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Mo(0 0 1) surface: A pseudopotential calculation
ARTICLE IN PRESS Journal of Magnetism and Magnetic Materials 320 (2008) 2629– 2634
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The structure and magnetism of Fe/Mo(0 0 1) surface: A pseudopotential calculation A.A. Ramanathan, J.M. Khalifeh , B.A. Hamad Department of Physics, University of Jordan, Amman 11942, Jordan
a r t i c l e in fo
abstract
Article history: Received 27 October 2007 Received in revised form 13 May 2008 Available online 27 May 2008
The effect of structure on the magnetism of iron monolayers (MLs) on molybdenum is investigated using the density functional theory (DFT) with norm conserving pseudopotentials and a plane wave basis, under the local spin density approximation (LSDA). Relaxation of 5 and 7 ML of Mo resulted in a contraction of 11.3% and 11.7%, respectively, for the top Mo–Mo interlayer spacing in close agreement with experimental results. In the case of one Fe overlayer, the top Fe–Mo interlayer spacing contracted by 15.8% for a ferromagnetic (FM) p(1 1) and 20.6% for an antiferromagnetic (AF) c(2 2) configuration. The magnetic moment of the surface (Fe) layer is enhanced from its theoretically calculated bulk value. Total energy calculations show that the AF c(2 2) is the stable state with a magnetic moment of 2.53 mB. The surface Fe atoms are AF coupled with each other and with the Mo layers below, showing layered AF coupling. The present study demonstrates the reliability of the pseudopotential approach under LSDA with core corrections included to the calculation of magnetic properties of combined transition metal systems. & 2008 Elsevier B.V. All rights reserved.
Keywords: Magnetism Fe Mo DFT LSDA GGA Pseudopotential XC
1. Introduction Structural and magnetic properties of Fe thin films on noble metals have been widely studied [1–3] due to their enhanced surface magnetic moment. Perpendicular surface anisotropy of magnetism is observed in all the above studies for a few monolayer of Fe. In the above studies different experimental techniques are used to show that for 1–3 monolayers (MLs) of Fe on different noble metals like Ag, Au and in addition on Cu and W there is a ferromagnetic order. Recently, Fe/W and Fe/Mo received great attention due to their thermo stability [4,5] and the pseudomorphic layer by layer growth [6,7] of Fe films up to 2 ML on these substrates. Magnetism is a very complex issue and is influenced by various factors like type of unit cell, lattice spacing, type and number of nearest neighbors, orientation, thickness of film and relaxation of surfaces, etc. The combination of these factors sometimes gives unexpected and contradictory results. Therefore, even though we have studied this system before [8] by the semi-empirical, tight binding recursion method in the Hartree–Fock approximation, it is not futile if the same system is studied again especially since the present study is an ab-initio calculation. In our previous work for
Corresponding author.
E-mail addresses: [email protected] (J.M. Khalifeh), [email protected] (B.A. Hamad). 0304-8853/$ - see front matter & 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2008.05.035
the (0 0 1) orientation we [8] had an unexpected huge reduction of the Fe surface layer magnetic moment from its bulk value to 1.29 mB, which is in qualitative agreement with the results of Mirbt et al. [9] for Fe/Mo/Fe sandwiches. Their paper deals with the ideal Fe–Mo interface. It can be seen from the paper that the magnetic moment of Fe is very sensitive to the value of lattice constant used in the calculations. Moreover, the magnetic properties at the interface do not necessarily represent those at the surface. The recent finding by Kubetzka et al. [10] of an enhancement of surface Fe magnetic moment from the bulk value for a similar system, namely Fe/W(0 0 1) motivated us to undertake a second study of the system for the (0 0 1) orientation, from first principles. The present study of Fe/Mo in the (0 0 1) orientation is also motivated by the fact that, to our knowledge, there has been no first-principles study of this system. Moreover, no pseudopotential calculation has been attempted for this system in any orientation. This paper is organized as follows: Section 2 outlines the computational method, Section 3 gives the results and discussion under two sub-headings, structural and magnetic.
2. Method of calculations All calculations are based on the density functional theory (DFT) as implemented in the ABINIT software package [11,12]. The pseudopotentials used are Trouiller–Martins (TM) with core corrections for local spin density approximation (LSDA) exchange
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correlations [13]. In order to obtain the theoretical lattice constants for Mo and Fe, convergence tests with respect to the cut-off energy and k-points are performed. Using the converged values, the lattice constants are optimized. Convergence for the self-consistent field (SCF) iterations is achieved when the total energy differences between two consecutive iterations is lees than 1 106 Ha. [14]. Both Mo and Fe have a Bcc ground state lattice structure. According to our calculations, the mismatch in the lattice constants of these two metals is about 13% under LSDA (the experimental mismatch is about 9%). This large mismatch in the case of LSDA is related to the typical behavior of this functional to underestimate the lattice constant of the 3d transition metals. This mismatch is a driving force for structural relaxation and possible magnetic re-orientation due to the magnetoelastic effect. The supercell for the slab calculation consists of seven atomic layers and seven layers of vacuum (54%). The amount of vacuum and the number of atomic layers are chosen carefully after making convergence tests for the number of vacuum and atomic layers with respect to the bulk density of states (DOS). This amount of vacuum is required to minimize any Coulomb and exchange interactions between the layers. Repeated slab geometries consisting of five Mo substrate layers and one Fe layer on each surface were employed in all calculations. The unit cell consists of one atom per layer for the (1 1) cell and two atoms per layer for the (2 2). Spin polarized calculations are applied for Fe/Mo(0 0 1). Spin–orbit coupling has not been taken into account; therefore, the magnetic direction is taken arbitrary. Because of the twodimensional (in plane) translational invariance in the supercell, there are no in-plane components of force and therefore no inplane relaxation. There are only vertical components of forces. All atoms are allowed to relax simultaneously and therefore the true minimum is reached. The vertical lattice spacings are allowed to relax simultaneously according to the sign and magnitude of forces. The structure is optimized when the total maximum force is o1 mRy/a.u. The SCF iterations have the same convergence criteria as mentioned before. For structural relaxation we used the Broyden–Fletcher–Goldfarb–Shanno (BFGS) minimization algorithm modified to take into account the total energy as well as the gradients [15]. The magnetic moments for the different atomic layers are calculated as the differences between the spin-up and spin-down integrated local density of states (LDOS). The tetrahedron method is used to obtain the angular momentum projected LDOS [16]. The integration is done over atomic spheres of radii 2.4 and 2.6 a.u. for Fe and Mo, respectively.
3. Results and discussion 3.1. Structural results The optimized lattice constants obtained for Mo and Fe along with details of kinetic energy cut-off and k-point mesh used is given in Table 1. First, we started the calculations for Mo clean
surface. We performed the calculations for five and seven atomic layer slabs. The two-dimensional lattice constant for the slab is taken to be the calculated lattice constant of Mo, namely 6.035 a.u. Structural relaxation is then performed with a force tolerance of 5 104 Ha/a.u. (1 mRy/a.u). All the atoms are allowed to relax simultaneously but only in the vertical direction. The structures are optimized when the force on each atom is o1 mRy/a.u. The direction and degree of relaxation for the vertical interlayer spacing depend on the magnitude and sign of the forces present. The optimized interlayer spacing for the Mo five and seven layer slabs are given in Table 2. The LEED experimental result [17] is also listed in the table. The top Mo–Mo interlayer spacing is contracted in both the five and seven ML by 11.3% and 11.7%, respectively. This is in good agreement with LEED experimental results of 11.5%, which has been obtained by a comparison of the experimental intensity spectra with the calculated intensity spectra using the layer-Korringa–Kohn–Rostoker method. Our results are also supported by the results obtained by Noguera et al. [18]. Photoemission experiments applied to the (0 0 1) face of Mo reveal two sharp peaks at 0.3 and 3 eV below the Fermi level. Noguera et al. [18] identified the upper peak due to normal emission as a combination of a surface resonance and a d band edge around the G point of the two-dimensional Brillouin zone. They had to include a 10–13% inward relaxation of the surface layer in the theoretical calculation in order to place the energy of this resonance at the energy of the experimental peak. Next the spin-polarized relaxation of the Fe/Mo(0 0 1) system is done. In Table 3 we present the relaxation for the case of 1 ML of Fe/Mo(0 0 1). The surface Fe–Mo contraction is 15.8% and 20.6% for the FM p(1 1) and the AF c(2 2) configurations, respectively. This is in good agreement with the results of Kubetzka et al. [10] for a similar system, namely Fe/W(0 0 1), which is also included in the table for comparison. They performed a first-principles FPLAPW study of Fe/W(0 0 1) using a nine layer slab and a GGA (PBE) exchange correlation within the FLEUR code. To find the equilibrium top interlayer spacing they calculated the total energy of the system as a function of the interlayer distance d between the Fe ML and the W layer for FM and AF configurations. They found the AF state to have the lowest energy at the equilibrium interlayer distance of 2.58 a.u. corresponding to an inward relaxation of 13.9% as quoted explicitly by them in Ref. [19]. Fig. 1 shows the charge density for the Fe/Mo(0 0 1) system interpolated along the [0 0 1] direction (z-axis). We notice from the figure how the charge density peaks at the surface, showing a marked change from that of the other layers. Also we see that there is a charge leak into the vacuum. This outward flow of charge into the vacuum region serves to screen the surface discontinuity and gives rise to a dipole layer of charge. The charge redistribution on relaxation is also clearly seen. The large values of inward relaxations obtained for our system can be explained by the following facts: (i) The bcc (0 0 1) surface is a relatively rough open surface and the Smoluchowski smoothing is more pronounced for rougher surfaces [20]. As pointed out by Smoluchowski, the difference
Table 1 Details of lattice constants Element
Pseudo potential
Ecut (Ha)
Grid
Lattice constant (Bohr)
Magnetic moment (mB) Bulk
Molybdenum Iron
42mo_pspnca 26fe.pspnca
40 70
10 10 10 12 12 12
6.0347 (1.3%)b 5.2432 (3%)b
2.1167 (3.3)c
a b c
Core corrections included. Difference with respect to experimental value. Magnetic moment of free-standing ML.
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Table 2 Interlayer spacing Dij for 5 and 7 Mo layers for clean Mo(0 0 1) surface
5 ML Moa 7 ML Moa Mo (Expt)b
D12
D23
D34
2.68 (11.3%) 2.66 (11.7%) (11.5%)
3.1 (2.8%) 3.12 (3.4%)
3.00 (0.57%)
The values in brackets are the percentage change in the interlayer spacings. a Present LDA calculations. b Ref. [17].
Table 4 The angular momentum decomposed layer charges for the p(1 1) configuration Layers
s
S (Fe) S-1 (Mo) S-2 (Mo) C (Mo)
0.46 0.37 0.38 0.38
d
0.29 0.35 0.37 0.37
(0.24) (0.33) (0.37) (0.37)
Total charge/layer
6.05 3.87 3.86 3.88
(5.85) (3.90) (3.86) (3.85)
6.82 4.63 4.65 4.67
(6.51) (4.58) (4.61) (4.60)
NON-MAGNETIC LDOS Fe/Mo(001) Fe Overlayer D23
D34
4
2.54 (15.8%)
3.00 (0.4%)
2
2.39 (20.6%)
3.07 (1.97%)
2.58 (13.9%)
The values in brackets are the percentage change in the interlayer spacings. a FM p(1 1)calculations. b AF c(2 2) calculations. c Tunneling microscopy and FLAPW, Ref. [10].
TOTAL CHARGE DENSITY (001)direction 70 Charge density (electrons/Bohr3)x10-2
(0.42) (0.35) (0.38) (0.38)
6
LDOS(States/eV/atom)x102
1ML Fe/Moa 3.04 (0.9%) 1ML Fe/Mob 3.03 (0.57%) 1ML Fe/W (0 0 1)c
p
The non-relaxed values are given in brackets.
Table 3 Interlayer spacing Dij for Fe/Mo (0 0 1) sLaboratory D12
2631
EF d=6a.u.
0 6
-2
0 EF
4
2 d=Non-Relaxed spacing
2 0 -2
0
2
6 EF
4
d=Relaxed spacing
2
60
0 -2
50
0
2
Energy E-EF(eV) Relaxed Non-Relaxed Relaxed atom positions Non-Relaxed atom positions
40 30 20 10 0 -1
1
3
5
7
9
11
13 15 17 19 21 23 25
Z coordinate (Bohr) Fig. 1. The total charge density along the [0 0 1] direction for the relaxed and nonrelaxed Fe/Mo(0 0 1) systems.
between non-relaxed and relaxed charge densities causes an additional electrostatic potential and gives rise to an inward electrostatic force on the top layer. (ii) For transition metals there is an additional contribution to surface relaxation from d electrons. Owing to the localized nature of the 3d electrons, a very small amount of charge transfer in or out of the surface amounts to a fairly large surface dipole moment. The additional inward force is proportional to the strength of the d bonds. The largest inward force would be found for a half-filled d band, since in this case all the bonding states and none of the antibonding states are filled [20].
The angular momentum decomposed layer charges are presented in Table 4, again the non-relaxed values are given in
Fig. 2. Non-magnetic LDOS for the Fe overlayer for three different Fe–Mo interlayer distances d12.
brackets. In the table, the angular momentum decomposed layer charge for the Fe overlayer shows a marked increase in s, p and d electrons on relaxation. This is as expected due to the d band Fe–Mo hybridization. Furthermore, the hybridization between Fe and Mo d bands can be understood qualitatively by plotting the non-magnetic LDOS for three different interlayer spacing ‘d’, Fig. 2. When d ¼ 6 a.u, there is hardly any interaction of the Fe overlayer with the Mo surface below. The LDOS has a strong peak at EF due to the fact that there is no hybridization and FM is favored. Next, we see that the LDOS has broadened due to the beginning of hybridization, when d is reduced to 3.068 a.u (the non-relaxed interlayer spacing). Finally on relaxation, when d is equal to the relaxed interlayer spacing, there is maximum hybridization and the d band has broadened considerably and resembles the Mo interface LDOS. The effect of the relaxation on the Fe overlayer magnetic moment will be discussed in the next section. 3.2. Magnetic results Due to the reduction in coordination number at the surface, the magnetic moments of 3d transition-metal surfaces are generally enhanced. The magnetic moment is particularly enhanced for bcc Fe(0 0 1) where the majority of spin band is not already saturated in the bulk. The values for surface Fe(0 0 1) magnetic moment found by various studies [21–24] range from 2.97 to 3.01 mB as compared to the experimental bulk value of 2.2 mB. This enhancement is attributed to narrowing of the d band width, shifts of peak
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100
Density of States (electron states/atom/ev)
LDOS Fe/Mo(001) Fe Overlayer
Spin up
-5
-4 -3 Spin Down
180 160 140 120 100 80 60 40 20 0 -2 -1 -20 0 -40 -60 -80 ENERGY E-EF (ev)
80 LDOS SURFACE Fe ATOM 13
60 40 20
-5
-3
-1
0
1
-20 -40
1
2
Majority Minority
-60 -80 -100 100 80
Fig. 3. The LDOS for majority and minority electrons of Fe over layer of the Fe/ Mo(0 0 1) system in p(1 1) configuration.
Majority Minority
60 40
positions with EF lying in the peak of the minority LDOS and surface states [23]. The effect of substrate on the Fe surface magnetism is very evident in our work. Since Mo is iso-electric to W, one expects to find similar behavior for our system Fe/Mo(0 0 1) as in Fe/W(0 0 1), Ref. [10] and it is indeed the case. The LDOS for the majority and minority electrons of the Fe overlayer in the case of the FM and AF configurations are shown in Figs. 3 and 4. From the figures we see that the minority DOS peak shifts position towards the Fermi level, which results in an enhancement of the Fe magnetic moment as compared to the calculated bulk value of 2.12 mB, unlike our previous work [8]. This is due to the fact that the earlier tight binding method used is an approximate method, dependent on many parameters. Moreover, it does not take into account the electron correlation, which is essential for magnetic calculations. Also it does not allow for optimization of geometry and the relaxation was roughly estimated by recovering the bulk volume. The layer projected magnetic moments for the clean and overlayer systems considered in our work is given in Table 5. The nonrelaxed values are given in brackets. Also included in the table are the results of surface layer magnetic moments for a similar system, namely Fe/W(0 0 1) for comparison from Ref. [10]. From the table we see that relaxation plays an important role in reducing the magnitude of the magnetic moment. The small magnetic moment of 0.75 mB on the clean Mo surface in the sevenlayer Mo slab is completely quenched on relaxation. The nonmagnetic state is found to be the stable state. Moreover, the relaxation of the system reduces the magnetic moment of the Fe overlayer from 3.23 to 2.79 mB in the case of the p(1 1) configuration. This is due to the band splitting reduction in the relaxed case owing to the smaller magnetic energy. The difference between the majority and minority occupations upon relaxation leads to a reduction of the magnetic moment. Furthermore, we notice from the table that the coupling is AF between Fe overlayer and the adjacent Mo layer. In the case of the AF calculations, the magnetic moment of the relaxed Fe/Mo(0 0 1) system is found to be 2.53 mB, which is related to the larger inward relaxation obtained in this case. By comparing the total energies of the FM and the AF configurations, one can see that both solutions are metastable at an interlayer distance of 2.46 a.u. as shown in Fig. 5. However, the AF configuration is the stable state at a relaxed spacing of 2.395 a.u. (Fig. 5). Table 6 shows the angular momentum decomposed integrated charges for the surface Fe atoms in the two channels. One can see clearly from the table that
20 0 -5
-3
-1
-20
1
-40 LDOS SURFACE Fe ATOM 14
-60 -80 -100
Fig. 4. The LDOS for majority and minority electrons of Fe over layer of the Fe/ Mo(0 0 1) system in AF c(2 2) configuration.
Table 5 Magnetic moments, in units of (mB), for the different layers in the p(1 1) configuration
7 ML Mo Fe/Mo (0 0 1)a Fe/Mo (0 0 1)b Fe/W (0 0 1)c
S
S1
S2
C
0.0 (0.75) 2.79 (3.23) 2.53 2.67
0.0 (0.17) 0.25 (0.07)
0.0 (0.03) 0.16 (0.08)
0.0 (0.0) 0.18 (0.17)
The non-relaxed values are given in brackets. a Present LSDA calculations. b AF c(2 2) calculations. c Tunneling microscopy and FLAPW Ref. [10].
the spin moment of the two surface atoms are in opposite directions and equal in magnitude to 2.53 mB. This AF coupling between the surface Fe atoms is also evident in Fig. 4 where, the majority and minority LDOS for the two surface Fe atoms are mirror images. Spin density is determined entirely by the difference in exchange and correlation potential between the two spin states, which is of much shorter range and weaker in intensity than is the spin independent Coulomb potential. Due to the strongly localized nature of the 3d electrons and the large overlap between the core electrons and the valence electrons, the accurate treatment of the non-linear core-valence exchange and correlation interaction is essential for the determination of the magnetic properties of 3d
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2633
Total energy vs interlayer distance
Total energy (Ha)
-200.440 -200.445 -200.450 -200.455 -200.460 -200.465 -200.470 -200.475 -200.480 -200.485 -200.490 -200.495 -200.500 -200.505 -200.510 -200.515 -200.520 -200.525 -200.530 2.30
AF FM
2.32
2.34
2.36
2.38 2.40 2.42 2.44 Interlayer spacing (a.u)
2.46
2.48
2.50
Fig. 5. The total energy vs. interlayer distance.
Table 6 The angular momentum decomposed charges for Fe surface for the AF c(2 2) configuration Surface (Fe) Atom 13
s
p
d
f
Spin up Spin down
0.22 0.24
0.13 0.16
1.75 4.38
0.01 0.01
Surface (Fe) atom 14 Spin up Spin down
0.24 0.22
0.16 0.13
4.38 1.75
0.01 0.01
transition metals. Therefore, the use of LSDA pseudopotential with core corrections has given us good qualitative agreement with the results of a similar system namely the work of Kubetzka et al. [10], studied using GGA, PBE form. Also, we note from Table 1 that, the lattice constant for Mo is overestimated and closer to experimental value giving values of magnetic moment for the bulk and free-standing Fe monolayer that are in agreement with full potential results. The main conclusions are given below:
Appreciable inward relaxations for Mo(0 0 1) surfaces of 11.3%
and 11.7% are obtained for the clean five and seven ML, respectively, is in good agreement with experiment. For p(1 1) magnetic calculations with one Fe overlayer, the top Fe–Mo interlayer spacing is contracted by 15.8% and the magnetic moment of Fe overlayer is enhanced from its theoretically calculated bulk value of 2.12–2.79 mB due to the decrease in coordination and lowering of symmetry at the surface. The Fe overlayer is AF coupled with the Mo interfacial layer, which has a small induced magnetic moment 0.25 mB. For AF c(2 2) the top Fe–Mo interlayer spacing is contracted by 20.6% and the magnetic moment of Fe overlayer is enhanced to 2.53 mB. This is the stable state from total energy calculations. Due to the AF coupling at the surface, the Mo interfacial layer has negligible induced magnetic moment. Magnetism is sensitive to the non-linearities in the XC potential when the core and valence contributions are
separated [25]. The accurate treatment of the non-linearity of the core-valence exchange and correlation interaction is essential for the proper description of the magnetic moment and magnetic energy as shown in Refs. [26,27]. We believe that the present LSDA calculation correctly predicts the relaxation and magnetic properties of the Fe overlayer and gives reliable results since, the ‘pspnc’ TM LSDA pseudopotential used contains core corrections. We think a FP-LAPW calculation and an experimental study of this system in the [0 0 1] direction using scanning tunneling microscopy will confirm our results.
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The structure and magnetism of Fe/Mo(0 0 1) surface: A pseudopotential calculation A.A. Ramanathan, J.M. Khalifeh , B.A. Hamad Department of Physics, University of Jordan, Amman 11942, Jordan
a r t i c l e in fo
abstract
Article history: Received 27 October 2007 Received in revised form 13 May 2008 Available online 27 May 2008
The effect of structure on the magnetism of iron monolayers (MLs) on molybdenum is investigated using the density functional theory (DFT) with norm conserving pseudopotentials and a plane wave basis, under the local spin density approximation (LSDA). Relaxation of 5 and 7 ML of Mo resulted in a contraction of 11.3% and 11.7%, respectively, for the top Mo–Mo interlayer spacing in close agreement with experimental results. In the case of one Fe overlayer, the top Fe–Mo interlayer spacing contracted by 15.8% for a ferromagnetic (FM) p(1 1) and 20.6% for an antiferromagnetic (AF) c(2 2) configuration. The magnetic moment of the surface (Fe) layer is enhanced from its theoretically calculated bulk value. Total energy calculations show that the AF c(2 2) is the stable state with a magnetic moment of 2.53 mB. The surface Fe atoms are AF coupled with each other and with the Mo layers below, showing layered AF coupling. The present study demonstrates the reliability of the pseudopotential approach under LSDA with core corrections included to the calculation of magnetic properties of combined transition metal systems. & 2008 Elsevier B.V. All rights reserved.
Keywords: Magnetism Fe Mo DFT LSDA GGA Pseudopotential XC
1. Introduction Structural and magnetic properties of Fe thin films on noble metals have been widely studied [1–3] due to their enhanced surface magnetic moment. Perpendicular surface anisotropy of magnetism is observed in all the above studies for a few monolayer of Fe. In the above studies different experimental techniques are used to show that for 1–3 monolayers (MLs) of Fe on different noble metals like Ag, Au and in addition on Cu and W there is a ferromagnetic order. Recently, Fe/W and Fe/Mo received great attention due to their thermo stability [4,5] and the pseudomorphic layer by layer growth [6,7] of Fe films up to 2 ML on these substrates. Magnetism is a very complex issue and is influenced by various factors like type of unit cell, lattice spacing, type and number of nearest neighbors, orientation, thickness of film and relaxation of surfaces, etc. The combination of these factors sometimes gives unexpected and contradictory results. Therefore, even though we have studied this system before [8] by the semi-empirical, tight binding recursion method in the Hartree–Fock approximation, it is not futile if the same system is studied again especially since the present study is an ab-initio calculation. In our previous work for
Corresponding author.
E-mail addresses: [email protected] (J.M. Khalifeh), [email protected] (B.A. Hamad). 0304-8853/$ - see front matter & 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2008.05.035
the (0 0 1) orientation we [8] had an unexpected huge reduction of the Fe surface layer magnetic moment from its bulk value to 1.29 mB, which is in qualitative agreement with the results of Mirbt et al. [9] for Fe/Mo/Fe sandwiches. Their paper deals with the ideal Fe–Mo interface. It can be seen from the paper that the magnetic moment of Fe is very sensitive to the value of lattice constant used in the calculations. Moreover, the magnetic properties at the interface do not necessarily represent those at the surface. The recent finding by Kubetzka et al. [10] of an enhancement of surface Fe magnetic moment from the bulk value for a similar system, namely Fe/W(0 0 1) motivated us to undertake a second study of the system for the (0 0 1) orientation, from first principles. The present study of Fe/Mo in the (0 0 1) orientation is also motivated by the fact that, to our knowledge, there has been no first-principles study of this system. Moreover, no pseudopotential calculation has been attempted for this system in any orientation. This paper is organized as follows: Section 2 outlines the computational method, Section 3 gives the results and discussion under two sub-headings, structural and magnetic.
2. Method of calculations All calculations are based on the density functional theory (DFT) as implemented in the ABINIT software package [11,12]. The pseudopotentials used are Trouiller–Martins (TM) with core corrections for local spin density approximation (LSDA) exchange
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correlations [13]. In order to obtain the theoretical lattice constants for Mo and Fe, convergence tests with respect to the cut-off energy and k-points are performed. Using the converged values, the lattice constants are optimized. Convergence for the self-consistent field (SCF) iterations is achieved when the total energy differences between two consecutive iterations is lees than 1 106 Ha. [14]. Both Mo and Fe have a Bcc ground state lattice structure. According to our calculations, the mismatch in the lattice constants of these two metals is about 13% under LSDA (the experimental mismatch is about 9%). This large mismatch in the case of LSDA is related to the typical behavior of this functional to underestimate the lattice constant of the 3d transition metals. This mismatch is a driving force for structural relaxation and possible magnetic re-orientation due to the magnetoelastic effect. The supercell for the slab calculation consists of seven atomic layers and seven layers of vacuum (54%). The amount of vacuum and the number of atomic layers are chosen carefully after making convergence tests for the number of vacuum and atomic layers with respect to the bulk density of states (DOS). This amount of vacuum is required to minimize any Coulomb and exchange interactions between the layers. Repeated slab geometries consisting of five Mo substrate layers and one Fe layer on each surface were employed in all calculations. The unit cell consists of one atom per layer for the (1 1) cell and two atoms per layer for the (2 2). Spin polarized calculations are applied for Fe/Mo(0 0 1). Spin–orbit coupling has not been taken into account; therefore, the magnetic direction is taken arbitrary. Because of the twodimensional (in plane) translational invariance in the supercell, there are no in-plane components of force and therefore no inplane relaxation. There are only vertical components of forces. All atoms are allowed to relax simultaneously and therefore the true minimum is reached. The vertical lattice spacings are allowed to relax simultaneously according to the sign and magnitude of forces. The structure is optimized when the total maximum force is o1 mRy/a.u. The SCF iterations have the same convergence criteria as mentioned before. For structural relaxation we used the Broyden–Fletcher–Goldfarb–Shanno (BFGS) minimization algorithm modified to take into account the total energy as well as the gradients [15]. The magnetic moments for the different atomic layers are calculated as the differences between the spin-up and spin-down integrated local density of states (LDOS). The tetrahedron method is used to obtain the angular momentum projected LDOS [16]. The integration is done over atomic spheres of radii 2.4 and 2.6 a.u. for Fe and Mo, respectively.
3. Results and discussion 3.1. Structural results The optimized lattice constants obtained for Mo and Fe along with details of kinetic energy cut-off and k-point mesh used is given in Table 1. First, we started the calculations for Mo clean
surface. We performed the calculations for five and seven atomic layer slabs. The two-dimensional lattice constant for the slab is taken to be the calculated lattice constant of Mo, namely 6.035 a.u. Structural relaxation is then performed with a force tolerance of 5 104 Ha/a.u. (1 mRy/a.u). All the atoms are allowed to relax simultaneously but only in the vertical direction. The structures are optimized when the force on each atom is o1 mRy/a.u. The direction and degree of relaxation for the vertical interlayer spacing depend on the magnitude and sign of the forces present. The optimized interlayer spacing for the Mo five and seven layer slabs are given in Table 2. The LEED experimental result [17] is also listed in the table. The top Mo–Mo interlayer spacing is contracted in both the five and seven ML by 11.3% and 11.7%, respectively. This is in good agreement with LEED experimental results of 11.5%, which has been obtained by a comparison of the experimental intensity spectra with the calculated intensity spectra using the layer-Korringa–Kohn–Rostoker method. Our results are also supported by the results obtained by Noguera et al. [18]. Photoemission experiments applied to the (0 0 1) face of Mo reveal two sharp peaks at 0.3 and 3 eV below the Fermi level. Noguera et al. [18] identified the upper peak due to normal emission as a combination of a surface resonance and a d band edge around the G point of the two-dimensional Brillouin zone. They had to include a 10–13% inward relaxation of the surface layer in the theoretical calculation in order to place the energy of this resonance at the energy of the experimental peak. Next the spin-polarized relaxation of the Fe/Mo(0 0 1) system is done. In Table 3 we present the relaxation for the case of 1 ML of Fe/Mo(0 0 1). The surface Fe–Mo contraction is 15.8% and 20.6% for the FM p(1 1) and the AF c(2 2) configurations, respectively. This is in good agreement with the results of Kubetzka et al. [10] for a similar system, namely Fe/W(0 0 1), which is also included in the table for comparison. They performed a first-principles FPLAPW study of Fe/W(0 0 1) using a nine layer slab and a GGA (PBE) exchange correlation within the FLEUR code. To find the equilibrium top interlayer spacing they calculated the total energy of the system as a function of the interlayer distance d between the Fe ML and the W layer for FM and AF configurations. They found the AF state to have the lowest energy at the equilibrium interlayer distance of 2.58 a.u. corresponding to an inward relaxation of 13.9% as quoted explicitly by them in Ref. [19]. Fig. 1 shows the charge density for the Fe/Mo(0 0 1) system interpolated along the [0 0 1] direction (z-axis). We notice from the figure how the charge density peaks at the surface, showing a marked change from that of the other layers. Also we see that there is a charge leak into the vacuum. This outward flow of charge into the vacuum region serves to screen the surface discontinuity and gives rise to a dipole layer of charge. The charge redistribution on relaxation is also clearly seen. The large values of inward relaxations obtained for our system can be explained by the following facts: (i) The bcc (0 0 1) surface is a relatively rough open surface and the Smoluchowski smoothing is more pronounced for rougher surfaces [20]. As pointed out by Smoluchowski, the difference
Table 1 Details of lattice constants Element
Pseudo potential
Ecut (Ha)
Grid
Lattice constant (Bohr)
Magnetic moment (mB) Bulk
Molybdenum Iron
42mo_pspnca 26fe.pspnca
40 70
10 10 10 12 12 12
6.0347 (1.3%)b 5.2432 (3%)b
2.1167 (3.3)c
a b c
Core corrections included. Difference with respect to experimental value. Magnetic moment of free-standing ML.
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Table 2 Interlayer spacing Dij for 5 and 7 Mo layers for clean Mo(0 0 1) surface
5 ML Moa 7 ML Moa Mo (Expt)b
D12
D23
D34
2.68 (11.3%) 2.66 (11.7%) (11.5%)
3.1 (2.8%) 3.12 (3.4%)
3.00 (0.57%)
The values in brackets are the percentage change in the interlayer spacings. a Present LDA calculations. b Ref. [17].
Table 4 The angular momentum decomposed layer charges for the p(1 1) configuration Layers
s
S (Fe) S-1 (Mo) S-2 (Mo) C (Mo)
0.46 0.37 0.38 0.38
d
0.29 0.35 0.37 0.37
(0.24) (0.33) (0.37) (0.37)
Total charge/layer
6.05 3.87 3.86 3.88
(5.85) (3.90) (3.86) (3.85)
6.82 4.63 4.65 4.67
(6.51) (4.58) (4.61) (4.60)
NON-MAGNETIC LDOS Fe/Mo(001) Fe Overlayer D23
D34
4
2.54 (15.8%)
3.00 (0.4%)
2
2.39 (20.6%)
3.07 (1.97%)
2.58 (13.9%)
The values in brackets are the percentage change in the interlayer spacings. a FM p(1 1)calculations. b AF c(2 2) calculations. c Tunneling microscopy and FLAPW, Ref. [10].
TOTAL CHARGE DENSITY (001)direction 70 Charge density (electrons/Bohr3)x10-2
(0.42) (0.35) (0.38) (0.38)
6
LDOS(States/eV/atom)x102
1ML Fe/Moa 3.04 (0.9%) 1ML Fe/Mob 3.03 (0.57%) 1ML Fe/W (0 0 1)c
p
The non-relaxed values are given in brackets.
Table 3 Interlayer spacing Dij for Fe/Mo (0 0 1) sLaboratory D12
2631
EF d=6a.u.
0 6
-2
0 EF
4
2 d=Non-Relaxed spacing
2 0 -2
0
2
6 EF
4
d=Relaxed spacing
2
60
0 -2
50
0
2
Energy E-EF(eV) Relaxed Non-Relaxed Relaxed atom positions Non-Relaxed atom positions
40 30 20 10 0 -1
1
3
5
7
9
11
13 15 17 19 21 23 25
Z coordinate (Bohr) Fig. 1. The total charge density along the [0 0 1] direction for the relaxed and nonrelaxed Fe/Mo(0 0 1) systems.
between non-relaxed and relaxed charge densities causes an additional electrostatic potential and gives rise to an inward electrostatic force on the top layer. (ii) For transition metals there is an additional contribution to surface relaxation from d electrons. Owing to the localized nature of the 3d electrons, a very small amount of charge transfer in or out of the surface amounts to a fairly large surface dipole moment. The additional inward force is proportional to the strength of the d bonds. The largest inward force would be found for a half-filled d band, since in this case all the bonding states and none of the antibonding states are filled [20].
The angular momentum decomposed layer charges are presented in Table 4, again the non-relaxed values are given in
Fig. 2. Non-magnetic LDOS for the Fe overlayer for three different Fe–Mo interlayer distances d12.
brackets. In the table, the angular momentum decomposed layer charge for the Fe overlayer shows a marked increase in s, p and d electrons on relaxation. This is as expected due to the d band Fe–Mo hybridization. Furthermore, the hybridization between Fe and Mo d bands can be understood qualitatively by plotting the non-magnetic LDOS for three different interlayer spacing ‘d’, Fig. 2. When d ¼ 6 a.u, there is hardly any interaction of the Fe overlayer with the Mo surface below. The LDOS has a strong peak at EF due to the fact that there is no hybridization and FM is favored. Next, we see that the LDOS has broadened due to the beginning of hybridization, when d is reduced to 3.068 a.u (the non-relaxed interlayer spacing). Finally on relaxation, when d is equal to the relaxed interlayer spacing, there is maximum hybridization and the d band has broadened considerably and resembles the Mo interface LDOS. The effect of the relaxation on the Fe overlayer magnetic moment will be discussed in the next section. 3.2. Magnetic results Due to the reduction in coordination number at the surface, the magnetic moments of 3d transition-metal surfaces are generally enhanced. The magnetic moment is particularly enhanced for bcc Fe(0 0 1) where the majority of spin band is not already saturated in the bulk. The values for surface Fe(0 0 1) magnetic moment found by various studies [21–24] range from 2.97 to 3.01 mB as compared to the experimental bulk value of 2.2 mB. This enhancement is attributed to narrowing of the d band width, shifts of peak
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100
Density of States (electron states/atom/ev)
LDOS Fe/Mo(001) Fe Overlayer
Spin up
-5
-4 -3 Spin Down
180 160 140 120 100 80 60 40 20 0 -2 -1 -20 0 -40 -60 -80 ENERGY E-EF (ev)
80 LDOS SURFACE Fe ATOM 13
60 40 20
-5
-3
-1
0
1
-20 -40
1
2
Majority Minority
-60 -80 -100 100 80
Fig. 3. The LDOS for majority and minority electrons of Fe over layer of the Fe/ Mo(0 0 1) system in p(1 1) configuration.
Majority Minority
60 40
positions with EF lying in the peak of the minority LDOS and surface states [23]. The effect of substrate on the Fe surface magnetism is very evident in our work. Since Mo is iso-electric to W, one expects to find similar behavior for our system Fe/Mo(0 0 1) as in Fe/W(0 0 1), Ref. [10] and it is indeed the case. The LDOS for the majority and minority electrons of the Fe overlayer in the case of the FM and AF configurations are shown in Figs. 3 and 4. From the figures we see that the minority DOS peak shifts position towards the Fermi level, which results in an enhancement of the Fe magnetic moment as compared to the calculated bulk value of 2.12 mB, unlike our previous work [8]. This is due to the fact that the earlier tight binding method used is an approximate method, dependent on many parameters. Moreover, it does not take into account the electron correlation, which is essential for magnetic calculations. Also it does not allow for optimization of geometry and the relaxation was roughly estimated by recovering the bulk volume. The layer projected magnetic moments for the clean and overlayer systems considered in our work is given in Table 5. The nonrelaxed values are given in brackets. Also included in the table are the results of surface layer magnetic moments for a similar system, namely Fe/W(0 0 1) for comparison from Ref. [10]. From the table we see that relaxation plays an important role in reducing the magnitude of the magnetic moment. The small magnetic moment of 0.75 mB on the clean Mo surface in the sevenlayer Mo slab is completely quenched on relaxation. The nonmagnetic state is found to be the stable state. Moreover, the relaxation of the system reduces the magnetic moment of the Fe overlayer from 3.23 to 2.79 mB in the case of the p(1 1) configuration. This is due to the band splitting reduction in the relaxed case owing to the smaller magnetic energy. The difference between the majority and minority occupations upon relaxation leads to a reduction of the magnetic moment. Furthermore, we notice from the table that the coupling is AF between Fe overlayer and the adjacent Mo layer. In the case of the AF calculations, the magnetic moment of the relaxed Fe/Mo(0 0 1) system is found to be 2.53 mB, which is related to the larger inward relaxation obtained in this case. By comparing the total energies of the FM and the AF configurations, one can see that both solutions are metastable at an interlayer distance of 2.46 a.u. as shown in Fig. 5. However, the AF configuration is the stable state at a relaxed spacing of 2.395 a.u. (Fig. 5). Table 6 shows the angular momentum decomposed integrated charges for the surface Fe atoms in the two channels. One can see clearly from the table that
20 0 -5
-3
-1
-20
1
-40 LDOS SURFACE Fe ATOM 14
-60 -80 -100
Fig. 4. The LDOS for majority and minority electrons of Fe over layer of the Fe/ Mo(0 0 1) system in AF c(2 2) configuration.
Table 5 Magnetic moments, in units of (mB), for the different layers in the p(1 1) configuration
7 ML Mo Fe/Mo (0 0 1)a Fe/Mo (0 0 1)b Fe/W (0 0 1)c
S
S1
S2
C
0.0 (0.75) 2.79 (3.23) 2.53 2.67
0.0 (0.17) 0.25 (0.07)
0.0 (0.03) 0.16 (0.08)
0.0 (0.0) 0.18 (0.17)
The non-relaxed values are given in brackets. a Present LSDA calculations. b AF c(2 2) calculations. c Tunneling microscopy and FLAPW Ref. [10].
the spin moment of the two surface atoms are in opposite directions and equal in magnitude to 2.53 mB. This AF coupling between the surface Fe atoms is also evident in Fig. 4 where, the majority and minority LDOS for the two surface Fe atoms are mirror images. Spin density is determined entirely by the difference in exchange and correlation potential between the two spin states, which is of much shorter range and weaker in intensity than is the spin independent Coulomb potential. Due to the strongly localized nature of the 3d electrons and the large overlap between the core electrons and the valence electrons, the accurate treatment of the non-linear core-valence exchange and correlation interaction is essential for the determination of the magnetic properties of 3d
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2633
Total energy vs interlayer distance
Total energy (Ha)
-200.440 -200.445 -200.450 -200.455 -200.460 -200.465 -200.470 -200.475 -200.480 -200.485 -200.490 -200.495 -200.500 -200.505 -200.510 -200.515 -200.520 -200.525 -200.530 2.30
AF FM
2.32
2.34
2.36
2.38 2.40 2.42 2.44 Interlayer spacing (a.u)
2.46
2.48
2.50
Fig. 5. The total energy vs. interlayer distance.
Table 6 The angular momentum decomposed charges for Fe surface for the AF c(2 2) configuration Surface (Fe) Atom 13
s
p
d
f
Spin up Spin down
0.22 0.24
0.13 0.16
1.75 4.38
0.01 0.01
Surface (Fe) atom 14 Spin up Spin down
0.24 0.22
0.16 0.13
4.38 1.75
0.01 0.01
transition metals. Therefore, the use of LSDA pseudopotential with core corrections has given us good qualitative agreement with the results of a similar system namely the work of Kubetzka et al. [10], studied using GGA, PBE form. Also, we note from Table 1 that, the lattice constant for Mo is overestimated and closer to experimental value giving values of magnetic moment for the bulk and free-standing Fe monolayer that are in agreement with full potential results. The main conclusions are given below:
Appreciable inward relaxations for Mo(0 0 1) surfaces of 11.3%
and 11.7% are obtained for the clean five and seven ML, respectively, is in good agreement with experiment. For p(1 1) magnetic calculations with one Fe overlayer, the top Fe–Mo interlayer spacing is contracted by 15.8% and the magnetic moment of Fe overlayer is enhanced from its theoretically calculated bulk value of 2.12–2.79 mB due to the decrease in coordination and lowering of symmetry at the surface. The Fe overlayer is AF coupled with the Mo interfacial layer, which has a small induced magnetic moment 0.25 mB. For AF c(2 2) the top Fe–Mo interlayer spacing is contracted by 20.6% and the magnetic moment of Fe overlayer is enhanced to 2.53 mB. This is the stable state from total energy calculations. Due to the AF coupling at the surface, the Mo interfacial layer has negligible induced magnetic moment. Magnetism is sensitive to the non-linearities in the XC potential when the core and valence contributions are
separated [25]. The accurate treatment of the non-linearity of the core-valence exchange and correlation interaction is essential for the proper description of the magnetic moment and magnetic energy as shown in Refs. [26,27]. We believe that the present LSDA calculation correctly predicts the relaxation and magnetic properties of the Fe overlayer and gives reliable results since, the ‘pspnc’ TM LSDA pseudopotential used contains core corrections. We think a FP-LAPW calculation and an experimental study of this system in the [0 0 1] direction using scanning tunneling microscopy will confirm our results.
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