tively it, the ordered state, and 0.34 and 3.00/~ B in i Permanent address and address for correspondence: Physics. Department, Brock University, St. Catharines, ...
Journal of Magnetism aad Magnetic Materials 87 (1990) 97-105 North-Holland
97
E L E C T R O N I C S T R U C T U R E O F O R D E R E D A N D D I S O R D E R E D Pd3Fe S.K. BOSE l j. K U D R N O V S K Y 2 M. van S C H I L F G A A R D E , P. B L O C H L , O. JEPSEN, M. M E T H F E S S E L , A.T. P A X T O N and O.K. A N D E R S E N Max-Planck-lnstitut fib" FestkSrperforschung~ Heisenbergstrasse 1, 7000 Stuttgart 80, Fed. Rep. Germany
Received 15 September 1989; in revised form 11 December 1989
We study :he electronic structure of ferromagnetic Pd3Fe in ordered and disordered phases using the Linear Muffin Tin Orbitals (LMTO) method. The ordered structure is studied in the atomic sphere approximation (ASA) using a basis of s, p, d and f orbitals and combined correction is included in the calculation. Self-consistent potential parameters from the ordered phase calculation are used in the calculation for the disordered phase which is studied using the LMTO-CPA method developed recently. Calculations for the disordered phase are performed with and without relaxation of the lattice. The relaxed-lattice calculation takes into account, in an approximate why, the possible deviation from the ideal lattice structure due to the difference in the sizes of the constituent atoms. The calculated magnetic moments per Fe and Pd sites in the disordered alloy agrees well with the experiment-,d value. Bloch spectral functions for the disordered phase are briefly discussed. This calculation is the first application of the LMTO-CPA technique to a spin-polarized system.
1. Introduction The intermetallic c o m p o u n d Pd3Fe exists in both ordered and disordered phases and exhibits interesting properties related to electronic structure [1-6]. The ordered phase is of C u 3 A u structure and the (substitutionally) disordered phase is fcc with Pd and Fe atoms randomly occupying the sites with probabilities proportional to their relative concentrations. Both the ordered and the disordered phases have the same lattice parameter, 3.849 .~. The electronic specific heat coefficient ~, is larger in the disordered phase than in the ordered phase [2], contrary to the suggestion by Flanagan et al. [7] based on a study of the hydrogen solubility in ordered and disordere~ Pd3Fe. MSssbauer studies [5] yield values for the magnetic moments at Pd and Fe sites to be 0.57 and 2.70# B, respectively it, the ordered state, and 0.34 and 3.00/~ B in i Permanent address and address for correspondence: Physics Department, Brock University, St. Catharines, Ontario L2S 3A1, Canada. 2 Permanent address: Institute of Physics, Czechoslovak Academy of Sciences, Na Slovance 2, 180 40 Prague 8, Czechoslovakia.
the disordered state. The spin wave dispersion for the ordered alloy shows an anisotropy [4] which is hard to explain on the basis of a purely localized or purely itinerant m o m e n t picture. However, a model based on a coupled system of itinerant and localized electron spins seems capable of explaining this anisotropy [8]. All this and some additional properties of Pd3Fe were pointed out by K u h n e n and da Silva who, in a series of recent publications [9,10], studied the electronic structure of ordered and disordered P d a F e using L M T O [11] and K K R C P A [12,13] methods, respectively. L M T O calculation for the ordered structure was performed using the atomic sphere approximation (ASA) and equal-sized overlapping atomic spheres were used for Pd and Fe. Self-consistent potentials from this L M T O - A S A calculation were carried over to the disordered phase where K K R - C P A method, based on non-overlapping muffin-tin spheres was applied. The electronic specific heat coefficient calculated for the ordered phase (7.61 m J / m o l K 2) agreed reasonably well with the experimental value (8.17 n d / m o l K2), while the calculated value for the disordered phase (13.30 m J / m o l K 2) was
0304-8853/90/$03.50 © 1990 - Elsevier Science Publishers B.V, (North-Holland)
98
S.K. Bose et al. / Electronic structure of Pd3Fe
somewhat larger than the experimental one (11.28 m J / m o l K2) [2]. More severe was the disagreement between the calculated magnetic moment (2.10tta) at the Fe site in the disordered alloy and the value (3.0/~B) obtained in the M~Sssbauer experiment [5]. The magnetic moment at Fe site in the ordered alloy calculated via the LMTO method (3.10/~B) was closer to the experimental value 2.70#8. Since the potentials used in the ordered and the disordered alloy calculations were the same, it is not clear why the Fe magnetic moment in the calculation of Kuhnen and da Silva [9,10] differed so significantly between the ordered and the disordered states. Disorder may induce large variations in the local magnetic moments at different sites, depending on the local environment. However, the average magnetic moment per Pd or Fe site is expected to change only a little as a result of disorder, if the cow,position of the alloy (i.e. the concentrations of the components) remains unchan£ed. Since many technical details of the KKR-C~r~A calculation of Kuhnen and da Silva [10] are not available to us, it is difficult for us to comment on the reason for disagreement in their result for the Fe magnetic moment in disordered Pd3Fe with the experimental value. The KKR- CPA method [12,13] is known to y~eld very reliable and accurate results for disordered alloys, if carried to self-consistency. It is our contention that the low value of Fe magnetic moment in disordered Pd3Fe obtained by these authors is likely due to an inappropriate transfer of potentials from overlapping atomic spheres in the L M T O - A S A calculation to non-overlapping (touching) muffin-tin spheres, occupying only two-thirds of the total volume, in ~he K K R - C P A calculation, and that an appropriate utilization of the ordered phase self-consistent L M T O - A S A potentials in the disordered phase would lead to an improved result. To show this we have calculated the electronic structure and local magnetic moments in disordered PdaFe using the L M T O CPA tecbrtique developed recently by Kudrnovsky and coworkers [14,15]. This calculation would be of interest even without the K K R - C P A results for comparison, since it is the first application of the L M T O - C P A technique to a spin-polarized system. As in the K K R - C P A calculations of ref. [10]
self-consistent L M T O - A S A potentials from the ordered phase calculation are used in the disordered phase, but here the treatment of the latter, described via overlapping atomic spheres, is consistent with that of the ordered phase. The L M T O - A S A calculation for ordered Pd3Fe is performed including the combined correction [11] which was neglected in ref. ~9]. In addition, we have included the "f'" orbitals in the basis i.e. our calculation involves 16 LMTOs per atom, compared with 9 LMTOs per atom in the calculation of ref. [9]. It is our experience that for binary alloys the inclusion of the f-orbitals leads to a small change (---0.5 eV) in the total band width, while the shape of the density of states stays essentially unchanged. The inclusion of "f"orbitals in the basis also has non-negligible effects on cohesive enert;ies and equilibrium lattice parameters, etc. Although the disordered phase of Pd3Fe is supposed to be of fcc structure, small deviations fr~z,man ideal fcc structure are expected on account of the difference between the sizes of the Fe and Pd atoms. Kudrnovsky and Drchal [15] have recently shown how this relaxation of the lattice can be easily incorporated in L M T O - C P A calculations in an approximate way. Thus for disordered Pd3Fe we present results for both an ideal fcc and a relaxed lattice and discuss the effect of lattice-relaxation on the electronic structure. The rest of this paper is divided into the following sections. In section 2 we present the results for ordered Pd3Fe. In section 3 we present the results for disordered Pd3Fe with and without lattice relaxation. Bloch spectral functions for the disordered phase are briefly discussed. In section 4 we summarize the results and present our conclusions.
2. Ordered Pd3Fe The LMTO method [11] has undergone substantial development in the past decade, most of which is discussed in detail in two recent publications [16,17]. For ordered Pd3Feo(Cu3Au structure with lattice parameter 3.849 A) we have performed a self-consistent scalar-relativistic LMTO
S.K. Bose et al. / Electronic structure of Pd3Fe
calculation in the atomic sphere approximation (ASA) using the L D A parametrization of von Barth and Hedin [18]. Errors due to the ASA can be minimized using the so-called "combined correction" [11]. This correction was not included in the calculation of Kuhnen and da Silva [9]. We have included the combined correction following a method suggested in [17], based on the energy-derivative of the screened (~i*,~t-binding) structure constant matrix. Our calculation also includes the "f"-orbitals, which were neglected in [9]. The calculations of Kuhnen ar, d da Silva [9] were performed with equal-sized atomic spheres (radius s = 2.843 a.u.) for Pd and Fe. We have chosen different sphere radii for Pd (s = 2.892 a.u.) and Fe ( s = 2.681 a.u.), which were determined following the prescription suggested in [17]. According to this prescription [19] if VA° and V° are the sphere volumes in pure A and B elements and VAOBis the volume available per atom in the binary (AB) alloy, then the sphere volume VA and VB for the alloy calculation are to be determined from
v2)/v2 ag (v _vo)/vo=
(1)
and the condition
cAv +
= v°.,
(2)
where flA° and t o are the bulk-modulii and eA and c B are the concentrations of the elements A and B, respectively. Here VOB- VA° -- V° is the deviation from Vegard's law. Thus the sphere volumes for the constituents in the alloy remain as close to that in the pure system as permitted by their compressibilities and the available volume per atom in the alloy. It is our experience that sphere radii determined via (1) and (2) guarantee small charge transfer in the disordered phase. Purely for technical reasons one should also check that the overlap of two neighbouring spheres remains small. For ASA results to be reliable one should keep the quantity (s 1 + s 2 - d l a ) / S ~, i = 1, 2 below 25%, where s~ and s 2 are the radii of two overlapping (i.e. s~ + s 2 > d~z ) spheres separated by a distance d~2. It is to he noted that for an ordered structure
99
calculation the choice of sphere radii is not crucial (as long as the sphere overlaps are within reasonable limits as discussed above), since all contributions to the potential, including the Madelung, can be calculated exactly for any arbitrary division of space into spheres which may or may not be neutral. However, for the disordered alloy there is no accurate way of calculating the Madelung contribution to the potential inside a given sphere. Hence the ideal choice of sphere radii for the disordered alloy is the one which leads to more or less neutral spheres. Eqs. (1) and (2) are geared towards that purpose [20]. For ordered Pd3Fe our results agree quite well with that of Kuhnen and da Silva [9]. There are small differences mainly due to the inclusion of "f"-orbitals in our calculation. One important difference between our results and that in ref. [9] is that the difference between the centers of the down and up spin " d " b?~nds for Fe in ref. [9], C~ - C r , is 0.162 Ryd, while in our calculation it is 0.204 Ryd. Here C = W ( - ) + E r This difference, due mainly to the inclusion of " f " orbitals in our calculation, leads to a higher " d " bandwidth for Fe by - 0 . 0 4 Ryd (0.5 eV). The spin magnetic moment m, which is mainly due to the polarization of the d band, should satisfy approximately the relation C+ - C T = m L where I is the Stoner parameter. Here we have taken C s C T as a measure of the " d " band splitting due to spin-polarization. For Fe, I is estimated to lie in the range 0.066 Ryd (as ealcul?,ted by the L M T O ASA method [21] and 0.063 Ryd (as calculated by Janak [22])° With C s-Cr equal to 0.204 Ryd and I in the range 0.066-0.068 Ryd [23] we get m = 3-3.09/x B. This is in excellent agreement with the exact value of m (3.098#B) in our calcT:tation obtained by taking the difference in the number of down- and up-spin electrons at the Fe site. The value 0.162 Ryd for C~ - C r as obtained in ref. [9] would yield m = 2.38-2.45/t B. However, the exact value of Fe magnetic moment calculated in ref. [9] (3.1#B) agrees very well with that in our calculation. This is probably not surprising since the relation C s - C r = m I is expected to hold only approximately and small differences in the shapes of the densities of states in the two calculations could account for the above discrepancies.
100
S.K. Bose et al. / Electronic structure of Pd3Fe
Further discussion related to this point is postponed till section 3. For Pd the difference C ~ - Cr aod for "d" orbitals in our calculation is 0.0184 Ryd. With I = 0 . 0 5 Ryd [21,22] for Pd, this yields m = 0.368#B, whereas the exact value of Pd magnetic moment in our calculation is 0.378#B. The calculation of [9] yields C~ - C~ = 0.016 Ryd and hence m = 0.32/x B via the relation C~ - C r = mI. How~ver, the exact value of Pd magnetic moment in ref. [9] is 0.38/z B, the same as in the present calculation. Total densities of states for ordered paramagnetic and ferromagnetic Pd3Fe are shown in figs. la and b, respectively. Fig. 2 shows the spin-resolved local densities of states per Pd and Fe sites. The general shapes and features of these densities of states are in very good agreement with the results of Kuhnen and Da Silva [9]. Total density of states per atom at the Fermi energy, N(EF), for the paramagnetic case is 36 states/Ryd, leading to a value of the electronic specific heat coefficient -/= 25 m J / t o o l K. These numbers are 38 states/Ryd atom and 26.4 m J / m o l K in ref. [9]. For the ferromagnetic case N ( E F ) = l l . 6
40
20
40
20
-0.8
1 ~
! ~} Pd spin down
(c) Fe spin
30
m
0
up
~ l ( d )
60
Fe spin down
3O
-0.8
-0.4
0.0
0.4
Energy fftyd,) Fig. 2. Spin-resolved local densitites of states of or,5~red Pd3Fe alloy.
states/(Ryd atom) and -f = 7 9 m J / m o l K, compared with 11.14 states/Ryd atom and 7.61 m J / m o l K in ref. [9]. The experimental value of y as reported in ref. [2] ;.s S.2 m J / m o l K. To summarize, our results for ordered Pd3Fe are in reasonably good agreement with the earlier calculation of Kuhnen and Da Silva [9], except that the approximate relation C~ - C T = rnI seems to hold better for the potential parameters obtained in our calculation than in ref. [9]. Both in our calculation and that of ref. [9] the Fe magnetic moment is --14% higher than the experimental value. As pointed out in ref. [9] the agreement could perhaps be improved by including the spin-orbit coupling. The discrepancy may also be a shortcoming of the local density approximation.
(a)
I
0 20
-0.4
0.0
0.4
Energy {Zy~l Fig. 1. Total density of states or ordered Pd3Fe alloy. (a) Paramagnetic state; (b) ferromagnetic state
3. Disordered Pd3Fe The electronic structure of disordered PD3Fe is studied using the L M T O - C P A method discussed
S.K. Bose et aL / Electronic structure of Pd3Fe
in refs. [14,15]; and only the fen'omagnetic state is considered. The starting point is the magnetic alloy Hamiltonian for spin-up ( a = 1") and spindown (o = J, ) orientations, expressed in the nearly orthogonal LMTO representation "t [24]: H~L,R'L'
= C LS ' LL' + (a
L) '/2
o × { S ° ( 1 - y°S ° ) - - 1 }RL.R,L,(AR_,L,) 1/2 .
(3)
Here R denotes a lattice site and L = (i, m) (l ~< 2) is the orbital index. The quantities X = C, A, y, called the potential parameters [16,17], are matrices diagonal with respect to the indices R, L and o and X~L( = X~t) are independent of the index m. In general the quantities X~L describe the scattering properties of the atoms placed at the lattice site R for electrons of spin orientation o. In the alloy Pd3Fe they take randomly the values X~a.L and X[¢,L with probabilities 0.75 and 0.25, respectively, each having two different values for a = $ and ~. The structure enters the Hamiltonian via a non-random spin-independent structure constant S O with elements S~_Lm_'L'. 0 Our Hamiltoalan takes into account properly the difference in atomic levels (via C) as well as in bandwidths (via A) and shapes (via "t) of the constituent crystals. We also take into account, in an approximate way, the effect of the relaxation of the lattice (from ideal fcc structure) due to the difference in the sizes of Pd and Fe atoms. To this end we use the scaling property of the structure constant ele0 ments S~L,~'L' with respect to the interatomic distance d = [ R . - .R' I. With the z-axis chosen along the interatomic vector R - _R', of length d, the elements S~L.R,t, o are proportional to (w/d)t" +l+ 1, where w is the averagc Wig;,cr Seitz radius for the lattice. For the ideal lattice d is independent of the type of the atoms occupying the sites R and R'. In the disordered alloy the nearest-ne~ghbour ~listances for Pd-Pd, P d - F e and F e - F e pairs are expected to be different, causing a deviation from the fcc structure. Kudruovsky and Drchal [15] argue that the resulting change in the elements Sg/.,~,L, 0 can be absorbed approximately in the potential parameters z~ and "t. The
101
prescription is to keep the structure elements S0.RL.6'L' unchanged (i.e. the same as in an fcc lattice) but use the values of A and 3' appropriate for the pure species determined for the equilibrium atomic volumes in the alloy. The appropriate equilibrium atomic volumes for the alloy components are to be determined from eqs. (1) and (2) discussed in section 2. For details of the arguments leading to this result readers are referred to ref. [15]. The random alloy Hamiltonian (3) is treated within the coherent potential approximation (CPA) using the method of ref. [14]. In short, we transform the Green's function corresponding to the Hamiltonian (3) from the "I-LMTO representation to an auxiliary representation, where the CPA configurational averaging can be done exactly within the single-site approximation (SSA). After performing the configurational averaging we revert back to the original physical LMTO representation, y. This procedure is followed separately for the spin-up and the spin-down Harniltonian. For further details see ref. [14]. The potential parameters entering (3) are the charge self-consistent spin-polarized potential parameters of the closely related ordered Pd3Fe compound, evaluated at the experimental lattice constant, 3.849 ,~. Thus our LMTO-CPA calculations for the disordered phase are not charge self-consistent. The LMTO potentials for the ordered phase contain a Madelung contribution which is expected to be zero in a random alloy. The transfer of these potentials to the disordered phase introduces some error. We have attempted to redace this error by trying to keep the spheres in the ordered phase almost charge neutral. The use of eqs. (1) and (2) is geared towards that purpose. The charge transfer from a single Fe sphere to three Pd spheres was found to be 0.2 electrons for the ordered phase, i.e. each Pd sphere gained about 0.07 electrons. The use of these ordered state potentials was found to give almost charge neutral spheres for the disordered alloy (~,ee table 1). If one varied the sphere radii in the ordered phase calculation to obtain totally charge neutral spheres and transferred the corresponding potentials to the disordered alloy, the error could be further reduced. This prescription would per-
102
S.K, Bose et a L / Electronic structure of PdjFe
Table 1 Some features of the total and local densities of states in spin-polarized disordered Pd3Fe. q represents charge in a given sphere
E F (Ryd) N(EF) (states/Ryd atom) N ~ (EF) (states/Ryd spin) N s (EF) (states/Ryd spin) Np~d (EF) (states/Ryd atom spin) Np~d (EF) (states/Ryd atom spin) N~e (EF) (states/Ryd atom spin) N~e (EF) (states/Ryd atom spin) qr,d qFe
Unrelaxed lattice
Relaxed lattice
-0.1493 14.3579 4.3599 9.9980 5.3225 7.6324 1.4728 17.0947 10.0032 7.9904
-0.1582 12.8297 3.8899 8.9398 4.5228 7.6020 1.9905 12.9529 10.0469 7.8592
haps be the next best thing to a fully charge self-consistent calculation for the diso~'dered alloy. Due to a weak ferromagnetism of Fe there is the possibility to fill some Fe-majority spin holes at the cost of the minority spin electrons. As a result, the Lgcal Fe magnetic moment increases with respect to that in pure Fe and the d-up level splitting, , C,vd,d t - C re,d ~ I, is significantly smaller than the d~5own level splitting, I Cp~d,a- CF~e,d I" Thus the spin-down electrons see a much stronger disorder than the spin-up electrons. This effect seems to be common to alloys of transition metals with iron, and was found for the COl_xFex system [25] as well as for the Ni0.asFe0.65 alloy [26]. The calculated magnetic moments on Fe and Pd atoms (table 2) are similar for both ordered Table 2 Magnetic moments, m, and electronic specific heat coefficients, y, in ordered and disordered Pd3Fe
disordered Pd 3Fe (unrelaxed lattice) disordered Pd 3Fe (relaxed lattice) disordered Pd 3Fe (experiment) ordered Pd3Fe ordered Pd 3Fe (experiment) a) Ref. [5]; b) ref. [2].
mpd (b;~)
mFe (/~)
Y (mJ/mol K 2)
0.3497
2.8375
9.81
0°3486
2.8629
8.76
0.34 a) 0.3780
3.00 a) 3.0978
11.28 b) 7.9
0.57 a)
2.70 a)
8.17 b)
and disordered phases. We note much better agreement with the experimental value for the Fe moment in the disordered alloy than in ref. [10]. Both ordered and disordered phase magnetic moments for Pd and Fe sites obey the approximate relation Cd$,i - Cd.i=mli ( i = Pd or Fe) as discussed in section 2. This is to be expected, irrespective of whether the system is ordered or not, if the local-spin-density (LSD) theory can be approximated by a Stoner formalism. Our calculations show that such a formalism is valid for Pd3Fe, ordered as well as disordered. It is to be noted that in the calculation of ref. [10] the relation Cd ~' - - C d )" = m I seems to hold for Pd for which the calculated magnetic moments for the ordered (0.38#B) and the disordered (0.35/xa) phases are very dose. For Fe, for which the above relation seems to be violated, the difference between the ordered and disordered state moment is larger and the agreement with the experimental value (for the disordered alloy) poorer. The main reason for the improved Fe moments in the disordered phase in our calculation is that the disordered and the ordered phases are treated consistently. Self-consistent L M T O - A S A potentials from the ordered phase calculation are used in the L M T O - C P A calculation for the disordered phase using single-site ASA. The accuracy of such single-site ASA c:aiculation has been discussed in ref. [15]. In ref. [~.0] the disordered alloy potentials are constructed from the L M T O - A S A calculation for the ordered alloy, but the CPA calculations are done using the K K R method. Atomic potentials in the L M T O - A S A method are determined for slightly overlapping W - S spheres and for use in the K K R method they should be adjusted to the touching muffin-tin spheres occupying approximate!y 2 / 3 of the volume of W - S sphere. Another uncertainly in this procedure concerns the choice of the alloy muffin-tin zero because of the mismatch of the muffin-tin zeroes of the constituents. An additional reason for the improved Fe moment in our calculation is more careful determination of the ordered alloy potential by using a larger basis set and incorporating combined correction terms. This resdts in a greater splitting of the up- and down-spin " d " bands of Fe by -- 0.02 Ryd, and thus in larger moments on Fe atoms.
S . K . B o s e et aL
/ Electronic
The spin-up and spin-down densitites of states projected on Pd and Fe atoms are presented in figs. 3 and 4 for the cases without and with lattice relaxations, respectively. For spin-up electrons for which C Pd.dt t Cv~.d nearly coincide the main effect of disorder is the difference in the hopping elements (parameter A) for Pd and Fe atoms. Th;s is a weak effect and hence sharp features in the spin-up densities of states, reminiscent of the ordered state, persist in the disordered alloy. The difference in the hopping dements is overestimated in the calculations that neglect the effect of lattice relaxation. The reason is that the F e - F e and P d - P d distances are the same in this case, dictated by the alloy lattice constant which is close to that for pure Pd for Pd3Fe alloy. For the relaxed lattice case we assume that the nearestneighbour distances for the F e - F e and P d - P d pairs are respectively close to the values in the pure Fe and pure Pd cry,stals. While for the ordered Pd3Fe alloy there is no nearest-neighbour F e - F c pair, in random alloy there is a finite probability for such pairs. Thus lattice relaxation brings the
(a) spin d o w n
/i A
ii
(b)spin up
/Ak,
-.6
-.6
-.4
-,2
0
Energy faya) Fig. 3. Spin-resolved densities of states in disordered Pd3Fe, calculated for an ideal (unrelaxed) lattice. For each spin the solid line represents the total density of states, the long- and the short-dashed lines represent the Fe and Pd projected densities of states, respectively.
103
s t r u c t u r e o f Pale F e
(a) spin d o w n
,-\
/ X
/
/
~
"
i
1
(b) spin up
!/,/
-,a
-.6
~.4
-.2
0
Energy ~y~)
Fig. 4. Spin-resolveddensities of states in disordered Pd3Fe. calculated for a relaxed lattice. For each spin the solid line represents the total densityof states, the long- and short-dashed lines represent the Fe and Pd projected densities of states. respectively.
atoms in~'~he F e - F e pair closer together, increasing the h o p ~ $ between such atoms. The effective a for the Fe a~b~ in the relaxed lattice is larger than in the o r d e r e a ~ t t i c e and is closer to that for Pd atoms. Here the r~'~ disorder in space closes the gap in the hopping eidiqents between different pairs i.e. reduces the disorder( in the potential parameters or the scattering ~:2perties of the atoms. The Fe-projected density~-~f states is broader for the relaxed lattice, while th~¢hange in the Pd-projected density of states is quit~",~,nall~ because the alloy is Pd-rich. The situation is s~'-~r, lar to that in the Cu-rich Cu7sPd25 random alloy, where the lattice relaxation has observable consequences [15,27]. For the spin-down electrons the diagonal disorder due to large [ Cp~.d ~ - CF~e.di splitting dominates over the weake~ off-diagonal disorder due to different nearest-neighbour hoppings; and the density of states curves are significantly smoothed. The peak in the Fe-projected density of states is broader for the relaxed lattice, due to the smaller F e - F e nearest-neighbour distance than in
S,K. Bose e: al. / Electronic structure of Pdj Fe
104
the unrelaxed lattice. Again the local density of states for Pd is similar in the relaxed and the unrelaxed lattice calculations. Note that the general shape and features of the density of states for our unrelaxed lattice calculation agree quite well with that of ref. [10]. In a recent paper dealing with the interrelation between compositional order and magnetism Staunton et al. [28] have presented K K R - C P A results for spin-polarized densities of states as well as some measures of ordering tendencies in Pd3Fe and several other alloys. Our densities of states show reasonable agreement with those presented in [281. The difference in the behaviour of the up- and down-spin electrons in the presence of disorder is even more clearly seen in the spin-resolved Bloch spectral functions defined as
A°(k, EI=-IImY',(GZL(k_,E+iO+)),
(4)
L
where G°(z)=(z-H") -i, and ( . . . ) denotes configurational averaging. In fig. 5 we present the results for the case where the effect of lattice
X
A
-.8
-.,5
-.4
-.2
0
Energy (Ryd) Fig. 5. Spin-resolved spectral functions in disordered Pd3Fe, calculated for a relaxed lattice along the X --* F --* L line in the irreducible Brillouin zone of an lcc structure. The solid and the dashed lines represent the spin-up a~.cl ~pin-down components, respectively.
relaxation is included, along the X - F - L line in the irreducible fcc Brillouin zone. We have retained a sma!l imaginary part ~ = 0.005 Ryd in the argument of G. We note sharp Lorentzian-like peaks for weakly disordered spin-up electrons and much broader, non-Lorentzian peaks of spin-down electrons, especially in the impurity region of Fe " d " states between - 0 . 3 and 0.0 Ryd. A direct consequence of larger diagonal disorder acting on the spin-down electrons is the shift of the corresponding spectral densities upwards in comparison with the spin-up ones. This behaviour is in qualitative agreement with similar study presented in ief. [10] for the unrelaxed lattice.
4. Conclusions We have studied the electronic structure of ordered and disordered PdaFe alloy using the LMTO method. The ordered structure is studied using a basis of s, p, d and f orbitals and incorporating the combined correction terms. We present a comparison of our results with an earlier LMTO calculation of Kuhnen and Da Silva [9], where the " f " orbitals and the combined correction terms were not taken into account. Self-consistent potentials from the ordered alloy calculation are used in an L M T O - C P A calculation for the disordered alloy using single-site atomic sphere approximation. As in the earlier study of disordered Pd3Fe [10] and other Fe-based alloys [25,26] spin-down electrons see a greater disorder than spin-up electrons, which is reflected in the spin-resolved local densities of states and spectral functions. Effect of lattice relaxation, taken into account in an approximate way in our calculation, is more pronounced for the Fe local density of states than for Pd, as is expected for this Pd-rich alloy. The main result of the L M T O - C P A calculation is an improved value for the Fe magnetic moment, in better agreement with the experimental value than in the K K R - C P A study of ref. [10]. This, we believe, is due to a consistent tre,atment of the ordered and the disordered alloy in our calculation as well as a more careful determination of the ordered state potentials than in ref. [10].
S.K. Bose et a L / Electronic structure of Pd3Fe
Acknowledgement One of us (S.K.B.) would like to acknowledge partial financial support provided by the Natural Sciences and Engineering Research Council of Canada.
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