The electronic structure of non-transition element impurity atoms in iron and nickel

162

THE IRON

ELECTRONIC AND

STRUCTURE

OF NON-TRANSITION

ELEMENT

IMPURITY

ATOMS

IN

NICKEL

K. T E R A K U R A * q 'alcndish l,ahor, t,~ry. Madin~lev Road. ('ambrid~c ("B3 ()HE', En,~,laml

lhe electronic structure of an impurit,, atom of at non tran',ition element is discussed on the basis of ab initio calculations. A detailed discus'don is given of the difl'crence belween iron and nickel with regard to the magnetic disturbance due to an impurity atom. An anti-resonance effect, which is an inlportant concept in d-band metals and allo}~, is discussed b) using a simple rnodel.

1. I n t r o d u c t i o n

We will d i s c u s s the e l e c t r o n i c s t r u c t u r e of an i m p u r i t y atom of a n o n - t r a n s i t i o n e l e m e n t in a t r a n s i t i o n metal, laying e m p h a s i s on the magnetic properties. A l t h o u g h m u c h e x p e r i m e n t a l data has a c c u m u l a t e d , there are only a few theoretical studies of these alloys b e c a u s e of the difficulty of setting up a simple model with a s o u n d basis. P r e v i o u s l y we have d e v e l o p e d a f o r m a l i s m , which is called the + ' p s e u d o - G r e e n i a n " m e t h o d , to carry out an ab initio c a l c u l a t i o n Ill. We applied this f o r m a l i s m to certain problems 12,3] and s u c c e e d e d in giving c o n s i s t e n t i n t e r p r e t a t i o n s of e x p e r i m e n t a l o b s e r v a t i o n s . A n o t h e r i m p o r t a n t f o r m a l i s m of an i m p u r i t y p r o b l e m was d e v e l o p e d on the basis of the K K R m e t h o d by B e e b y [4] and Harris 151, and r e c e n t l y e x t e n d e d by H a m a z a k i [61. S o m e of our r e c e n t work [7,81 is b a s e d on H a m a z a k i ' s formalism. O n e of the i m p o r t a n t c o n c e p t s d e r i v e d from o u r c a l c u l a t i o n s is an a t t t i - r e s o n a n c e effect which was originally suggested by F a n o [9]. A detailed d i s c u s s i o n of this will be given in section 4 on the basis of a c l u s t e r c a l c u l a t i o n . 2. A s u r v e y

of t h e p r o b l e m

O n e of the i n t e r e s t i n g d i f f e r e n c e s in the m a g n e t i s m of iron and nickel lies in their resp o n s e to the a d d i t i o n of a n o n - t r a n s i t i o n element. T a b l e 1 s h o w s some e x a m p l e s of the c h a n g e of the s a t u r a t i o n m a g n e t i z a t i o n A M of iron 1101 and nickel [ I l l due to an i m p u r i t y atom. AM values for Ti and V in nickel are also s h o w n . In this table, ZL is the v a l e n c e n u m b e r of an i m p u r i t y atom. The most striking d i f f e r e n c e * Work supported in part by the SR(' grant RG.0727. " Permanent address: l)epartmenl of Physics. ()saka [Inivcrsit~,, l'oyonaka, Osaka. Japan. I>hy~'icd 91B (1977) 162 166 (if! North-Holland

Fable I ('hange of magnetization per impurit~ utom On/.~.}

Z,

AM (in Nil"

AM (in Fe)"'

('u

I

I. 14

2.(}0

AI (;a Si C;e Sn

:~ 3

2.g0

4

3.77 ~;.71} 4.22

2.27 (2. ~,7} 1.41 (I.55} 2.28 (2.15) 1.36 (I.g3)

Ti

V

4 4

4 "~

0.97 (0.g7)

1.8'

5.2"

' ref. I I: '+ref. 10: ' ref. I~'~.

Figures in parentheses are calculated ,.alues Iglb e t w e e n iron and nickel is that in n i c k e l - b a s e d alloys A M is nearly equal to Z,, while in i r o n - b a s e d alloys AM does not strongly d e p e n d on the kind of i m p u r i t y atom and is n e a r l y equal to m i n u s the m a g n e t i c m o m e n t of pure iron per atom. in a d d i t i o n to the m a g n e t i z a t i o n m e a s u r e m e n t s + n e u t r o n diffraction e x p e r i m e n t s were carried out for these alloys [12, 131, and it was s h o w n that the m a g n e t i c d i s t u r b a n c e is e x t e n d e d in n i c k e l - b a s e d alloys, while it is fairly localized in i r o n - b a s e d alloys. For some impurities in iron, such as Sm AM is not about 2.2p.+ but a b o u t 1/., H, and a m a g n e t i c d i s t u r b a n c e was o b s e r v e d at the s u r r o u n d i n g sites. H o w e v e r the d i s t u r b a n c e is still very small c o m p a r e d with that in n i c k e l - b a s e d alloys. A naive e x p l a n a t i o n for the case of nickel. which was originally p r o p o s e d by Mott 1141 by using a simple rigid b a n d model, is that the extra s, p e l e c t r o n s of i m p u r i t y a t o m s flow into the d - b a n d of m i n o r i t y spin, r e d u c i n g the n u m b e r of holes and the m a g n e t i c m o m e n t on the surr o u n d i n g nickel atoms. It is difficult, h o w e v e r , to imagine that an i m p u r i t y atom is in such a highly ionized state in nickel.

163 In his review article in 1964 Mutt [15] proposed three alternative explanations, in all of which the fundamental difference between ironbased and nickel-based alloys was attributed to the states with s or p s y m m e t r y taken about the impurity site. These explanations are not consistent with our recent calculation [8] and will be criticized in section 3. It may be interesting to point out a close relation between transition metal impurities and non-transition metal impurities in nickel. Friedel [161 gave a beautiful explanation for the former case. Let us confine ourselves to a case of a strongly repulsive perturbation such as for Ti and V in nickel. An impurity atom of Ti or V brings about a split off state a b o v e the Fermi level in the majority spin band. The number of electrons is adjusted in the minority spin d-band to satisfy Friedel's sum rule. Friedel gave the following expression for AM (in t~n): AM = - 1 0 - AZ,

(1)

where A Z = Z~--Zh, and Z i and Zh are the valence numbers of an impurity atom and a host atom, respectively. We notice that eq. (1) can be rewritten as AM= -Zi,

(2)

because Zh : 10 for nickel. Eq. (2) agrees well with the observed AM values (see table I) and it is rather surprising that AM for non-transition metal impurities like A1 and Si behaves in the same way as that for Ti and V. It is implicitly assumed in Friedel's argument that the number of occupied states of s and p symmetries does not change with alloying. If this is justified even for a non-transition metal impurity, AM will be given by the same equation as eq. (2), because the perturbing potential is strongly repulsive with regard to d s y m m e t r y in this case too. T e r a k u r a and K a n a m o r i [2] pointed out that Fano's anti-resonance theory [9] can be used to prove the smallness of AN~ and AN~, the change of the number of occupied states of s and p s y m m e t r y , respectively, and they proved it numerically. The magnetization change AM in iron-based alloys is more difficult to understand but has been successfully explained [8]. A detailed discussion will be given in the next section.

3. Calculated results and discussion

3.1. Magnetization change in iron First we discuss the magnetization change in iron-based alloys. The most important quantity in the present problem is the change in the number of states below a given energy due to the presence of an impurity atom, AN~(E), where or denotes the spin state and F refers to the s y m m e t r y of the state taken about the impurity atom. Another important quantity is the local density of states within the WignerSeitz sphere of an impurity site, n~(E). The calculated AM values for impurities of AI, Si, Ga, Ge and Sn are listed in the fifth column of table 1. (Ref. 8 should be consulted for some details in the calculations of AM.) We also calculated n~(E) and found that the magnetic m o m e n t localized at the impurity site is very small ( < 0 . 1 /~B), SO the variation of AM values amongst elements comes from the disturbance at the surrounding sites. This is consistent with the neutron diffraction experiment

[12]. 3.2. The difference nickel-based alloys

between

iron-based

and

We can guess to some extent what is happening at the surrounding sites by using AN~ and n~. This helps us to understand the difference between iron and nickel. There are three important effects associated with the disturbance at the neighbouring iron sites. The first one is a narrowing effect. In fig. 1, we show AN~,(E) for an impurity atom of AI in iron by a full curve. A broken curve shows A/~d~(E) = Fe

f

n dFe, ( E ) d E ,

(3)

where rid, is the density of states of down spin with de s y m m e t r y for pure iron and the dotted curve shows E

X2,(E) = f

nd,(E) A~ dE,

(4)

where nda~ is for an impurity atom of AI in iron. The corresponding figure for up spin can roughly be obtained by shifting the origin of energy, and E ~ shows the corresponding Fermi energy. The sum of A/Q~, and X~, should be equal to AN2, if an impurity atom of AI did not disturb the surrounding sites at all. The discrepancy between AN2, and (A/Q~,+ Xd~) can

164 2[ ,

• 4

:!

}{dE

!4 ~

.

[

[1

i

l'>

Fig. 1. T h e full c u r v e s h o w s AN,~,(E) for an i m p u r i t ~ a t o m of AI. T h e definitions of AN,,~,(E) a n d X,,~,(EI are g i v e n b,, eqs. (3) a n d (4). A r o u g h e s t i m a t i o n of l h e s e quantilie~, for tip spin c a n be m a d e by shifting lhe F e r m i e n e r g y H, to HI.

be interpreted as the narrowing of the d-bands at the s u r r o u n d i n g sites due to the r e m o v a l of d-orbitals at the impurity site. If the Fermi level lies in the upper half of the d-band, the narrowing results in the filling of the d-band and vice versa in the lower half of the d-band. This effect will p r o b a b l y be playing a m a j o r role in the charge t r a n s f e r of d-electrons in the alloys of M n - , F e - , C o - , N i - and C u - A I [17]. The difference b e t w e e n A N d and (AN a + X,l) gives a c o r r e c t order of magnitude for the o b s e r v e d variation in this charge t r a n s f e r f r o m Mn through Cu and predicts the c h a n g e of sign in Ti-, V- and C r - b a s e d alloys. In f e r r o m a g n e t i c iron, the narrowing effect is effective in the filling of the d-band in the up spin band where the Fermi level lies just below the u p p e r edge of the d-band. The s e c o n d effect is a resonance one. The n a r r o w i n g effect is not important in the d o w n spin band, b e c a u s e the Fermi level lies just at the middle of the d-band. H o w e v e r , fig. I indicates the filling of the d-band at the s u r r o u n d i n g sites in this case also. This is due to the p r e s e n c e of an impurity r e s o n a n c e state just at the Fermi level where the host d-density of states has a sharp dip (see fig. 2 of [8]). The w a v e f u n c t i o n of the r e s o n a n c e state spreads widely o v e r the s u r r o u n d i n g sites. The reso n a n c e e n e r g y does not d e p e n d on the impurity potential very strongly b e c a u s e of the small amplitude of the w a v e f u n c t i o n at the impurity site. This is quite a n a l o g o u s to the a r g u m e n t s of H a l d a n e and A n d e r s o n [18]. H o w e v e r , as the r e s o n a n c e state lies just at the Fermi level, a slight c h a n g e in the r e s o n a n c e e n e r g y is fairly sensitively reflected in A N ~,(E~)

• [ ,'

7,

I

~ • • .<

" 7 .... , 6c

0 ':~

r ,=r

Fig. 2 . . ~ N {([~/I for an i m p u r i l y a l o m of AI (full c u r v e ) and S;i ( b r o k e n c u r v c ) . A d o t t e d c u r v e s h o w s the d e n q l ~ of ~lates per iltL)ll] of do~,'l] spin e l e c t r o n s of p u r e Jl'oll

and is a m a j o r cause of the variation of AM amongst elements The third effect is an anti-resonance one. Fig. 2 s h o w s AN,~(E) for an impurity atom of AI (full curve) and Si l b r o k e n c u r v e t in iron. A striking feature of AN'f (H) is thai il is ver~ small in the e n e r g y range of the d-band. AN'~[(EI has also a similar structure [2, 7, g]. Accordingly the c h a n g e in the n u m b e r of o c c u p i e d states with s and p s y m m e t r i e s is very small, while there a c c u m u [ a t e a considerable n u m b e r of s and p electrons at the impurity site to maintain the local charge neutrality. It will be shown in the next section that the dip in ANT results from the loss of d-states at the surrounding sites. In iron-based alloys the filling of the d-bands (due to the narrowin~ effect in the up spin band and the resonance effect in the d o w n spin band) and the loss of d-stales due to the antiresonance effect cancel to some extent in each spin d-band and the net magnetic d i s t u r b a n c e ~.tt the s u r r o u n d i n g sites b e c o m e s small. On the other hand in nickel-based alloys the narrowin~ effect and the resonance effect have no effect at all on the n u m b e r of d-electrons with up spin, b e c a u s e the up spin d-bands arc already fully occupied. ()nly the anti-resom:nce effect is effective. This is the reason w h y eq. (2) holds even for non-transition metal impurities. F u r t h e r m o r e , the a n t i - r e s o n a n c e results in an electron deficit at the s u r r o u n d i n g sites, which is then s c r e e n e d by the d o w n spin d-electrons to bring about an appreciable reduction in magnetization at the s u r r o u n d i n g nickel sites [2, 13l. It should be noted here that the m a j o r

165 difference between iron-based and nickel-based alloys lies in the absence of the narrowing effect and the resonance effect in nickel-based alloys. These two effects are related to states with d symmetry with respect to the impurity site. On the other hand the anti-resonance effect, which is related to states with s and p symmetries, is effective in both cases. This implies that there is no substantial difference between the two systems regarding the interaction between sand p-states at the impurity site and the d-states at the surrounding sites. Some experimental results support this conclusion [8]. As the anti-resonance effect is a very common effect in d-band metals and alloys, AM will be given by eq. (1) so long as the matrix is a strong ferromagnet, that is the majority spin d-band is filled, and the impurity potential is not too strong (see a comment at the end of section 4). This was pointed out in [8] for some ternary alloys. 4. S u p p l e m e n t a r y effect

d i s c u s s i o n of an a n t i - r e s o n a n c e

In order to understand the underlying mechanism of the dip in AN~ and ANp we have to calculate the disturbance at the surrounding sites due to an impurity. It has recently been shown that the local density of states is essentially determined by the neighbouring environment [19]. Important aspects can be obtained with a fairly small cluster of atoms. Our cluster is composed of 13 atoms of Cu with a configuration of a central atom and its nearest neighbours in a fcc crystal. The cluster is embedded in a free electron sea. The bottom of the free electron band is taken at the muffintin zero of the Cu potential. This cluster is designated as Cuj3. Then the central Cu atom is replaced with a Si atom in order to discuss an impurity problem. This cluster is designated as Cu~2Si. We will show some of the results related to states with s symmetry taken about the centre. Details of the calculation and other results will be presented in a forthcoming paper [20]. We can construct four symmetrized orbitals with s symmetry (strictly speaking, it is not s but A~ symmetry), one from each of s, p, d¢ and d y orbitals at the surrounding sites. The local densities of states associated with these symmetrized orbitals are written as n~.r(E) for

Cu~3 and n,,r(E) for Cu~2Si. The subscript F specifies the symmetry of the constituent orbitals at the surrounding sites. ANt, the change in the number of states associated with the replacement of Cu with Si is shown in fig. 3 (full curve). This AN, should be compared with the one in fig. 1 in [7] where the results for an infinite crystal are shown. The discrepancy between the two cases near the bottom of the band is related to the fact that the wavelength is infinite at the bottom of the band [21]. This discrepancy does not matter, because our main purpose is to elucidate the physical meaning of the dip in AN~(E) at about 0.26 Ry. The present calculation shows that although the number of occupied states of s symmetry at the central site below Eo, the dip position, is larger for Si than for Cu, AN~ is nearly zero at E0. In order to understand this in detail, we show n~.d~ o (broken curve) and n~.d~ (dotted curve) in fig. 3. An important point is that the central peak in n °~,d~ becomes considerably smaller and a shoulder grows in the higher energy region as we replace Cu with Si. dy also behaves similarly, though its effect is smaller than that of de. These changes in the spectra associated with d-orbitals at the surrounding sites explain the rapid decrease of AN~(E) in the energy range from 0.15 to 0.26Ry and the gradual increase above 0.26 Ry. If we make the impurity potential much deeper, a substantial change will occur. We can reasonably expect the appearance of a sharp resonance state just at the dip position E0 and I/Ryd r,

AN, (E) 1.0-

i, ii

,~

10

';J

5

h II [i h ~r

.5-

0 .4

-.

12 '

' .0

.2

,',-.4

0 .6

E(Ryd) Fig. 3. The full curve shows AN,(E) from a cluster calculation associated with the replacement of a Cu atom with a Si atom at the central site. Bound states are present at about 0.02Ry (Cu,0 and -0.27Ry (Cu,:Si). The broken curve and dotted curve are for n'2,,~ (Cu,~) and n,d, (Cu,2Si).

166

then ANt(E)is no l o n g e r s m a l l n e a r E0. r h e c h a n g e s in t h e m a g n e t i z a t i o n a n d t h e s p e c i f i c heat d u e t o an i n t e r s t i t i a l i m p u r i t y

a t o m o f c a r b o n in

nickel suggest the presence of a resonance slate o f this k i n d [20, 22,231. Other examples where the anti-resonance e f f e c t p l a y s an i m p o r l a n ! r o l e are the K n i g h t shift a n d t h e s p i n - l a t t i c e r e l a x a t i o n l i m e 1"~ of N M R o f a n o n - t r a n s i t i o n e l e m e n ! in a t r a n s i t i o n metal, and soft X-ray emission spectra. The a n t i - r e s o n a n c e e f f e c t r e s u l t s in a v e r y s m a l l K n i g h t s h i f t a n d a v e r y l o n g T, [3,241. T h e dip in t h e local d e n s i t y o f s t a t e s has b e e n o b s e r v e d by s o f t X - r a y e m i s s i o n s p e c t r a [7, 25, 26].

expresses

P r o f e s s o r J. K a n a m o r i

his s i n c e r e t h a n k s

to

for valuable suggestions

and discussions throughout

the w o r k

presented

in t h i s p a p e r . T h a n k s are a l s o g i v e n t o P r o f e s s o r V.

H e i n e a n d Dr. D . G .

Pettifor

f o r t h e i r hos-

p i t a l i t y a n d s t i m u l a t i n g d i s c u s s i o n s t)f t h e a n t i r e s o n a n c e e f f e c t . T h e a u t h o r is i n d e b t e d to M r . C . M . M . N e x , M r . D.J. T i t t e r i n g t o n a n d Dr. H. Ohiwa

(Tokyo

Ill J. Kanamori. K. lerakura and K Yammhl tqo~l Theor. F'h~s.41 (1969) 1426:Progr. Theor. Ph'~, Suppl no. 46 (1970) 221 121 K. rerakura and .I Kanamori. Piogr. /hoot Ph),~ 4t~ (1971) 1007,

131 K. Terakura and J. Kanafnori, J. Phys NoL .lap ",4 (1973) 1~20. 141 J.l.. Beth}. Proc. Roy. %oc. ,~,~()2 (1967) IlL 151 R. Harris. J. Phys. (': Solid Slate Phys. 3 (197()) 172:4 (1971) ~69. 161 M. Hamazaki, Masler Thesis, 1 okyo Univcrxily, Japan 171 K. Terakura, J. Phys. Soc. Jap. 40 (1976) 4~,0

W e h a v e d i s c u s s e d t h e e l e c t r o n i c s t r u c t u r e of an i m p u r i t y a t o m o f a n o n - t r a n s i t i o n e l e m e n t in a t r a n s i t i o n m e t a l on t h e b a s i s of s e v e r a l ab in(rio c a l c u l a t i o n s . O u r m a i n s u b j e c t w a s the m a g n e t i c d i s t u r b a n c e d u e to an i m p u r i t y a t o m in ferromagnetic iron and nickel. We have given a d e t a i l e d d i s c u s s i o n o f an a n t i - r e s o n a n c e e f f e c t w h i c h p l a y s an i m p o r t a n t r o l e in a w i d e r a n g e o f p r o b l e m s in d - b a n d m e t a l s and a l l o y s . W e b e l i e v e that s o m e o f t h e q u a l i t a t i v e a s p e c t s o f o u r r e s u l t s c a n be a p p l i e d It) an i n t e r s t i t i a l impurity and a chemisorbed atom. Detailed d i s c u s s i o n s f o r t h e s e p r o b l e m s will be g i v e n in the future. author

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5. C o n c l u s i o n

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their

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in

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14] ]

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