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Anisotropic dielectric response of ferromagnetic cobalt
Journal of Magnetism and Magnetic Materials 148-144 (1995) 89-N
EISEVIER
Anisotropicdielectricresponseof ferromagneticcobalt R. Ahuja aq*, S. Auluck b, B. Johansson a, M.A. Khan ’ a Condensed Matter 7heoty Group, Department of Physics, Uppsala Uniuersiry, Box 530 Uppsala, Sweden b Department of Physics, University of ho&-e, Roorkee-247 667, India ’ Institut de Physique et Chemi des Materiawr de Strassimurg, Uniuersite Louis Pasteur, 23, rue du Loess, 67037 Strassbourg, France
Abstract A s&-cnn&~~n! &r!mnic ~tnwt~~r~ knvnnnn-8 C~~~~-p.4~h~~ -l--n ---I--> L+------A ..“_.Y.W fnv LV. .nmugULAaL re~~uura~~lriic cubait is obtained through the linear muffin tin orbital (LMTO) method. Using the derived band scheme, the frequency (0) dependent dielectric function for the electric field parallel (& and perpendicular (e I) to the c-axis for the majority and minority spin bands have been calculated. It is observed that the main contribution to the total E(W) comes from the minority spin band, and that e ‘-(WI contains more structure than ell( w). These observations are rather well confirmed by the experimental results.
The electronic structure of Co has been extensively studied experimentally [l-5]. Among the experiments, there are some optica measurements [4,5] for the ferromagnetic Co which show anisotropic effects due to the hexagonal close-packed (hcp) structure. Most of the experimental results [l-3] can be explained with the help of existing energy band structure calculations [6-83. As regarding the optical properties, the only available theoretical calculation was the one based on a joint density of states (JDOS) as derived by Kulikov and Kulatov [S]. Unfortunately, in this type of calculation the selection rules for the interband transitions and anisotropic effect, due to the hcp structure, are not taken into account. Due to these inherent limitations of the JDOS approach, it is not possible to explain the origin of the measured anisotropic properties. In the present work, the above mentioned deficiencies are properly taken care of. First we calculate the energy bands for hcp Co in the ferromagnetic phase by means of the linear muffin tin orbital (LMTO) method [9,1Oj. The self-consistent eigenvalues and eigenvectors thus obtained are used to calculate the imaginary part of the dielectric function (E(O)) where the band to band transition matrix etments are included in the calculation. The elf(w) and E J, (01 are catculated for the case when the electric field vector d is parallel and perpendicular to the c-axis, respectively. The obtained results are compared with the experimental data.
Corresponding author. [email protected], l
0304~8853/95/$09.50
Fax:
+ 46-18-183524;
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We consider hcp Co with lattice constant a = 2.515 A and c = 4.082 A. The 3d7 4s’ electrons are treated as condurtion electrons and the remaining electrons are kept frozen. The parameterization of the local-exchange-correlation potential by von Barth and Hedin [Ill and an LMTO basis set including s, p, d and f partial waves were used. For the DOS and E(O) calculations we use 196 k-points in the irreducible Brillouin zone (IBZ). In Fig. 1 we show the obtained DOS decomposed into contributions from the majority and minority spin bands, Some relevant properties such as the position of the minimum (r,) of the band with respect to the Fermi level (E,), DOS at E, (n(E,)) and magnetic moment as obtained in the present work and in some earlier studies are given in Table 1. Once the ab initio energy bands are obtained the imagi-
..*” I co
Fig. 1. Calcuhted totat density of states (DOS) for ferromagnetic hcp Co for both the majority and the minority spin bands. The Fermi level is set at zem energy and marked by a vertical line.
8 1995 Elsevicr Science B.V. All rights reserved
SSDI 0304-8853(94)01131-l
I
90
R. Ahuju et al./Journal
of Magnet&m and Magnetic MateriuIs 140-144 fl99S) 89-90
Table 1 General characteristics of the electronic structure of hco Co EF-~,
(Ry) Present work Singal and Das [7] Kulikov and Kulatov [S] Experiment
0.645 0.63 0.71 0.647 111
nary part of the dielectric ?he re!etion I1 21
function
n(EF) (states/ Ry/atom) 26.2 17.9 ;*I
is catulated
In Fig. 2(a) we show for majority and E”“(O),
l ( 0) = +0)+2&(&J)3 1.55 1.58 1.69 1.56 [3]
through
where e and m are the charge and mass of the electron, respectively. The quantity w is the photon energy and F,,,+(k) is the dipolar matrix element between the initial I nk) and final I n’k) states. S, is the constant energy (w) surface. e is the unit vector in the electric field direction. The energy difference o,,,,(k) is defined as %W =Km -w)~ (2) where E,,(k) and E,(k) are the respective eigenvalues for the n and n’ bands. The main relation (1) decomposes [12,13] into e”(o) and E’(W) according to when E is parallel or perpendicular to the c-axis of the hcp structure, The same relation can be also used to calculate the JDOS by putting ~&c) = 1 for all the interband transitions. Here we will present only the calculation when Fn,,(k) is properly included.
spin tl! a), E l(w)
where the latter is given by
*
(3)
Fig. 2(b) presents the same quantities for the minority spin. Comparing the two figures one notices that the major contribution to the optical structures comes from the minority spin band. This seems evident already from the DOS curve (Fig. 1) where the majority band is almost filled and therefore there are no high density of states regions above E, for this spin, whereas the partially filled minority band has high density of states peaks on both sides of EF which favours more interband transitions. The same rrend was also obU1. --ledI by Kulikov and Kulatov [8] in their JDOS calculation. From Fig. 2(b), it is obvious that the main contribution to E(W) comes from t ‘( 01. The calculated main peak structures are at 0.087, 0.105, 0.117 and 0.129 Ry and these agree quite well with those observed by Weaver et al. [4] and Kirillova et al. [5] at 0.088, 0.107, 0.121 and 0.135 Ry. In the high energy range of the calculation we also observe two important interband transitions at 0.438 and 0.462 Ry (Fig+ 2(b)). The first is due to &l(o) and the second is due to E ‘(~1. In this range the experiment by Kirillova et al. [5] showed a structure at 0.404 Ry in E”(W). In conclusion, our energy bands for hcp ferromagnetic Co are in agreement with experimental measurements such as photo-emission spectra [l], specific hear [2] and neutron diffraction [3] experiments. Moreover these btiulds give an appropriate representation of the observed interband optical properties when the dipolar selection rules are properly taken into account. References [I) P. Himpsel and D.E. Eastman, Phys. Rev. i3 21 (1980) 3207. [2] C.H. Chang, CT. Wei and P.A. Beck, Phys. Rev. 120 (1960) 426. 131 R.M. Moon, Phys. Rev. A 136 (1964) 195, [Q] J.H. Weaver, E. Colavita, D.W. Lynch and R. Rosei, Phys. Rev. B 19 (1979) 3850. [5] M.M. Kirillova, L.V. Nomerovannaya and G.A. Bolotin, Opt. Spectrosk. 49 (1980) 742. [6] S. Wakoh and J. Yamashita, 1. Phys. Sot. Jpn. 28 (1970) 1151.
Fig. 2. Different (parallel, perpendicular and total) components of the calculated imaginary part of l (a) for the majority spin up bands (a) and the minority spin down bands (b) in Co.
[7] CM. Singal and T.P. Das, Phys. Rev. B 16 (1977) 5068. [8] N.I. Kulikov and E.T. Kulatov, I. Phys. F 12 (1982) 2267. 191 O.K. Andersen, Phys. Rev, B 12 (1975) 3060. 1103 H.L. Skriver, The LMTO Method (Springer, Berlin, 1984). [ll] U. von Barfh and L. Hedin, J. Phys. C 5 (11972) 1629. [12] M.A. Khan, J. Phys. Sot. Jpn. 62 (1993) 1682. [13] R. Ahuja, S. Auluck, B. Johanssol and M.A. Khan, Phys. Rev. B 50 (1994) 2128.
EISEVIER
Anisotropicdielectricresponseof ferromagneticcobalt R. Ahuja aq*, S. Auluck b, B. Johansson a, M.A. Khan ’ a Condensed Matter 7heoty Group, Department of Physics, Uppsala Uniuersiry, Box 530 Uppsala, Sweden b Department of Physics, University of ho&-e, Roorkee-247 667, India ’ Institut de Physique et Chemi des Materiawr de Strassimurg, Uniuersite Louis Pasteur, 23, rue du Loess, 67037 Strassbourg, France
Abstract A s&-cnn&~~n! &r!mnic ~tnwt~~r~ knvnnnn-8 C~~~~-p.4~h~~ -l--n ---I--> L+------A ..“_.Y.W fnv LV. .nmugULAaL re~~uura~~lriic cubait is obtained through the linear muffin tin orbital (LMTO) method. Using the derived band scheme, the frequency (0) dependent dielectric function for the electric field parallel (& and perpendicular (e I) to the c-axis for the majority and minority spin bands have been calculated. It is observed that the main contribution to the total E(W) comes from the minority spin band, and that e ‘-(WI contains more structure than ell( w). These observations are rather well confirmed by the experimental results.
The electronic structure of Co has been extensively studied experimentally [l-5]. Among the experiments, there are some optica measurements [4,5] for the ferromagnetic Co which show anisotropic effects due to the hexagonal close-packed (hcp) structure. Most of the experimental results [l-3] can be explained with the help of existing energy band structure calculations [6-83. As regarding the optical properties, the only available theoretical calculation was the one based on a joint density of states (JDOS) as derived by Kulikov and Kulatov [S]. Unfortunately, in this type of calculation the selection rules for the interband transitions and anisotropic effect, due to the hcp structure, are not taken into account. Due to these inherent limitations of the JDOS approach, it is not possible to explain the origin of the measured anisotropic properties. In the present work, the above mentioned deficiencies are properly taken care of. First we calculate the energy bands for hcp Co in the ferromagnetic phase by means of the linear muffin tin orbital (LMTO) method [9,1Oj. The self-consistent eigenvalues and eigenvectors thus obtained are used to calculate the imaginary part of the dielectric function (E(O)) where the band to band transition matrix etments are included in the calculation. The elf(w) and E J, (01 are catculated for the case when the electric field vector d is parallel and perpendicular to the c-axis, respectively. The obtained results are compared with the experimental data.
Corresponding author. [email protected], l
0304~8853/95/$09.50
Fax:
+ 46-18-183524;
email:
We consider hcp Co with lattice constant a = 2.515 A and c = 4.082 A. The 3d7 4s’ electrons are treated as condurtion electrons and the remaining electrons are kept frozen. The parameterization of the local-exchange-correlation potential by von Barth and Hedin [Ill and an LMTO basis set including s, p, d and f partial waves were used. For the DOS and E(O) calculations we use 196 k-points in the irreducible Brillouin zone (IBZ). In Fig. 1 we show the obtained DOS decomposed into contributions from the majority and minority spin bands, Some relevant properties such as the position of the minimum (r,) of the band with respect to the Fermi level (E,), DOS at E, (n(E,)) and magnetic moment as obtained in the present work and in some earlier studies are given in Table 1. Once the ab initio energy bands are obtained the imagi-
..*” I co
Fig. 1. Calcuhted totat density of states (DOS) for ferromagnetic hcp Co for both the majority and the minority spin bands. The Fermi level is set at zem energy and marked by a vertical line.
8 1995 Elsevicr Science B.V. All rights reserved
SSDI 0304-8853(94)01131-l
I
90
R. Ahuju et al./Journal
of Magnet&m and Magnetic MateriuIs 140-144 fl99S) 89-90
Table 1 General characteristics of the electronic structure of hco Co EF-~,
(Ry) Present work Singal and Das [7] Kulikov and Kulatov [S] Experiment
0.645 0.63 0.71 0.647 111
nary part of the dielectric ?he re!etion I1 21
function
n(EF) (states/ Ry/atom) 26.2 17.9 ;*I
is catulated
In Fig. 2(a) we show for majority and E”“(O),
l ( 0) = +0)+2&(&J)3 1.55 1.58 1.69 1.56 [3]
through
where e and m are the charge and mass of the electron, respectively. The quantity w is the photon energy and F,,,+(k) is the dipolar matrix element between the initial I nk) and final I n’k) states. S, is the constant energy (w) surface. e is the unit vector in the electric field direction. The energy difference o,,,,(k) is defined as %W =Km -w)~ (2) where E,,(k) and E,(k) are the respective eigenvalues for the n and n’ bands. The main relation (1) decomposes [12,13] into e”(o) and E’(W) according to when E is parallel or perpendicular to the c-axis of the hcp structure, The same relation can be also used to calculate the JDOS by putting ~&c) = 1 for all the interband transitions. Here we will present only the calculation when Fn,,(k) is properly included.
spin tl! a), E l(w)
where the latter is given by
*
(3)
Fig. 2(b) presents the same quantities for the minority spin. Comparing the two figures one notices that the major contribution to the optical structures comes from the minority spin band. This seems evident already from the DOS curve (Fig. 1) where the majority band is almost filled and therefore there are no high density of states regions above E, for this spin, whereas the partially filled minority band has high density of states peaks on both sides of EF which favours more interband transitions. The same rrend was also obU1. --ledI by Kulikov and Kulatov [8] in their JDOS calculation. From Fig. 2(b), it is obvious that the main contribution to E(W) comes from t ‘( 01. The calculated main peak structures are at 0.087, 0.105, 0.117 and 0.129 Ry and these agree quite well with those observed by Weaver et al. [4] and Kirillova et al. [5] at 0.088, 0.107, 0.121 and 0.135 Ry. In the high energy range of the calculation we also observe two important interband transitions at 0.438 and 0.462 Ry (Fig+ 2(b)). The first is due to &l(o) and the second is due to E ‘(~1. In this range the experiment by Kirillova et al. [5] showed a structure at 0.404 Ry in E”(W). In conclusion, our energy bands for hcp ferromagnetic Co are in agreement with experimental measurements such as photo-emission spectra [l], specific hear [2] and neutron diffraction [3] experiments. Moreover these btiulds give an appropriate representation of the observed interband optical properties when the dipolar selection rules are properly taken into account. References [I) P. Himpsel and D.E. Eastman, Phys. Rev. i3 21 (1980) 3207. [2] C.H. Chang, CT. Wei and P.A. Beck, Phys. Rev. 120 (1960) 426. 131 R.M. Moon, Phys. Rev. A 136 (1964) 195, [Q] J.H. Weaver, E. Colavita, D.W. Lynch and R. Rosei, Phys. Rev. B 19 (1979) 3850. [5] M.M. Kirillova, L.V. Nomerovannaya and G.A. Bolotin, Opt. Spectrosk. 49 (1980) 742. [6] S. Wakoh and J. Yamashita, 1. Phys. Sot. Jpn. 28 (1970) 1151.
Fig. 2. Different (parallel, perpendicular and total) components of the calculated imaginary part of l (a) for the majority spin up bands (a) and the minority spin down bands (b) in Co.
[7] CM. Singal and T.P. Das, Phys. Rev. B 16 (1977) 5068. [8] N.I. Kulikov and E.T. Kulatov, I. Phys. F 12 (1982) 2267. 191 O.K. Andersen, Phys. Rev, B 12 (1975) 3060. 1103 H.L. Skriver, The LMTO Method (Springer, Berlin, 1984). [ll] U. von Barfh and L. Hedin, J. Phys. C 5 (11972) 1629. [12] M.A. Khan, J. Phys. Sot. Jpn. 62 (1993) 1682. [13] R. Ahuja, S. Auluck, B. Johanssol and M.A. Khan, Phys. Rev. B 50 (1994) 2128.