SPIN SUSCEPTIBILITY OF METALLIC BINARY ALLOYS

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SPIN SUSCEPTIBILITY OF METALLIC BINARY ALLOYS

H. Fukuyama

To cite this version:

H. Fukuyama. SPIN SUSCEPTIBILITY OF METALLIC BINARY ALLOYS. Journal de Physique Colloques, 1974, 35 (C4), pp.C4-141-C4-144. �10.1051/jphyscol:1974424�. �jpa-00215615�

JOURNAL DE PHYSIQUE Colloque C4, suppfdnzent au no 5 , Tome 35, Mai 1974, page C4-14 1

SPIN SUSCEPTIBILITY OF METALLIC BINARY ALLOYS

H. FUKUYAMA Bell Laboratories

Murray Hill, New Jersey 07974, USA

Resume. ^- On a tttudie la susceptibilitt dynamique de spin de I'alliage binaire dans un &tat paramagnetique ou ferromagnktique. On a discute en detail le spectre des ondes de spin.

Abstract. - Dynamical spin susceptibility in metallic substitutional binary alloys is discussed both in paramagnetic and ferromagnetic state. Spin wave spectrum is discussed in detail.

1 . Introduction. - This paper discusses the theory mined self-consistently. By this simplification the of the alloys of magnetic metals at the finite concen- electronic structure can be discussed by applying the tration. The model is a substitutional binary alloy coherent potential approximation (CPA) [5]. As is of the type A, B, -, represented by seen in (2. I), the contributions of the mutual Coulomb interaction to the one-particle potential are taken into X ⁼ 1 tij a& a j , , + C ^{E: ni,a} + 2-' U i ni,o ni,-a,

i J , c i,o i,a account. Since the CPA treats the dilute limit exactly,

this scheme reproduces the virtual bound state of

. Friedel-Anderson-Wolff-Clogston-Moriya [6]. Static where t i j is the transfer integral which is assumed to and uniform spin susceptibility is discussed in [l]

be independent of the type of atoms. E: and U i are the and [2].

potential energy and the mutual Coulomb interaction at the ith site, which can take either ^82, U A or E:, UB.

There have been proposed two different schemes to (1.1) ; one by Hasegawa and Kanamori (HK) [I]

and Levin, Bass and Bennemann [2], the other by Harris and Zuckermann (HZ) [3] and Kato and Shimizu [4]. The difference between these two is briefly discussed in the next section. The main purpose of this paper is to examine the structure of the dyna- mica1 spin susceptibility in the HK scheme, which enables us to see the possibility of the instability toward the spin density wave state and to examine the elementary excitation in the paramagnetic state (paramagnons) and in the ferromagnetic state (spin wave). Especially the spin wave spectrum and its damping in the system with the excluded volume is examined.

2. Difference of the two schemes. - 2.1 HK

SCHEME. - In the Hartree-Fock approximation the atomic energy at the ith site can be approximated as

where < n i , - , > is the thermal average of the electron number. This scheme assumes either n , , - , or n B , - , for < n i , - , > depending on the kind of atoms at this site, requiring that these values are to be deter-

2 . 2 HZ ^SCHEME. - In contrast to 2.1, this scheme focusses on the dynamical effects of the Coulomb interaction, neglecting the spatial variation ofthepoten- tial. Thus such one-particle quantities as the local density of state are uniform in space, excluding the possible existence of the virtual bound state. Formally this scheme is self-consistent only if ^{E A , , = E ~} in ^{, ~} (2. I), in which case it takes into account some of the higher order processes of the electron-hole scattering

I X \

'f /' '\

( b ) x X imp =

FIG. 1. - Comparison of the two schemes for the excess spin susceptibility due to dilute magnetic A atoms in nonmagne- tic B atoms. HK scheme is equivalent to (a) in this case.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1974424

C4-142 H. FUKUYAMA

than 2.1 does. As an illustration to see the difference between 2.1 and 2.2, figure 1 shows the processes which are included in the calculation of the excess spin susceptibility xxi,, in the system with dilute magnetic A atoms (UA # 0, x is the concentration of A atoms) immersed in nonmagnetic B atoms (UB = 0).

Solid line is the electron Green's function and the cross denote A atoms over which averaging procedures are made. The wavy line and the square are UA and

0 0

= &A,U - &B,U = &A - EB + UA HA,-,. As is seen in figure 1, 2.1 includes more of the physically impor- tant processes than 2.2 does in this case of the dilute limit.

In this paper we are interested in the effects of the potential scattering as well as the random electron- hole interaction at the finite alloy concentration as a natural extension of the theory of the virtual bound state, and then we will use the KH scheme in the following.

3. ^x(Q,o) in the HK scheme [7], [8]. - We first derive the longitudinal susceptibility in the parama- gnetic state since it is mathematically transparent.

The transverse susceptibility in the ferromagnetic state is easily deduced from this. In the presence of the oscillatory magnetic field in z-direction, the Hamiltonian in the Hartree-Fock decoupling is given by

where g and 11, are the g-factor and the Bohr magneton respectively.

S: = (alT ^{ai,? -} a l l aiSl) and < ^{ni,-, > H}

are the average electron number in the presence of the magnetic field. < n,,, >* can be written as follows t o the linear order of the field.

< ni,, > in (3.2) is given by (2.1) and xi(= xA or xB) is the local susceptibility at A or B site to be deter- mined later. By (3.2), (3.1) is rearranged as

where ^{E ~}is defined by (2.1) and S, ^{, ~} = XS; + y ~ E ,

( y = 1 - x). By employing the linear response theory

[9] to the last term in (3.3), we get the equation for XA

By inserting (3.4) into (3.6), we have

where

The equation for X, similar to (3.6) yields

Solving linear eq. (3.7) and (3.9), we obtain the total susceptibility as

x(Q, W ) = XXA YXB . (3.10) The correlation functions in (3.8) are calculated as those in simple alloys represented by the first two terms in (3.3) by use of similar techniques in [lo].

The transverse susceptibility is obtained by simply changing

KmB and 6, in this case are given in ref. [8]. Instead of giving rather complicated mathematical expressions for these, we note that the structure of (3.7) and (3.9) is physically understandable, since, for example, the enhancement factor for xA includes KAA, the corre- lation function of spins located at A site only. If 6, = - E ~ is not negligible compared to the , ~

bandwidth, KAA can appreciably differ from KBB.

4. Discussions on x(Q, ^{o). -} 4.1 PARAMAGNETIC

STATE. - 4.1.1 Instability toward the spin density wave state. - Parallel to the discussions by Penn [I I], the possible instability is examined by looking at the function K defined by x-'(Q, 0) cc (1 - K(Q)) ; the paramagnetic state is unstable if K(Q) 3 1. K i n the case of the Wolff model (UA # 0, UB = 0) in the simple cubic crystal is shown in figure 2. Note that big difference between p (average density of states) and pA (local density of states at a magnetic A site). Simi- larly qualitative differences between KAA and KAB are found.

4.1.2 Paramagnons. - The existence of a long- lived diffusive spin fluctuation is shown near the ferromagnetic instabilities ; i. e.,

~ ( Q w ) = xo DQ'[DQ" id1-' (4.1)

SPIN SUSCEPTIBILITY OF METALLIC BINARY ALLOYS C4- 143

WOLFFMODEL X = ~ M B ' [ T O + E ~ ] ; K ' U A ( ~ + * Z )

"0 : [S-,,Sol

*, ^{: [s!a.s,l}

9 : [s",.s::]

DIFF: SITE

a : [s%.s%l

SAME SITE

So : xs:: + ysB

of the term of the order of Q 4 to z-' in disordered metallic ferromagnets is pointed out by Yamada and Shimizu [I41 in the case of a dilute alloy where the scattering is treated within the Born approximation.

If the system is saturated, the damping, zY1, starts from the higher order in Q.

D M and z-l are numerically evaluated and shown in figure 3 in the system with the excluded volume ;

E: + CQ and then electrons hop through A atoms whose contraction is x.

FIG. 2. - An example showing the instability toward the spin density wave state in the Wolff model (UA # 0, U B = 0). V O is the half of the bandwidth of the pure system with the simple cubic symmetry (with some extra simplification of the density of states curve) and 60 = E: - 8:. p and pa are the average den- sity of states and the local density of states at magnetic A sites

whose concentration is x . n is the carrier number.

where X, and D are the static uniform susceptibility of alloys and the diffusion constant proportional to respectively. In the case of weak scatterings caused by dilute impurities, X, and D reduce to those of Fulde and Luther [12].

4.2 ORDERED STATE. - 4.2.1 Bulk spin wave. - Within the present approximation, mB ⁼ n,.? - nBl is nonzero if m, is. Thus it could happen that m

-

However, this is highly accidental in the present case of metals. In the more general case of m # 0, the spin wave of the type ^o = D M Q 2 exists and its damping at zero temperature is proportional to Q4 when the system is unsaturated. More explicitly,

FIG. 3. - Spin wave in the system with the excluded volume (1 - x).

are the magnetization per an A atom, the stiffness constant and the ratio of the damping to the spin wave frequency.

V o is the same as in figure 2 and a is the lattice spacing.

where

D M = (2 nm)-I 1m 4 . 2 . 2 Local spin wave. - In the dilute alloy of A

- rn atoms, there could occur that 1 + K A A ( ~ , 0) = 0 (4.2) has a real solution, o = o,. This corresponds to the

2 Larmour frequency of an A spin under the exchange

= (nm)-' ^{D M} Q' Z

($)

k Although this mode is physically the same as that

(4'3) discussed by Wolfram 1151, the equation for o = w, 9, (1) = Q t (1) (k, ^E + i 0) is the retarded Green function is different.-The effects'ofthe scattering are

neglected in [15]. (For example, if U, ⁼ U,, there and ~ ( k ) = C ^eCkRijtij is the band energy. The existence

i exists no solution in eq. (55) of [15].)

C4- 144 H. FUKUYAMA

References

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