- Email: [email protected]
Electronic structure of hexagonal iron layers
Journal of Magnetism and Magnetic Materials 93 (1991) 285-289 North-Holland
285
Electronic structure of hexagonal iron layers M . O . S e l m e a n d P. P e c h e u r Laboratoire de Physique du Solide, URA 155, Ecole des Mines, Parc de Saurupt, 54042 NANCY Cedex, France
Using a tight-binding Hamiltonian, the recursion method and a self-consistent Stoner criterion to calculate the magnetic m o m e n t s we have investigated the magnetism of 6 hexagonal layers of Fe between r u t h e n i u m layers on each sides. With ferromagnetic spin-polarization we obtain a magnetic moment of about 2/x B per atom in the 4 inner iron layers and one dead magnetic iron layer at the interface, but the densities of states of this layer shows an antiferromagnetic tendency. Allowing a reverse polarization of the interface layer, a magnetic m o m e n t of 1.7/x B is obtained for this layer. The two dead layers found experimentally by M. Maurer et al. and C. Liu and S.D. Bader may be related to interface defects such as substitutional Ru atoms or steps.
1. Introduction
Hexagonal-closed-packed FexRuy superlattices with the modulation along the [0001] axis have been grown recently by Maurer et al. [1], by molecular beam epitaxy. Fe thickness x up to 7 monolayers are stabilized in the HCP structure by a y thickness of only two Ru layers. The mean Wigner-Seitz radius of Fe (i.e. the radius of the atomic sphere of Fe) is 2.72au, close to the ruthenium W.S. radius (2.79 au) but much larger than that of the fcc or high-pressure hexagonal phases of bulk iron. This should favour magnetism in the iron layers. Indeed ferromagnetic behaviour has been found by M6ssbauer spectroscopy and bulk magnetic measurements, provided that the Fe layers are thicker than about 4 monolayers. Beyond this value, the fraction of magnetic sites can be explained as if all the additional layers were fully magnetic with a magnetic moment of about 2/x B per iron atom [1]. More recently, Liu and Bader [2] have obtained f c c - F e ( l l l ) films, grown on a basal Ru(0001) plane, up to 8 monolayers of Fe. Here again ferromagnetism was observed only for films thicker than about 2 monolayers. The dead layers at the interface were attributed to d-band hybridizatiorr between Fe and Ru, with reference to the HCP FexRUl_ x
solid solutions, where no stable ferromagnetism is found up to Fe concentration as high as x = 75%. To investigate this interface problem we have performed calculations on a sandwich of 6 iron hexagonal layers between 11 ruthenium layers on each sides. We use a simple procedure based on the recursion method, a tight-binding Hamiltonian and the Stoner criterion to obtain spinpolarized density of states for the iron layers. The hopping integrals of the Hamiltonian are obtained from L M T O calculations for the pure metals, while the diagonal elements for each layer are determined from a charge neutrality condition. The Stoner criterion is written A E = m I , where AE is the shift between spin-up and spindown bands, m the corresponding atomic moment and I the Stoner parameter. I has been calculated from first principles [3] for metallic elements. The Stoner model has been used successfully to calculate the zero temperature magnetic properties of transition metals [4] and intermetallic compounds [5]. In the present calculation the Stoner criterion is applied self-consistently to each iron layer even when, after the first iteration the up and down densities of states differ from each other and from their paramagnetic form. The scheme is similar to that used in ref. [5].
0304-8853/91/$03.50 © 1991 - E l s e v i e r Science Publishers B.V. (North-Holland)
286
M.O. Selme, t~ Pecheur / Eh'etronie structure of hexagonal iron layers"
2. Procedure
We have used a tight-binding Hamiltonian which includes s, p and d orbitals and first neighbours interactions since these are much larger than the second neighbours ones in the fcc or H C P structures. For F e - F e and R u - R u interactions we obtained the hopping integrals from L M T O calculations for the pure metals. It has been shown indeed that the L M T O m e t h o d could be recast in a tight-binding form [6], leading to the following Hamiltonian: Hm,R, r = CRI6R,R,61, l, +
d~mSm.n,rCdR,r ,
where R stands for the site and l for the s, p or d orbital. Sin, n,r are structure constants which depend only on the structure. T h e y have been shown to be well approximated by a universal exponential law as a function of distance IR' - RI normalized to the W i g n e r - S e i t z radius [7]. T h e on-site term Sin, m has been tabulated for the c o m m o n structures [7]. cRl and d m are potential p a r a m e ters which d e p e n d on the atom and the W i g n e r - S e i t z radius. They are also tabulated for metallic elements [7]. W e have used rw. s = 2.72 au for iron and 2.79 au for ruthenium. T h e hopping integrals between Fe and R u at the interface have been taken as the square root m e a n of the corresponding ones between F e - F e and R u - R u . In the sandwich, to locate the center of the bands in each layer, we shifted the bands rigidly (i.e. s, p and d bands by the same amount) so as to obtain local neutrality in each layer. This shift was close to 2 eV for all the iron layers, while the first R u layer at the interface had to be shifted by only 0.28 eV. To obtain the magnetic m o m e n t we used the Stoner criterion in the form of the geometric construction introduced by A n d e r s e n et al. [8]. The up and down bands are shifted by A E while maintaining the total n u m b e r of electrons constant (local neutrality). For each A E the quanity m/AE is plotted as a function of the magnetic m o m e n t m (i.e. the difference between the number of up and down spin electrons). The intersec-
tion of this curve with the horizontal line 1 / I gives the magnetic moment. The Stoner parameter was taken from ref. [3] ( I = 0.92 eV). For bulk iron, where all sites are equivalent the p r o c e d u r e corresponds to a rigid shift of the up and down spin densities of states in a g r e e m e n t with the results of first principle treatments [9]. First principle calculations for bulk fcc [10] or H C P [11] iron show a transition between p a r a m a g n e t i s m and strong ferromagnetism in a narrow range of the W i g n e r - S e i t z radius (between about 2.66 to 2.72au). It is important to obtain this transition in the same range of the W.S. radius with our simple tight-binding model since the W.S. radius in the Fe,Ru~. layers are close to these values. So we have further shifted rigidly the s and p Fe diagonal elements relative to the d ones. This led for bulk H C P Fe to a transition between a paramagnetic state of 2.66au and a ferromagnetic state at 2.72 au with a magnetic m o m e n t of 2.3/x~. The corresponding n u m b e r of d electrons in the tight-binding model of bulk Fe was then close to 7. In the sandwich case, since the layers are nonequivalent, starting from the paramagnetic densities of states we obtain different m, and A E i values for each iron layer (fig. 1). We no longer are in a rigid band case so we p r o c e e d iteratively toward self-consistency. We split the diagonal elements of layer i by A E i (for each layer i) and we recalculate the local densities of states for each layer. From these, using the same geometrical construction as before, we obtain a new set of m i and A E i. The process is iterated until the mi values settle down. A b o u t four runs are necessary to obtain self-consistency. In some cases, for bulk iron or in the sandwich, the m / A E versus m curves have three intersections with 1 / 1 (fig. 1). T h e middle one with a positive slope corresponds to an instable state, the lower one to a low spin state and the u p p e r one to a high spin state. However, it has been shown by Krasko [12] that the low spin states in bulk fcc iron are highly d e p e n d e n t on the detail of the density of states a r o u n d the Fermi level, since they correspond to small A E. We cannot expect to describe these
M.O. Selme, P. Pecheur / Electronic structure of hexagonal iron layers
287
ua
1"l12.,¢ 1.1"
0.5
1.0
1.5
;t.O
M(Pb)
Fig. 1. Magnetization curves from the paramagnetic densities of states for: (a) the central iron layers: (b) the intermediate layers; (c) the interface iron layers. The figure shows the intersection of the m / b E curves versus m with the 1/1 straight line.
2.0 1.8 a
i, o
~
C
E
0
1.81
0.~
1.E
QE
2
E
i
b
d
i
i i
0
E
Fig. 2. Local density of states in the paramagnet=c calculation for: (a) the central iron layers; (b) the intermediate iron layers; (c) the interface iron layers; (d) the interface ruthenium layers.
M.O. Selme, P. Pecheur / Electronic structure of hexagonal iron layers
288
states reliably for we use a moment method to obtain the densities of states. The high spin states are much less sensitive to the details of the densities of states [12]. In the iterative procedure only these states were considered. All the densities of states were obtained with the recursion method using a cluster with more than 20000 atoms, and a 13 levels approximation
to the continuous fraction. The square root terminator was used.
3. Results and discussion
Fig. 2 shows the resulting densities of states for the paramagnetic calculation in the sandwich.
1.8L. 1.E
1.6 L
a
,
ii
I
,
0
2
~
I
II
E
1.8 1.6
b
,.~J
I
¢~
I
0
2
, , j ,
,
16
,
2
E
1.4
1.2
C
/ 0
2
0
2
E
Fig. 3. Local density of states for majority (left) and minority (right) spin states in the ferromagnetic calculation for: (a) the central iron layers; (b) the intermediate iron layers; (c) the interface iron layers (no net magnetic moment).
M.O. Selme, P. Pecheur / Electronic structure of hexagonal iron layers
The central iron layers are very similar to the results for bulk H C P iron. The density of states in the intermediate layer is somewhat wider and lower due to the increased influence of ruthenium. This effect becomes obvious in the interface iron layer. No interface states are found. The second ruthenium layer (not shown) is already quite similar to bulk ruthenium. Fig. 1 shows the magnetization curves from the paramagnetic densities of states. The interface layer is found to be non-magnetic although not far from the transition. The lowering of the density of states in this layer due to the 3 ruthenium first neighbours is responsible for this behaviour. Fig. 3 shows the densities of states in the iron layers after completion of the self-consistent procedure in the ferromagnetic case. The resulting moments per atom are 2.2 and 2.1p~B in the central and intermediate layers instead of 2.1 and 1.8/.t B obtained from the paramagnetic densities of states in fig. 1. The interface layers remain non-magnetic, but although there is no net ferromagnetic moment, there is a clear antiferromagnetic tendency in the density of states due to hybridization with the intermediate layer. So we repeated the self-consistent procedure with a reverse polarization for the interface layers. This time, the interface layers were found magnetic with a net moment of 1.7gB. The moments of the central and intermediate layers were similar to those of the ferromagnetic case.
4. Conclusions
The main conclusion of the calculation is that a perfect interface like the one considered does not lead to the two dead layers found in the experiments. At most one dead layer is obtained (in the
289
ferromagnetic calculation). It seems necessary to take into account interface defects (such as substitutional Ru atoms in the Fe interface layer or interface steps) in further work. The treatment of such defects are certainly possible with the recursion method since periodicity is not required to obtained the local densities of states. Another interesting point is the tendency for the interface to have a reverse magnetization compared to the inner layers. This may increase the influence of defects like interface steps in suppressing magnetism in the intermediate layers. Further possible magnetic order should be investigated, such as complete antiferromagnetic order of the layers. A total energy calculation would be useful to discriminate between these various possibilities but it is not clear whether this can be performed meaningfully within our simple model.
References [1] M. Maurer, J.C. Ousset, M. Piecuch, M.F. Ravet and J.P. Sanchez, Mater. Res. Soc. Symp. Proc. 151 (1989) 99. [2] C. Liu and S.D. Bader, Phys. Rev. B 41 (1990) 553. [3] J.F. Janak, Phys, Rev. B 16 (1977) 255. [4] O. Gunnarsson, J. Phys. F 6 (1976) 587. [5] S. Frota-Pess6a, Phys. Rev. B 36 (1987) 904. [6] O.K. Andersen and O. Jepsen, Phys. Rev. Lett. 53 (1984) 2571. [7] O.K. Andersen, O. Jepsen and D. Gl6tzel, in: Highlights of Condensed Matter Theory, eds. F. Bassani, F. Fumi and M.P. Tosi (North-Holland, Amsterdam, 1985). [8] O.K. Andersen, J. Madsen, U.K. Poulsen, O. Jepsen and J. Koll~r, Physica B 86-88B (1977) 249. [9] V.L. Moruzzi, J.F. Janak and A.R. Williams, Calculated Electronic Properties of Metals (Pergamon, New York, 1978). [10] V.L. Moruzzi, P.M. Marcus, K. Schwarz and P. Mohn, Phys. Rev. B 33 (1986) 1784. [11] J. Kfibler, Solid State Commun. 72 (1989) 631. [12] G.L. Krasko, Phys. Rev. B 36 (1987) 8565.
285
Electronic structure of hexagonal iron layers M . O . S e l m e a n d P. P e c h e u r Laboratoire de Physique du Solide, URA 155, Ecole des Mines, Parc de Saurupt, 54042 NANCY Cedex, France
Using a tight-binding Hamiltonian, the recursion method and a self-consistent Stoner criterion to calculate the magnetic m o m e n t s we have investigated the magnetism of 6 hexagonal layers of Fe between r u t h e n i u m layers on each sides. With ferromagnetic spin-polarization we obtain a magnetic moment of about 2/x B per atom in the 4 inner iron layers and one dead magnetic iron layer at the interface, but the densities of states of this layer shows an antiferromagnetic tendency. Allowing a reverse polarization of the interface layer, a magnetic m o m e n t of 1.7/x B is obtained for this layer. The two dead layers found experimentally by M. Maurer et al. and C. Liu and S.D. Bader may be related to interface defects such as substitutional Ru atoms or steps.
1. Introduction
Hexagonal-closed-packed FexRuy superlattices with the modulation along the [0001] axis have been grown recently by Maurer et al. [1], by molecular beam epitaxy. Fe thickness x up to 7 monolayers are stabilized in the HCP structure by a y thickness of only two Ru layers. The mean Wigner-Seitz radius of Fe (i.e. the radius of the atomic sphere of Fe) is 2.72au, close to the ruthenium W.S. radius (2.79 au) but much larger than that of the fcc or high-pressure hexagonal phases of bulk iron. This should favour magnetism in the iron layers. Indeed ferromagnetic behaviour has been found by M6ssbauer spectroscopy and bulk magnetic measurements, provided that the Fe layers are thicker than about 4 monolayers. Beyond this value, the fraction of magnetic sites can be explained as if all the additional layers were fully magnetic with a magnetic moment of about 2/x B per iron atom [1]. More recently, Liu and Bader [2] have obtained f c c - F e ( l l l ) films, grown on a basal Ru(0001) plane, up to 8 monolayers of Fe. Here again ferromagnetism was observed only for films thicker than about 2 monolayers. The dead layers at the interface were attributed to d-band hybridizatiorr between Fe and Ru, with reference to the HCP FexRUl_ x
solid solutions, where no stable ferromagnetism is found up to Fe concentration as high as x = 75%. To investigate this interface problem we have performed calculations on a sandwich of 6 iron hexagonal layers between 11 ruthenium layers on each sides. We use a simple procedure based on the recursion method, a tight-binding Hamiltonian and the Stoner criterion to obtain spinpolarized density of states for the iron layers. The hopping integrals of the Hamiltonian are obtained from L M T O calculations for the pure metals, while the diagonal elements for each layer are determined from a charge neutrality condition. The Stoner criterion is written A E = m I , where AE is the shift between spin-up and spindown bands, m the corresponding atomic moment and I the Stoner parameter. I has been calculated from first principles [3] for metallic elements. The Stoner model has been used successfully to calculate the zero temperature magnetic properties of transition metals [4] and intermetallic compounds [5]. In the present calculation the Stoner criterion is applied self-consistently to each iron layer even when, after the first iteration the up and down densities of states differ from each other and from their paramagnetic form. The scheme is similar to that used in ref. [5].
0304-8853/91/$03.50 © 1991 - E l s e v i e r Science Publishers B.V. (North-Holland)
286
M.O. Selme, t~ Pecheur / Eh'etronie structure of hexagonal iron layers"
2. Procedure
We have used a tight-binding Hamiltonian which includes s, p and d orbitals and first neighbours interactions since these are much larger than the second neighbours ones in the fcc or H C P structures. For F e - F e and R u - R u interactions we obtained the hopping integrals from L M T O calculations for the pure metals. It has been shown indeed that the L M T O m e t h o d could be recast in a tight-binding form [6], leading to the following Hamiltonian: Hm,R, r = CRI6R,R,61, l, +
d~mSm.n,rCdR,r ,
where R stands for the site and l for the s, p or d orbital. Sin, n,r are structure constants which depend only on the structure. T h e y have been shown to be well approximated by a universal exponential law as a function of distance IR' - RI normalized to the W i g n e r - S e i t z radius [7]. T h e on-site term Sin, m has been tabulated for the c o m m o n structures [7]. cRl and d m are potential p a r a m e ters which d e p e n d on the atom and the W i g n e r - S e i t z radius. They are also tabulated for metallic elements [7]. W e have used rw. s = 2.72 au for iron and 2.79 au for ruthenium. T h e hopping integrals between Fe and R u at the interface have been taken as the square root m e a n of the corresponding ones between F e - F e and R u - R u . In the sandwich, to locate the center of the bands in each layer, we shifted the bands rigidly (i.e. s, p and d bands by the same amount) so as to obtain local neutrality in each layer. This shift was close to 2 eV for all the iron layers, while the first R u layer at the interface had to be shifted by only 0.28 eV. To obtain the magnetic m o m e n t we used the Stoner criterion in the form of the geometric construction introduced by A n d e r s e n et al. [8]. The up and down bands are shifted by A E while maintaining the total n u m b e r of electrons constant (local neutrality). For each A E the quanity m/AE is plotted as a function of the magnetic m o m e n t m (i.e. the difference between the number of up and down spin electrons). The intersec-
tion of this curve with the horizontal line 1 / I gives the magnetic moment. The Stoner parameter was taken from ref. [3] ( I = 0.92 eV). For bulk iron, where all sites are equivalent the p r o c e d u r e corresponds to a rigid shift of the up and down spin densities of states in a g r e e m e n t with the results of first principle treatments [9]. First principle calculations for bulk fcc [10] or H C P [11] iron show a transition between p a r a m a g n e t i s m and strong ferromagnetism in a narrow range of the W i g n e r - S e i t z radius (between about 2.66 to 2.72au). It is important to obtain this transition in the same range of the W.S. radius with our simple tight-binding model since the W.S. radius in the Fe,Ru~. layers are close to these values. So we have further shifted rigidly the s and p Fe diagonal elements relative to the d ones. This led for bulk H C P Fe to a transition between a paramagnetic state of 2.66au and a ferromagnetic state at 2.72 au with a magnetic m o m e n t of 2.3/x~. The corresponding n u m b e r of d electrons in the tight-binding model of bulk Fe was then close to 7. In the sandwich case, since the layers are nonequivalent, starting from the paramagnetic densities of states we obtain different m, and A E i values for each iron layer (fig. 1). We no longer are in a rigid band case so we p r o c e e d iteratively toward self-consistency. We split the diagonal elements of layer i by A E i (for each layer i) and we recalculate the local densities of states for each layer. From these, using the same geometrical construction as before, we obtain a new set of m i and A E i. The process is iterated until the mi values settle down. A b o u t four runs are necessary to obtain self-consistency. In some cases, for bulk iron or in the sandwich, the m / A E versus m curves have three intersections with 1 / 1 (fig. 1). T h e middle one with a positive slope corresponds to an instable state, the lower one to a low spin state and the u p p e r one to a high spin state. However, it has been shown by Krasko [12] that the low spin states in bulk fcc iron are highly d e p e n d e n t on the detail of the density of states a r o u n d the Fermi level, since they correspond to small A E. We cannot expect to describe these
M.O. Selme, P. Pecheur / Electronic structure of hexagonal iron layers
287
ua
1"l12.,¢ 1.1"
0.5
1.0
1.5
;t.O
M(Pb)
Fig. 1. Magnetization curves from the paramagnetic densities of states for: (a) the central iron layers: (b) the intermediate layers; (c) the interface iron layers. The figure shows the intersection of the m / b E curves versus m with the 1/1 straight line.
2.0 1.8 a
i, o
~
C
E
0
1.81
0.~
1.E
QE
2
E
i
b
d
i
i i
0
E
Fig. 2. Local density of states in the paramagnet=c calculation for: (a) the central iron layers; (b) the intermediate iron layers; (c) the interface iron layers; (d) the interface ruthenium layers.
M.O. Selme, P. Pecheur / Electronic structure of hexagonal iron layers
288
states reliably for we use a moment method to obtain the densities of states. The high spin states are much less sensitive to the details of the densities of states [12]. In the iterative procedure only these states were considered. All the densities of states were obtained with the recursion method using a cluster with more than 20000 atoms, and a 13 levels approximation
to the continuous fraction. The square root terminator was used.
3. Results and discussion
Fig. 2 shows the resulting densities of states for the paramagnetic calculation in the sandwich.
1.8L. 1.E
1.6 L
a
,
ii
I
,
0
2
~
I
II
E
1.8 1.6
b
,.~J
I
¢~
I
0
2
, , j ,
,
16
,
2
E
1.4
1.2
C
/ 0
2
0
2
E
Fig. 3. Local density of states for majority (left) and minority (right) spin states in the ferromagnetic calculation for: (a) the central iron layers; (b) the intermediate iron layers; (c) the interface iron layers (no net magnetic moment).
M.O. Selme, P. Pecheur / Electronic structure of hexagonal iron layers
The central iron layers are very similar to the results for bulk H C P iron. The density of states in the intermediate layer is somewhat wider and lower due to the increased influence of ruthenium. This effect becomes obvious in the interface iron layer. No interface states are found. The second ruthenium layer (not shown) is already quite similar to bulk ruthenium. Fig. 1 shows the magnetization curves from the paramagnetic densities of states. The interface layer is found to be non-magnetic although not far from the transition. The lowering of the density of states in this layer due to the 3 ruthenium first neighbours is responsible for this behaviour. Fig. 3 shows the densities of states in the iron layers after completion of the self-consistent procedure in the ferromagnetic case. The resulting moments per atom are 2.2 and 2.1p~B in the central and intermediate layers instead of 2.1 and 1.8/.t B obtained from the paramagnetic densities of states in fig. 1. The interface layers remain non-magnetic, but although there is no net ferromagnetic moment, there is a clear antiferromagnetic tendency in the density of states due to hybridization with the intermediate layer. So we repeated the self-consistent procedure with a reverse polarization for the interface layers. This time, the interface layers were found magnetic with a net moment of 1.7gB. The moments of the central and intermediate layers were similar to those of the ferromagnetic case.
4. Conclusions
The main conclusion of the calculation is that a perfect interface like the one considered does not lead to the two dead layers found in the experiments. At most one dead layer is obtained (in the
289
ferromagnetic calculation). It seems necessary to take into account interface defects (such as substitutional Ru atoms in the Fe interface layer or interface steps) in further work. The treatment of such defects are certainly possible with the recursion method since periodicity is not required to obtained the local densities of states. Another interesting point is the tendency for the interface to have a reverse magnetization compared to the inner layers. This may increase the influence of defects like interface steps in suppressing magnetism in the intermediate layers. Further possible magnetic order should be investigated, such as complete antiferromagnetic order of the layers. A total energy calculation would be useful to discriminate between these various possibilities but it is not clear whether this can be performed meaningfully within our simple model.
References [1] M. Maurer, J.C. Ousset, M. Piecuch, M.F. Ravet and J.P. Sanchez, Mater. Res. Soc. Symp. Proc. 151 (1989) 99. [2] C. Liu and S.D. Bader, Phys. Rev. B 41 (1990) 553. [3] J.F. Janak, Phys, Rev. B 16 (1977) 255. [4] O. Gunnarsson, J. Phys. F 6 (1976) 587. [5] S. Frota-Pess6a, Phys. Rev. B 36 (1987) 904. [6] O.K. Andersen and O. Jepsen, Phys. Rev. Lett. 53 (1984) 2571. [7] O.K. Andersen, O. Jepsen and D. Gl6tzel, in: Highlights of Condensed Matter Theory, eds. F. Bassani, F. Fumi and M.P. Tosi (North-Holland, Amsterdam, 1985). [8] O.K. Andersen, J. Madsen, U.K. Poulsen, O. Jepsen and J. Koll~r, Physica B 86-88B (1977) 249. [9] V.L. Moruzzi, J.F. Janak and A.R. Williams, Calculated Electronic Properties of Metals (Pergamon, New York, 1978). [10] V.L. Moruzzi, P.M. Marcus, K. Schwarz and P. Mohn, Phys. Rev. B 33 (1986) 1784. [11] J. Kfibler, Solid State Commun. 72 (1989) 631. [12] G.L. Krasko, Phys. Rev. B 36 (1987) 8565.