Electronic structure and magnetic properties of random alloys: Fully relativistic spin-polarized linear muffin-tin-orbital method. A. B. Shick. Institute of Physics ...
PHYSICAL REVIEW B
VOLUME 54, NUMBER 3
15 JULY 1996-I
Electronic structure and magnetic properties of random alloys: Fully relativistic spin-polarized linear muffin-tin-orbital method A. B. Shick Institute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, CZ-180 40 Praha 8, Czech Republic and Institute of Solid State Chemistry, Ural Division of RAS, GSP-145 Pervomaiskaia 91, Ekaterinburg, Russia
V. Drchal and J. Kudrnovsky´ Institute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, CZ-180 40 Praha 8, Czech Republic and Institute for Technical Electrochemistry, Technical University, Getreidemarkt 9, A-1060 Vienna, Austria
P. Weinberger Institute for Technical Electrochemistry, Technical University, Getreidemarkt 9, A-1060 Vienna, Austria ~Received 6 March 1996! A Green’s function method is developed to calculate the electronic and magnetic properties of random alloys containing heavy elements. Based on the local spin density approximation, the all-electron spin-polarized version of the fully relativistic linear muffin-tin-orbital method in the tight-binding representation is used to describe disorder within the coherent potential approximation. The method is first tested on ferromagnetic 3d metals ~Fe, Co, Ni! and then applied to study the electronic and magnetic properties of the fcc alloy systems Co 50Ni 50 and Co 50Pt 50 . @S0163-1829~96!06427-2#
ˆ 5c a• @ p2 ~ e/c ! A~ r!# 1 ~ b 2I ! mc 2 1V ~ r! I . ~1! H D 4 4
I. INTRODUCTION
Relativistic effects become important in heavier elements, particularly for 5d transition metals and for the actinides. The interplay of spin polarization and relativistic effects ~mass-velocity and Darwin correction, and particularly the spin-orbit coupling! can affect the ground state ~lattice parameter and magnetic structure, values of local magnetic moments, etc.! and can lead to a number of important effects, such as magnetic anisotropy energies, polarization dependence of photoemission spectra, and magneto-optical phenomena. We present a fully relativistic generalization of the tightbinding ~TB! linear muffin-tin-orbital ~LMTO! method to the case of disordered magnetic alloys. This approach is based on a relativistic generalization of the local spin density approximation and employs the Kohn-Sham-Dirac equations in the presence of a magnetic field. The alloy disorder is treated within the coherent potential approximation ~CPA! by using a Green’s function formalism. The relativistic LMTO method1–3 for nonmagnetic systems is a straightforward generalization of the nonrelativistic LMTO method.4,5 It turns out that rather simple modifications are needed in the nonmagnetic case, whereas in magnetically polarized systems one encounters certain conceptual as well as technical problems. Originally, the relativistic LMTO method for the spin-polarized case was developed by Ebert6 and by Solovyev et al.7–9
A ~ r! 5A ext~ r! 1
1 c
E
d E xc@ j # j ~ r8 ! d 3r 1c , u r2r8u d j ~ r!
~2!
where A ext(r) corresponds to an external field. This formulation, however, leads to serious problems because the actual form of E xc@ j # is unknown. In order to avoid these difficulties, a simplified theory12 that captures most of the effects of magnetic fields is commonly used. It is based on the Gordon decomposition16 of the four-current ~see also Ref. 17, or Ref. 10!. In this simplified theory the spin part of the four-current is retained, while the orbital part, which would lead to diamagnetic effects, is neglected. Only the coupling of the effective magnetic field to the spin density is considered. The Kohn-Sham-Dirac equations are then given by ˆ C 5E C , H i i i
ˆ 5H ˆ 1H ˆ , H 0 1
~3!
ˆ consists of a nonmagnetic term where the Hamiltonian H ˆ 22–24 ˆ and a magnetic part H H 0 1
II. RELATIVISTIC TB-LMTO METHOD FOR SPIN-POLARIZED SYSTEMS A. Dirac equation in the presence of a magnetic field
The Dirac Hamiltonian in the presence of an electrostatic field, described by the potential V(r), and a magnetic field B(r)5¹3A(r), as given by its vector potential A(r), is of the form10 0163-1829/96/54~3!/1610~12!/$10.00
Exchange and correlation effects can be included, at least in principle, within a relativistic generalization of the density functional formalism.11–15 The exchange and correlation energy E xc@ j # has to be a Lorentz scalar and a functional of the relativistic four-current j(r)5„c%(r),j(r)… which includes both the spin and the orbital contributions. The effective four-potential A(r)5„V(r),A(r)… is then defined as
54
ˆ 5c a•p1 ~ b 2I ! mc 2 1V eff~ r! I ,H ˆ 5 b S•Beff@ r ,m# , H 0 4 4 1 ~4!
a5 1610
S D 0
s
s 0
,
b5
S
I2
0
0
2I 2
D
,
S5
S D s
0
0
s
.
~5!
© 1996 The American Physical Society
54
ELECTRONIC STRUCTURE AND MAGNETIC PROPERTIES OF . . .
Here, s is the vector of Pauli matrices, p is the momentum operator, and I n denotes an (n3n) unit matrix.10 The energy E is related to the full relativistic energy W by W5E1mc 2 . In the following Rydberg atomic units, \51, m51/2, c5274.071 979, and e 2 52, are used. According to the local ~spin-polarized! density functional theory,12,13,18–20 the effective potential V eff@ %,m# is given in terms of the functional derivative with respect to the particle density %(r) V eff@ %,m#~ r! 5V ext~ r! 1
E
2% ~ r8 ! d E xc@ %,m# 1 . u r2r8u d % ~ r!
d 3r
d E xc@ %,m# B @ %,m#~ r! 5 m B B ~ r! 1 , d m~ r! ext
~8!
Its only constant of motion is J z . The solutions of the corresponding Dirac equation ˆ F ~ E,r! 5EF ~ E,r! H m m
~9!
for a given energy E and quantum number m corresponding to J z are of the form F m ~ E,r! 5
S
2
(k F km~ E,r! ,
(
~11!
S
D
12 k d 1 c f km ~ r ! 1 @ E2V ~ r !# g km ~ r ! dr r
(
^ km u s z u k 8 m 8 & g k 8 m 8 ~ r ! 50,
k8m8
F km ~ E,r! 5
S
i f km ~ E,r ! V 2 km ~ rˆ !
^ km u s z u k 8 m 8 & 5 d l l 8 d mm 8
,
~10!
where V km (rˆ) are the spin-spherical harmonics.10,21 The radial amplitudes g km (r) and f km (r) satisfy the following set of coupled differential equations ~see, in particular, Refs. 21–25!:
(
s561/2
sgn~ s !
3c ~ l j 21 ; m 2s,s ! c ~ l j 8 21 ; m 2s,s ! 52 d mm 8 1
F
2m d 2 k 11 kk 8
A~ k 2 k 8 ! 2 24 m 2 u k 2 k 8u
G
d k 1 k 8 ,21 ,
~13!
are expressed in terms of Clebsch-Gordan coefficients. Since the coupling between solutions with different quantum numbers l corresponds to the so-called magnetic spin-orbit interaction „}c 22 r 21 @ dB(r)/dr # L•S…, 25 this term is usually neglected as it is of order c 22 and since B(r) varies only slowly with respect to r. However, quite recently, Jenkins and Strange26 pointed out that this approximation might lead to incorrect results for magnetocrystalline anisotropies. In the present work the approximation D l 50 is assumed. The infinite set of coupled radial Dirac equations thus blocks into coupled equations for each pair l m . We will denote these blocks by the index L5( l , m ). When u m u 5 l 1 21 , there is no coupling and there is only one solution with quantum numbers km as in the nonmagnetic case. When u m u , l 1 21 , there are four coupled equations ~11! and ~12! for the four unknown functions g km , f km ,g k 8 m , and f k 8 m . The indices k and k 8 52 k 21 will be denoted by k 1 and k 2 . Equations ~11! and ~12! yield two linearly independent regular solutions, the initial conditions of which are discussed in Appendix A. The solutions of ~9! can therefore be classified with respect to their behavior at r→0, i.e., Eq. ~10! can be rewritten as F l m ~ E,r! 5
D
~12!
where the matrix elements of s z ,
( k5k ,k 1
g km ~ E,r ! V km ~ rˆ !
D
B~ r ! ^ 2 km u s z u 2 k 8 m 8 & c f k 8 m 8 ~ r ! 50, c 2 k8m8
~7!
where m B is the Bohr magneton. It should be noted that Bext is a true external magnetic field, while Beff is rather an effective spin-dependent potential. Within the atomic sphere approximation ~ASA!, the effective potential V eff@ %,m# is represented by a set of spherically symmetric potentials V R(r) in individual Wigner-Seitz spheres. Similarly, the effective spin-dependent potential Beff is represented in each Wigner-Seitz sphere by a ‘‘magnetic field’’ BR(r)5B R(r)nR , where B R(r) is the ~spherically symmetric! amplitude of the effective magnetic field and nR is a unit vector which defines its direction. For a single sphere, the direction of the magnetic field can be assumed to point along the z direction, n5zˆ, where zˆ is a unit vector parallel to the z axis. The Kohn-Sham-Dirac Hamiltonian ~3! reduces then to ˆ 5c a•p1 ~ b 2I ! mc 2 1V ~ r ! 1 m B ~ r ! b S . H 4 B z
D
11 k d E2V ~ r ! 1 g km ~ r ! 2 11 c f km ~ r ! dr r c2
2B ~ r ! ~6!
Similarly, the effective magnetic field Beff@ %,m# is expressed via the functional derivative with respect to the spin density m, eff
S
1611
2
S
g k l m ~ E,r ! V km ~ rˆ ! i f k l m ~ E,r ! V 2 km ~ rˆ !
D
,
~14!
where two independent solutions are labeled by indices l 1 5 k 1 , l 2 5 k 2 . Although the indices l are numerically equal to the k ’s, they carefully should be distinguished because the l’s are connected with the behavior at r→0, while the k ’s refer to spin-spherical harmonics, and, consequently, control the matching condition at the Wigner-Seitz sphere. Solutions for different l 1 and l 2 are linearly independent and have to be orthonormalized. In what follows, we will
´ , AND WEINBERGER SHICK, DRCHAL, KUDRNOVSKY
1612
assume that this orthonormalization is performed, i.e., ^ F l (E) u F l 8 (E) & 5 d ll 8 . By differentiating with ˙ (E) u F (E) & respect to the energy E, one gets ^ F l l8 ˙ 1 ^ F l (E) u Fl 8 (E) & 50, from which follows that ˙ (E) & 50, i.e., orthogonality for l5l 8 , whereas ^ F l (E) u F l ˙ (E) & are not orthogofor lÞl 8 the states u F l (E) & and u F l8 nal. In view of the linearized muffin-tin orbitals, however, it is suitable to perform a further orthogonalization transformation ~see Appendix A!. In the following the resulting orthogonal functions will be denoted by u C l (E) & . In the ASA limit of the LMTO theory4 the solutions of the Dirac equation for E5V(r) and B(r)50 that are regular (r) and irregular (i) at the origin, respectively, can be formulated as
S
1 F km ~ r! 5 2 ~ 2 l 11 ! ~r!
~i!
F km ~ r! 5
S
SD SD r w
i
k 111 l wc
SD SD w r
k2l i wc
l
V km ~ rˆ!
r w
l 21
V 2 km ~ rˆ!
l 11
w r
V km ~ rˆ! l 12
V 2 km ~ rˆ!
D
D
,
~15!
.
~16!
The construction of canonical muffin-tin orbitals proceeds in two steps. First, auxiliary muffin-tin orbitals are defined such that their angular dependence at the Wigner-Seitz sphere is particularly simple, namely, the large component is proportional to V km , and the small component is proportional to V 2 km at u ru 5s. Since they refer to km -like partial waves, these orbitals are denoted with indices of the type k and are defined in terms of a general solution of the Dirac equation regular at r50, which is given by a linear combination of solutions ~14!, and a suitable linear combination of regular free-space solutions ~15! inside the Wigner-Seitz sphere, and the irregular free-space solution ~16! outside the Wigner-Seitz sphere,
(
l5 k 1 , k 2
1
8
1
~r!
2
0m
F Rk 8 m ~ E,r! P Rk 8 k ~ E !~ r Rs R! .
N l0 mk
N l0 mk
N l0 mk
1 1 2 1
1 2 2 2
D
,
P 0L5
S
P k0 mk
P k0 mk
P k0 mk
P k0 mk
1 1
1 2
2 1
2 2
D
. ~18!
Using the following matrix notation for the radial amplitudes: g L5
S
g k1l1m
g k1l2m
g k2l1m
g k2l2m
D
,
f L5
S
f k1l1m
f k1l2m
f k2l1m
f k2l2m
D
, ~19!
and a similar notation for the analog of the logarithmic derivative: D L~ E ! 5sc f L~ E ! g 21 L ~ E ! 2 k 2I 2 , k 5
S
k1
0
0
k2
D
, ~20!
the potential functions and the normalization functions can be written as the following matrices: P 0L~ E ! 52 ~ 2 l 11 !
SD w s
2 l 11
@ D L~ E,s ! 1 l 11 #
N 0L~ E ! 52 ~ 2 l 11 !
SD w s
l 11
~21!
21 g 21 . L ~ E,s !@ D L~ E,s ! 2 l #
~22!
As follows from the second Green’s theorem ~see Appendix A!, the potential function matrix ~21! is symmetric. The canonical muffin-tin orbitals are connected with a condition of regularity at the origin, i.e., they refer to the independent solutions of Dirac equation, and thus are denoted by indices of the type l. Formally, they are obtained by multiplying ~17! from the right with @ N 0Rm (E) # 21 0 x Rl m ~ E,r ! 5C Rl m ~ E,r ! 1
3 @ N 0Rm ~ E !# k21 l 5
(
k5k1 ,k2
(
k,k85k1 ,k2
0m
~r!
F Rk 8 m ~ E,r! P Rk 8 k ~ E !
~ r Rs R! .
~23! The radial amplitudes are expanded up to linear terms in «5E2E n ,L around an energy E n ,L ,
0m C Rl m ~ E,r! N Rl k~ E !
( k 5k ,k
S
N l0 mk
3 @ D L~ E,s ! 2 l # 21 ,
B. Muffin-tin orbitals and linearization
˜ x Rkm ~ E,r! 5
N 0L5
54
g L~ E,s ! 'g n ,L1«g˙ n ,L , f L~ E,s ! ' f n ,L1« ˙f n ,L ,
~24!
where the following notation is used:
~17!
g n ,L5g k ~ E n ,L ,s ! ,
g˙ n ,L5g˙ L~ E n ,L ,s ! ,
It is required that ˜ x Rkm (E,r) is continuous at the Wigner0m Seitz sphere ( u ru 5s). Here, N R k l (E) are the normalization 0m functions, and P Rkk 8 (E) are the potential functions. Note that the matching condition is formulated for each value of the quantum number m separately. Due to the linear independence of V km and V 2 km the matching condition yields eight equations for the eight elements of the unknown matrices N 0L and P 0L ,
f n ,L5 f L~ E n ,L ,s ! ,
˙f 5 ˙f ~ E ,s ! . n ,L L n ,L
~25!
The Lth block of the LMTO canonical potential function matrix is given by P 0L~ E ! 5 @ g L1W TL~ EI 2 2C L! 21 W L# 21 ,
~26!
where the superscript T denotes a transposed matrix, while the Lth block of the normalization function is
54
ELECTRONIC STRUCTURE AND MAGNETIC PROPERTIES OF . . .
N 0L~ E ! 5
A
w ~ EI 2 2C L! 21 W LP 0L~ E ! . 2
Here, the C L , W L , and g L are matrices of potential parameters as given by
SD
s 1 g L5 2 ~ 2 l 11 ! w
2 l 11
~ D n ,L1A n ,L2 l !
3 ~ D n ,L1A n ,L1 l 11 ! 21 ,
SDSD
w W L5 2
1/2
s w
l 11
~28!
g Tn ,LA n ,L~ D n ,L1A n ,L1 l 11 ! 21 , ~29!
C L5E n ,LI 2 1sg Tn ,LA n ,Lg n ,L ~30! and where A n ,L52 ~ sg n ,Lg˙ Tn ,L! 21 . ~31!
The matrices g, f ,D, g ,W,C,A,N , and P do not commute with each other. Some are symmetric (D, g ,A,C, and P 0 ), while others (g, f , W, and N 0 ) are nonsymmetric. This is connected with the different meaning of the indices k and l. For the matrices D, g ,A, and P 0 both indices are of type k . For the matrix C both indices are of type l, for the matrices g and f the left index is of type k and the right one is of type l, while for the matrices W and N 0 the left index is of type l and the right one is of type k . The potential parameter W ~in matrix formulation nonsymmetric! corresponds to the square root of the parameter D such that D5W T W. Let us note that the matrix of normalization functions N 0 corresponds to a square root of P˙ 0 in the following sense: 0
P˙ 0L~ E ! 5
~32!
The matrices of radial functions g L(E,r) and f L(E,r) have dimension 232 or 131 according to whether u m u , l 1 21 or u m u 5 l 1 21 . The canonical potential function matrix is then constructed L-blockwise, P RL,R8L8 ~ E ! 5 P 0RL~ E ! d RR8 d LL8 , 0
c ~ l 8 j 8 1/2; m 8 2 s , s !
0,nr
3S R8 L 8 ,RL c ~ l j1/2; m 2 s , s ! ,
~34!
where L5( l , m 2 s ) and L 8 5( l 8 , m 8 2 s ). It is defined via 0,nr its non-relativistic counterpart S R8 L 8 ,RL . The large components of the irregular and regular free-particle solutions ~15! and ~16! fulfill the identity
S
w u r2R8 u 52
D
l 8 11
V k 8 m 8 ~d r2R8!
S
1 u r2Ru 2 ~ 2 l 11 ! w
( km
D
l
d! S V km ~ r2R Rkm ,R8 k 8 m 8 , 0
which follows from a similar identity valid in the nonrelativistic case,5 and from the definition ~34!. The small component of any free-space solution F of the Dirac equation @i.e., for V(r)5E and B(r)50# can be expressed via its large component ~see, e.g., Ref. 10! as F5
0
2 0 @ N ~ E !# T N 0L~ E ! . w L
s 561/2
~35!
2sg Tn ,LA n ,L~ D n ,L1A n ,L1 l 11 ! 21 A n ,Lg n ,L ,
D n ,L5sc f n ,Lg 21 n ,L2 k 2I 2 ,
(
0
S R8 k 8 m 8 ,Rkm 5
~27!
1613
~33!
where the P 0RL(E) are the potential functions as defined in Eq. ~26! for the atomic species located at R. The potential functions and the potential parameters are diagonal in R and block diagonal in L. The potential functions, originally defined for real energies, can be continued analytically into the complex plane. C. Relativistic structure constants and screening transformation
The other important quantity in the LMTO formalism is the relativistic ~canonical! structure constant matrix
SD f x
,
i x 52 s•pf . c
~36!
Applying the identity ~36! to both sides of ~35!, we find an expansion of the same type to be valid also for the small components. Consequently, ~i!
F R8 k 8 m 8 ~ r! 52
0 ! F ~Rrkm ~ r! S Rkm ,R8k 8 m 8 , ( km
~37!
which is the relativistic analog of the nonrelativistic relation.5 It should be noted that Eq. ~37! is exact. In the spin-polarized case, the relativistic ~canonical! structure constant matrix is the same as in the nonmagnetic case because in both cases it describes the geometry of an empty lattice. Formally this follows from the fact that the structure constants are defined in terms of solutions of the Dirac equation in the absence of external fields. An important feature of the LMTO method is that it can be formulated in various representations27 characterized by a diagonal screening matrix a with elements a Rkm ,R8 k 8 m 8 5 d RR8 d kk 8 d mm 8 a Rkm , a Rkm [ a Rl . This screening transformation remains valid also in the relativistic case, because of the completeness and orthogonality relations for the Clebsch-Gordan coefficients ~see Ref. 28!. Consequently, the concept of the screening transformation remains valid also in the spin-polarized case without any further modification. When switching from the canonical representation ~superscript 0) to a screened representation a , the potential function matrix is transformed according to a P RL ~ z ! 5 P 0RL~ z !@ 12 a RLP 0RL~ z !# 21
5 @ g L2 a L1W TL~ zI 2 2C L! 21 W L# 21 ,
~38!
and similarly the structure constants transform by a
S Rkm ,R8k 8 m 8 5 @ S 0 ~ 12 a S 0 ! 21 # Rkm ,R8k 8 m 8 .
~39!
´ , AND WEINBERGER SHICK, DRCHAL, KUDRNOVSKY
1614
The canonical structure constant matrix S 0 is long ranged because for large u R2R8 u its elements behave like u R2R8 u 2 l 2 l 8 21 . On the other hand, by a proper choice of a screening matrix a one can achieve fast decay of S a in real space. It has been found27 that for close-packed lattices, the so-called tight-binding muffin-tin-orbital ~TB-MTO! representation b , which is site independent but l dependent, and specified by b s 50.3485, b p 50.053 03, and b d 50.010 71 ( b l 50 for l .2) gives the fastest and an essentially monotonic decay in real space. In fact, S b vanishes beyond the second coordination shell for close-packed lattices. The Hamiltonian H and the overlap matrix O in the socalled orthogonal MTO representation g are of the form8,27 g
H Rl m ,R8 l 8 m 8 5C Rll 8 d RR8 d mm 8 1
g W Rl mkm S Rkm ,R8 k 8 m 8 ~ W T ! R8 k 8 m 8 l 8 m 8 , ( kk
8
g
O Rl m ,R8 l 8 m 8 5 d RR8 d ll 8 d mm 8 ,
~40!
where the potential parameters C,W, and g were defined in ~28!–~29!. The geometry of the problem enters the Hamiltonian via the structure constant matrix S g defined in ~39! with a 5 g . Considering only the simplest possible case, namely, an ideal periodic lattice randomly occupied by atoms A and B ~neglect of lattice relaxations!, the ideal bulk structure constant matrix S 0 is used in ~39! to define S g .
The Green’s function ~GF! of ~40! is given by g
~41!
In the following the superscript g is omitted. By using the relation between the GF’s in different MTO representations,27 one can express G(z) in terms of an arbitrary screened MTO representation a as a
( kk
8
a a m Rl mkm ~ z ! g Rkm ,R8 k 8 m 8 ~ z !
3~ m a ! R8k 8 m 8 l 8 m 8 ~ z ! , T
~42!
~43!
a
a 21 m Rl W P a ~ z !# Rl mkm . mkm ~ z ! 5 @~ z2C !
E
~ BZ!
d 3 k@ Pa ~ z ! 2S a ~ k!# 21 .
~46!
In the case of homogeneous bulk alloys, all site-diagonal quantities are independent of the actual lattice site. We shall thus omit the site index R. In ~46!, Pa (z) is the coherent potential function which is a site-diagonal matrix with indices km . The scattering of electrons by individual atoms is described by the sitediagonal t matrices t a Q ~ z ! 5 @ P a ,Q ~ z ! 2Pa ~ z !# $ 11F a ~ z !@ P a Q ~ z ! 2Pa ~ z !# % 21
~47!
~ Q5A,B ! ,
such that within the single-site approximation ~CPA!
(
Q5A,B
c Q t a Q ~ z ! 50.
~48!
F a Q ~ z ! 5 $ 11F a ~ z !@ P a Q ~ z ! 2Pa ~ z !# % 21 F a ~ z ! ~49! is used to calculate the conditionally averaged on-site element of the physical Green’s function29 aQ
^ G ~ z ! & Rl m ,Rl 8 m 8 5l l m l 8 m ~ z ! d mm 8 Q
aQ Q m lamkm ~ z ! F km , k 8 m 8 ~ z ! ( kk
8
T
is the so-called auxiliary or nonphysical Green’s function, a a (z) and m R (z) are matrices diagonal in R, which can and l R be expressed in terms of potential parameters, l Rl m l 8 m ~ z ! 5 $ @ z2C1W ~ g 2 a ! 21 W T # 21 % Rl m l 8 m ,
1 V BZ
3@ m a Q ~ z !# k 8 m 8 l 8 m 8 ,
where a g Rkm ,R8k 8 m 8 ~ z ! 5 @~ P a ~ z ! 2S a ! 21 # Rkm ,R8k 8 m 8
F a~ z ! 5
1
G Rl m ,R8 l 8 m 8 5l Rl m l 8 m ~ z ! d RR8d mm 8 1
CPA ~for details see Ref. 29!. The site-diagonal potential parameter matrices C R ,W R , and g R randomly take on two values, C A ,W A , and g A with the probability c A , or C B ,W B , and g B with the probability c B 512c A . The on-site element of the configurationally averaged auxiliary Green’s function F a (z)5 ^ g a (z) & RR is a matrix in km and can be expressed as an integral over the first Brillouin zone of volume V BZ ,
This CPA equation has to be solved for the effective medium characterized by the coherent potential function Pa (z). The conditionally averaged on-site block of the auxiliary Green’s function29
III. GREEN’S FUNCTIONS AND THE COHERENT POTENTIAL APPROXIMATION
G Rl m ,R8l 8 m 8 5 @~ z2H g ! 21 # Rl m ,R8l 8 m 8 .
54
~44! ~45!
The configurational averaging of the Green’s function for a homogeneous disordered alloy is performed within the
~50!
which is needed, e.g., to calculate component resolved and averaged density matrices, D Q (E) and D(E), respectively, 1 Q D Q ~ E ! 52 Im^ G ~ E1i0 ! & RR , p
D~ E !5
(Q c Q D Q~ E ! . ~51!
These density matrices are used for charge self-consistent calculations and to calculate the densities of states. So far we have considered only the case that the effective magnetic field Beff is oriented parallel to the z axis. This is an unnecessary restriction that can be easily removed by a transformation to a new frame of reference. One can either transform the potential functions ~but then their block-diagonal form is lost!, or one can perform an inverse transformation on the structure constants. From a practical point of view, the former approach is more convenient.8
54
ELECTRONIC STRUCTURE AND MAGNETIC PROPERTIES OF . . . IV. CHARGE AND SPIN DENSITIES
The spherically averaged particle density for a site occupied by an atomic species of type Q is given by
1615
The projection of the total angular momentum onto the z axis Q can be found in terms of these Q km , J Q,z 5
Q m Q km . ( km
~58!
1 Q, ~ gg ! Q, ~ f f ! % ˜ Q~ r ! 5 @ Rkmkm ~ r,r ! 1Rkmkm ~ r,r !# 1% Q,core~ r ! , 4 p km ~52!
The corresponding projection of the orbital angular momentum is then given as the difference
where the R(gg) and the R( f f ) are the contributions from the large and small components, respectively,
L Q,z 5J Q,z 2S Q,z
(
because J5L1S. The z projection of the total magnetic dipole moment of an atom Q is then given by Q,z MQ 12S Q,z ), while the spin and orbital magnetic tot5 m B (L Q,z and moments are calculated as MQ s 52 m B S Q Q,z M l 5 m B L , respectively. The spectroscopic g factor is Q Q then g Q s 52M tot/M s .
Q, ~ gg !
Rkmk 8 m 8 ~ r,r 8 ! 5
( ll
8
E
EF
2`
dEg kQl m ~ E,r ! D l m l 8 m 8 ~ E ! g k 8 l 8 m 8 ~ E,r 8 ! , Q
Q
Q, ~ f f !
Rkmk 8 m 8 ~ r,r 8 ! 5
( ll
8
E
EF
2`
V. TOTAL ENERGY
dE f kQl m ~ E,r ! D l m l 8 m 8 ~ E ! f k 8 l 8 m 8 ~ E,r 8 ! . Q
Q
~53! g kQl m (E,r)
f kQl m (E,r)
The radial amplitudes and are regular solutions of the radial Dirac equations ~11! and ~12!, orQ thonormalized according to ~67!, and the D l m l 8 m 8 (E) are defined via the ~physical! Green’s function in ~51!. The integrals in ~53! are evaluated in terms of the moments of the Q D l m l 8 m 8 (E). The % Q,core is the core electron charge density. For core states the Dirac equation in the presence of a magnetic field is solved using the method developed by Ebert.30 The total number of electrons in the atomic sphere is given by Q Q 54 p
E
s
0
˜ Q ~ r ! 5N Q, ~ core! 1 drr 2 %
(E lm
EF
2`
dED lQm l m ~ E ! , ~54!
where N Q,(core) is the number of core electrons for atomic species Q. The spherical average of the spin density in an atomic sphere occupied by an atomic species of type Q is given by ˜ Q~ r ! 5 m
1 Q, ~ gg ! @ Rkmk 8 m 8 ~ r,r ! ^ km u su k 8 m 8 & 4 p km k 8 m 8
( ( Q, ~ f f !
˜ Q,core~ r ! . 2Rkmk 8 m 8 ~ r,r ! ^ 2 km u su 2 k 8 m 8 & # 1m ~55! The projection of the spin moment corresponding to the atomic species Q onto the z axis is defined as S Q,z 54 p
E
s
0
˜ Q,z ~ r ! . drr 2 m
~56!
From ~52! it follows immediately that the number of km -like valence electrons is given by Q Q km 5
E
s
0
Q, ~ gg ! Q, ~ f f ! drr 2 @ Rkmkm ~ r,r ! 1Rkmkm ~ r,r !# .
~59!
~57!
The external magnetic field Bext is generated by magnetic dipole moments from all other sites. In systems with inversion symmetry ~as are, e.g., cubic lattices! it is zero. The total energy for spin-polarized systems within the ASA is given by the usual expression occ
E tot5
(i « i 2 (R 2 1 2
E
sR
E (E E ( E (R
sR
d 3 rmR~ r! BR~ r! d 3r
R
R
d 3 r% R~ r! V R~ r!
sR
2Z R
% R~ r! % R~ r8… d 3 r8 zr2r8u sR q Rq R8 % R~ r! 1E xc@ %,m# 1 u ru u R2R 8u ” R8 R5
d 3r
sR
(
~60! where q R5Q R2Z R is the net charge at site R. The first three terms correspond to the kinetic energy, the fourth term is the Hartree energy, the fifth term is the Coulomb interaction energy of electrons and nuclei, the sixth term is the exchangecorrelation energy, and the last term is the Madelung electrostatic energy. It should be recalled that the correct form of the exchange-correlation energy in the spin-polarized relativistic theory is still an open question. Various fits to E xc@ %,m# of a relativistic spin-polarized electron gas are employed within the local spin density approximation ~LSDA!.19,31,32 Very often the nonrelativistic expressions for exchange and correlation energy ~for example, that of Vosko, Wilk, and Nusair33! are used even in the relativistic calculations. This approximation is also assumed in the present work. The Madelung energy vanishes on average within the single-site CPA for homogeneous alloys. The short-range order charge correlations can be approximated either by varying the radii of alloy components such that the charge neutrality and volume conservation conditions are fulfilled,34 or by keeping equal radii of alloy components, but including
´ , AND WEINBERGER SHICK, DRCHAL, KUDRNOVSKY
1616
screening charges induced on nearest neighbors.35 This leads to the following expression for averaged alloy Madelung energy ~per one site!: E Mad52 2b
c A q 2A 1c B q 2B R1
,
~61!
where R 1 is the nearest neighbor distance. The coefficient b can be varied to achieve better agreement with experimental data. If the screening charge is located on nearest neighbors b 5 21 . 36 It should be noted that b is an additional parameter that does not follow from the density functional theory, or from the theory of random alloys. The core electron charge densities are either kept fixed ~frozen-core approximation!, or determined self-consistently together with the valence charge densities ~all-electron theory!. The all-electron treatment is generally more correct. It becomes particularly important if the sphere radii of the alloy constituents are adjusted so as to achieve charge neutrality. In such a case, the core states should be modified in accordance with changes of atomic radii and corresponding variations in atomic potentials and normalization conditions. VI. RESULTS AND DISCUSSION
Here we present results obtained by the relativistic spinpolarized version of the TB-LMTO-CPA method described above for 3d transition metals ~bcc Fe, fcc Co, and fcc Ni! and the magnetic alloy systems Co 50Ni 50 and Co 50Pt 50 . We first used the frozen-core approximation in order to test the method and to compare with experimental data and previous theoretical results. Only then did we perform also allelectron calculations. The k-space integration covers the full Brillouin zone ~BZ! because the usual restriction to the irreducible wedge of the BZ is no longer valid.37 We used an equidistant grid in the full BZ. The energy integration was performed along a contour in the upper complex half plane by employing the Gauss quadrature with 12 points. The potential parameters determined self-consistently then served as an input for a calculation of densities of states at real energies. The matrix CPA equations were solved by using a simple iteration procedure.34 A standard mixing scheme was employed to solve the LSDA part, based on the exchange-correlation function of Vosko, Wilk, and Nusair.33 In order to analyze the numerical accuracy of our computational scheme, we have performed self-consistent calculations of E tot for Co 50Ni 50 using different numbers of k points (N k ) in the full BZ. Using equal values of the Wigner-Seitz radii ~2.6 a.u.! for Co and Ni, the total energy E tot as a function of N k is given in Table I for Co 50Ni 50 . It turns out that E tot converges well with increasing N k . The values of the spin and the orbital magnetic moments turned out to be quite insensitive to the choice of N k ~see Table I!. According to Table I, reasonable accuracy is achieved for N k 54600, which was then used in all further calculations. A. Transition metals
The purpose of the frozen-core calculations was to compare with existing results of relativistic spin-polarized Korringa-Kohn-Rostoker ~KKR! and LMTO band-structure
54
TABLE I. Convergence of the total energy, spin (M s ), orbital (M l ), and total magnetic moments (M tot) for Co 50Ni 50 with respect to the number of k points (N k ) used in the Brillouin zone integration. The magnetic moments are in units of m B and the total energy difference DE tot(N k )5E tot(N k )2E tot(17 920) is in mRy. Nk 2432 4600 7776 12152 17920
DE tot(N k )
Ms
Ml
M tot
0.207 -0.128 -0.028 0.030 0.000
1.117 1.143 1.165 1.165 1.159
0.054 0.056 0.057 0.057 0.057
1.171 1.199 1.222 1.222 1.216
calculations.6,8 The same values of Wigner-Seitz radii (2.67 a.u. for bcc Fe, 2.6 a.u. for fcc Co, and 2.6 a.u. for fcc Ni! as used in Refs. 6 and 8 have been assumed. The values of the spin and the orbital magnetic moments are given in Table II and are compared to other calculations and to available experimental data.38 Our results agree quite well with those in Refs. 6 and 8. The small differences in the magnetic moments are due to the different types of exchange-correlation potentials used and other computational details. The results of Ref. 6 are based on non-self-consistent calculations using the potentials taken from Ref. 39, while the self-consistent calculations of Ref. 8 used the exchange-correlation potential of von Barth and Hedin.40 Let us note that similarly to nonrelativistic or scalar-relativistic calculations, the relativistic theory also does not reproduce exactly the experimental values of the magnetic moments for 3d metals. The total energy as a function of the Wigner-Seitz radius for Ni has been calculated for 4600 k points in the full BZ. The results of the frozen-core approximation and the allelectron calculation are given in Table III. The all-electron results are closer to the experimental values. With the exception of bulk moduli which differ from the experimental value38 ~1.836 Mbar! by almost 40%, there is quite a good agreement with experimental data. It seems to be a rather TABLE II. Spin (M s ), orbital (M l ), and total magnetic moments (M tot) ~in m B ) for Fe, Co, and Ni. Fe
Co
Ni
Present work
Ms Ml M tot
2.23 0.043 2.273
1.63 0.069 1.699
0.64 0.049 0.689
SPRKKR a
Ms Ml M tot
2.12 0.041 2.161
1.54 0.068 1.608
0.57 0.049 0.619
SPRLMTO b
Ms Ml M tot
2.22 0.040 2.260
1.61 0.080 1.690
0.60 0.050 0.650
Experiment c
M tot
2.216
1.751
0.616
Reference 6, spin-polarized relativistic KKR ~SPRICKR!. Reference 8, spin-polarized relativistic LMTO ~SPRLMTO!. c Reference 38. a
b
ELECTRONIC STRUCTURE AND MAGNETIC PROPERTIES OF . . .
54
TABLE III. Equilibrium Wigner-Seitz radii (r WS , in a.u.!, bulk moduli (B, in Mbar!, magnetic moments ~in m B ), and g s factors for Ni. Frozen core
All electron
Experiment a
2.538 2.511 0.677 0.043 2.13
2.581 2.478 0.682 0.048 2.15
2.602 1.836 0.616 0.050 2.21
r WS B M tot Ml gs a
Reference 38.
common feature that relativistic effects stiffen the bulk moduli by reducing the calculated equilibrium lattice constant.41,34 B. fcc Co 50Ni 50 alloy
We studied this system in order to compare the results of our method with the scalar-relativistic calculations in Ref. 42 and available experimental data,38 and also to analyze numerically the influence of spin-orbit coupling on magnetic and total energy properties of 3d-metal alloys. The total energy as a function of the Wigner-Seitz radius for the Co 50Ni 50 alloy was calculated for N k 54600. Charge neutrality and volume conservation conditions were fulfilled by varying the atomic sphere radii of the alloy components. The results of the frozen-core approximation and the allelectron calculation are given in Table IV. Again, the allelectron calculation is in better agreement with the experimental lattice constant. The average alloy total magnetic moment as well as the magnetic moments for Co and Ni are in reasonably good agreement with the experimental data. The difference in g factors is caused by an overestimation of the Ni spin magnetic moment using the Vosko, Wilk, and Nusair exchange-correlation potential. For the saturation magnetization42 the scalar-relativistic result of 1.17m B , which was obtained from spin magnetic moments corrected for g s factors, agrees quite well with our results; the agreement with the experimental equilibrium lattice constant, however, is better in the present case. TABLE IV. Equilibrium Wigner-Seitz radii (r WS , in a.u.!, bulk moduli (B, in Mbar!, magnetic moments ~in m B ), and g s factors for Co 50Ni 50 .
r WS B alloy M tot alloy Ml g alloy s Co M tot Co Ml g Co s Ni M tot Ni Ml g Ni s a
Reference 38.
Frozen core
All electron
Experiment a
2.537 2.647 1.182 0.050 2.09 1.654 0.059 2.07 0.711 0.041 2.12
2.583 2.573 1.194 0.055 2.10 1.692 0.066 2.08 0.696 0.045 2.14
2.606 1.14 2.17 1.7
0.58
1617
TABLE V. Equilibrium Wigner-Seitz radii (r WS , in a.u.!, bulk moduli (B, in Mbar!, magnetic moments ~in m B ), and g s factors for Co 50Pt 50 . Frozen core All electron Experiment a Model A Model A Model B Model C r WS B alloy M tot alloy Ml g alloy s Co M tot Co Ml g Co s Pt M tot M Pt l g Pt s a
2.755 2.721 1.066 0.051 2.10 1.787 0.046 2.05 0.345 0.055 2.38
2.801 2.488 1.121 0.061 2.12 1.859 0.052 2.06 0.384 0.070 2.41
2.833 2.741 1.161 0.066 2.12 1.909 0.052 2.05 0.412 0.080 2.42
2.828 2.658 1.118 0.061 2.12 1.824 0.046 2.05 0.411 0.075 2.45
2.775 1.05
Reference 38. C. fcc Co 50Pt 50 alloy
In contrast to Co-Ni alloys, in which both constituents have approximately the same atomic volume, the situation in Co-Pt alloys is different and charge transfer due to the mismatch in atomic radii has to be expected. Various treatments of charge-transfer effects have been put forward. Model A: Charge neutrality and volume conservation conditions are fulfilled by a variation of atomic sphere radii.29,34 Model B: The Wigner-Seitz radii of atoms A and B are fixed and set equal to the average alloy Wigner-Seitz radius. The effect of Madelung potentials is neglected. Model C: The Wigner-Seitz radii of A and B atoms are fixed and set equal to the average alloy Wigner-Seitz radius. The local Madelung contributions are included using ~61!. In order to see how the various treatments of charge transfer influence the calculated equilibrium properties of the alloy, we have compared models A, B, and C ~with b 51/2). The ground state properties as calculated within the frozen-core approximation and assuming neutral spheres ~model A), and as obtained in all-electron calculations for models A, B, and C are summarized in Table V together with the experimental data. The total energy as a function of the Wigner-Seitz radius is shown in Fig. 1 for these three models. According to Table V and Fig. 1, including shortrange charge correlations either by variation of the component radii in such a way that the atomic spheres are neutral, or by adding local Madelung corrections, leads to a lowering of the total energy and equilibrium lattice constant of the alloy. As follows from the comparison with the results of spinpolarized relativistic KKR-CPA calculations and spinpolarized scalar-relativistic KKR-CPA calculations,43 as well as with the experimental values of total magnetic alloy moments, the present results are in a reasonable accordance with all these data. Let us note that there is a small discrepancy in the values of the Co orbital magnetic moment between our results and the spin-polarized relativistic KKR-CPA calculations. The reason probably lies in different computational schemes that can lead to differences in the relatively small values of orbital moments for 3d-metal atoms. For example,
1618
´ , AND WEINBERGER SHICK, DRCHAL, KUDRNOVSKY
FIG. 1. Total energy ~in Ry! as a function of the average alloy Wigner-Seitz radius ~in a.u.! for Co 50Pt 50 alloy. The results of model A ~neutral spheres! are compared to those of model B ~equal spheres, no Madelung corrections! and model C ~equal spheres, local Madelung corrections with b 51/2).
the calculations in Ref. 43 employed potentials based on scalar-relativistic LMTO calculations and therefore are not fully self-consistent as in the present paper. In all cases, we find reasonable agreement with experimental data.38 The equilibrium alloy Wigner-Seitz radius obtained for model A in the frozen-core approximation is slightly smaller than the experimental value ~2.775 a.u.!, while all radii found from the all-electron calculations are larger. The total densities of states for Co and Pt ~model C, equilibrium lattice constant! are shown on Fig. 2, and spinresolved densities of d states for Co and Pt are given in Figs. 3 and 4. Finally, the j-resolved densities of d states for Co and Pt are shown on Figs. 5 and 6. The Co 3d bands are quite narrow and their shapes are similar to those of fcc Co. Consequently, the change of the Co magnetic moment upon alloying is small. The exchange splitting between the majority and minority bands is quite pronounced ~Fig. 3!, but the spin-orbit splitting between the d 5/2 and d 3/2 states of Co is
FIG. 2. Local density of states at Co atoms ~full line! and at Pt atoms ~dashed line! in the Co 50Pt 50 alloy ~model C). The vertical dotted line denotes the Fermi energy E F .
54
FIG. 3. Spin-resolved local densities of 3d states at Co atoms ~majority spin: full line; minority spin: dashed line! in the Co 50Pt 50 alloy ~model C). The vertical dotted line denotes the Fermi energy E F .
small ~Fig. 5!. The behavior of Pt is different. The Pt 5d bands are broad, and their upper parts are affected by hybridization ~Fig. 4! with Co 3d states lying in the same energy region. The magnetic moment of Pt is thus induced by interaction with Co atoms. The exchange splitting for Pt is energy dependent, being negligible at the bottom of the band, and quite strong in the upper part of the band. The spin-orbit splitting for Pt is strong; the d 3/2 states are near the bottom of the band, while the d 5/2 states extend over the whole d band. D. Exchange and spin-orbit parameters
The relative strengths of exchange and spin-orbit splitting discussed above can be analyzed using the simple model described in Appendix B. By means of this model we extracted the exchange splitting and spin-orbit interaction parameters for d electrons in Fe, Co, Ni, and Pt and for Co and Pt in Co 50Pt 50 alloy, and also for Co and Pt as impurities. These parameters are summarized in Table VI. For all ferromagnetic 3d metals the exchange splitting parameter is al-
FIG. 4. Spin-resolved local densities of 5d states at Pt atoms ~majority spin: full line minority spin: dashed line! in the Co 50Pt 50 alloy ~model C). The vertical dotted line denotes the Fermi energy E F .
54
ELECTRONIC STRUCTURE AND MAGNETIC PROPERTIES OF . . .
1619
TABLE VI. Exchange I and spin-orbit j parameters ~in mRy! for metallic Fe, Co, Ni, and Pt, and for Co and Pt in Co 50Pt 50 , as well as for Co as impurity in Pt and Pt as impurity in Co. For a comparison estimates obtained by the LMTO band structure calculations for elemental metals are given.
m
FIG. 5. Local densities of 3d states at Co atoms resolved according to j ( j53/2: full line; j55/2: dashed line! in the Co 50Pt 50 alloy ~model C). The vertical dotted line denotes the Fermi energy E F .
most constant and lies in a narrow interval between 71 and 75 mRy, while the variations in the spin-orbit parameter are larger. For Co 50Pt 50 the exchange parameter for Co is lower than in pure Co, while the spin-orbit coupling remains essentially unchanged. Pt shows a considerably smaller exchange splitting than 3d metals, while—not surprising—the spinorbit interaction parameter is considerably larger and comparable to the exchange splitting parameter. The concentration dependence of the parameters is rather weak. VII. CONCLUSION
It was shown that theoretical concepts of the LMTO method4,27,29 such as the form of the structure constants, screening, Green’s functions, and the coherent potential approximation, etc., remain valid also in the spin-polarized relativistic case. In comparison with the nonmagnetic case, however, several modifications are needed.8 ~1! The only remaining good quantum number is m . ~2! A more complicated set of coupled radial Dirac equations has to be solved. ~3! Formally, the main difference between the spin-polarized
I
5/2 3/2 1/2
j
3/2 1/2
Ia ja a
Fe 71 72 71
Metal Co Ni 73 75 73
Pt
74 75 74
4.1 8.7
5.8 6.5
7.2 7.2
65 5.5
68 6.5
70 7.6
41.6 41.6
Co 50Pt 50 Co Pt
Impurities Co Pt
70 71 70
69 69 68
48 48 51
5.7 6.7
43 42
47 42 46
6.5 3.8
45 44
68 44
Reference 44.
and the nonmagnetic cases is that the logarithmic derivative, potential function, and potential parameters are non-diagonal matrices with respect to the indices ( km ). They are block diagonal of dimension (232) or (131) depending on k and m . These blocks are denoted by indices L5( l , m ). ACKNOWLEDGMENTS
The authors thank the Grant Agency of the Czech Republic ~Project No. 202/93/0688!, the Austrian Ministry of Science ~Project No. GZ 45.384!, and the Austrian Science Foundation ~P10231!, and one of them ~A.B.S.! also thanks Exxon Research ~Annandale, NJ! for financial support. APPENDIX A: PROPERTIES OF SINGLE-SPHERE SOLUTIONS
For the radial Dirac equations ~11! and ~12! the initial conditions at r50 can be formulated in the same way as in the nonmagnetic case because the effective magnetic field B(r) remains finite at r50 and thus does not affect the initial conditions:
S D SDS D SD g k 1 ,l 1 f k 1 ,l 1
g k 2 ,l 1 f k 2 ,l 1
g k 1 ,l 2
1
5A 1 r a 1
q1 0
,
0
f k 1 ,l 2
g k 2 ,l 2 f k 2 ,l 2
0
5A 2 r a 2
0 1
.
q2
~A1!
Here A 1 and A 2 are normalization constants, and the exponents and quotients are defined as a i 5211 @ l 2i 2 ~ 2Z/c ! 2 # 1/2, FIG. 6. Local densities of 5d states at Pt atoms resolved according to j ( j53/2: full line; j55/2: dashed line! in the Co 50Pt 50 alloy ~model C). The vertical dotted line denotes the Fermi energy E F .
q i5
c ~ l 111a i !~ i51,2 ! . 2Z i ~A2!
The single-site solutions can be orthogonalized to their first energy derivatives using the transformation45
´ , AND WEINBERGER SHICK, DRCHAL, KUDRNOVSKY
1620
S
u C l 1~ E ! & u C l 2~ E ! &
DS 5
cos@ Q ~ E !#
2sin@ Q ~ E !#
sin@ Q ~ E !#
cos@ Q ~ E !#
DS
u F l 1~ E ! &
D
,
u F l 2~ E ! & ~A3!
such that ^ C l 1 (E) u C l 2 (E) & 5 d l 1 ,l 2 and ^ C l 1 (E) u ˙ (E) & 50. The first condition is fulfilled automatically C l2 due to the form of the transformation matrix, whereas the second condition leads to the following ordinary differential equation: ˙ ~ E !& 5 ^ F ~ E !u F ˙ ~ E !&. ˙ ~ E ! 52 ^ F ~ E ! u F Q l1 l2 l2 l1
~A4!
u C l 1 ~ E n ,L! & 5 u F l 1 ~ E n ,L! & ,
˙ ~ E ! & 5 @ 12 u F ~ E ! &^ F ~ E ! u # u F ˙ ~ E !&, uC l1 n ,L l2 n ,L l2 n ,L l1 n ,L ˙ ~ E ! & 5 @ 12 u F ~ E ! &^ F ~ E ! u # u F ˙ ~ E !&, uC l2 n ,L l1 n ,L l1 n ,L l2 n ,L ~A5! i.e., at E n ,L the states remain unchanged, but their energy derivatives are modified by projecting out their nonorthogonal parts. In this paper we assume that the radial amplitudes are transformed according to ~A5!, i.e.,
ˆ 2E u C ~ E ! & 50, 2 ^ C l8m~ E !u H lm
˙ ˆ 2^C l 8 m ~ E ! u H 2E u C l m ~ E ! & 5 d ll 8 ,
~A8!
where the overdot denotes a derivative with respect to energy, leads to the following important identities: 21 T f L~ E,s ! g 21 L ~ E,s ! 5 @ f L~ E,s ! g L ~ E,s !# ,
~A9!
g˙ TL~ E,s ! ˙f L~ E,s ! 5 ˙f TL~ E,s ! g˙ L~ E,s ! .
~A11!
For example, it follows from ~A9! that the logarithmic derivative is a symmetric matrix, D TL(E)5D L(E). APPENDIX B: ANALYSIS OF SPIN-ORBIT AND EXCHANGE SPLITTING
It is tempting to characterize the magnetic properties of alloy constituents in terms of approximate spin-orbit and exchange splitting parameters. Consider therefore the following simple single-site Hamiltonian:
l5l 1 ,l 2 , H5H 0 2 21 IM s z 1 j L•S,
˙ ~ E !g0 g˙ n , k l 1 m 5g˙ n0 , k l m 2Q n ,L n , k l 2 m , 1 0 ˙f ˙0 ˙ n , k l 1 m 5 f n , k l 1 m 2Q~ E n ,L ! f n , k l 2 m ,
˙ ~ E !g0 g˙ n , k l 2 m 5g˙ n0 , k l m 1Q n ,L n , k l 1 m , 2 0 ˙f ˙0 ˙ n , k l 2 m 5 f n , k l 2 m 1Q~ E n ,L ! f n , k l 1 m ,
~A6!
where the superscript zero denotes the functions g and f before this additional orthogonalization. The subscript n indicates that g and f are evaluated at the energy E n ,L . The second Green’s theorem allows one to simplify many formulas. It is derived similarly as in the nonrelativistic case. (2) (1) Let F km (r) and F k 8 m 8 (r) be two bispinors of the form ~10!, not necessarily solutions of the Dirac equation, and let ^ F (1) u Hˆ u F (2) & denote the integral over the Wigner-Seitz sphere. Then ~2!
^ C l m ~ E ! u Hˆ 2E u C l 8 m ~ E ! &
cs 2 @ f TL~ E,s ! g˙ L~ E,s ! 2g TL~ E,s ! ˙f L~ E,s !# 5I 2 , ~A10!
u C l 2 ~ E n ,L! & 5 u F l 2 ~ E n ,L! & ,
f n , k l m 5 f n0 , k l m ,
The right-hand side of ~A7! is the relativistic analog of a Wronskian functional determinant. Suppose that C l m (E) and C l 8 m (E) are orthonormal solutions of the coupled Dirac equations for a particular energy E. Combination of ~A7! with the identities
˙ ^ C l m ~ E ! u Hˆ 2E u C l8m~ E ! &
The initial condition for ~A4! is to some extent arbitrary. The simplest choice of Q(E n ,L)50, where E n ,L is the linearization energy for the subblock L4~l m ), yields
g n , k l m 5g n0 , k l m ,
54
~2!
1! ˆ ˆ u F ~ 1 !& u H u F k 8m 8& 2 ^ F k 8m 8u H ^ F ~km km 1! 2! 2! 1! 5 d kk 8 d mm 8 cs 2 @ f ~km ~ s ! g ~km ~ s ! 2 f ~km ~ s ! g ~km ~ s !# .
~A7!
~B1!
where I is the exchange splitting parameter, M is the magnetic moment of an atom, j is the spin-orbit splitting parameter, L and S are the orbital and spin angular momentum operators, and H 0 contains neither magnetic effects nor spinorbit coupling. The Hamiltonian in ~B1! has two constants of motion, namely, L2 and J z . Consequently, in an angular momentum representation, only states with the same l and m are coupled. If l and l 8 denote 2 l 21 and l , respectively, such a representation is given by
^ l m u H u l m & 5E 0 ~ l ! 2 21 IM sgnm 1 21 j l ,
~B2!
if u m u 5 l 11/2, while for u m u , l 11/2
^ l m u H u l m & 5E 0 ~ l ! 2IM
1 m 1 jl , 2 l 11 2
1 ^ l 8 m u H u l m & 5 ^ l m u H u l 8 m & 5 IM 2
^ l 8 m u H u l 8 m & 5E 0 ~ l ! 1IM
A S lm D 12
2
2 11
m 1 2 j ~ l 11 ! , 2l 2
2
,
~B3!
ELECTRONIC STRUCTURE AND MAGNETIC PROPERTIES OF . . .
54
where E 0 ( l ) is defined by
^ l m u H 0 u l 8 m 8 & 5E 0 ~ l ! d ll 8 d mm 8 .
~B4!
Quite clearly, because L2 is a constant of motion, H is not a ~true! relativistic Hamiltonian. Its angular momentum representation ~B3! and ~B4!, however, is of the same formal Q structure as the matrix of potential parameters C ll 8 m which
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1621
characterize the properties of atomic species Q provided that coupling to other atoms @see Eq. ~40!# is neglected. This similarity arises from the approximation D l 50, assumed in order to decouple the infinite hierarchy of radial Dirac equations for radial amplitudes. Fitting, therefore, corresponding blocks of the angular momentum representation of H ~B3! and ~B4! to those of the Q potential the parameters C ll 8 m , the parameters I and j can be extracted.
25
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