THE INFLUENCE OF DEFECTS ON MAGNETIC PROPERTIES OF fcc-Pu
A. O. Shorikova b, V. I. Anisimova'b, M. A. Korotin", V. V. Dremovc* Ph. A. Sapozhnikov€
" Institute of Metal Physics, Russian Academy of Sciences 620990, Ekaterinburg, Russia
b Theoretical Physics and Applied Mat-hematic Department, Urals State Technical University
620002, Ekaterinburg, Russia
cRussian Federal Nuclear Center "Institute of Technical Physics" 456770, Snezhinsk, Chelyabinsk Region, Russia
Received February 6, 2013
The influence of vacancies and interstitial atoms on magnetism in Pu is considered in the framework of the density functional theory. The crystal structure relaxation arising due to different types of defects is calculated using the molecular dynamics method with a modified embedded atom model. The local density approximation with explicit inclusion of Coulomb and spin-orbit interactions is applied in matrix invariant form to describe correlation effects in Pu with these types of defects. The calculations show that both vacancies and intersti-tials give rise to local moments in the /-shell of Pu in good agreement with experimental data for aged Pu. Magnetism appears due to the destruction of a delicate balance between spin-orbit and exchange interactions.
DOI: 10.7868/S004445101310012X
1. INTRODUCTION
Band structure calculations of ¿»-Pu predict the static magnetic order of /-electrons with the full magnetic moment values 0.25 5 fiB with a substantial impact of the spin moment fl 4]. These results contradict the experimental measurements of magnetic properties of non-aged Pu without impurities. These data indicate the absence of any ordered or disordered, static or dynamic magnetic moments in Pu at low temperatures [5, 6].
Recent progress in calculation methods allows correctly describing the ground state of pure Pu in the ¿-phase and the model <i-phase [7,8]. It was shown in Ref. [7] that the delicate balance between spin orbit (SO) and exchange interactions determines the nonmagnetic ground state in pure Pu. These interactions have the magnitude close to each other in actinides and its compounds and the balance could be easily broken by crystal field of legands. Also, Soderlind [9] confirms the important role of SO and orbital polarization
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in formation of the nonmagnetic ground state of plu-toniuni in the framework of model density functional theory (DFT) calculation. Impurities like A1 and Ga that are used to stabilize the fee-phase of Pu act in the same way. Several groups report the presence of the ordered magnetic moment in aged Pu-Al and Pu-Ga alloys [10 12]. The magnitude of moments is small, <10-3 'fi.b (Ref. [101) 0-15 t'B (Ref. [11, 12]), and these moments could arise due to distortion of the crystal structure near interstitial Pu atoms and vacancies.
A substantial drawback of the local (spin) density approximation (L(S)DA) is the underestimation of the orbital moment [13, 14]. As a result, the DFT in description of 4/- and 5/-metals fails, since the orbital moment in them can overcome the spin one. Taking the Coulomb repulsion U and SO interactions in full matrix rotation-invariant form into account in the LDA—U—SO method could improve the results. An achievement of this method is that the exact magnetic order does not have to be set at the start of iterations. Both the magnitude and the direction of the magnetic moment are calculated for each atom. Magnetic order and the "easy axis" direction are the result of a self-consistent interaction procedure.
In the LDA—U method [15], the energy functional
795
8*
Elda+u depends, in addition to the charge density p(r), on the occupation matrix n™m, for a particular orbital for which correlation effects are taken into account (in our case, it is the 5/ plutoniuni orbital). The LDA—U method in the general form nondiagonal in spin variables was defined in Ref. [16]:
Elda+u[p(t), {n}] =
= ELDA[p(v)} + Ev[{n}] - Edc[{n}}, (1)
where p(r) is the charge density and Elda[p(r)] is the standard LDA functional. The occupation matrix is defined as
Ep
l r
nmm.' = / ,(E)dE, (2)
7T J
where G™ ,(E) = {ms\(E — Hloa+u)^1]'"!1 «') are the elements of the Green's function matrix in a local orbital basis set (m is the magnetic quantum number and h is the spin index for the correlated orbital). In this paper, this basis set is formed of LMT-orbitals from the tight-binding LMTO method based on the atomic sphere approximation (TB-LMTO-ASA) [17]. In Eq. (1), the Coulomb interaction energy term Eu [{}i}] is a function of the occupation matrix n™m,:
£r[{u}] = 1
= 2 Z-, {<m,m"|V'cc|m',m'")n™mlnsJlrnlll -
{m},ss'
- (»».»»"irool»»'".»/»')»^/»^^,,}, (3)
where I",, is the screened Coulomb interaction between correlated electrons. Finally, the last term in Eq. (1) correcting for double counting is a function of the total number of electrons in the spirit of the LDA and is a functional of the total charge density,
Edc[{"'}} = y.Yl.Y - 1) - \JhN(N - 2), (4)
where N = Tr(n™'m,) is the total number of electrons in a particular shell, and U and Jh are the screened Coulomb and Hund exchange parameters, which can be determined in the constrain LDA calculations [18, 19]. The screened Coulomb interaction matrix elements (ii).ii)"\\~,, \in'.in'") can be expressed in terms of the parameters U and Jh (see Ref. [15]).
The functional in Eq. (1) defines the effective single-particle Hamiltonian with an orbital-dependent potential added to the usual LDA potential:
HlDA+U = HLDj 1+ I m«)l/mm'im's'l> (5)
In S.IU '' s/
where V""' / =
■mm.
= Y, i '"" I V"\'"'' m"')nm"m*" +
in'' .in
+ ((ll). il)"\\",, \ll)' .11)'"} — - ill). 11)" \\], \ll)'", iii') )/c"m"' } ~
- (1 - ¿W ) '»"lr" I'»'"-™')n"m"m'" -
m" ,m//;
-U(n-^+±Jh(N- 1). (6)
In this paper, we use the LDA—U—SO method, which includes the LDA—U Hamiltonian (6), nondiagonal in spin variables, and the spin orbit coupling term
Hlda+u+so = Hlda+u + El so-.
(7)
Hso = AL • S,
where A is the spin orbit coupling parameter. In the LS basis, the SO coupling matrix has nonzero matrix elements that are diagonal ((Hso)^1 m) as as diagonal ((Hsom and in spin variables
(complex spherical harmonics) [20]:
(Hsotlm = ^s/(l + ^)(l-w + l)(Sml.1,m). <8> (HSo)m',m = Am«d"TO/,TO,
where I, m are orbital quantum numbers and the spin index is h = +1/2,-1/2. The peculiarities of the LDA—U—SO method and its implementation to the problem of pure Pu and several plutoniuni compounds were described in detail in Ref. [7].
In this paper, four different fcc-Pu supercells are investigated: one interstitial (IS) Pu atom in a 32-atom supercell, a vacancy in an 8-atom supercell, and two 32-atom supercells with both an IS and a vacancy at minimal and large distances. Due to the presence of defects, the perfect fee structure was to be distorted, and therefore the relaxation of the crystal structure for all supercells under investigation should be taken into account. Because the LMTO method does not allow performing structure relaxation correctly, we use the classical molecular dynamics (CMD) with the modified embedded atom model (MEAM) by Baskes [21 23] as the interatomic potential. The MEAM is a many-body potential, i.e., interaction between a pair of atoms depends on the local structure (on positions of their common neighbors). The parameterization of the MEAM
for pure plutonium and plutoniuni gallium alloys was given in Ref. [21] and the potential is currently widely-used in CMD simulations of plutoniuni properties and processes in Pu caused by self-irradiation [21 25].
Adding an IS or a vacancy to initial supercell makes the Pu atoms inequivalent. That is why the different types of atoms in the tables below have additional numbers (e.g., Pul, etc). The crystal structure relaxation lowers the symmetry again, and the new Pu classes are divided into subclasses (see Table 4). All calculations of the electronic structure and magnetic properties were made using the tight-binding linear muffin-tin orbitals method with the atomic sphere approximation (the TB-LMTO-ASA computation scheme). In the LDA—U calculation scheme, the values of the direct Coulomb (U) and Hund exchange (Jh) parameters should be determined as the first step of the calculation procedure. This can be done in an ab initio way by constrained LDA calculations [18,19]. In our calculations, the Hund exchange parameter Jh was found to be Jh = 0.48 eV. The value of the Coulomb parameter U was set to 2.5 oV because this value provides the correct equilibrium volume of <">-Pu (see Ref. [7] for the details).
2. INTERSTITIAL PLUTONIUM ATOM IN A 32-ATOM SUPERCELL
First, the 32-atom supercell of fcc-Pu with one additional Pu atom was considered. The supercell has three coordination spheres around the defect, which is sufficient for describing relaxation of the position of neighboring atoms. The classical molecular dynamics method was used to describe distortion of the crystal structure. New positions of Pu atoms in the supercell were used in the subsequent calculation of the electronic structure. Adding one additional Pu atom lowers the symmetry of the cell. Four new inequivalent classes of plutoniuni belonging to four different coordination spheres around the IS arise. Moreover, the Pu atoms within each new class become inequivalent due to different local neighborhoods. To take this lowering of symmetry into account, no symmetrization was applied in our electronic structure calculation. Because no additional symmetry conditions were imposed on the electronic subsystem, the magnitude of local moments at Pu sites and their directions can be arbitrary-arid correspond to the minimum of the total energy.
The LDA—U—SO calculations for metallic Pu in the 6 phase gave a nonmagnetic ground state with zero values of the spin S, orbital L, and total J moments [7, 8].
Our calculation for the 32-atom supercell with one IS shows that small local magnetic moments develop at the Pu sites. The local
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