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Possible route to d0 magnetism in α-PbO compound
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Possible route to d0 magnetism in α-pbo compound J. Berashevich, A. Reznik
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S0022-3697(14)00078-X http://dx.doi.org/10.1016/j.jpcs.2014.04.002 PCS7292
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Journal of Physics and Chemistry of Solids
Received date: 20 November 2013 Revised date: 13 February 2014 Accepted date: 3 April 2014 Cite this article as: J. Berashevich, A. Reznik, Possible route to d0 magnetism in α-pbo compound, Journal of Physics and Chemistry of Solids, http://dx.doi.org/ 10.1016/j.jpcs.2014.04.002 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Possible route to d0 magnetism in α-PbO compound J. Berashevich
Thunder Bay Regional Research Institute, 290 Munro St., Thunder Bay, ON, P7A 7T1, Canada and Max Planck Institute for the Physics of Complex Systems, Nthnitzer Str. 38, 01187 Dresden, Germany
A. Reznik
Thunder Bay Regional Research Institute, 290 Munro St., Thunder Bay, ON, P7A 7T1, Canada and Department of Physics, Lakehead University, 955 Oliver Road, Thunder Bay, ON, P7B 5E1, Canada With help of the first-principle methods we investigate a possibility to induce d0 magnetism in the α-PbO compound. The Pbi interstitial defect is found to gain the magnetic properties upon incorporation into α-PbO crystal structure. The Pbi interstitial generates the p localized state with two electrons on-site whose spin alignment is governed by the Hund’s rule giving a rise to formation of the local magnetic moment 2μB . The magnetic coupling between defects is a function of occupancy (0) of the defect states. For two defects in their zero charge state Pbi the antiferromagnetic (AFM) (1+) show already the coupling is more stable. The Pb interstitial defects in the charge state Pbi ferromagnetic (FM) ground state. The magnetic coupling is found to be driven by the long-range order interactions. The critical limit on the defect concentration required to establish the magnetic percolation above room temperature is about 1020 cm−3 which may become accessible when the Pbi defects are formed on the surface of a single crystal. The substitution of the Pb interstitial with the non-magnetic elements of s2 px outer shell (the local magnetic moment is defined by 1 ≤ x ≤ 3) is revealed to be a perspective way to tune the magnetic behavior. I.
INTRODUCTION
Spintronics has recently emerged as a widely successful technology that exploits the principles of magnetism. Classical metal ferromagnets do not meet all requirements of spintronics because semiconducting behavior is an essential condition for operation of the electronic devices. It has led to development of a new branch of research that is focused on semiconductors exhibiting magnetism at room temperature [1]. The first success was in so-called ”diluted magnetic semiconductors” which are the conventional semiconductors doped with magnetic ions. Magnetic ions with partially filled 3d or 4f shells allow the spin alignment of on-site electrons [2–5]. Interest in ferromagnetic semiconductors was further generated when magnetism was demonstrated in materials composed of ’non-magnetic’ elements (i.e. without unpaired electrons) [6–9]. The origin of so-called d0 magnetism was proposed to be due to localized sp states. The interacting vacancies create a network of unpaired electrons, which with sufficient long-range order effect exhibit the magnetic coupling [9]. Here we show that d0 magnetism can be induced by the 2D network of the interstitial defects Pbi (Pbi :6s2 6p2 outer shell) incorporated between layers of the tetragonal lead oxide α-PbO. The Pbi defect forms the localized state occupied by two unpaired electrons Pbi :6p2 . The on-site ordering of the localized electrons Pbi :6p2 obeys the first Hund’s rule that results in formation of the local magnetic moment 2.0 μB . In our model compound, the triplet state is found to be stable as defined by the spin polarization energy Epol =0.235 eV. The experimental observation of the magnetic signal from the α-PbO samples at room temperature [10] verifies formation of
the local magnetic moment. Therefore, otherwise than the magnetic moment occurs due to the partially filled p shell instead of the 3d or 4f shells, Pbi works exactly as the magnetic impurity and it provides several advantages over a doping with magnetic ions. Thus, the Pbi state demonstrates the high on-site spin stability due to localized Pbi :6p2 electrons, the extended tails of their wavefunctions induce the long-range electron spin ordering. In addition, the interstitial defect is almost non-invasive to the electronic and crystal structures of the host (the host Pb:6s2 electrons are only involved in the defect state formation). Moreover, we found that the magnetic behavior of the interacting defects can be tuned by changing the charge state of Pbi or alternatively, through substitution with impurities of the specific s2 p1≤x≤3 outer shell configuration: x = 2 works for 2.0 μB , while x = 1 or x = 3 for 1.0 μB . We suggest that from technological point of view, the layered crystal structure of α-PbO promises the superior advantages in establishing the magnetic percolation and more importantly, a practical way for its control: i) the α-PbO compound grows in polycrystalline form such as the large surface area offers enormous potential for doping (the formation energy of the Pbi interstitial defect drastically decreases on the surface of a single crystal [18]); ii) a solubility of the magnetic centers and their properties can be tuned applying the substitutional doping; iii) any compounds of the tetragonal PbO type can be used as a semiconductor matrix. II.
METHODS
In our study we applied the generalized gradient approximation (GGA) with the PBE parametrization [12]
2 provided by WIEN2k package for the density functional theory (DFT) calculations [13]. The supercell approach with sufficiently large supercell of 108-atom size (3×3×3 array of the primitive unit cells) for single defect and of 160-atom size (5×4×2) for two interacting defects has been used. The Pb:5p, 5d, 6s, 6p and O:2s, 2p electrons have been treated as the valence electrons (the energy cut-off was -8 Ry). The geometry optimization procedure for supercell containing the Pbi defect was performed based on minimization of the total energies and forces [14]: the energy convergence limit was set to 0.0001 Ry while the residual forces did not exceed 0.5 mRyd/Bohr. For integration of the Brillouin-zone, the MonkhorstPack scheme using a (5×5×4) k-mesh was applied. The product of the atomic sphere radius and plane-wave cutoff in k-space RKmax was equal to 7. The localization of the defect wavefunctions has been additionally examined with Hartree-Fock (HF) applied directly to the unpaired electrons that allowed to preserve accuracy provided by DFT at the same time correcting the unpaired electrons self-interaction [15]. Moreover, since the effect of the spin-orbit interactions can not be neglected for the heavy atom as Pb, influence of the spin-orbit coupling (so) on the energetic position of the defect states inside the band gap is investigated with (GGA+so). The tetragonal lead oxide α-PbO possesses the layered structure leading to formation of platelets upon compound growth and each platelets can be considered as a single crystal. The layers within a single crystal are held together by the interlayer interaction of the Pb:6s2 electrons [16]. In terms of the electronic properties, the interlayer interactions generate a dipping of the conduction band at M∗ point [17] thus inducing an effect of the band gap shrinkage. GGA tends to overestimate the interlayer separation in the layered structures, but it is also well known to underestimate the band gap size. For α-PbO system, it results in a compensation effect [18]: for the lattice parameters optimized with GGA the band gap is only slightly underestimated giving 1.8 eV against the experimental value of 1.9 eV [19] (application of the experimental lattice constants shrinks the band gap by 0.22 eV). The correct band gap size is important to place the defect states inside the band gap: when the band gap size is underestimated, the hole-carrying defect state occurs closer to the host conduction band thus causing the longer defect tails. In order to avoid the spurious long-range order interactions [5], all band structure calculations have been performed with the GGA optimized lattice parameters. However, the formation energy of the interstitial defects have been defined applying the experimental lattice constants [18].
III.
THE DEFECT-INDUCED LOCAL MAGNETIC MOMENT
The interstitial defects have never been considered in origin of d0 magnetism because they do not belong to a
class of the most common defects in crystalline materials: the interstitials induce a significant perturbation of the host lattice that results in their rather large formation energy [20]. The layered structure of α-PbO is different as it allows the foreign atoms to squeeze between layers causing a minimal lattice deformation to the host immediate neighborhood (see Fig. 1 (a)). As a result, the formation energy of Pbi appearing between layers is not too high. For the Pb-rich/vacuum conditions, the neutral charge state is characterized by the formation energy of 1.23 eV (for details on calculation of the formation energy of defects in the α-PbO crystal structure see [18]). The band diagram and the density of states (DOS) separatly for the up-spin and down-spin states are presented in Fig. 1 (b) and (c) for our model system containing Pbi (the band diagram of the defect-free system is shown in Fig. 1 (b) as well). The Pb interstitial makes a bond with Pb atom from the host. The length of the Pbi -Pb bond is fairly short, 2.9 ˚ A, that indicates a double bond formation. With respect to modification induced to the host, an out-of-plane displacement of 0.54 ˚ A occurs for the Pb atom involved in bonding. The antibonding orbitals of the Pbi -Pb bond are presented by the 2u and 2d bands in Fig. 1 (b), while the bonding states appear deeper in the valence band (see ’antibonding’ notation in Fig. 1 (c)). The on-site defect states generate the 1u and 1d bands. The 1u state is filled with two electrons and appears just above the midgap at 0.99 eV + EV , where EV is a top of the valence band. In contrast, the spin-down 1d state is empty and in such, it is shifted to the conduction band edge EC . In order to understand an asymmetry in filling of the spin-up and spin-down states, a mechanism of bonding of Pbi with the host lattice has to be explored. The integration of the Pb interstitial into the crystal lattice requires an excitation of its Pbi :6s2 6p2 ground state to Pbi :6s1 6p3 . The Pbi :6s1 and Pbi :6p1z electrons participate in formation of the Pbi -Pb bond with the host and in such, the host shares its Pb:6s2 electrons (see Fig. 1 (c)). Since the Pb:6s2 electrons from the host participate in bonding with the defect, the interlayer interaction generated by overlap of the Pb:6s2 electrons is disturbed. For the supercell size 3×3×3 employed to calculate a single defect, this effect is rather significant. Corresponding disturbance can be seen in a dispersion of the 2u and 2d bands between Γ∗ and M∗ points. Moreover, a behavior of the host bands near the conduction band top is also altered through a suppression of the band dipping at the M∗ point being also controlled by the interlayer interactions of the Pb:6s2 electrons [17]. The similar alteration in the band behavior upon substitutional doping accompanied by depletion of the electron and hole density is observed for another compound possessing the αPbO crystal structure [21]. The depletion effect is argued to appear due to the enhanced hopping via substituted sites thus contributing in larger splitting of the conduction and the valence bands (the α-PbO crystal structure is well recognized in superconductivity topic). The 1u and 1d bands are induced by a state localized
3
FIG. 2: (a) The positions of the 1u and 1d orbitals within the band gap calculated with the different methods. (b) The electron density map for the triplet state (GGA) at the Pbi site is plotted for isovalues of ±0.005 e/˚ A3 with help of Xcrysden [23]. The density map demonstrates the spin density calculated for the energy range (1.0 eV+EV )±0.15 eV.
FIG. 1: (a) The Pb interstitial in the layered crystal structure of α-PbO. (b) The band diagram for a defect-free single crystal (green dotted line) and system hosting Pbi : 1u and 1d are for the on-site defect states, while 2u and 2d are the antibonding orbitals of the Pbi -Pb bond. (c) Total and partial density of states for Pbi and host Pb atoms (’top’ is referred to Pb atom forming the Pbi -Pb bond, while ’bottom’ for Pb atoms in the bottom layer). The ’bonding’ and ’antibonding’ in the B and C panels should be referred to the Pbi -Pb bond.
at the defect site, namely by two Pbi :6px+y electrons left on-site after bonding. The Hund’s rule dictates both unpaired Pbi :6px+y electrons to occupy the 1u spin-up band. The empty 1d spin-down band is pushed up to EC that causes a considerably large spin-exchange splitting between 1u and 1d of order 0.52 eV as presented in Fig. 2 (a). The two localized Pbi :6px+y electrons occu-
pying the 1u spin-up state are responsible for the ground triplet state followed by formation of the local magnetic moment 2.0 μB . The spin-polarization energy defined as the energy difference between the spin-unpolarized and spin-polarized states [22] is found to be Epol =0.235 eV, i.e. the triplet state is expected to be stable at room temperature that is confirmed experimentally [10]. The Pbi :6s2 6p2 electronic configuration is clearly a key for the Pb interstitial to act as a magnetic impurity: the Pbi :6s2 electrons are used for attachment to the host while partially filled Pbi :6p2 shell contributes in developing of the local magnetic moment. Therefore, other chemical elements possessing the s2 p1≤x≤3 outer shell can be used as the magnetic impurity instead of Pbi . The impurities of x = 1 or x = 3 configurations have been found to induce the local magnetic moments of 1 μB , while x = 2 works for 2μB [11]. Because GGA approach is well known to delocalize the wavefunction of the hole-like defect state due to selfinteraction of the unpaired electrons [15], the limitations and reliability of this method has to be examined. As expected, an application of the HF approach to the unpaired electrons has induced the stronger on-site localization of the defect wavefunction emerging in stabilization of the triplet state (Epol =0.490 eV). Thus, an increase in the splitting of the 1u and 1d energies up to 1.12 eV is observed (see Fig. 2 (a)). However, an application of the experimental lattice parameters has opposite effect on the splitting which is suppressed because of stronger hybridization of the impurity state with the host lattice due to the interlayer distance reduction. Therefore, when HF is combined with the experimental lattice parameters, they compensate each other such as the splitting of the 1u and 1d energies is reduced to 0.68 eV. This value is very close to that found with GGA originally (see Fig. 2 (a)) that indicates its reliability when the optimized lattice
4 parameters are used. Moreover, for the heavy elements like Pb atom, the spin-orbit coupling plays an important role and, therefore its effect on the 1u and 1d energies has to be examined. The GGA+so approach has been applied directly to the Pbi :6px+y electrons. The spin-orbit coupling is found to break a degeneracy of the 1u and 1d states (see GGA+so in Fig. 2 (a)). It reduces the splitting of the occupied 1u and unoccupied 1d levels to 0.2 eV, i.e. the stability of the triplet state decreases while a shift of 1u state towards the conduction band contributes in enlargement of the defect tails. Since the magnetic signal is observed at room temperature experimentally [10], we believe that effect of the spin-orbit coupling on the on-site stability is not crucial. IV.
THE LONG-RANGE ORDER INTERACTIONS
Formation of the local magnetic moment is the first step in achieving the magnetism. For the magnetic percolation to manifest at room temperature, the effective interactions of the Pbi :6p2x+y electrons between the defect sites is essential. Our studies on formation of the defect tails has revealed a duality in their behavior. Since occupied band 1u is located deep inside the band gap, the Pbi :6p2x+y electrons show localization on the defect site thus allowing to form the stable local magnetic moment 2μB (see the electron density map in Fig. 2 (b)). However, the long defect tails are also formed as a result of the bonding/hybridization of the Pbi :6s1 and Pbi :6p1z electrons with the host Pb:6s and Pb:6p, O:2p electrons (see partial DOS in Fig. 1 (c)). In order to establish the spin ordering of the localized Pbi :6p2x+y electrons between the defect states, FM or AFM, the defects have been considered on the nearestneighbor sites. For these calculations, the spins of the 6p2x+y electrons have been always aligned on-site, while the inter-defect spin ordering has been changed from antito ferromagnetic. The 6p2x+y state is exactly half filled and, therefore, for two interacting defects the virtual hopping is allowed only when their spins are antiferromagnetically ordered. It contributes in a lowering of the total energy in the AFM state which becomes the ground state [5]. The ground state stability is defined by the energy difference between the AFM and FM states as EM = EAFM − EFM (a negative sign refers to the AFM ground state). For two Pb interstitials placed at a distance 4.1 ˚ A the magnetic energy difference is found to be EM =-0.96 eV. To reach the FM arrangement to be an essential requirement for spintronics, the virtual hopping between defects has to be allowed in the FM state. It becomes possible when two interacting defects appear in (1+) (0) their charged states Pbi , i.e. each Pbi has to drop one electron. The local magnetic moment induced by the (1+) (1+) Pbi state is 1 μB . The interaction of two Pbi states placed at a distance 4.1 ˚ A gives the magnetic energy dif-
ference EM =1.46 eV. The most common way to change the charge state of defect is to consider the electron exchange with another defect. The growth conditions required to favor a formation of the Pbi defect is Pb-rich/O-poor limit (oxygen deficiency). At these growth conditions, the O vacancy (VO ) is another defect possessing the low formation energy. For the defects in a bulk, the formation energy of the O vacancy in its neutral charge state is 0.85 eV (the formation energy of the Pbi defect at the same conditions is 1.23 eV). Therefore, we would examine an interaction of Pbi with the VO defect with respect to a stability of its local magnetic moment and to explore a possibility to (1+) switch its charge state to Pbi . In the neutral charge state, VO is occupied by two electrons possessing the strong localization on the vacancy site [18]. Therefore, VO can be an electron donor. Since the charge exchange is allowed between Pbi and VO , they would act as the compensating centers to each other. It is (0) found that for the VO and Pbi defects appearing in close proximity to each other (the defect separation is 2.27 ˚ A), the strong interactions between defects accompanied by their hybridization destroy the local magnetic moment at Pbi as shown in Fig. 3 (Pbi in a singlet state induces the defect state at EV +1.22 eV in the band gap). The Pbi :6px+y electrons from the interstitial and the Pb:6s and Pb:6p and O:2p electrons from the host lattice, all have been found to contribute equally in formation of the Pbi -VO pair. The interaction between defects manifest itself in lowering of the formation energy of the Pbi -VO pair by 0.84 eV in comparison to the non-interacting defects. Provided the defects are separated by the sufficient distance of 6.2 ˚ A, the local magnetic moment at Pbi site recovers. However, because the interaction between Pbi with VO does not vanish completely, the spin polarization energy of this state is found to be considerably low as defined by Epol =0.06 eV. Therefore, the interaction of the Pbi and VO defects is not viable way to convert the (1+) Pb interstitial into the Pbi charge state. Moreover, in order to suppress the destabilizing effect of the VO vacancies on formation of the local magnetic moment at Pbi site, a post-growth annealing treatment in atmosphere of oxygen would be required. The strong spin-orbit interactions at the defect site of(0) fers another way to convert Pbi into the charge state (1+) Pbi . Since the spin-orbit coupling breaks a degeneracy of the 1u orbitals (see GGA+so in Fig. 2(a)), the selective optical excitation of the electrons localized at EV +1.11 eV can be implemented (α-PbO is known as a good photoconductor [24] for which a mechanism of the optical excitation would be plausible). After selective excitation, the interactions between defects would promote the FM ordering of their localized electrons left at EV +0.60 eV. The next step is to examine the realistic defect concentration required to establish the magnetic percolation between the magnetic defects above room temperature.
5 TABLE I: The value of the stabilization energy EM = EAFM − EFM for two interacting defects placed at the next-nearestneighbor sites (the matching defect concentration is 1020 cm−3 ). The defects have been considered to appear in the (0) (1+) charged states Pbi and Pbi . The positive sign of EM indicates the FM ground state, while negative is for AFM state. (0)
010 or 100 111
FIG. 3: Schematic band diagram demonstrating a hybridization of the VO state with Pbi destroying the triplet state at the Pbi site. The electron density is plotted for the energy range (0.9 eV+EV )±0.4 eV with help of Xcrysden [23].
The value of the formation energy of the Pb interstitial in a bulk is 1.23 eV at the Pb-rich/O-poor conditions. For this formation energy, the thermodynamic limit on the defect concentration is very low [18] to enable the effective interactions between defects. However, provided the defect appears on the surface of a single crystal (platelet), its formation energy is reduced already by ∼ 1.0 eV [18]. The defect concentrations at the platelet’s surface calculated for the deposition temperature 570 K can reach up to 1020 cm−3 at the Pb-rich/O-poor conditions. This is relevant to the defects formed at the next-nearestneighbor sites. Therefore, an efficiency of the magnetic coupling between defects is considered for the Pbi defects placed at the next-nearest-neighbor sites. We have examined several locations of the defects in the lattice. One is superposition of the defects, i.e. the defects are attached to the same layer along 010 or 100 directions. This arrangement provides a maximum overlap of the defect tails (the defect separation is 8.1 ˚ A). The second is oblique arrangement for which the defects are considered to be attached to the opposite layers but they are in the same interlayer space (called here 111 arrangement with the characteristic defect separation 8.66 ˚ A). The value of the stabilization energy EM calculated with GGA for the defects in the (0) (1+) charged states Pbi and Pbi are presented in Table I. It is worth mentioning that when the spin-orbit interactions are included into account (GGA+so), it increases the defect tails due to a shift of the 1u level towards the EC and, therefore, enhances the magnetic coupling between defects. The weakest interactions between the defect tails is observed in the 111 arrangement because the defects are attached to the different layers. In order to receive the more realistic results on Curie temperature TC , this path-
EM for Pbi , eV -0.192 -0.021
(1+)
EM for Pbi 0.311 0.067
, eV
way characterized by the weakest coupling between defects is considered [25]. The Curie temperature TC can be roughly estimated with help of simplified mean-field approximation for the Heisenberg model TC = 2/3kB EM [26] (kB is the Boltzmann constant). (1+) placed at For the defects in the charged state Pbi the next-nearest-neighbour sites in the 111 arrangement for which the exchange interaction strength is accounted by EM =0.067 eV, the Curie temperature TC is found to be way above room temperature. For the defects in the (0) charge state Pbi , a magnitude of their magnetic interactions in the AFM state is weaker and in such, it is predicted to manifest at room temperature only for the 010 or 100 arrangements. As separation between the defects increases, the magnetic coupling is drastically suppressed. For example, for two defects in their charged (0) states Pbi separated by a distance 12.3 ˚ A(010 or 100 arrangements), the magnitude of the stabilization energy is found to be only EM =-0.006 eV. Under the same con(1+) ditions, the stabilization energy for two Pbi defects is almost EM =0.01 eV. In both cases, the Curie temperature TC is not higher than 100 K. Therefore, our simulations have predicted that for the magnetic percolation to manifest above room temperature the defects have to appear at the next-nearest-neighbour sites. It imposes the critical limit on the defect concentration to be 1020 cm−3 . For the Pb interstitial defect, such defect concentration is viable only on the platelet’s surface.
V.
CONCLUSION
With help of the first-principle calculations, we have predicted an appearance of the p local orbital magnetism in polycrystalline α-PbO compound induced by the non-magnetic Pb atoms incorporated between layers as the interstitial defects. The magnetism is generated by the Pb interstitial defect through its partially filled p-shell that allows to apply the Hund’s rule to on-site ordering of the p electrons and in such, this defect acts as the magnetic center. Therefore, the specific s2 p2 outer shell configuration of the defect, namely the p2 electrons, is a key for appearance of the local magnetic moment of 2 μB . The combination of the s2 p2 outer shell with
6 the α-PbO crystal structure is unique because the Pbi defect is almost ’non-invasive’ to the host lattice and its electronic properties. The defect-host hybridization allows to establish the extended defect tails enabling the long-range order interactions between defects. The critical limit on the defect concentration required for obtaining the magnetic coupling at room temperature is found to be 1020 cm−3 . It can be reached when the defects are formed on the platelet’s surface. In order to generate the 2D magnetic network of the interstitial defects on the platelet’s surface, the post-deposition annealing in lead vapor atmosphere can be applied.
VI.
and thoughtful insights in our work. The computational facilities have been acquired through membership in Shared Hierarchical Academic Research Computing Network (SHARCNET:www.sharcnet.ca) and Compute/Calcul Canada. Authors are also thankful to Dr. O. Rubel for sharing his clusters. Financial support of Ontario Ministry of Research and Innovation through a Research Excellence Program Ontario is highly acknowledged.
ACKNOWLEDGEMENT
We would like to thank Prof. P. Fulde, Prof. T. Chakraborty and Dr. L. Hozoi for their guidance
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Possible route to d0 magnetism in α-pbo compound J. Berashevich, A. Reznik
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Received date: 20 November 2013 Revised date: 13 February 2014 Accepted date: 3 April 2014 Cite this article as: J. Berashevich, A. Reznik, Possible route to d0 magnetism in α-pbo compound, Journal of Physics and Chemistry of Solids, http://dx.doi.org/ 10.1016/j.jpcs.2014.04.002 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting galley proof before it is published in its final citable form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Possible route to d0 magnetism in α-PbO compound J. Berashevich
Thunder Bay Regional Research Institute, 290 Munro St., Thunder Bay, ON, P7A 7T1, Canada and Max Planck Institute for the Physics of Complex Systems, Nthnitzer Str. 38, 01187 Dresden, Germany
A. Reznik
Thunder Bay Regional Research Institute, 290 Munro St., Thunder Bay, ON, P7A 7T1, Canada and Department of Physics, Lakehead University, 955 Oliver Road, Thunder Bay, ON, P7B 5E1, Canada With help of the first-principle methods we investigate a possibility to induce d0 magnetism in the α-PbO compound. The Pbi interstitial defect is found to gain the magnetic properties upon incorporation into α-PbO crystal structure. The Pbi interstitial generates the p localized state with two electrons on-site whose spin alignment is governed by the Hund’s rule giving a rise to formation of the local magnetic moment 2μB . The magnetic coupling between defects is a function of occupancy (0) of the defect states. For two defects in their zero charge state Pbi the antiferromagnetic (AFM) (1+) show already the coupling is more stable. The Pb interstitial defects in the charge state Pbi ferromagnetic (FM) ground state. The magnetic coupling is found to be driven by the long-range order interactions. The critical limit on the defect concentration required to establish the magnetic percolation above room temperature is about 1020 cm−3 which may become accessible when the Pbi defects are formed on the surface of a single crystal. The substitution of the Pb interstitial with the non-magnetic elements of s2 px outer shell (the local magnetic moment is defined by 1 ≤ x ≤ 3) is revealed to be a perspective way to tune the magnetic behavior. I.
INTRODUCTION
Spintronics has recently emerged as a widely successful technology that exploits the principles of magnetism. Classical metal ferromagnets do not meet all requirements of spintronics because semiconducting behavior is an essential condition for operation of the electronic devices. It has led to development of a new branch of research that is focused on semiconductors exhibiting magnetism at room temperature [1]. The first success was in so-called ”diluted magnetic semiconductors” which are the conventional semiconductors doped with magnetic ions. Magnetic ions with partially filled 3d or 4f shells allow the spin alignment of on-site electrons [2–5]. Interest in ferromagnetic semiconductors was further generated when magnetism was demonstrated in materials composed of ’non-magnetic’ elements (i.e. without unpaired electrons) [6–9]. The origin of so-called d0 magnetism was proposed to be due to localized sp states. The interacting vacancies create a network of unpaired electrons, which with sufficient long-range order effect exhibit the magnetic coupling [9]. Here we show that d0 magnetism can be induced by the 2D network of the interstitial defects Pbi (Pbi :6s2 6p2 outer shell) incorporated between layers of the tetragonal lead oxide α-PbO. The Pbi defect forms the localized state occupied by two unpaired electrons Pbi :6p2 . The on-site ordering of the localized electrons Pbi :6p2 obeys the first Hund’s rule that results in formation of the local magnetic moment 2.0 μB . In our model compound, the triplet state is found to be stable as defined by the spin polarization energy Epol =0.235 eV. The experimental observation of the magnetic signal from the α-PbO samples at room temperature [10] verifies formation of
the local magnetic moment. Therefore, otherwise than the magnetic moment occurs due to the partially filled p shell instead of the 3d or 4f shells, Pbi works exactly as the magnetic impurity and it provides several advantages over a doping with magnetic ions. Thus, the Pbi state demonstrates the high on-site spin stability due to localized Pbi :6p2 electrons, the extended tails of their wavefunctions induce the long-range electron spin ordering. In addition, the interstitial defect is almost non-invasive to the electronic and crystal structures of the host (the host Pb:6s2 electrons are only involved in the defect state formation). Moreover, we found that the magnetic behavior of the interacting defects can be tuned by changing the charge state of Pbi or alternatively, through substitution with impurities of the specific s2 p1≤x≤3 outer shell configuration: x = 2 works for 2.0 μB , while x = 1 or x = 3 for 1.0 μB . We suggest that from technological point of view, the layered crystal structure of α-PbO promises the superior advantages in establishing the magnetic percolation and more importantly, a practical way for its control: i) the α-PbO compound grows in polycrystalline form such as the large surface area offers enormous potential for doping (the formation energy of the Pbi interstitial defect drastically decreases on the surface of a single crystal [18]); ii) a solubility of the magnetic centers and their properties can be tuned applying the substitutional doping; iii) any compounds of the tetragonal PbO type can be used as a semiconductor matrix. II.
METHODS
In our study we applied the generalized gradient approximation (GGA) with the PBE parametrization [12]
2 provided by WIEN2k package for the density functional theory (DFT) calculations [13]. The supercell approach with sufficiently large supercell of 108-atom size (3×3×3 array of the primitive unit cells) for single defect and of 160-atom size (5×4×2) for two interacting defects has been used. The Pb:5p, 5d, 6s, 6p and O:2s, 2p electrons have been treated as the valence electrons (the energy cut-off was -8 Ry). The geometry optimization procedure for supercell containing the Pbi defect was performed based on minimization of the total energies and forces [14]: the energy convergence limit was set to 0.0001 Ry while the residual forces did not exceed 0.5 mRyd/Bohr. For integration of the Brillouin-zone, the MonkhorstPack scheme using a (5×5×4) k-mesh was applied. The product of the atomic sphere radius and plane-wave cutoff in k-space RKmax was equal to 7. The localization of the defect wavefunctions has been additionally examined with Hartree-Fock (HF) applied directly to the unpaired electrons that allowed to preserve accuracy provided by DFT at the same time correcting the unpaired electrons self-interaction [15]. Moreover, since the effect of the spin-orbit interactions can not be neglected for the heavy atom as Pb, influence of the spin-orbit coupling (so) on the energetic position of the defect states inside the band gap is investigated with (GGA+so). The tetragonal lead oxide α-PbO possesses the layered structure leading to formation of platelets upon compound growth and each platelets can be considered as a single crystal. The layers within a single crystal are held together by the interlayer interaction of the Pb:6s2 electrons [16]. In terms of the electronic properties, the interlayer interactions generate a dipping of the conduction band at M∗ point [17] thus inducing an effect of the band gap shrinkage. GGA tends to overestimate the interlayer separation in the layered structures, but it is also well known to underestimate the band gap size. For α-PbO system, it results in a compensation effect [18]: for the lattice parameters optimized with GGA the band gap is only slightly underestimated giving 1.8 eV against the experimental value of 1.9 eV [19] (application of the experimental lattice constants shrinks the band gap by 0.22 eV). The correct band gap size is important to place the defect states inside the band gap: when the band gap size is underestimated, the hole-carrying defect state occurs closer to the host conduction band thus causing the longer defect tails. In order to avoid the spurious long-range order interactions [5], all band structure calculations have been performed with the GGA optimized lattice parameters. However, the formation energy of the interstitial defects have been defined applying the experimental lattice constants [18].
III.
THE DEFECT-INDUCED LOCAL MAGNETIC MOMENT
The interstitial defects have never been considered in origin of d0 magnetism because they do not belong to a
class of the most common defects in crystalline materials: the interstitials induce a significant perturbation of the host lattice that results in their rather large formation energy [20]. The layered structure of α-PbO is different as it allows the foreign atoms to squeeze between layers causing a minimal lattice deformation to the host immediate neighborhood (see Fig. 1 (a)). As a result, the formation energy of Pbi appearing between layers is not too high. For the Pb-rich/vacuum conditions, the neutral charge state is characterized by the formation energy of 1.23 eV (for details on calculation of the formation energy of defects in the α-PbO crystal structure see [18]). The band diagram and the density of states (DOS) separatly for the up-spin and down-spin states are presented in Fig. 1 (b) and (c) for our model system containing Pbi (the band diagram of the defect-free system is shown in Fig. 1 (b) as well). The Pb interstitial makes a bond with Pb atom from the host. The length of the Pbi -Pb bond is fairly short, 2.9 ˚ A, that indicates a double bond formation. With respect to modification induced to the host, an out-of-plane displacement of 0.54 ˚ A occurs for the Pb atom involved in bonding. The antibonding orbitals of the Pbi -Pb bond are presented by the 2u and 2d bands in Fig. 1 (b), while the bonding states appear deeper in the valence band (see ’antibonding’ notation in Fig. 1 (c)). The on-site defect states generate the 1u and 1d bands. The 1u state is filled with two electrons and appears just above the midgap at 0.99 eV + EV , where EV is a top of the valence band. In contrast, the spin-down 1d state is empty and in such, it is shifted to the conduction band edge EC . In order to understand an asymmetry in filling of the spin-up and spin-down states, a mechanism of bonding of Pbi with the host lattice has to be explored. The integration of the Pb interstitial into the crystal lattice requires an excitation of its Pbi :6s2 6p2 ground state to Pbi :6s1 6p3 . The Pbi :6s1 and Pbi :6p1z electrons participate in formation of the Pbi -Pb bond with the host and in such, the host shares its Pb:6s2 electrons (see Fig. 1 (c)). Since the Pb:6s2 electrons from the host participate in bonding with the defect, the interlayer interaction generated by overlap of the Pb:6s2 electrons is disturbed. For the supercell size 3×3×3 employed to calculate a single defect, this effect is rather significant. Corresponding disturbance can be seen in a dispersion of the 2u and 2d bands between Γ∗ and M∗ points. Moreover, a behavior of the host bands near the conduction band top is also altered through a suppression of the band dipping at the M∗ point being also controlled by the interlayer interactions of the Pb:6s2 electrons [17]. The similar alteration in the band behavior upon substitutional doping accompanied by depletion of the electron and hole density is observed for another compound possessing the αPbO crystal structure [21]. The depletion effect is argued to appear due to the enhanced hopping via substituted sites thus contributing in larger splitting of the conduction and the valence bands (the α-PbO crystal structure is well recognized in superconductivity topic). The 1u and 1d bands are induced by a state localized
3
FIG. 2: (a) The positions of the 1u and 1d orbitals within the band gap calculated with the different methods. (b) The electron density map for the triplet state (GGA) at the Pbi site is plotted for isovalues of ±0.005 e/˚ A3 with help of Xcrysden [23]. The density map demonstrates the spin density calculated for the energy range (1.0 eV+EV )±0.15 eV.
FIG. 1: (a) The Pb interstitial in the layered crystal structure of α-PbO. (b) The band diagram for a defect-free single crystal (green dotted line) and system hosting Pbi : 1u and 1d are for the on-site defect states, while 2u and 2d are the antibonding orbitals of the Pbi -Pb bond. (c) Total and partial density of states for Pbi and host Pb atoms (’top’ is referred to Pb atom forming the Pbi -Pb bond, while ’bottom’ for Pb atoms in the bottom layer). The ’bonding’ and ’antibonding’ in the B and C panels should be referred to the Pbi -Pb bond.
at the defect site, namely by two Pbi :6px+y electrons left on-site after bonding. The Hund’s rule dictates both unpaired Pbi :6px+y electrons to occupy the 1u spin-up band. The empty 1d spin-down band is pushed up to EC that causes a considerably large spin-exchange splitting between 1u and 1d of order 0.52 eV as presented in Fig. 2 (a). The two localized Pbi :6px+y electrons occu-
pying the 1u spin-up state are responsible for the ground triplet state followed by formation of the local magnetic moment 2.0 μB . The spin-polarization energy defined as the energy difference between the spin-unpolarized and spin-polarized states [22] is found to be Epol =0.235 eV, i.e. the triplet state is expected to be stable at room temperature that is confirmed experimentally [10]. The Pbi :6s2 6p2 electronic configuration is clearly a key for the Pb interstitial to act as a magnetic impurity: the Pbi :6s2 electrons are used for attachment to the host while partially filled Pbi :6p2 shell contributes in developing of the local magnetic moment. Therefore, other chemical elements possessing the s2 p1≤x≤3 outer shell can be used as the magnetic impurity instead of Pbi . The impurities of x = 1 or x = 3 configurations have been found to induce the local magnetic moments of 1 μB , while x = 2 works for 2μB [11]. Because GGA approach is well known to delocalize the wavefunction of the hole-like defect state due to selfinteraction of the unpaired electrons [15], the limitations and reliability of this method has to be examined. As expected, an application of the HF approach to the unpaired electrons has induced the stronger on-site localization of the defect wavefunction emerging in stabilization of the triplet state (Epol =0.490 eV). Thus, an increase in the splitting of the 1u and 1d energies up to 1.12 eV is observed (see Fig. 2 (a)). However, an application of the experimental lattice parameters has opposite effect on the splitting which is suppressed because of stronger hybridization of the impurity state with the host lattice due to the interlayer distance reduction. Therefore, when HF is combined with the experimental lattice parameters, they compensate each other such as the splitting of the 1u and 1d energies is reduced to 0.68 eV. This value is very close to that found with GGA originally (see Fig. 2 (a)) that indicates its reliability when the optimized lattice
4 parameters are used. Moreover, for the heavy elements like Pb atom, the spin-orbit coupling plays an important role and, therefore its effect on the 1u and 1d energies has to be examined. The GGA+so approach has been applied directly to the Pbi :6px+y electrons. The spin-orbit coupling is found to break a degeneracy of the 1u and 1d states (see GGA+so in Fig. 2 (a)). It reduces the splitting of the occupied 1u and unoccupied 1d levels to 0.2 eV, i.e. the stability of the triplet state decreases while a shift of 1u state towards the conduction band contributes in enlargement of the defect tails. Since the magnetic signal is observed at room temperature experimentally [10], we believe that effect of the spin-orbit coupling on the on-site stability is not crucial. IV.
THE LONG-RANGE ORDER INTERACTIONS
Formation of the local magnetic moment is the first step in achieving the magnetism. For the magnetic percolation to manifest at room temperature, the effective interactions of the Pbi :6p2x+y electrons between the defect sites is essential. Our studies on formation of the defect tails has revealed a duality in their behavior. Since occupied band 1u is located deep inside the band gap, the Pbi :6p2x+y electrons show localization on the defect site thus allowing to form the stable local magnetic moment 2μB (see the electron density map in Fig. 2 (b)). However, the long defect tails are also formed as a result of the bonding/hybridization of the Pbi :6s1 and Pbi :6p1z electrons with the host Pb:6s and Pb:6p, O:2p electrons (see partial DOS in Fig. 1 (c)). In order to establish the spin ordering of the localized Pbi :6p2x+y electrons between the defect states, FM or AFM, the defects have been considered on the nearestneighbor sites. For these calculations, the spins of the 6p2x+y electrons have been always aligned on-site, while the inter-defect spin ordering has been changed from antito ferromagnetic. The 6p2x+y state is exactly half filled and, therefore, for two interacting defects the virtual hopping is allowed only when their spins are antiferromagnetically ordered. It contributes in a lowering of the total energy in the AFM state which becomes the ground state [5]. The ground state stability is defined by the energy difference between the AFM and FM states as EM = EAFM − EFM (a negative sign refers to the AFM ground state). For two Pb interstitials placed at a distance 4.1 ˚ A the magnetic energy difference is found to be EM =-0.96 eV. To reach the FM arrangement to be an essential requirement for spintronics, the virtual hopping between defects has to be allowed in the FM state. It becomes possible when two interacting defects appear in (1+) (0) their charged states Pbi , i.e. each Pbi has to drop one electron. The local magnetic moment induced by the (1+) (1+) Pbi state is 1 μB . The interaction of two Pbi states placed at a distance 4.1 ˚ A gives the magnetic energy dif-
ference EM =1.46 eV. The most common way to change the charge state of defect is to consider the electron exchange with another defect. The growth conditions required to favor a formation of the Pbi defect is Pb-rich/O-poor limit (oxygen deficiency). At these growth conditions, the O vacancy (VO ) is another defect possessing the low formation energy. For the defects in a bulk, the formation energy of the O vacancy in its neutral charge state is 0.85 eV (the formation energy of the Pbi defect at the same conditions is 1.23 eV). Therefore, we would examine an interaction of Pbi with the VO defect with respect to a stability of its local magnetic moment and to explore a possibility to (1+) switch its charge state to Pbi . In the neutral charge state, VO is occupied by two electrons possessing the strong localization on the vacancy site [18]. Therefore, VO can be an electron donor. Since the charge exchange is allowed between Pbi and VO , they would act as the compensating centers to each other. It is (0) found that for the VO and Pbi defects appearing in close proximity to each other (the defect separation is 2.27 ˚ A), the strong interactions between defects accompanied by their hybridization destroy the local magnetic moment at Pbi as shown in Fig. 3 (Pbi in a singlet state induces the defect state at EV +1.22 eV in the band gap). The Pbi :6px+y electrons from the interstitial and the Pb:6s and Pb:6p and O:2p electrons from the host lattice, all have been found to contribute equally in formation of the Pbi -VO pair. The interaction between defects manifest itself in lowering of the formation energy of the Pbi -VO pair by 0.84 eV in comparison to the non-interacting defects. Provided the defects are separated by the sufficient distance of 6.2 ˚ A, the local magnetic moment at Pbi site recovers. However, because the interaction between Pbi with VO does not vanish completely, the spin polarization energy of this state is found to be considerably low as defined by Epol =0.06 eV. Therefore, the interaction of the Pbi and VO defects is not viable way to convert the (1+) Pb interstitial into the Pbi charge state. Moreover, in order to suppress the destabilizing effect of the VO vacancies on formation of the local magnetic moment at Pbi site, a post-growth annealing treatment in atmosphere of oxygen would be required. The strong spin-orbit interactions at the defect site of(0) fers another way to convert Pbi into the charge state (1+) Pbi . Since the spin-orbit coupling breaks a degeneracy of the 1u orbitals (see GGA+so in Fig. 2(a)), the selective optical excitation of the electrons localized at EV +1.11 eV can be implemented (α-PbO is known as a good photoconductor [24] for which a mechanism of the optical excitation would be plausible). After selective excitation, the interactions between defects would promote the FM ordering of their localized electrons left at EV +0.60 eV. The next step is to examine the realistic defect concentration required to establish the magnetic percolation between the magnetic defects above room temperature.
5 TABLE I: The value of the stabilization energy EM = EAFM − EFM for two interacting defects placed at the next-nearestneighbor sites (the matching defect concentration is 1020 cm−3 ). The defects have been considered to appear in the (0) (1+) charged states Pbi and Pbi . The positive sign of EM indicates the FM ground state, while negative is for AFM state. (0)
010 or 100 111
FIG. 3: Schematic band diagram demonstrating a hybridization of the VO state with Pbi destroying the triplet state at the Pbi site. The electron density is plotted for the energy range (0.9 eV+EV )±0.4 eV with help of Xcrysden [23].
The value of the formation energy of the Pb interstitial in a bulk is 1.23 eV at the Pb-rich/O-poor conditions. For this formation energy, the thermodynamic limit on the defect concentration is very low [18] to enable the effective interactions between defects. However, provided the defect appears on the surface of a single crystal (platelet), its formation energy is reduced already by ∼ 1.0 eV [18]. The defect concentrations at the platelet’s surface calculated for the deposition temperature 570 K can reach up to 1020 cm−3 at the Pb-rich/O-poor conditions. This is relevant to the defects formed at the next-nearestneighbor sites. Therefore, an efficiency of the magnetic coupling between defects is considered for the Pbi defects placed at the next-nearest-neighbor sites. We have examined several locations of the defects in the lattice. One is superposition of the defects, i.e. the defects are attached to the same layer along 010 or 100 directions. This arrangement provides a maximum overlap of the defect tails (the defect separation is 8.1 ˚ A). The second is oblique arrangement for which the defects are considered to be attached to the opposite layers but they are in the same interlayer space (called here 111 arrangement with the characteristic defect separation 8.66 ˚ A). The value of the stabilization energy EM calculated with GGA for the defects in the (0) (1+) charged states Pbi and Pbi are presented in Table I. It is worth mentioning that when the spin-orbit interactions are included into account (GGA+so), it increases the defect tails due to a shift of the 1u level towards the EC and, therefore, enhances the magnetic coupling between defects. The weakest interactions between the defect tails is observed in the 111 arrangement because the defects are attached to the different layers. In order to receive the more realistic results on Curie temperature TC , this path-
EM for Pbi , eV -0.192 -0.021
(1+)
EM for Pbi 0.311 0.067
, eV
way characterized by the weakest coupling between defects is considered [25]. The Curie temperature TC can be roughly estimated with help of simplified mean-field approximation for the Heisenberg model TC = 2/3kB EM [26] (kB is the Boltzmann constant). (1+) placed at For the defects in the charged state Pbi the next-nearest-neighbour sites in the 111 arrangement for which the exchange interaction strength is accounted by EM =0.067 eV, the Curie temperature TC is found to be way above room temperature. For the defects in the (0) charge state Pbi , a magnitude of their magnetic interactions in the AFM state is weaker and in such, it is predicted to manifest at room temperature only for the 010 or 100 arrangements. As separation between the defects increases, the magnetic coupling is drastically suppressed. For example, for two defects in their charged (0) states Pbi separated by a distance 12.3 ˚ A(010 or 100 arrangements), the magnitude of the stabilization energy is found to be only EM =-0.006 eV. Under the same con(1+) ditions, the stabilization energy for two Pbi defects is almost EM =0.01 eV. In both cases, the Curie temperature TC is not higher than 100 K. Therefore, our simulations have predicted that for the magnetic percolation to manifest above room temperature the defects have to appear at the next-nearest-neighbour sites. It imposes the critical limit on the defect concentration to be 1020 cm−3 . For the Pb interstitial defect, such defect concentration is viable only on the platelet’s surface.
V.
CONCLUSION
With help of the first-principle calculations, we have predicted an appearance of the p local orbital magnetism in polycrystalline α-PbO compound induced by the non-magnetic Pb atoms incorporated between layers as the interstitial defects. The magnetism is generated by the Pb interstitial defect through its partially filled p-shell that allows to apply the Hund’s rule to on-site ordering of the p electrons and in such, this defect acts as the magnetic center. Therefore, the specific s2 p2 outer shell configuration of the defect, namely the p2 electrons, is a key for appearance of the local magnetic moment of 2 μB . The combination of the s2 p2 outer shell with
6 the α-PbO crystal structure is unique because the Pbi defect is almost ’non-invasive’ to the host lattice and its electronic properties. The defect-host hybridization allows to establish the extended defect tails enabling the long-range order interactions between defects. The critical limit on the defect concentration required for obtaining the magnetic coupling at room temperature is found to be 1020 cm−3 . It can be reached when the defects are formed on the platelet’s surface. In order to generate the 2D magnetic network of the interstitial defects on the platelet’s surface, the post-deposition annealing in lead vapor atmosphere can be applied.
VI.
and thoughtful insights in our work. The computational facilities have been acquired through membership in Shared Hierarchical Academic Research Computing Network (SHARCNET:www.sharcnet.ca) and Compute/Calcul Canada. Authors are also thankful to Dr. O. Rubel for sharing his clusters. Financial support of Ontario Ministry of Research and Innovation through a Research Excellence Program Ontario is highly acknowledged.
ACKNOWLEDGEMENT
We would like to thank Prof. P. Fulde, Prof. T. Chakraborty and Dr. L. Hozoi for their guidance
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