of the band gap. Such attempts for 4f compounds have not been done except for the works of Szotek et al. [7]. They showed that the occupied 4f states of Pr metal are pulled down to 8 eV below the Fermi energy (EF). However, the unoccupied levels appear at about 0.7 eV above Ev, which is lower compared with 3 eV in BIS [8]. As the self-consistent charge distribution is determined by the occupied states in the L S D - S I C method, we need corrections to calculate the unoccupied states, especially for localized states such as the 4f orbits. Anisimov et al. have introduced the unoccupied states potential correction (USPC) for 3d compounds [9]. In this paper, we perform a selfconsistent calculation for La metal and LaSb including the SIC and USPC. Correspondence to: O. Sakai, Department of Physics, Tohoku University, Sendai 980, Japan.
The self-consistent equation for the localized states is given by [3] [H 0 + V~m~(r - Rv)]Wnm~(r - Ru) = E elan,m,. . . . w , , ~ ( r - R , ) . ~n'ret'
(1)
Here, H o is the standard LSD Hamiltonian and V~,,,,(r- R~) is the SIC and UPSC potential associated with the localized Wannier orbit at site v, band index n and spin tT. The local symmetry of the orbit is specified by m. The quantity e~,,,,, . . . . is the Lagrange multiplier to ensure the orthogonality among localized orbits. The SIC potential is constructed following Perdew's formulation [4]. The USPC potential is constructed in the following way [9]. First we make the charge distribution of the localized orbit p,,.~ ( r - R , ) = ] w , , ~ ( r - R~)I 2. Next, we calculate the integrated number of the charge in each Wigner-Seitz (WS) cell. We remove this number from the charge distribution of the occupied valence electrons in the WS cell. Next we add p , m , ( r - R ) to construct the potential acting on the added electron in the WS cell. The reduction of the occupied electron charge corresponds to the screening effect by the valence electrons. We made a spherical averaging of the charge in each WS cell to simplify the calculation. Harrison et al. have shown that it is equivalent to diagonalize the following unified Hamiltonian [10]: n u : O n o O ~- E
( P i ( a o ~- v i ) e i ~- o ( a 0 ~- Vi)P i
+ e , ( n o + V~)0},
(2)
to solve eq. (1). Here i denotes the Wannier orbit and Pi is the projection operator on the ith state, and
0921-4526/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
180
O. Sakai, Y. Kaneta / 4 f energy levels in band theory
0 = 1 - E i Pi. When we do simplifying approximations, V~P~ ~- P , ~ P , , P,~ ~ PiVgPi and Pg(V~ + Vj)Pj 0, which may be applicable to very localized orbits, the unified Hamiltonian is rewritten as follows: U . ~- H o - ~?, P, HoP j + E
1.1
7 .6 c-" v
As noted in ref. 5, H u can be diagonalized based on the Bloch representation. In this paper, we use the L M T O - A S A band calculation method [11]. The Wannier orbit for the occupied states is constructed from the occupied Bloch states imposing that it has atomic symmetry [12,13]. The Wannier orbit of the unoccupied f state is made from the unoccupied bands. In fig. l(a), we show the band structures of La metal calculated by including the USPC term. The f bands lie at about 6 eV above E r consistently with the observed value 5.5 eV in BIS [8]. This is in contrast to 2.7 eV in the LSD calculation of fig. l ( b ) [14]. When we estimate the energy shift from the charge distribution of the LSD calculation, it is above 6 eV. This value is reduced to 3.3 eV in the self-consistent calculation. The occupation number of the f symmetry charge in the WS cell decreases to 0.15 from the 0.48 of the LSD calculation. The Wannier orbit of f symmetry has amplitude 93% on its site. In fig. 2, we show the band structures calculated by including SIC for LaSb. Hasegawa has carried out a detailed LSD calculation for La pnictides based on the A P W method [15,16]. He noted that the overlap between the top of the valence p(Sb) band at F and the bottom of the conduction d(La) band at X is large
L(USPC)
.5 .4 .3
E~
.2 .1 .0 -.1
xZw
q
L
^
Fig. 2. Band structure of LaSb SIC term. We use a = 6.500,~ ture, and chose the WS radii of SIC term is included for 5s and
F
a
X
2
F
calculated by including the for NaCl-type crystal strucLa and Sb to be equal. The 5p of Sb.
in calculation when compared with FSS experiments [171. The d - p band overlap is largely reduced by the SIC term in fig. 2. The energy shift of the p band is about - 0 . 3 7 e V when one estimates it by using the charge distribution given by the LSD calculation. The Wannier orbit of the p-type has amplitude 61% on its site and the intersite Coulomb terms is important. In previous studies, we have shown that FSS of LaSb and CeSb is reproduced well when we shift up d bands relatively 0.3eV [2,16]. The present self-consistent
(b) 1.1
1.0
1.0
.9
.9
L(LSD)
.8
.8 e~
L Sb(SIC)
.8
(3)
P,V~P,.
i~j
(a)
.9
.7
r~ v
.7 .6
.6
.5
r-,
r-~
Er
.4
.5 EF
.4 .3
.3 .2
.2
.1
.1
XZw
Q L
^
F
a
X
~
F
xZw
Q L
A
P
A
X
E
P
Fig. 1. (a) Band structure of FCC La calculated by including the USPC term. We use a = 5.310 ,~. Fermi energy is determined from about 365 sampling points in the first BZ and is not so correct. (b) Band structure of La calculated by LSD [19] theory based on the L M T O - A S A method with the combined correction term [11]. The 5p states of La are treated as the frozen core and relativistic effects except the spin-orbit interaction are included.
O. Sakai, Y. Kaneta / 4 f energy levels in band theory
calculation gives 0.28eV for this value. In a strict sense, the present calculation is inconsistent because small hole pockets appear around the F point, while we have treated the p bands as completely occupied. We expect that the small holes will not change the charge distribution drastically. In the calculation including the USPC terms, we were unable to obtain stable f(La) states. The f component of the valence bands increases more and more beyond one electron/cell and the f bands become very wide. When we tentatively drop the second term of eq. (3) for the f component, we obtain relatively narrow f bands at about 5.5 eV above E F [18]. But the d - p overlap increases again even when we include the SIC term. These results seem to relate the delocalization transition of the f states [20]. To check this point, improvements in the calculations, such as using eq. (2) directly, are necessary. The calculation including the SIC and USPC terms seems to improve the results of the standard LSD method. Detailed comparisons with dHvA experiments on LaSb and CeSb, will be given in the near future.
Acknowledgements The authors would like to thank T. Kasuya, K. Takegahara, H. Harima, H. Yamagami, A. Hasegawa and H.K. Yoshida for stimulating discussions. This work was partly supported by grant-in-aid No. 04231105 from the Ministry of Education, Science and Culture of Japan. The numerical computation was partly performed at the Computer Centers of the Institute for Molecular Science and of Tohoku University.
181
References [1] H. Harima, N. Kobayashi, K. Takegahara and T. Kasuya, J. Magn. Magn. Mater. 52 (1985) 367. [2] T. Kasuya, O. Sakai, J. Tanaka, H. Kitazawa and T. Suzuki, J. Magn. Magn. Mater. 63 & 64 (1987) 9. [3] H. Harima, O. Sakai, T. Kasuya and A. Yanase, Solid State Commun. 66 (1988) 603. [4] J.P. Perdew and A. Zunger, Phys. Rev. B 23 (1981) 5048. [5] R.A. Heaton, J.G. Harrison and C.C. Lin, Phys. Rev. B 28 (1983) 5992. [6] A. Svane, Phys. Rev. Lett. 68 (1992) 1900. [7] Z. Szotek, W.M. Temmerman and H. Winter, Physica B 172 (1991) 19. [8] J.K. Lang, Y. Baer and P.A. Cox, J. Phys. F 11 (1981) 121. [9] V.I. Anisimov, M.A. Korotin and E.Z. Kurmaev, J. Phys.: Condens. Matter 2 (1990) 3973. [I0] J.G. Harrison, R.A. Heaton and C.C. Lin, J. Phys. B 16 (1983) 2079. [11] H.L. Skriver, The LMTO Method (Springer-Verlag, Berlin, 1984). [12] W. Kohn, Phys. Rev. B 7 (1973) 4388. [13] O.K. Anderson, Z. Pawlowska and O. Jepson, Phys. Rev. B 34 (1986) 5253. [14] The present LMTO-ASA calculation gives f levels higher by about 0.5 eV than with previous APW calculations, see for example, W.E. Pickett, A.J. Freeman and D.D. Koeiling, Phys. Rev. B 22 (1980) 2695. [15] A. Hasegawa, J. Phys. C 13 (1980) 6147. [16] A. Hasegawa, J. Phys. Soc. Jpn. 54 (1985) 677. [17] H. Kitazawa, T. Suzuki, M. Sera, I. Oguro, A. Yanase, A. Hasegawa and T. Kasuya, J. Magn. Magn. Mater. 31-34 (1983) 421. [18] S. Ogawa and S. Suga, private communication; S. Ogawa, Ph.D. Thesis, Univ. of Tokyo (1990). [19] O. Gunnarsson and B.I. Lundqvist, Phys. Rev. B 13 (1976) 4274. [20] See for example, R.D. Cowan, Nuclear Instrum. Methods 110 (1973) 173.
Physica B 186-188 (1993) 179-181 North-Holland
4f energy levels in band theory and the Fermi surface structure of La and Ce compounds Osamu Sakai and Yasunori Kaneta Department of Physics, Tohoku University, Sendai 980, Japan We calculate 4f energy levels by a non-empirical band theory including the self-interaction correction and the unoccupied states potential correction. The 4f levels of La compounds appear at about 6 eV above the Fermi energy consistently with Bremsstrahlung isochromat spectroscopy experiments, but further improvements are necessary for the Fermi surface structure calculation.
In this paper, we attempt to calculate the 4f energy levels by a non-empirical method in the band calculation. The standard band theory based on the local spin density (LSD) approximation often fails to predict the 4f energy levels in rare earth compounds [1]. The unoccupied 4f energy levels usually appear to be lower compared with those obtained by Bremsstrahlung isochromat spectroscopy (BIS), while the occupied levels appear to be shallower compared with photoemission spectroscopy (PES). Previous works on La and Ce compounds show that the Fermi surface structures (FSS) are reproduced fairly well by calculations, shifting the 4f levels to be consistent with BIS and PES from those of the LSD band calculation [2,3]. Recently, importance of the self-interaction correction ( S I C ) [ 4 , 5 ] for the occupied states has been extensively discussed for 3d compounds [6]. It drastically improves the LSD results for the magnitude of the band gap. Such attempts for 4f compounds have not been done except for the works of Szotek et al. [7]. They showed that the occupied 4f states of Pr metal are pulled down to 8 eV below the Fermi energy (EF). However, the unoccupied levels appear at about 0.7 eV above Ev, which is lower compared with 3 eV in BIS [8]. As the self-consistent charge distribution is determined by the occupied states in the L S D - S I C method, we need corrections to calculate the unoccupied states, especially for localized states such as the 4f orbits. Anisimov et al. have introduced the unoccupied states potential correction (USPC) for 3d compounds [9]. In this paper, we perform a selfconsistent calculation for La metal and LaSb including the SIC and USPC. Correspondence to: O. Sakai, Department of Physics, Tohoku University, Sendai 980, Japan.
The self-consistent equation for the localized states is given by [3] [H 0 + V~m~(r - Rv)]Wnm~(r - Ru) = E elan,m,. . . . w , , ~ ( r - R , ) . ~n'ret'
(1)
Here, H o is the standard LSD Hamiltonian and V~,,,,(r- R~) is the SIC and UPSC potential associated with the localized Wannier orbit at site v, band index n and spin tT. The local symmetry of the orbit is specified by m. The quantity e~,,,,, . . . . is the Lagrange multiplier to ensure the orthogonality among localized orbits. The SIC potential is constructed following Perdew's formulation [4]. The USPC potential is constructed in the following way [9]. First we make the charge distribution of the localized orbit p,,.~ ( r - R , ) = ] w , , ~ ( r - R~)I 2. Next, we calculate the integrated number of the charge in each Wigner-Seitz (WS) cell. We remove this number from the charge distribution of the occupied valence electrons in the WS cell. Next we add p , m , ( r - R ) to construct the potential acting on the added electron in the WS cell. The reduction of the occupied electron charge corresponds to the screening effect by the valence electrons. We made a spherical averaging of the charge in each WS cell to simplify the calculation. Harrison et al. have shown that it is equivalent to diagonalize the following unified Hamiltonian [10]: n u : O n o O ~- E
( P i ( a o ~- v i ) e i ~- o ( a 0 ~- Vi)P i
+ e , ( n o + V~)0},
(2)
to solve eq. (1). Here i denotes the Wannier orbit and Pi is the projection operator on the ith state, and
0921-4526/93/$06.00 © 1993 - Elsevier Science Publishers B.V. All rights reserved
180
O. Sakai, Y. Kaneta / 4 f energy levels in band theory
0 = 1 - E i Pi. When we do simplifying approximations, V~P~ ~- P , ~ P , , P,~ ~ PiVgPi and Pg(V~ + Vj)Pj 0, which may be applicable to very localized orbits, the unified Hamiltonian is rewritten as follows: U . ~- H o - ~?, P, HoP j + E
1.1
7 .6 c-" v
As noted in ref. 5, H u can be diagonalized based on the Bloch representation. In this paper, we use the L M T O - A S A band calculation method [11]. The Wannier orbit for the occupied states is constructed from the occupied Bloch states imposing that it has atomic symmetry [12,13]. The Wannier orbit of the unoccupied f state is made from the unoccupied bands. In fig. l(a), we show the band structures of La metal calculated by including the USPC term. The f bands lie at about 6 eV above E r consistently with the observed value 5.5 eV in BIS [8]. This is in contrast to 2.7 eV in the LSD calculation of fig. l ( b ) [14]. When we estimate the energy shift from the charge distribution of the LSD calculation, it is above 6 eV. This value is reduced to 3.3 eV in the self-consistent calculation. The occupation number of the f symmetry charge in the WS cell decreases to 0.15 from the 0.48 of the LSD calculation. The Wannier orbit of f symmetry has amplitude 93% on its site. In fig. 2, we show the band structures calculated by including SIC for LaSb. Hasegawa has carried out a detailed LSD calculation for La pnictides based on the A P W method [15,16]. He noted that the overlap between the top of the valence p(Sb) band at F and the bottom of the conduction d(La) band at X is large
L(USPC)
.5 .4 .3
E~
.2 .1 .0 -.1
xZw
q
L
^
Fig. 2. Band structure of LaSb SIC term. We use a = 6.500,~ ture, and chose the WS radii of SIC term is included for 5s and
F
a
X
2
F
calculated by including the for NaCl-type crystal strucLa and Sb to be equal. The 5p of Sb.
in calculation when compared with FSS experiments [171. The d - p band overlap is largely reduced by the SIC term in fig. 2. The energy shift of the p band is about - 0 . 3 7 e V when one estimates it by using the charge distribution given by the LSD calculation. The Wannier orbit of the p-type has amplitude 61% on its site and the intersite Coulomb terms is important. In previous studies, we have shown that FSS of LaSb and CeSb is reproduced well when we shift up d bands relatively 0.3eV [2,16]. The present self-consistent
(b) 1.1
1.0
1.0
.9
.9
L(LSD)
.8
.8 e~
L Sb(SIC)
.8
(3)
P,V~P,.
i~j
(a)
.9
.7
r~ v
.7 .6
.6
.5
r-,
r-~
Er
.4
.5 EF
.4 .3
.3 .2
.2
.1
.1
XZw
Q L
^
F
a
X
~
F
xZw
Q L
A
P
A
X
E
P
Fig. 1. (a) Band structure of FCC La calculated by including the USPC term. We use a = 5.310 ,~. Fermi energy is determined from about 365 sampling points in the first BZ and is not so correct. (b) Band structure of La calculated by LSD [19] theory based on the L M T O - A S A method with the combined correction term [11]. The 5p states of La are treated as the frozen core and relativistic effects except the spin-orbit interaction are included.
O. Sakai, Y. Kaneta / 4 f energy levels in band theory
calculation gives 0.28eV for this value. In a strict sense, the present calculation is inconsistent because small hole pockets appear around the F point, while we have treated the p bands as completely occupied. We expect that the small holes will not change the charge distribution drastically. In the calculation including the USPC terms, we were unable to obtain stable f(La) states. The f component of the valence bands increases more and more beyond one electron/cell and the f bands become very wide. When we tentatively drop the second term of eq. (3) for the f component, we obtain relatively narrow f bands at about 5.5 eV above E F [18]. But the d - p overlap increases again even when we include the SIC term. These results seem to relate the delocalization transition of the f states [20]. To check this point, improvements in the calculations, such as using eq. (2) directly, are necessary. The calculation including the SIC and USPC terms seems to improve the results of the standard LSD method. Detailed comparisons with dHvA experiments on LaSb and CeSb, will be given in the near future.
Acknowledgements The authors would like to thank T. Kasuya, K. Takegahara, H. Harima, H. Yamagami, A. Hasegawa and H.K. Yoshida for stimulating discussions. This work was partly supported by grant-in-aid No. 04231105 from the Ministry of Education, Science and Culture of Japan. The numerical computation was partly performed at the Computer Centers of the Institute for Molecular Science and of Tohoku University.
181
References [1] H. Harima, N. Kobayashi, K. Takegahara and T. Kasuya, J. Magn. Magn. Mater. 52 (1985) 367. [2] T. Kasuya, O. Sakai, J. Tanaka, H. Kitazawa and T. Suzuki, J. Magn. Magn. Mater. 63 & 64 (1987) 9. [3] H. Harima, O. Sakai, T. Kasuya and A. Yanase, Solid State Commun. 66 (1988) 603. [4] J.P. Perdew and A. Zunger, Phys. Rev. B 23 (1981) 5048. [5] R.A. Heaton, J.G. Harrison and C.C. Lin, Phys. Rev. B 28 (1983) 5992. [6] A. Svane, Phys. Rev. Lett. 68 (1992) 1900. [7] Z. Szotek, W.M. Temmerman and H. Winter, Physica B 172 (1991) 19. [8] J.K. Lang, Y. Baer and P.A. Cox, J. Phys. F 11 (1981) 121. [9] V.I. Anisimov, M.A. Korotin and E.Z. Kurmaev, J. Phys.: Condens. Matter 2 (1990) 3973. [I0] J.G. Harrison, R.A. Heaton and C.C. Lin, J. Phys. B 16 (1983) 2079. [11] H.L. Skriver, The LMTO Method (Springer-Verlag, Berlin, 1984). [12] W. Kohn, Phys. Rev. B 7 (1973) 4388. [13] O.K. Anderson, Z. Pawlowska and O. Jepson, Phys. Rev. B 34 (1986) 5253. [14] The present LMTO-ASA calculation gives f levels higher by about 0.5 eV than with previous APW calculations, see for example, W.E. Pickett, A.J. Freeman and D.D. Koeiling, Phys. Rev. B 22 (1980) 2695. [15] A. Hasegawa, J. Phys. C 13 (1980) 6147. [16] A. Hasegawa, J. Phys. Soc. Jpn. 54 (1985) 677. [17] H. Kitazawa, T. Suzuki, M. Sera, I. Oguro, A. Yanase, A. Hasegawa and T. Kasuya, J. Magn. Magn. Mater. 31-34 (1983) 421. [18] S. Ogawa and S. Suga, private communication; S. Ogawa, Ph.D. Thesis, Univ. of Tokyo (1990). [19] O. Gunnarsson and B.I. Lundqvist, Phys. Rev. B 13 (1976) 4274. [20] See for example, R.D. Cowan, Nuclear Instrum. Methods 110 (1973) 173.