- Email: [email protected]
Effects of band structure and matrix element on RKKY interaction
Journal of Magnetism and Magnetic Materials 43 (1984) 21-42 North-Holland, Amsterdam
21
EFFECTS OF BAND STRUCTURE AND MATRIX ELEMENT ON RKKY INTERACTION A. NARITA * and T. KASUYA Department of Physics, Tohoku University, Sendai 980, Japan Received 1 December 1983
The effects of band structure and matrix elements on the RKKY interaction J(R) are separately investigated. When the Fermi surface has planes perpendicular to R, effects appear on the period of oscillation, the phase shift and the amplitude of J(R). The applicable region of the asymptotic form for large R and the validity of the free electron approximation are also examined. If there are no tangential planes perpendicular to R, it is found that: 1) when two interacting localized spins are on lattice points in the crystal, exponential damping appears even for the constant matrix element model and the matrix element effects introduce competing terms causing a sign change; 2) when one of the spins is at an interstitial position, the constant matrix model gives a weaker J ( R ) t x R -2 damping, but the character of this term changes into the exponential damping by taking into account matrix elements.
1. Introduction
Recently, interesting experiments concerning the exchange interaction between magnetic atoms in Heusler alloys were performed by the group of Ishikawa [1-3]. Their experiments produced the following results. 1) The nearest and next nearest neighbour exchange interactions are strongly ferromagnetic, 2) the long range interactions far from the third neighbour are the oscillating type. Furthermore, it seems that they are interpreted very well by the RKKY interaction, treating the s- and p-electrons as free. Result 1) was interpreted in terms of the virtual double exchange mechanism by Kasuya [4]. On the other hand, it was shown from neutron scattering experiments that the d-electrons are quite localized and the magnetic moment per Mn atom is about 4/~B [5]. Accordingly, Heusler alloys can be considered to be typical s-d systems, similar to the rare-earth system for the s-f and d - f interactions. Here, the matter in question is why the long range d - d exchange interaction can be interpreted very well by treating the conduction electron as free. Because, for Heusler alloys, the Fermi surface in the free electron model includes many Brillouin zone planes in its sphere because of the many conduction electrons per unit cell (from about four to seven) the Fermi surface will accordingly be cut into many zone boundaries. Thus, the real Fermi surface is predicted to be deformed considerably from the sphere. A similar situation should exist in many other magnetic materials in which the character of the interactions is analyzed by a method based on the RKKY interaction. The RKKY interaction in the free electron model is exactly calculated [6-10] and is well known. This interaction in metals with nonspherical Fermi surfaces was investigated by Roth et al. [11,12]. They derived the asymptotic form for the large distance between the two interacting localized spins (abbreviated as f-spins). But its range of validity was not clarified and the matrix element effect was not considered explicitly. For the indirect s-f exchange interaction in nonmetals, the asymptotic form with exponential damping was derived for the parabolic conduction and valence bands [13,14]. But the region where it can be applied and the matrix element effect are again unknown. * Present address: Department of Applied Mathematics, Akita Technical College, Akita 011, Japan.
0304-8853/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
A. Narita, T. Kasuya / R K K Y interaction
22
Our purpose is to investigate accurately the effects of the Fermi surface deformation and of the matrix elements to the R K K Y interactions, in order to give physical insight into the questions mentioned above. We will calculate this interaction rigorously assuming the simple periodic potential and will try to examine and clarify the validity of the asymptotic form of its interaction comprehensively and the validity of the free electron approximation itself in the s-d or s - f model (referred to as the s - f model below).
2. M o d e l
We investigate the effects of a periodic potential on the R K K Y interaction based on the following Hamiltonian,
H= Ho + H~_ f,
(1)
• [2m +Vc°sQz i ,
(2)
Hs_f=N ~i ~n S(ri-R.,)oi*Sn,,
(3)
where the usual notations in the s - f model are used. H o denotes the band energy in the conduction electron system. In H 0, the simplified one-dimensional periodic potential, the direction of which is defined as z-axis, is introduced. V and 2,rr/Q are the strength and the period of the potential, respectively. Therefore, the conduction electron is free for the x- and the y-directions. H~_ f denotes the exchange interaction between the conduction electron spin aj and the f-spin S.,. The s - f coupling is assumed to be the 6-function type in real space. The f-spin distribution is assumed in the following way. The lattice points are specified by the vector R, = (anx, an v, 2"nn./Q)(n x, ny, n=: integer), in which a is the lattice constant for the x-y plane and 2~r/Q is the latiice constant for the z-direction and is equal to the period of the periodic potential. Each lattice point is occupied by an f-spin. For the z direction, a particular f-spin can be also situated as an impurity, on an interstitial position between the neighbouring lattice points. Then the position of such an f-spin is represented by the vector R,, = (an~, anu, 2~(n z + t)/Q) (0 ~< t < 1). S., in eq. (3) denotes the spin operator of the f-spin situated at the position specified by R.,. N is the number of the unit cell. The eigenfunctions, {q,~k(r)}, of the one body Hamiltonian H 0 are taken as the basis set. Since the periodic potential exists only for the z-direction, q~,k(r) can be written as q~,k(r)= eikru,k:(z), in which u~k(z) has the periodicity u,k(z + 2~/Q)= u,k(z). Similarly, the eigenvalues E,k of H o can be written as
Evk=(hZ/2rn)[k~ +e,(k.)],
( k 2 = k 2 +k~).
(4)
We treat H 0 and H~_f as the unperturbed and the perturbation Hamiltonians, respectively. As is well known from second order perturbation theory [7], the expression for the R K K Y interaction between two arbitrary f-spins is obtained in general as follows, =
2
~',p" k,k'
where f~k is the Fermi distribution function and the matrix element I,(p'k', pk) is given by
I, ( p'k', uk ) = Iu,. k, (2 ~rt/Q ) * u,k:(2"~t/ Q ).
(6)
In later sections, the potential effects to the RKKY interaction will be investigated, assuming that one of two interacting f-spins is necessarily situated on the lattice point, i.e. R,,,, = 0. Since the Fermi surface is
A. Narita, 7", Kasuya / R K K Y interaction
23
deviated from a sphere, J(Rnt, t) [ R , , , , - - 0 in eq. (5)] shows the anisotropic property concerning the direction of the vector Rnt. On the other hand, owing to the nonuniform charge density of the conduction electron, the behaviour of J(R.,, t) depends on t through the matrix element. Thus there exist two effects, due to the deviation of the Fermi surface from the sphere and due to the matrix elements. In the calculations of J(R.t, t) given by eq. (5), we must have knowledge about u,k:.(2~rt/Q ) and %(k~). In the next section, the solution of the one body Hamiltonian H 0 will be described.
3. Fermi surface and matrix elements
The periodic part of the Bloch function can be expanded into the Fourier series, Uuk ( Z ) = E c.(uk:)e inQ =.
(7)
n
Eq. (7) is inserted into the Schr6dinger equation HoeP.k(r ) = [email protected](r) and then we get the equation for
c.(.kz),
[%(k~)-(k~ + nQ) 2] c . ( . k . )
)[
= ( m V / h = c . _ , ( , k z ) + c.+,
(~,k~)].
(8)
u . k ( z ) and ~.(kz) can be determined by solving eq. (8).
3.1. Fermi surface The Fermi energies Ef determined numerically are shown in fig. 1 as a function of V / E ° . There, ~1= Q / 2 k °, Ef° = h2k°2/2m, in which k ° is the radius of the Fermi sphere for V = 0 and E 0 denotes the
Ef~E; I
Eo/E~ I
I
I
I
[
J
.. | e-o_
08 Oo T
",
~ 1~02 ,.Vo
Fig. 1. The Fermi energy Ef (solid lines) and the bottom energy E 0 (dashed lines) reduced by the free Fermi energy Ee° are shown as functions of the reduced potential energy VIE ° for various values of , / = Q/2k°f.
A. Narita, T. Kasuya / R K K Y interaction
24
bottom energy of the first band. In our calculations, as will be seen in later sections, ~/ is an importanl parameter. When ~ > 1, the free electron Fermi sphere is contained in the first Brillouin zone completely When ~/= 1, it touches the zone boundary and for ~/< 1, it extends into the other BriUouin zones. The shape of the Fermi surface for each 71 value (~/= 1.125, 1 and 0.75) was calculated numerically and is draw~ in fig. 2. Since the Fermi surface in the present model is symmetrical around the kz-axis, only a fourth par1 of its cross section is shown. The quantities q0, ql, q2 and ~ are also shown in fig. 3. q0 and q2 are the diameters of the bellies formed in the first and second zones, respectively, q~ is the diameter of the neck formed in the first zone. ~ is the Fermi wave number along the kz-axis. As seen from fig. 1, for 7/= 1 and 1.125, the lowering of the Fermi energy is larger than that of the bottom in the first band, because the energy of any state at the Fermi energy decreases more than that for bottom. For ~/= 0.75, in the
q.=1.0 "z/K;
q,/2
.
z.
1.o
V/E=;
1.(
=~.4
0
.V/E;=O
1.12-
=o.8
0.8
~. = 1.125
Kz/K
0.8
V/E'~=0'6
" ~
0.6 0.4 0.4
a
0.2
Oo
012
Kz/K;
0.'4
/1 0.'6
0:8
--L- "o/2
O.2
1.0 KVK;
0
0:2
o4
o6
o:8
lO K"/K;
't : 0 . 7 5
1.0
o,
~
qqv ~
0.2
°o
ql/2
--~
I
qo/2
o12
oI~
J
o16
0:8
~.o K.,/K~
Fig. 2. The deformation of the Fermi surface due to the potential is shown for various values of V / E °. (a) ~ (c) "e = 0.75. For various notations, see text.
= 1.0, (b) ~ = 1.125
and
A. Narita, 7'. Kasuya / R K K Y interaction
25
Vc/E~'=0.8'=
(b) ,1.=1.125 1.2
( a ) "I. = 1.o I
I
I
I
qo/2k~
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
t
00
t
0.2
1
0.4
1
0.6
0.8
(~) ~.= o.T5
1.0 V/E~
|
=
w
0.6
0.8
.o/2~ 2 ~/Q
|
|
0.2
0.4
1.0
V/E;
v~.o.73
1.00B 0
1.004
'
'
'
1.00( 0.991 |
[
I
I
,
,
=
i
0.4
0.6
1-
1.0 0.8 0.( 0.,~ 0.2
00
0.2
0.8
v/E;
1.0
Fig. 3. The potential dependences of qo, q~, q2 and ~ are shown for various values of T/. (a) ~/= 1.0, (b) 7/= 1.125 and (c) 7/= 0.75.
A. Narita, T. Kasuya / RKKY interaction
26
beginning, the lowering of the bottom in the first band is larger than that of the Fermi energy, because the increase of energy for the state at the Fermi energy of the second zone is large. Since the Fermi surface in the second zone disappears for IV1 larger than IV~I= 0.73E °, the situation becomes the same as that for ~/= 1, for IVI sufficiently larger than IV~I. From the potential dependences of Er, E 0, Cl(k~) and c 2(kz) for k= near Q/2, the behaviour of qo, ql, q2 and ~ shown in fig. 3 can be easily understood. Since the potential dependences of Ef and E0are weak, that of q0 is also weak. qt, q2 and ~ have relatively stronger potential dependences, since those of c~(k=) and c2(k. ) for k. near Q / 2 are larger than that of El. Thus, when ~/= 1.125, the Fermi surface extends towards the z-direction and begins to have the gap on the zone boundary at IVI = IV~I= 0.64E °, while for ~/= 0.75, the Fermi surface in the second band shrinks as a whole and disappears at IVI = IV~l = 0.73E °.
3.2. Matrix element As seen from eqs. (5) and (6), the matrix element is written as
It( ~k, p'k') Io( v'k', ~k ) = I2B~,k,( t ) B~k.( t ) *,
(9)
B~k=(t ) = u ~k=(2"nt/ Q ) U~k:(O)*,
(10)
in which B~_(t) is written as the sum of the real part C,(~, k~) and the imaginary part D,(~, k..). The numerically calculated values are shown in fig. 4 for V I E ° = + 0.4. It follows from fig. 4 that the matrix element is strongly deviated from the value for the free electron model, particularly near the zone boundary. The behaviour at the origin and the zone boundary can be easily understood from the symmetry of the Bloch function [15].
C [~',k) 2 (b)
C. (~',k)
I
- -
(a) V/E¢ = 0
1.5
- - - 1"=2
~'= 1
--
V / E ~ = O. 4
- )'= 2 //
,
t=O, 0,25, 0.5,0.75(M=I) t=O ( u = 2 )
.- ...........
,'
/
//
o5
0.5
0.5 . . . . .
o. 25, o1~__- --""
\
t =0.25,0.75
/
-0.5
-0.5
iI / / ,/
0.5
o
or2
___-"
-I
t=0.5
-1
o.'~
o.'6
o~ 2k/Q
;.o
0
' 0.2
0.14
' 0.6
i 0.8 1.0 Zk/Q
A. Narita, T. Kasuya / RKKY interaction
27
Ct(u,k)
(c) . 2.0
. ~ 1
.
.
/
V/E; : - 0 . 4
D (~,k)
/
=
I
1.5
i
(d)
--- ~'--=2
V/E; : 0
~
t=O 1.0
M
=
1
-----
0.5
"". . . . . . . . . . .
1.0
J.-'o.
0.5
.
.
.
.
.
.
.
0.75 . . .
.
.
.
.
.
.
.
.
.
.
.
2J=
.
2
.
0.5
0
t= O, 0 . 2 5 , 0 . 5 , 0 . 7 5 t=0,0.5(~)=2)
0.25 o 0.75
/
-0.5
(~,,= I )
I
/
- 0.5
iI /
',.
~"
0.5
-1.0
0.25 ...................
-1.0
0
0.2
0.14
I
0.6
0.'8
1.0
0
2~
I
I
0.2
04-
I
I
0.8
0.6
1.0
2klQ
D.(u,k) Dt(~,, k )
!
(e) Y/E; = O. 4
~U= ---~=
I
,
!
i
(I)
2
1.5
VY E I•
=-0'4
---
=
~
2
1.5 0.75
....
- "',
1.0 0.75
1.0
0.5 0.5
0.25 0.75 0
t = O, O. 5(~=1 and u = 2 ) -
]
0.5
t -
......
-1.0
0.5
.:\ /
0.25
0.25 -
0
: 0,O.5(u:land u:'2) ~ , /
o~2 o~' o~6 0~8 2~0
1.0
--'~
1.0 0
0.'2
0~4
0~6
I
0.8 1.0 2k/Q
Fig. 4. The wave vector dependences of the matrix elements are shown for various values of V/E°. (a) and (d) V/E° = 0, (b) and (e) V/E° = 0.4 and (c) and (f) V/Ef° = - 0 . 4 , while (a), (b)and (c) are for the real part and (d), (e) and (O for the imaginary part.
A. Narita, T. Kasuya/ RKKY interaction
28
In the following, we classify the situation into two cases; cases A and B. In case A, there is no Fermi surface perpendicular to the k,-axis. In case B, the Fermi surface has sections perpendicular to the k~-axis. The temperature is taken to be zero.
4. Basic formulae
Firstly, we perform the integrals in eq. (5) with respect to q~, if' and k~_ analytically, by using eq. (4) and the cylindrical coordinates (k x = k I cos if, ky = k± sin q,, k'x = k~_ cos q¢ and k~, = k~ sin q~'), The result is as follows [15],
J(R., , t ) = J ( R
±
,R,,)=~ C
~
, fdk~O(,f-,~(k~) ) B ,,.(t)
*
x fdk'..B.,k;(t)ei'k;-k:'",,'fo~dkiklJo(R±k±) × [0(
mI 2 C=~r3NZh ~
B)Ko( R± v ~ ) - ( ' ~ / 2 ) O ( - B ) N o ( R . ~ - S ~ ) ] ,
(11)
19¢r [ IZ][Ue] 2 2mEf 2k°' l E° ]~ N I ' "f= h2 ,
(12)
B =,.,(k')-,.(k.)-kl,
(13)
where J0(x), No(x ) and Ko(x ) are the Bessel, Neumann and modified Bessel functions, respectively, O(x) is the step function and Ne is the number of conduction electrons. In eq. (11), the vector R,, is represented as R . , = (R j_, 0, R,,), since the Fermi surface is symmetrical around the kz-axis. The lattice constant a for the x-y plane can be any value. For a ~ 0, the f-spins distribute uniformly for the x-y plane and thus R± is continuous. On the other hand, R n t = R z 2~(n + t)/Q (n = integer and 0 ~< t < 1). R z may be often used for Rnt, in order to emphasize that Rn, is the z-component of R.,. When t = 0, R,, may be simply written as R,. Secondly, we get another expression by performing the integrals in eq. (5) with respect to ff and ~' in the cylindrical coordinates and by transforming k± and k', into ~ and c' [c --- k~ +%(k~), c' = k~2 +%,(k~)]. =
J(R.,,/)=1-~c
E f fd, pp,
d, '/(')
-/(~')] I,,(,)*I,,t(c'), C
--
(14)
I[
L,(,)= f
).
(15)
In eq. (15), the k~ integral is bounded by ~ - %(k~) > 0. Since % ( - k ~ ) = %(k~), f(c)[1 - f(~')] in eq. (14) can be replaced by f(c). In the limit R 1 ---,0, the k±-integral in eq. (11) is performable, using the asymptotic forms for R.L ~ 0. After the integration, we get
J(R~)
C E fdkfl(,f_%(k~))B.k(t).e_i.:k:fdk.,B,k,(t)ei.:k;
16~ ~'
:
× [(q-%,(k'))lnl,f-%,(k')l+(%,(k'~)-%(k~) In later sections, formulae (11), (14) and (16) will be used.
"
) lnl%,(k:)-%(k~)l].
(16)
A. Narita, T. Kasuya / R K K Y interaction
5. J(R:)
29
for t = 0 in case A
5.1. Constant matrix element The meaning of the constant matrix element is that the matrix element is the same as that in the free electron model; therefore B~k(t ) is given by fig. 4a and d and only the band structure effect on J(Rz) is considered. The numerically calculated results for , / = 1 based on eq. (16) are shown in fig. 5, where C O= Ck~'. From fig. 5, the following facts can be listed: 1) J ( R , ) always has a negative sign and 2) as V increases, J ( R , ) damps faster with increasing n. These facts can be explained as follows, based on eqs. (14) and (15). When the equal energy surface for the energy ~ makes an open orbit in the z-direction, I]t(c ) defined by eq. (15) becomes zero for t = 0. The equal energy surface with the energy larger than the Fermi energy makes the open orbit for the z-direction for ~/= 1. The final states in the intraband scattering thus belong to the open orbit equal energy surfaces and thus there is no contribution due to the intraband scattering. The contributions to J ( R , ) must occur through the interband scattering from the first band into the higher bands. Taking into account again the above mentioned fact concerning l~0(c), the only possible contribution is due to the process from the states belonging to the closed surface in the first zone to those belonging to the closed surface in the higher zone. In this process, there is a finite energy gap IV I between the initial and the final states, which causes exponential damping on J(R,). The largest contribution comes from scattering from the vicinity of the top of the first band for k± = 0 with energy c] into the bottom of the second band for k~_ = 0 with energy E 2 ( ( 2 -- E1 IvI). I]0(0 for c _< c 1 and I2o(C' ) for ~' > ¢2 always have =
.3.0J(Rz)/C°
constant
matrix element
v/E
X
o '~ []
=I.0
x
0.2 0.4 0.6 0.8
2.0
D O
1.0
10~-3 I 1O-/"A
10-2/
F'-,/ -1.o[
o
v/E = o
, -ak Rz
Fig. 5. The numerically calculated values of J ( R Z)/C o in the constant matrix element are shown for various values of V / E f°. The c a s e for V / E f° = 0 is shown by the solid curve. The positions marked by upward arrows correspond to t = 0 and the positions by downward arrows to t ~ 1/2. Note the change of the multiplication factor in the different ranges of 2k°Rz.
30
A. Narita, T. Kasuya / R K K Y interaction
opposite sign. Thus, J ( R , ) have negative sign, as easily seen from eq. (14). The present consideration does not depend on the value of */explicitly, but depends only on the shape of the Fermi surface. In fact, from the evaluation of the contribution due to the above scattering, we can obtain an analytic expression for J(R,) for an arbitrary value of ~, when the Fermi surface has an open orbit along the z-direction [15].
J(R,)=
16,~2n 2C°~12lvlKl(z), ( v = V / E ° , z
=~rnlvl )27/2 ,
where Kl(x ) is the modified Bessel function of the first order. The asymptotic forms of x ~ 0 and (~r/2x) 1/2 e x p ( - x ) for x ~ oo. From these asymptotic forms, we get
Jn0
6°'~4
(17)
Kl(x)
are
1/x
for
(z ~ 0),
(18)
(
(19)
J(R°) =
C°Lvl'/erl3
" " 2n5/--------16~r ~ exp~ - z )
z
~).
Note that eq. (18) agrees with the result of the free electron model. It followed from the numerical analysis [15] that the asymptotic form [eq. (19)] can be very well applied for z > 2.5. Therefore, it can be concluded that J(R,) in the constant matrix element can be described very well by the simple single term evaluated as the contribution due to the interband scattering from the states near the top of the first band into the states near the bottom of the second band.
5.2. Matrix element effect The matrix element effect can be investigated by adding its effect upon the results for the constant matrix element. The numerically calculated results in this case are shown in figs. 6 and 7. The following characteristics are seen: 1) J, can take a positive value for large values of n and V/Ef °. 2) While J, shows the damping with increasing values of n and V/Ef °, the damping is not so simple compared to the case for the constant matrix element, because J, for V > 0 changes sign on the way. It depends also on the sign of I7.
5.2.1. Interpretation of the numerical calculations Firstly, fact 1) listed above is considered. It must be noted that 1,0(E) defined by eq. (15) remains finite owing to the nonconstant matrix element, even if the equal energy surface makes an open orbit. Therefore, a contribution from the intraband scattering also occurs. We divide J, into contributions from the intraband (Jint . . . . ) and the interband scatterings (Jint ..... ), which are shown in fig. 8. As seen from these figures, the interband contribution is always negative, but the intraband contribution can be any sign. The intraband scattering may be divided into two processes by the initial state; 1) c o < ( < q and 2) q < ( < of, in which c o is the bottom energy of the first band. The sign change in J~n,.... can be understood as the competition between them [15]. For the interband contribution, the scattering from the first band into the second band is important. There are four possible processes: 1) % < c < q ~ % < c ' < c 3, 2) q <( 0, as already discussed, the sign of I10(c) depends only on n. The sign of Izo((' ) can be easily judged to be opposite to Ilo(C) for the same n, because the contribution into the states with energy ( ' - - - % is large. Therefore, the contributions from both 1) and 2) give negative sign on J~nte.... making it larger than that of the constant matrix element,
A. Narita, 7". Kasuya / R K K Y interaction
THE EFFECT OF MATRIX ELEMENT mr= 1.0 a
J (Rz)~C o I .00
0.80
x lo -2
0.60
v/E;
;/
,
o
0.2
A
0.4 0.6 0.8
D ,'
x lo -3
31
x l O -/-*
0.40 0.20 O.
oW,s
-0.20
-
-0.40 -0.60
v/E; : 0
-0.80
y./E)
:
0
~, 2k~R z -I
.00
THE EFFEC,r OF I'IR,rRIX ELEMEN'r
J(Rz)~C ° I .00
b
-
0.80
x10-3
\
i
, 0
•
o
-0.2
A
-0.4
|
o.6o - x10 -2 ~ 0.40
V/E;
q.= 1.0
'--
xlO -~
., .
.
.
.
I
~
.
-0,40
-0.60 -0.80 -I
I~'Y~"\--,~V/E 0"y / v/E;: o
,
2,;Rz
.OO
Fig. 6. The numerically calculated J ( R z ) / C o are shown, by taking into account the matrix element effect. The meaning of upward and downward arrows is the same as in fig. 5. (a) V > 0 and (b) V < 0.
32
A. Narita, T. Kasuya / R K K Y interaction
the effect of matrix element
in/Illd
v[= 1.0
,V/E;
Fig. 7. The potential dependence of J , / I J ° l , taking into account the matrix element effect, jo is defined by eq. (18).
except for n = 1. For V < 0, from similar considerations for the case V > 0, process 1) gives negative sign and 2) positive sign on Ji,t .... . Because the former contribution is larger, the sign is always negative, but the absolute value decreases rapidly with increasing n and IVI c o m p a r e d to that of the constant matrix element. The signs of Ilo(C ) and I20((') are listed in table 1 and the signs of Jint .... and Ji,t .... are listed in table 2. 5. 2.2. N u m e r i c a l a n a l y s e s
Here, fact 2) listed at the beginning of this subsection is considered. As is clear from the above consideration, when the matrix element effect is included, various contributions begin to compete with each other and the situation becomes complicated. Even if we consider each Jim .... and Jim .... separately, it is impossible to analyze the data by a single exponential term such as eq. (17). The most favourable case, Ji,t .... for positive V, is shown in fig. 9 as an example. The data can be fitted by the form Jintra.,,/Co = A × 10-3 × Ivl x n - ~ e x p ( - B n l o l ) . However, from fig. 9a, A = 7, a = 1.5 and B = 1.6 give the best fit, while from fig. 9b, A = 5, a = 1 and B = 1.7 give the best fit. This means that we need a more complicated form.
~=I.0
3intra,n/-Ip-r~ I
1.o[ 5
~.= 1.0
Jinter. n/IJ~ll
0,8
n=l
0.6
0.4
- 076 L
a
Fig. 8. The potential dependences of (a) Ji.tra,. and (b) Jint.... are shown for n = 1-5. jo is the value for V = 0.
' V/E;'
A. Narita, T. Kasuya / RKKY interaction Table 1 The signs of llo(Q and
33
120(4 )
11o(4) c
Sign of V Odd n Even n
12o(O
1
4>c 1
+
Table 2 The signs of J i m . . . . and Jinter,
¢>__4 2
+
--
+
--
+
+
--
+
--
_
--
+
--
+
+
n
inlra,n
Jinter,n
1 ) 4 0 < 4 < 41 ---~.c f < c '
2 ) cl < E < ~ f - - - * E f < ~ t
1 ) 40 < ~ < Cl ---* 42 < ¢ p
2 ) 41 < 4
Sign of V
-
+
-
+
-
+
-
+
11o(C)11o(4')
-
+
+
+ --
--
+
--
110(4) 120(4')
6. J(Rz)
for
t ~ 0 in case
<4f--*c2
<¢ t
A
6.1. Constant m a t r i x element F o r t * 0, one of the f-spins is situated on the lattice p o i n t a n d the o t h e r on the p o s i t i o n d e v i a t i n g from the lattice point. Results o f the n u m e r i c a l calculations for t = 1 / 2 are shown in fig. 5. S u r p r i s i n g l y enough, J ( R z) have c o n s t a n t positive sign, b u t b e c o m e larger t h a n that in the free electron m o d e l with increasing
164
Ji~w~ n
(a)
=1.0
Go
V>O
ld 3
(b)'
q=1.0
V>O
id ~.4
1@ i#
L7
1J o
¥./t-~= 0.8
n=5 '
o12
o14
' ,
~,t~
'
o:6
'
o:a
Id~ ....
pll
Fig. 9. J i n t . . . . for the positive potential and 71= 1 is shown for the various values of n and IVI/E °.
A. Narita, T. Kasuya / R K K Y interaction
34
potential. This is quite different from the case for t = 0 and is against common sense accepted thus far. Let us consider the result based on eqs. (14) and (15). It must be noted that the integral of the phase factor for t 4= 0 in eq. (15) is not zero, even for the open orbit. Therefore, the contribution from the intraband scattering is now not zero. It is shown that the contribution for the initial state existing in the region { < { 1 gives a damping term similar to the case for t = 0 and the main contribution comes from the initial state for {f > { > {~, which gives the R72 term. For the interband scattering too, the contribution from the same initial state become finite and gives the same R~-z dependent term. The analytic calculation gives the following asymptotic form for large R , [15], J( R_ ) = C sin2vt . 4~R----T[ ( { 2 - { ] ) l n l { 2 - { , ] - ( { 2 - { f ) l n [ { 2 - { r l - ( { f - { , ) l n l { f -
(20)
{,I],
in which C is a numerical factor given by eq. (12). As shown in fig. 10, the numerical results are well fitted by eq. (20).
6.2. Matrix element effect Numerical calculations taking into account the matrix element effect are shown in fig. 6. In this case, since two f-spins are mutually located on the positions of the anti-phase of the potential, J ( R ~ ) for t = 1 / 2 does not depend on the sign of V. The figure indicates that J ( R z ) damps with increasing potential and shows completely different behaviour from the result for the constant matrix element. This is indicated more clearly in fig. 11. In this figure, TI (l = intra, inter) is defined as Tt = [Jt(R~)lhS/a/Colvl l/z, where h = n + 1/2. From this figure, it follows that the matrix element effect causes R72 damping in the constant matrix element to change into exponential damping [15].
J(Rnt)//Co
lO-a
,
x ~ ~
r
i
i
i
constant matrix element
~ =1,0
165
t =112
' '~-o 4 2 =
5 I
10 20 *h I
--
"
50 I
100
Fig, 10. J ( R , , ) / C o for the constant matrix element is plotted as a function of h for V / E ° = 0.4 and 0.8. Here, R . ,
h=n+t.
= 2"~h/Q
and
35
A. Narita, T. Kasuya / R K K Y interaction Tintra
Tinter i
I
I
I
ettect of matrix
[o(a)
I
I
I
I
element
10.2
o
~=1,0
10-2_ ~':. .~\ % \ °o
~.
\
\
z,\~
\\
•
\
,
\\
\\\\\\\~\
I 2
I
I 4
I
I
-~~
= h=1.5
N\
\\NN~NN\~
2__!
\ I
I
t=1/2
1(54 .
164
~d5 0
~.= 1.0
~
hgto5
X \
10"3... ~
I
t.
h=O,5 |Q
I
effect 0t matrix element
(b)
,~N~ _
°
I
o
t =1/2
-
\
I
I 6
x
~l
16 5 I
~-
, 0
I
I 2
I
I 4
I
I 6
''XX'N~ Xl
Fig. 11. (a) Ti.tra and (b) Ti.ter are shown as functions of ( ~ / 2 ) h I V I / E °. The dashed lines show those for t = 0 in the constant matrix model.
7. J( R= ) in case B The asymptotic form of J(R z) in case B was derived from the method of expanding it by R~-1 using eq. (16). The main approximations used were as follows: 1) the contribution 0nly from the intraband scattering is collected; 2) the U-integral in eq. (16) is extended to infinity and the contribution from the poles is extracted. The expression obtained is [15]
~,~ ,0) [B2
J(R)/C=(-1)
ei2R~ + CC] + ( - - 1 ) p,
i
[eiZRt~(3BB(I),~(])+½B2,~(2))_cc],
(21)
where ~'r is the band number in which the Fermi level stands, and the following abbreviated notations are used, at R = R z, B = B,,~(t),, = %f(~) and F (') = ~-TF, (22) where F = B or c. Using eqs. (10) and(7), we can represent eq. (21) as the superposition of the harmonic waves,
J(Rz)/Co=-lEA. X
cos(2kr-nQ)R~+lEB, n
X
n
sin(2kr-nQ)Rz,
(x=2k°R~),
(23)
A. Narita, T. Kasuya / R K K Y interaction
36
where k f is the radius of the Fermi surface in the extended zone scheme (kf = ~ for 71 = 1.125, k f = Q for 7/= 0.75). A n and Bn depends in a complicated way on the potential. The values for A~ and B, obtained from the two wave approximation are shown in table 3. In this approximation, A, and B, for n > 2 and n < 0 are zero. Results for the constant matrix element are also shown. In fig. 12, the results for 7/= 0.75 obtained by the numerical calculations using eq. (16) and those obtained by the asymptotic form eq. (23) with table 3 are shown. It is seen that the asymptotic form is excellent in the whole region for the constant matrix element model (fig. 12a) and satisfactorily good even including the matrix element effect. The shift of the oscillation period with increasing field strength is simply due to the change of the Fermi wave vector 4. The increasing amplitude with increasing positive value of V is the matrix element effect seen in fig. 4b. As V increases, the Fermi surface in the second zone approaches the zone boundary, where C0(2, k) becomes large. The opposite happens for negative V, as seen in fig. 4c. The agreement for *1= 1.125 is less satisfactory, in particular, for small R,. This is because the interband contribution with exponential damping is more important for 7/= 1.125. Now we introduce the model to write J(Rz) in a similar form to the free electron model. J(Re)/C
0 = ( Is_f/l)2(kf/k°)4[sin(x
- (p) -- x c o s ( x
-- t ~ ) ] / x
(24)
4,
where x = 2krR z, / s - f is the modified s - f exchange coupling constant, q~ is the phase shift and k r is the effective radius of the Fermi surface. I S_r, q~ and k f are parameters. This model is referred to as the
Table 3 The values of A , and B, o b t a i n e d by the two wave a p p r o x i m a t i o n . The values in ( ) i n d i c a t e those in the c o n s t a n t m a t r i x element
VIE °
Ao
A1
Az
Bo
B1
B2
0.587 (0.945) 1.223 (0.945) 0.207 (0.733) 1.067 (0.733) 0.010 (0.310) 0.397 (0.310)
- 0.213 (0) 0.443 (0) - 0.161 (0) 0.829 (0) - 0.015 (0) 0,573 (0)
0.019 (0) 0.040 (0) 0.031 (0) 0.161 (0) 0.005 (0) 0.207 (0)
- 7.990 (0.094) 5.081 (0.094) - 9.278 (2.893) - 3.358 (2.893) - 1.660 (6.215) - 12.475 (6.215)
- 2.148 (0) 12.34 (0) 3.546 (0) 16.270 (0) 2.078 (0) - 5.765 (0)
0.652 (0) 2.070 (0) 0.023 (0) 6.827 (0) - 0.634 (0) 2.339 (0)
1.199 (0.964) 0.694 (0.964) 1.196 (0.845) 0.363 (0.845) 0.853 (0.593) 0.092 (0.593)
0.326 (0) - 0.188 (0) 0.692 (0) - 0.210 (0) 0.861 (0) - 0.093 (0)
0.022 (0) 0.013 (0) 0.100 (0) 0.030 (0) 0.217 (0) 0.024 (0)
- 0.889 (1.214) 4.095 (1.214) 0.619 (1.857) 5.705 (1.857) 5.704 (2.954) 3.639 (2.954)
- 4.150 (0) 1.150 (0) - 7.963 (0) - 0.775 (0) - 4.632 (0) - 2.547 (0)
- 0.547 (0) - 0.232 (0) - 2.356 (0) - 0.254 (0) - 3.792 (0) 0.359 (0)
1)~ =1.125 0.2 - 0.2 0.4 - 0.4 0.6 -0.6
2) n = 0.75 0.2 -
0.2 0.4
- 0.4 0.6 - 0.6
37
A. Narita, T. Kasuya / R K K Y interaction CONSTRNT M R T R I X ~1= 0 . 7 5 0
J (RzVc o
ELEMENT
v/E;
a
1.00
0.80
x10-2
0.60
i
&
0.2
Q
0.4
0
0.6
(3 x 104) -1
(6 x 10Z')
x 10-3
/
0.40 0.20
•.
O.
l/\..
•
-0.20
~0.40
0.2
-0.60
-1
X',o, ~.2
!o 7 1
I/R o 0.6
-0.80
0,4 0.6 ~ 2ktR z o
i 0
0.2
- I ,00 J(Rz)/Co 1.5 i 10 - 2 1.0
:0.75
b
10_z
(6x104)_ 1
/."~
,,!
10-3
(3xlOt') -1
V/E~ J(Rz~/C* 1.5r
~ : 0.75 ~ ~"
|
0.~
/
I
.~,
"
"I1" I/ ,"
-1 .C
-1.5
/ •
"~ - .......
i~ WE~' / oo
10-~
0.0 -0.2 0.4
..... :i. ...... 0.6 (3x10") i
.
,
(6x10")-'
I, Wti
H i/ ',,'e
0.2
it .o . ............. . . 0.4 0.6
10-3
1.0]-
- ff i.~11.\!
1, -0.5
10-2
~ = ...... =1 . . . .
II~ / '/~v / i / ; t..." I!/,' ; II
ii"=
~,
.c~lt~z~i . . . ~il
i \./
o| , ~ / , ~ .L!...~...! t/~['-...///I~":T"~'.[~:,."; [ ~If ~,1 ~/"~51 """"'~r/ I,,",.:.~. li :,1~,35~,~11;i~. \45
ll'
!~ o i "v"
_1.0L
L /
~' 2k~Rz
Fig. 12. The values for J ( R z ) / C o obtained by the asymptotic form (various lines) and by the numerical calculations (zx, ru, O) are shown as functions of 2 k ° R z for 7/= 0.75. Upward and downward arrows denote the positions specified by 2 k ° R z = 2~rh/71 (h = n for 1' and h = n + ½ for $, and n = 1-5), where the numerical calculations were carried out. (a) constant matrix element, (b) positive potential and (c) negative potential.
pseudo-free electron model in this paper. In the present case, kf is chosen to be equal to k f in eq. (23). J ( R z) in various cases are c o m p a r e d in fig. 13. It is seen clearly that J ( R , ) for the constant matrix element can be fitted by the pseudo-free electron model by choosing 1~_ f properly. Even when the matrix element effect is included, the m a i n terms are usually A 0 and B o in eq. (23) and effects of other terms are included in the phase shift ~ and the renormalized amplitude ls_ f. A typical example is the case for ~ = 1.125 and v = 0.4. W h e n the other terms are large, the pseudo-free electron model cannot fit the results. A typical example is the case for ~ = 1.125 and v = - 0 . 4 , where, due to a large B] term, a large oscillation with a long period is added. N o t e that 2kf = 1012k f - QI. The same situation exists for ~ = 0.75 and o = 0.4, but the period of the B 1 term is shorter than the former case, because now 2kf = 4.512k t - Q{.
38
A. Narita, T. Kasuya / R K K Y interaction
~= 1.125
J(Rz)/C o
................
1.5
k|
...... const,mat.ele ~-~v=0.4 ..... v= -0.4
V/E~'= v
!.".,.
1.0
10-2
10-3
/~
1
o.s
//'i il/'
~ I
O
:...... //'k "/,...%.
.....
lo -4
r,,
"' / "~\25t / ! ;j,'/""~:,:" \ , :"J', "".~.-..V~i
i'V] ii.,.;y
~.4
/
,
~:: !j ~.~ !(\1/
'/
/
/
i ~'-,~
'.
1.o i o - 2
,0-3
/!
,i~
io-~
(s,lofl-1
/
"
i i
-,.o
( 6 x 1 0 4 ) -1
0' //.7 "~
iI :i v
ill
~/ "~'.
~0~5
I -~.o / ..............,,, /
-2.C -1.5
......
/it ~' ,"
~ ~
--.--
°o,,,t.mot.~,, = .4 "O
. . . .
V= -0.4
\/
1 '~.,',! i /
,_~),~,:,
i!
!/
b
Fig. 13. Comparison of various J(R,.) for [VI/Er° = 0.4. The dotted line denotes J(R~.) given by eq. (24), with Is_ f = 1 and ~ = 0. for the constant matrix element (the dashed line), with the matrix element effect for positive V (the dot dashed line) and for negative V (the dot dot dashed line) are calculated by the asymptotic form of eq. (23) with the coefficients given in table 3. (a) ~1=1.125 and (b) ~ = 0.75.
J(R:)
8. J(Rl) T h e asymptotic form valid for large R~ was derived from eqs. (14) and (15), based on the method of the stationary phase by Roth et al. [11]. For the case under consideration, for example case B in ~/= 0.75,
J ( R ± ) / C = ba(O)2q~ F(qoR.) 8c~'(0)
bl(Q/2)2q4 F(qtR_L ) ~ b2(O/2)2q4 F(q2Rj_ ) 81(;'(Q/2) 1
+ b'(O'b2(Q/Z)[(q°+41%(O)(z(Q/2)]l/zq2)/214 [1-
8(~'(Q/Z) k(q°-qo + q2q2)2] F ( - ~
bl(O)bl(Q/2)[(q°+ql)/2]a[1-(q°-q']2]G(q°+q]R
R±/]
),
(25)
4 [ (~'(0)[(~'(Q/2)l] ,/2 where the n u m b e r of primes denotes the times of the differentiation with respect to k z, F(x) = (sin x x cos x)/x 4, G(x) = (cos x + x sin x)/x 4 and b,,(kz) = B,,k(O). The first term comes from scattering process a) from the belly(l) to the belly(l), in which (1) means the first zone, the second term b) from the neck(l) to the neck(l), the third term c) from the belly(2) to the belly(2), the fourth d) from the belly(l) to the belly(2) and its reversal, and the fifth e) from the belly(l) to the neck(l) and its reversal (see fig. 2c).
A. Narita, T. Kasuya / RKKY interaction
39
Note that there is no contribution due to the process from the neck(l) to the belly(2) and its reversal, since the product of the matrix element b ] ( Q / 2 ) b 2 ( Q / 2 ) is equal to zero. For case A in 77= 0.75 and 1.125, the terms related to the belly(2) vanish. For case B in ~/= 1.125, only process a) remains. When the potential V is positive, the second and fifth terms vanish, because b I ( Q / 2 ) = 0 and when the potential is negative, the third and the fourth terms vanish, becauseb2(Q/2 ) = 0. In eq. (25), the fourth and fifth terms are called the cross term. The behaviour of J(Rj_) for 7/= 0.75, 1 and 1.125 are shown as functions of 2k°R± in fig. 14. The case for T/= 1.125 is first considered. [VILE ° = 0.4 corresponds to case B and IVI/Ef° = 0.8 to case A. In all cases, the asymptotic form can reproduce the numerical results very well, even for the nearest neighbour distance, using the cubic lattice model. As was mentioned above, except for V / E ° = - 0 . 8 , only the first term of eq. (25) makes a contribution. Because the interband effect with the exponential damping form is small, in the present case as was mentioned before for J ( R z), the free electron model is applicable. The weak potential dependence of q0 leads to almost the same period as the free electron model. Change of amplitude is due to the matrix element near k z = 0, as is clearly seen in fig. 4. For V / E ° = - 0 . 8 , the first, second and fifth terms in eq. (25) make a contribution. In the whole region under consideration, the first term is the main term, but the second term becomes more important with increasing R~_, because of the smaller value of ql than q0- The effect of the second term is the phase shift type for a smaller R . region and the amplitude modulation type for larger R ± . The same argument is applicable for ~ = 1. For positive V, only the first term of eq. (25) contributes. The case of V / E f ° = - 0 . 4 is similar to the case of V / E f ° = - 0 . 8 for , / = 1.125. For V / E ° = - 0 . 8 , q] is about a half of q0 and thus the behaviour of J ( R . ) is well understood by these two wave models modulated by the interference term. For ~/= 0.75, the Fermi surface exists in the second zone for f V I / E ° = 0.4, but disappears for IVI/Ef° = 0.8. Therefore, the case for JVI/E ° = 0.8 is in the extension of ~ = 1.125 and 1. The behaviour for V / E ° = 0.8 is different from that of the free electron model, because the cylindrical feature of the Fermi surface is now important. For
V/E
J(R )/Co
0.60 I 0.40
+ -0.~
~.= 1.125 a
* A
-0.4 0.4
o
÷
0.8
x~°-3
x 1o -2
0.20 O. i
-
'
15
'
'
-0.20 -0.40
-0.60 -0.80
2:
-1.00
-1.20 -I
.40
+
> 2
R_L
40
A. Narita, T. Kasuya / R K K Y interaction
J(R-)/'Co
It= 1.000
+ .
b
-0.8 -0.4
0.4
a
0.60 I 0.40
o X(50) - I
0.8
x 1 0.3
0.20 O.
,,o
-0.20
-0.40 •8
-0.60
-0.80
V/Ef
0.8 0
0
-1.00 -1,20
-I
~
J(R.)/C, 1.5 -
"[ = 0.75 \
v/ET C
o.o
~% 1.0
\
0
~.--/~ --- 0,4 /" "~ '~. . . . . . . . . . . . 04 11A a-'0"8 . ---0.8
: .-.
'./
0.5
"i /
,
-0.5 ~ j "
2 k; R.L
>
.40
I
I I I
/
."
'
xlO-2 I -I.0----xlO-2 I u .......... xlo-1 I - x10-3 __.__x10-5 ..... x10-4 .... x10-1 ............. x10-2 -1.5 --"x10-4 " --"-- x10-2
I
/ I
~
"ll .:
l:*..
I
~ ~t I
d / ...."
~
I
/
j
"k// i . ..... '~
i
7"
xlO-4 xlO-4 .............xlO- ~, ----xlo -s .... x10-3 a
o >
2E~R.I"
Fig. 14. J ( R ± ) / C O are shown as functions of 2 k ° f R L . The results from numerical calculations ( + , *, zx, [3) for the cubic lattice model are compared with the asymptotic forms (solid lines in (a) and (b) and various lines of different types in (c)). (a) "q = 1.125, (b) ~1= 1.0 and (c) "1 = 0.75. For (c), note the different multiple factors on J ( R ± ) / C O for different values of V.
A. Narita, T. Kasuya/ RKKY interaction
41
V I E ° = - 0 . 8 , ql is nearly equal to q0 and thus the behaviour is similar to that of the free electron model, but the amplitude is very much enhanced. For V / E ° - 0.4, the contribution from the q2 term is added. Because of its large contribution, a complicated amplitude modulation type distortion is seen. For V / E ° = - 0 . 4 , t h e situation is similar to that for V / E ° = - 0 . 8 in , / = 1, because of the noncontribution from the q2 term.
9. Conclusions In the three-dimensional crystal, a one-dimensional simple periodic potential was introduced along the z-direction and its effect to the R K K Y interaction was investigated. This potential gives rise to two effects, that is; 1) the deviation of the Fermi surface from the sphere and 2) the matrix element effect due to the modification of the wave function of the conduction electron. A summary of the results obtained from the calculations carried out thus far are as follows.
9.1. J(Rz) 9.1.1. (Case A) No Fermi surfaces perpendicular to the z-axis 1) When both f-spins are located on the lattice points, the asymptotic form of the R K K Y interaction for a large distance shows exponential damping. This fundamental property does not depend upon the inclusion of the matrix element, but the detailed property is largely modified in quality by taking into account the matrix element. In the constant matrix element, the interaction is described very well by a simple single term with constant sign, because the only possible contribution is due to the interband process with finite energy gap IVI. The behaviour is well described by the analytic function given by eq. (17). When the matrix element effect is taken into account, the situation becomes much more complicated, because the intraband process is now not zero. Furthermore, in the interband process too, the contribution from the pair states with energy difference smaller than IVI can contribute. The sign of J ( R , ) depends on the sign of V. Because of competition among these different processes with different Rz-dependence, the sign of J ( R z) can change as a function of Rz. The sign change becomes evident as V decreases connecting to the R K K Y type continuously and also as the neck diameter q~ decreases connecting to case B. The matrix element effect is particularly important in this region. 2) When one of two interacting f-spins is located on the lattice point and the other on an interstitial position, the matrix element effect plays a crucial role. When the matrix element effect is taken into account accurately, the asymptotic form of J ( R , ) for a large distance indicates exponential damping similar to the previous case, including the matrix element effect. But in the constant matrix element, the asymptotic form for large R, shows a much slower damping, proportional to the inverse square of the distance. This is due to the fact that in the intraband contribution, for example, the energy difference between the pair states can be zero. By taking into account the matrix element effect correctly, the character of this R~-2 term changes into the exponential damping. Therefore, to obtain exponential damping, it is essential to take into account the matrix element effect correctly. In general, compared with the case 1), the R K K Y type effect appears more rapidly with decreasing V and q~. 9.1.2. (Case B) The Fermi surface has perpendicular planes to the z-axis 1) The asymptotic form, eq. (23), which is valid for a large distance, was derived up to the order of R~-a. Note that R z can be any value, including the interstitial point. This asymptotic form was shown to be applied practically from the nearest neighbour distance, except in the special case where the perpendicular part of the Fermi surface approaches the zone boundary extremely closely, because then the interband contribution considered in case A becomes relatively important.
42
,4. Narita, T. Kasuya / RKKY interaction
2) The formula called the pseudo-free electron model seen in the extended zone scheme is used together with that most cases, including the matrix element effect as fitted fairly well by the pseudo-free electron model, but with different frequencies become large.
was introduced by eq. (24), in which the single kf the phase shift and the amplitude. It was shown well as the constant matrix element case, can be some cases cannot be fitted, because other terms
9.2. J ( R j_) In this case, there is always a Fermi surface perpendicular to R ~ , the contribution of which is very similar to that of the free electron model. Therefore, the problem is the competition with other terms, the neck part, the Fermi surface in the second zone and the interband effect and the cross term a m o n g them. The general asymptotic form was derived and shown in eq. (25). The numerical results can be fitted very well by eq, (25), in most cases. W h e n the interband effect is large, it is possible to write J ( R ± ) in two terms, eq. (25) and the exponential d a m p i n g term due to the interband effect. Again the asymptotic form was checked to fit the pseudo-free electron model described above. W h e n the potential is small, it is possible to fit eq. (25) with the pseudo-free electron model, that is, the effects of other terms are approximately included in the phase shift and the effective amplitude. W h e n the potential becomes stronger, the behaviour of J ( R l ) becomes more complicated and thus cannot be fitted by the pseudo-free electron model. The results for the Heusler alloys should be considered, based on the present calculations. Comparison and interpretation of the results in actual materials will be carried out in separate papers.
Acknowledgement One of the authors (A.N.) would like to express his sincere thanks to Professor A. Yanase for helpful advice in the numerical calculations.
References [1] Y. Ishikawa and Y. Noda, Solid State Commun. 15 (1974) 833. [2] Y. Noda and Y. Ishikawa, J. Phys. Soc. Japan 40 (1976) 690, 699. [3] S. Tajima, Y. Ishikawa, D.J. Webster, N.W. Stringfellow and D. Tocchetti, J. Phys. Soc. Japan 43 (1977) 483. [4] T. Kasuya, Solid State Commun. 15 (1974) 1119. [5] D.J. Webster, Contemp. Phys. 10 (1969) 1559. [6] T. Kasuya, Progr. Theor. Phys. 16 (1956) 45. [7] T. Kasuya, in: Magnetism, vol. 2B, eds. G.T. Rado and H. Suhl (Academic Press, New York, 1967) p. 215. [8] A.J. Freeman, in: Magnetic Properties of Rare Earth Metals, ed. R.J. Elliot (Plenum Press, New York, 1972) chap. 6, p. 245. [9] M.A. Ruderman and C. Kittel, Phys. Rev. 96 (1954) 99. [10] K. Yosida, Phys. Rev. 106 (1957) 893. [11] L.M. Roth, H.J. Zeiger and T.A. Kaplan, Phys. Rev. 149 (1966) 519. [12] A. Blandin, J. Phys. Radium 22 (1961)507. [13] N. Bloembergen and T.J. Rowland, Phys. Rev. 97 (1955) 679. [14] H.J. Zeiger and G.W. Pratt, Magnetic Interactions in Solids (Clarendon Press, Oxford, 1973) p. 516. [15] A. Narita and T. Kasuya, Sci. Rep. Tohoku Univ. (submitted).
21
EFFECTS OF BAND STRUCTURE AND MATRIX ELEMENT ON RKKY INTERACTION A. NARITA * and T. KASUYA Department of Physics, Tohoku University, Sendai 980, Japan Received 1 December 1983
The effects of band structure and matrix elements on the RKKY interaction J(R) are separately investigated. When the Fermi surface has planes perpendicular to R, effects appear on the period of oscillation, the phase shift and the amplitude of J(R). The applicable region of the asymptotic form for large R and the validity of the free electron approximation are also examined. If there are no tangential planes perpendicular to R, it is found that: 1) when two interacting localized spins are on lattice points in the crystal, exponential damping appears even for the constant matrix element model and the matrix element effects introduce competing terms causing a sign change; 2) when one of the spins is at an interstitial position, the constant matrix model gives a weaker J ( R ) t x R -2 damping, but the character of this term changes into the exponential damping by taking into account matrix elements.
1. Introduction
Recently, interesting experiments concerning the exchange interaction between magnetic atoms in Heusler alloys were performed by the group of Ishikawa [1-3]. Their experiments produced the following results. 1) The nearest and next nearest neighbour exchange interactions are strongly ferromagnetic, 2) the long range interactions far from the third neighbour are the oscillating type. Furthermore, it seems that they are interpreted very well by the RKKY interaction, treating the s- and p-electrons as free. Result 1) was interpreted in terms of the virtual double exchange mechanism by Kasuya [4]. On the other hand, it was shown from neutron scattering experiments that the d-electrons are quite localized and the magnetic moment per Mn atom is about 4/~B [5]. Accordingly, Heusler alloys can be considered to be typical s-d systems, similar to the rare-earth system for the s-f and d - f interactions. Here, the matter in question is why the long range d - d exchange interaction can be interpreted very well by treating the conduction electron as free. Because, for Heusler alloys, the Fermi surface in the free electron model includes many Brillouin zone planes in its sphere because of the many conduction electrons per unit cell (from about four to seven) the Fermi surface will accordingly be cut into many zone boundaries. Thus, the real Fermi surface is predicted to be deformed considerably from the sphere. A similar situation should exist in many other magnetic materials in which the character of the interactions is analyzed by a method based on the RKKY interaction. The RKKY interaction in the free electron model is exactly calculated [6-10] and is well known. This interaction in metals with nonspherical Fermi surfaces was investigated by Roth et al. [11,12]. They derived the asymptotic form for the large distance between the two interacting localized spins (abbreviated as f-spins). But its range of validity was not clarified and the matrix element effect was not considered explicitly. For the indirect s-f exchange interaction in nonmetals, the asymptotic form with exponential damping was derived for the parabolic conduction and valence bands [13,14]. But the region where it can be applied and the matrix element effect are again unknown. * Present address: Department of Applied Mathematics, Akita Technical College, Akita 011, Japan.
0304-8853/84/$03.00 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
A. Narita, T. Kasuya / R K K Y interaction
22
Our purpose is to investigate accurately the effects of the Fermi surface deformation and of the matrix elements to the R K K Y interactions, in order to give physical insight into the questions mentioned above. We will calculate this interaction rigorously assuming the simple periodic potential and will try to examine and clarify the validity of the asymptotic form of its interaction comprehensively and the validity of the free electron approximation itself in the s-d or s - f model (referred to as the s - f model below).
2. M o d e l
We investigate the effects of a periodic potential on the R K K Y interaction based on the following Hamiltonian,
H= Ho + H~_ f,
(1)
• [2m +Vc°sQz i ,
(2)
Hs_f=N ~i ~n S(ri-R.,)oi*Sn,,
(3)
where the usual notations in the s - f model are used. H o denotes the band energy in the conduction electron system. In H 0, the simplified one-dimensional periodic potential, the direction of which is defined as z-axis, is introduced. V and 2,rr/Q are the strength and the period of the potential, respectively. Therefore, the conduction electron is free for the x- and the y-directions. H~_ f denotes the exchange interaction between the conduction electron spin aj and the f-spin S.,. The s - f coupling is assumed to be the 6-function type in real space. The f-spin distribution is assumed in the following way. The lattice points are specified by the vector R, = (anx, an v, 2"nn./Q)(n x, ny, n=: integer), in which a is the lattice constant for the x-y plane and 2~r/Q is the latiice constant for the z-direction and is equal to the period of the periodic potential. Each lattice point is occupied by an f-spin. For the z direction, a particular f-spin can be also situated as an impurity, on an interstitial position between the neighbouring lattice points. Then the position of such an f-spin is represented by the vector R,, = (an~, anu, 2~(n z + t)/Q) (0 ~< t < 1). S., in eq. (3) denotes the spin operator of the f-spin situated at the position specified by R.,. N is the number of the unit cell. The eigenfunctions, {q,~k(r)}, of the one body Hamiltonian H 0 are taken as the basis set. Since the periodic potential exists only for the z-direction, q~,k(r) can be written as q~,k(r)= eikru,k:(z), in which u~k(z) has the periodicity u,k(z + 2~/Q)= u,k(z). Similarly, the eigenvalues E,k of H o can be written as
Evk=(hZ/2rn)[k~ +e,(k.)],
( k 2 = k 2 +k~).
(4)
We treat H 0 and H~_f as the unperturbed and the perturbation Hamiltonians, respectively. As is well known from second order perturbation theory [7], the expression for the R K K Y interaction between two arbitrary f-spins is obtained in general as follows, =
2
~',p" k,k'
where f~k is the Fermi distribution function and the matrix element I,(p'k', pk) is given by
I, ( p'k', uk ) = Iu,. k, (2 ~rt/Q ) * u,k:(2"~t/ Q ).
(6)
In later sections, the potential effects to the RKKY interaction will be investigated, assuming that one of two interacting f-spins is necessarily situated on the lattice point, i.e. R,,,, = 0. Since the Fermi surface is
A. Narita, 7", Kasuya / R K K Y interaction
23
deviated from a sphere, J(Rnt, t) [ R , , , , - - 0 in eq. (5)] shows the anisotropic property concerning the direction of the vector Rnt. On the other hand, owing to the nonuniform charge density of the conduction electron, the behaviour of J(R.,, t) depends on t through the matrix element. Thus there exist two effects, due to the deviation of the Fermi surface from the sphere and due to the matrix elements. In the calculations of J(R.t, t) given by eq. (5), we must have knowledge about u,k:.(2~rt/Q ) and %(k~). In the next section, the solution of the one body Hamiltonian H 0 will be described.
3. Fermi surface and matrix elements
The periodic part of the Bloch function can be expanded into the Fourier series, Uuk ( Z ) = E c.(uk:)e inQ =.
(7)
n
Eq. (7) is inserted into the Schr6dinger equation HoeP.k(r ) = [email protected](r) and then we get the equation for
c.(.kz),
[%(k~)-(k~ + nQ) 2] c . ( . k . )
)[
= ( m V / h = c . _ , ( , k z ) + c.+,
(~,k~)].
(8)
u . k ( z ) and ~.(kz) can be determined by solving eq. (8).
3.1. Fermi surface The Fermi energies Ef determined numerically are shown in fig. 1 as a function of V / E ° . There, ~1= Q / 2 k °, Ef° = h2k°2/2m, in which k ° is the radius of the Fermi sphere for V = 0 and E 0 denotes the
Ef~E; I
Eo/E~ I
I
I
I
[
J
.. | e-o_
08 Oo T
",
~ 1~02 ,.Vo
Fig. 1. The Fermi energy Ef (solid lines) and the bottom energy E 0 (dashed lines) reduced by the free Fermi energy Ee° are shown as functions of the reduced potential energy VIE ° for various values of , / = Q/2k°f.
A. Narita, T. Kasuya / R K K Y interaction
24
bottom energy of the first band. In our calculations, as will be seen in later sections, ~/ is an importanl parameter. When ~ > 1, the free electron Fermi sphere is contained in the first Brillouin zone completely When ~/= 1, it touches the zone boundary and for ~/< 1, it extends into the other BriUouin zones. The shape of the Fermi surface for each 71 value (~/= 1.125, 1 and 0.75) was calculated numerically and is draw~ in fig. 2. Since the Fermi surface in the present model is symmetrical around the kz-axis, only a fourth par1 of its cross section is shown. The quantities q0, ql, q2 and ~ are also shown in fig. 3. q0 and q2 are the diameters of the bellies formed in the first and second zones, respectively, q~ is the diameter of the neck formed in the first zone. ~ is the Fermi wave number along the kz-axis. As seen from fig. 1, for 7/= 1 and 1.125, the lowering of the Fermi energy is larger than that of the bottom in the first band, because the energy of any state at the Fermi energy decreases more than that for bottom. For ~/= 0.75, in the
q.=1.0 "z/K;
q,/2
.
z.
1.o
V/E=;
1.(
=~.4
0
.V/E;=O
1.12-
=o.8
0.8
~. = 1.125
Kz/K
0.8
V/E'~=0'6
" ~
0.6 0.4 0.4
a
0.2
Oo
012
Kz/K;
0.'4
/1 0.'6
0:8
--L- "o/2
O.2
1.0 KVK;
0
0:2
o4
o6
o:8
lO K"/K;
't : 0 . 7 5
1.0
o,
~
qqv ~
0.2
°o
ql/2
--~
I
qo/2
o12
oI~
J
o16
0:8
~.o K.,/K~
Fig. 2. The deformation of the Fermi surface due to the potential is shown for various values of V / E °. (a) ~ (c) "e = 0.75. For various notations, see text.
= 1.0, (b) ~ = 1.125
and
A. Narita, 7'. Kasuya / R K K Y interaction
25
Vc/E~'=0.8'=
(b) ,1.=1.125 1.2
( a ) "I. = 1.o I
I
I
I
qo/2k~
1.0
1.0
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
t
00
t
0.2
1
0.4
1
0.6
0.8
(~) ~.= o.T5
1.0 V/E~
|
=
w
0.6
0.8
.o/2~ 2 ~/Q
|
|
0.2
0.4
1.0
V/E;
v~.o.73
1.00B 0
1.004
'
'
'
1.00( 0.991 |
[
I
I
,
,
=
i
0.4
0.6
1-
1.0 0.8 0.( 0.,~ 0.2
00
0.2
0.8
v/E;
1.0
Fig. 3. The potential dependences of qo, q~, q2 and ~ are shown for various values of T/. (a) ~/= 1.0, (b) 7/= 1.125 and (c) 7/= 0.75.
A. Narita, T. Kasuya / RKKY interaction
26
beginning, the lowering of the bottom in the first band is larger than that of the Fermi energy, because the increase of energy for the state at the Fermi energy of the second zone is large. Since the Fermi surface in the second zone disappears for IV1 larger than IV~I= 0.73E °, the situation becomes the same as that for ~/= 1, for IVI sufficiently larger than IV~I. From the potential dependences of Er, E 0, Cl(k~) and c 2(kz) for k= near Q/2, the behaviour of qo, ql, q2 and ~ shown in fig. 3 can be easily understood. Since the potential dependences of Ef and E0are weak, that of q0 is also weak. qt, q2 and ~ have relatively stronger potential dependences, since those of c~(k=) and c2(k. ) for k. near Q / 2 are larger than that of El. Thus, when ~/= 1.125, the Fermi surface extends towards the z-direction and begins to have the gap on the zone boundary at IVI = IV~I= 0.64E °, while for ~/= 0.75, the Fermi surface in the second band shrinks as a whole and disappears at IVI = IV~l = 0.73E °.
3.2. Matrix element As seen from eqs. (5) and (6), the matrix element is written as
It( ~k, p'k') Io( v'k', ~k ) = I2B~,k,( t ) B~k.( t ) *,
(9)
B~k=(t ) = u ~k=(2"nt/ Q ) U~k:(O)*,
(10)
in which B~_(t) is written as the sum of the real part C,(~, k~) and the imaginary part D,(~, k..). The numerically calculated values are shown in fig. 4 for V I E ° = + 0.4. It follows from fig. 4 that the matrix element is strongly deviated from the value for the free electron model, particularly near the zone boundary. The behaviour at the origin and the zone boundary can be easily understood from the symmetry of the Bloch function [15].
C [~',k) 2 (b)
C. (~',k)
I
- -
(a) V/E¢ = 0
1.5
- - - 1"=2
~'= 1
--
V / E ~ = O. 4
- )'= 2 //
,
t=O, 0,25, 0.5,0.75(M=I) t=O ( u = 2 )
.- ...........
,'
/
//
o5
0.5
0.5 . . . . .
o. 25, o1~__- --""
\
t =0.25,0.75
/
-0.5
-0.5
iI / / ,/
0.5
o
or2
___-"
-I
t=0.5
-1
o.'~
o.'6
o~ 2k/Q
;.o
0
' 0.2
0.14
' 0.6
i 0.8 1.0 Zk/Q
A. Narita, T. Kasuya / RKKY interaction
27
Ct(u,k)
(c) . 2.0
. ~ 1
.
.
/
V/E; : - 0 . 4
D (~,k)
/
=
I
1.5
i
(d)
--- ~'--=2
V/E; : 0
~
t=O 1.0
M
=
1
-----
0.5
"". . . . . . . . . . .
1.0
J.-'o.
0.5
.
.
.
.
.
.
.
0.75 . . .
.
.
.
.
.
.
.
.
.
.
.
2J=
.
2
.
0.5
0
t= O, 0 . 2 5 , 0 . 5 , 0 . 7 5 t=0,0.5(~)=2)
0.25 o 0.75
/
-0.5
(~,,= I )
I
/
- 0.5
iI /
',.
~"
0.5
-1.0
0.25 ...................
-1.0
0
0.2
0.14
I
0.6
0.'8
1.0
0
2~
I
I
0.2
04-
I
I
0.8
0.6
1.0
2klQ
D.(u,k) Dt(~,, k )
!
(e) Y/E; = O. 4
~U= ---~=
I
,
!
i
(I)
2
1.5
VY E I•
=-0'4
---
=
~
2
1.5 0.75
....
- "',
1.0 0.75
1.0
0.5 0.5
0.25 0.75 0
t = O, O. 5(~=1 and u = 2 ) -
]
0.5
t -
......
-1.0
0.5
.:\ /
0.25
0.25 -
0
: 0,O.5(u:land u:'2) ~ , /
o~2 o~' o~6 0~8 2~0
1.0
--'~
1.0 0
0.'2
0~4
0~6
I
0.8 1.0 2k/Q
Fig. 4. The wave vector dependences of the matrix elements are shown for various values of V/E°. (a) and (d) V/E° = 0, (b) and (e) V/E° = 0.4 and (c) and (f) V/Ef° = - 0 . 4 , while (a), (b)and (c) are for the real part and (d), (e) and (O for the imaginary part.
A. Narita, T. Kasuya/ RKKY interaction
28
In the following, we classify the situation into two cases; cases A and B. In case A, there is no Fermi surface perpendicular to the k,-axis. In case B, the Fermi surface has sections perpendicular to the k~-axis. The temperature is taken to be zero.
4. Basic formulae
Firstly, we perform the integrals in eq. (5) with respect to q~, if' and k~_ analytically, by using eq. (4) and the cylindrical coordinates (k x = k I cos if, ky = k± sin q,, k'x = k~_ cos q¢ and k~, = k~ sin q~'), The result is as follows [15],
J(R., , t ) = J ( R
±
,R,,)=~ C
~
, fdk~O(,f-,~(k~) ) B ,,.(t)
*
x fdk'..B.,k;(t)ei'k;-k:'",,'fo~dkiklJo(R±k±) × [0(
mI 2 C=~r3NZh ~
B)Ko( R± v ~ ) - ( ' ~ / 2 ) O ( - B ) N o ( R . ~ - S ~ ) ] ,
(11)
19¢r [ IZ][Ue] 2 2mEf 2k°' l E° ]~ N I ' "f= h2 ,
(12)
B =,.,(k')-,.(k.)-kl,
(13)
where J0(x), No(x ) and Ko(x ) are the Bessel, Neumann and modified Bessel functions, respectively, O(x) is the step function and Ne is the number of conduction electrons. In eq. (11), the vector R,, is represented as R . , = (R j_, 0, R,,), since the Fermi surface is symmetrical around the kz-axis. The lattice constant a for the x-y plane can be any value. For a ~ 0, the f-spins distribute uniformly for the x-y plane and thus R± is continuous. On the other hand, R n t = R z 2~(n + t)/Q (n = integer and 0 ~< t < 1). R z may be often used for Rnt, in order to emphasize that Rn, is the z-component of R.,. When t = 0, R,, may be simply written as R,. Secondly, we get another expression by performing the integrals in eq. (5) with respect to ff and ~' in the cylindrical coordinates and by transforming k± and k', into ~ and c' [c --- k~ +%(k~), c' = k~2 +%,(k~)]. =
J(R.,,/)=1-~c
E f fd, pp,
d, '/(')
-/(~')] I,,(,)*I,,t(c'), C
--
(14)
I[
L,(,)= f
).
(15)
In eq. (15), the k~ integral is bounded by ~ - %(k~) > 0. Since % ( - k ~ ) = %(k~), f(c)[1 - f(~')] in eq. (14) can be replaced by f(c). In the limit R 1 ---,0, the k±-integral in eq. (11) is performable, using the asymptotic forms for R.L ~ 0. After the integration, we get
J(R~)
C E fdkfl(,f_%(k~))B.k(t).e_i.:k:fdk.,B,k,(t)ei.:k;
16~ ~'
:
× [(q-%,(k'))lnl,f-%,(k')l+(%,(k'~)-%(k~) In later sections, formulae (11), (14) and (16) will be used.
"
) lnl%,(k:)-%(k~)l].
(16)
A. Narita, T. Kasuya / R K K Y interaction
5. J(R:)
29
for t = 0 in case A
5.1. Constant matrix element The meaning of the constant matrix element is that the matrix element is the same as that in the free electron model; therefore B~k(t ) is given by fig. 4a and d and only the band structure effect on J(Rz) is considered. The numerically calculated results for , / = 1 based on eq. (16) are shown in fig. 5, where C O= Ck~'. From fig. 5, the following facts can be listed: 1) J ( R , ) always has a negative sign and 2) as V increases, J ( R , ) damps faster with increasing n. These facts can be explained as follows, based on eqs. (14) and (15). When the equal energy surface for the energy ~ makes an open orbit in the z-direction, I]t(c ) defined by eq. (15) becomes zero for t = 0. The equal energy surface with the energy larger than the Fermi energy makes the open orbit for the z-direction for ~/= 1. The final states in the intraband scattering thus belong to the open orbit equal energy surfaces and thus there is no contribution due to the intraband scattering. The contributions to J ( R , ) must occur through the interband scattering from the first band into the higher bands. Taking into account again the above mentioned fact concerning l~0(c), the only possible contribution is due to the process from the states belonging to the closed surface in the first zone to those belonging to the closed surface in the higher zone. In this process, there is a finite energy gap IV I between the initial and the final states, which causes exponential damping on J(R,). The largest contribution comes from scattering from the vicinity of the top of the first band for k± = 0 with energy c] into the bottom of the second band for k~_ = 0 with energy E 2 ( ( 2 -- E1 IvI). I]0(0 for c _< c 1 and I2o(C' ) for ~' > ¢2 always have =
.3.0J(Rz)/C°
constant
matrix element
v/E
X
o '~ []
=I.0
x
0.2 0.4 0.6 0.8
2.0
D O
1.0
10~-3 I 1O-/"A
10-2/
F'-,/ -1.o[
o
v/E = o
, -ak Rz
Fig. 5. The numerically calculated values of J ( R Z)/C o in the constant matrix element are shown for various values of V / E f°. The c a s e for V / E f° = 0 is shown by the solid curve. The positions marked by upward arrows correspond to t = 0 and the positions by downward arrows to t ~ 1/2. Note the change of the multiplication factor in the different ranges of 2k°Rz.
30
A. Narita, T. Kasuya / R K K Y interaction
opposite sign. Thus, J ( R , ) have negative sign, as easily seen from eq. (14). The present consideration does not depend on the value of */explicitly, but depends only on the shape of the Fermi surface. In fact, from the evaluation of the contribution due to the above scattering, we can obtain an analytic expression for J(R,) for an arbitrary value of ~, when the Fermi surface has an open orbit along the z-direction [15].
J(R,)=
16,~2n 2C°~12lvlKl(z), ( v = V / E ° , z
=~rnlvl )27/2 ,
where Kl(x ) is the modified Bessel function of the first order. The asymptotic forms of x ~ 0 and (~r/2x) 1/2 e x p ( - x ) for x ~ oo. From these asymptotic forms, we get
Jn0
6°'~4
(17)
Kl(x)
are
1/x
for
(z ~ 0),
(18)
(
(19)
J(R°) =
C°Lvl'/erl3
" " 2n5/--------16~r ~ exp~ - z )
z
~).
Note that eq. (18) agrees with the result of the free electron model. It followed from the numerical analysis [15] that the asymptotic form [eq. (19)] can be very well applied for z > 2.5. Therefore, it can be concluded that J(R,) in the constant matrix element can be described very well by the simple single term evaluated as the contribution due to the interband scattering from the states near the top of the first band into the states near the bottom of the second band.
5.2. Matrix element effect The matrix element effect can be investigated by adding its effect upon the results for the constant matrix element. The numerically calculated results in this case are shown in figs. 6 and 7. The following characteristics are seen: 1) J, can take a positive value for large values of n and V/Ef °. 2) While J, shows the damping with increasing values of n and V/Ef °, the damping is not so simple compared to the case for the constant matrix element, because J, for V > 0 changes sign on the way. It depends also on the sign of I7.
5.2.1. Interpretation of the numerical calculations Firstly, fact 1) listed above is considered. It must be noted that 1,0(E) defined by eq. (15) remains finite owing to the nonconstant matrix element, even if the equal energy surface makes an open orbit. Therefore, a contribution from the intraband scattering also occurs. We divide J, into contributions from the intraband (Jint . . . . ) and the interband scatterings (Jint ..... ), which are shown in fig. 8. As seen from these figures, the interband contribution is always negative, but the intraband contribution can be any sign. The intraband scattering may be divided into two processes by the initial state; 1) c o < ( < q and 2) q < ( < of, in which c o is the bottom energy of the first band. The sign change in J~n,.... can be understood as the competition between them [15]. For the interband contribution, the scattering from the first band into the second band is important. There are four possible processes: 1) % < c < q ~ % < c ' < c 3, 2) q <(
A. Narita, 7". Kasuya / R K K Y interaction
THE EFFECT OF MATRIX ELEMENT mr= 1.0 a
J (Rz)~C o I .00
0.80
x lo -2
0.60
v/E;
;/
,
o
0.2
A
0.4 0.6 0.8
D ,'
x lo -3
31
x l O -/-*
0.40 0.20 O.
oW,s
-0.20
-
-0.40 -0.60
v/E; : 0
-0.80
y./E)
:
0
~, 2k~R z -I
.00
THE EFFEC,r OF I'IR,rRIX ELEMEN'r
J(Rz)~C ° I .00
b
-
0.80
x10-3
\
i
, 0
•
o
-0.2
A
-0.4
|
o.6o - x10 -2 ~ 0.40
V/E;
q.= 1.0
'--
xlO -~
., .
.
.
.
I
~
.
-0,40
-0.60 -0.80 -I
I~'Y~"\--,~V/E 0"y / v/E;: o
,
2,;Rz
.OO
Fig. 6. The numerically calculated J ( R z ) / C o are shown, by taking into account the matrix element effect. The meaning of upward and downward arrows is the same as in fig. 5. (a) V > 0 and (b) V < 0.
32
A. Narita, T. Kasuya / R K K Y interaction
the effect of matrix element
in/Illd
v[= 1.0
,V/E;
Fig. 7. The potential dependence of J , / I J ° l , taking into account the matrix element effect, jo is defined by eq. (18).
except for n = 1. For V < 0, from similar considerations for the case V > 0, process 1) gives negative sign and 2) positive sign on Ji,t .... . Because the former contribution is larger, the sign is always negative, but the absolute value decreases rapidly with increasing n and IVI c o m p a r e d to that of the constant matrix element. The signs of Ilo(C ) and I20((') are listed in table 1 and the signs of Jint .... and Ji,t .... are listed in table 2. 5. 2.2. N u m e r i c a l a n a l y s e s
Here, fact 2) listed at the beginning of this subsection is considered. As is clear from the above consideration, when the matrix element effect is included, various contributions begin to compete with each other and the situation becomes complicated. Even if we consider each Jim .... and Jim .... separately, it is impossible to analyze the data by a single exponential term such as eq. (17). The most favourable case, Ji,t .... for positive V, is shown in fig. 9 as an example. The data can be fitted by the form Jintra.,,/Co = A × 10-3 × Ivl x n - ~ e x p ( - B n l o l ) . However, from fig. 9a, A = 7, a = 1.5 and B = 1.6 give the best fit, while from fig. 9b, A = 5, a = 1 and B = 1.7 give the best fit. This means that we need a more complicated form.
~=I.0
3intra,n/-Ip-r~ I
1.o[ 5
~.= 1.0
Jinter. n/IJ~ll
0,8
n=l
0.6
0.4
- 076 L
a
Fig. 8. The potential dependences of (a) Ji.tra,. and (b) Jint.... are shown for n = 1-5. jo is the value for V = 0.
' V/E;'
A. Narita, T. Kasuya / RKKY interaction Table 1 The signs of llo(Q and
33
120(4 )
11o(4) c
Sign of V Odd n Even n
12o(O
1
4>c 1
+
Table 2 The signs of J i m . . . . and Jinter,
¢>__4 2
+
--
+
--
+
+
--
+
--
_
--
+
--
+
+
n
inlra,n
Jinter,n
1 ) 4 0 < 4 < 41 ---~.c f < c '
2 ) cl < E < ~ f - - - * E f < ~ t
1 ) 40 < ~ < Cl ---* 42 < ¢ p
2 ) 41 < 4
Sign of V
-
+
-
+
-
+
-
+
11o(C)11o(4')
-
+
+
+ --
--
+
--
110(4) 120(4')
6. J(Rz)
for
t ~ 0 in case
<4f--*c2
<¢ t
A
6.1. Constant m a t r i x element F o r t * 0, one of the f-spins is situated on the lattice p o i n t a n d the o t h e r on the p o s i t i o n d e v i a t i n g from the lattice point. Results o f the n u m e r i c a l calculations for t = 1 / 2 are shown in fig. 5. S u r p r i s i n g l y enough, J ( R z) have c o n s t a n t positive sign, b u t b e c o m e larger t h a n that in the free electron m o d e l with increasing
164
Ji~w~ n
(a)
=1.0
Go
V>O
ld 3
(b)'
q=1.0
V>O
id ~.4
1@ i#
L7
1J o
¥./t-~= 0.8
n=5 '
o12
o14
' ,
~,t~
'
o:6
'
o:a
Id~ ....
pll
Fig. 9. J i n t . . . . for the positive potential and 71= 1 is shown for the various values of n and IVI/E °.
A. Narita, T. Kasuya / R K K Y interaction
34
potential. This is quite different from the case for t = 0 and is against common sense accepted thus far. Let us consider the result based on eqs. (14) and (15). It must be noted that the integral of the phase factor for t 4= 0 in eq. (15) is not zero, even for the open orbit. Therefore, the contribution from the intraband scattering is now not zero. It is shown that the contribution for the initial state existing in the region { < { 1 gives a damping term similar to the case for t = 0 and the main contribution comes from the initial state for {f > { > {~, which gives the R72 term. For the interband scattering too, the contribution from the same initial state become finite and gives the same R~-z dependent term. The analytic calculation gives the following asymptotic form for large R , [15], J( R_ ) = C sin2vt . 4~R----T[ ( { 2 - { ] ) l n l { 2 - { , ] - ( { 2 - { f ) l n [ { 2 - { r l - ( { f - { , ) l n l { f -
(20)
{,I],
in which C is a numerical factor given by eq. (12). As shown in fig. 10, the numerical results are well fitted by eq. (20).
6.2. Matrix element effect Numerical calculations taking into account the matrix element effect are shown in fig. 6. In this case, since two f-spins are mutually located on the positions of the anti-phase of the potential, J ( R ~ ) for t = 1 / 2 does not depend on the sign of V. The figure indicates that J ( R z ) damps with increasing potential and shows completely different behaviour from the result for the constant matrix element. This is indicated more clearly in fig. 11. In this figure, TI (l = intra, inter) is defined as Tt = [Jt(R~)lhS/a/Colvl l/z, where h = n + 1/2. From this figure, it follows that the matrix element effect causes R72 damping in the constant matrix element to change into exponential damping [15].
J(Rnt)//Co
lO-a
,
x ~ ~
r
i
i
i
constant matrix element
~ =1,0
165
t =112
' '~-o 4 2 =
5 I
10 20 *h I
--
"
50 I
100
Fig, 10. J ( R , , ) / C o for the constant matrix element is plotted as a function of h for V / E ° = 0.4 and 0.8. Here, R . ,
h=n+t.
= 2"~h/Q
and
35
A. Narita, T. Kasuya / R K K Y interaction Tintra
Tinter i
I
I
I
ettect of matrix
[o(a)
I
I
I
I
element
10.2
o
~=1,0
10-2_ ~':. .~\ % \ °o
~.
\
\
z,\~
\\
•
\
,
\\
\\\\\\\~\
I 2
I
I 4
I
I
-~~
= h=1.5
N\
\\NN~NN\~
2__!
\ I
I
t=1/2
1(54 .
164
~d5 0
~.= 1.0
~
hgto5
X \
10"3... ~
I
t.
h=O,5 |Q
I
effect 0t matrix element
(b)
,~N~ _
°
I
o
t =1/2
-
\
I
I 6
x
~l
16 5 I
~-
, 0
I
I 2
I
I 4
I
I 6
''XX'N~ Xl
Fig. 11. (a) Ti.tra and (b) Ti.ter are shown as functions of ( ~ / 2 ) h I V I / E °. The dashed lines show those for t = 0 in the constant matrix model.
7. J( R= ) in case B The asymptotic form of J(R z) in case B was derived from the method of expanding it by R~-1 using eq. (16). The main approximations used were as follows: 1) the contribution 0nly from the intraband scattering is collected; 2) the U-integral in eq. (16) is extended to infinity and the contribution from the poles is extracted. The expression obtained is [15]
~,~ ,0) [B2
J(R)/C=(-1)
ei2R~ + CC] + ( - - 1 ) p,
i
[eiZRt~(3BB(I),~(])+½B2,~(2))_cc],
(21)
where ~'r is the band number in which the Fermi level stands, and the following abbreviated notations are used, at R = R z, B = B,,~(t),, = %f(~) and F (') = ~-TF, (22) where F = B or c. Using eqs. (10) and(7), we can represent eq. (21) as the superposition of the harmonic waves,
J(Rz)/Co=-lEA. X
cos(2kr-nQ)R~+lEB, n
X
n
sin(2kr-nQ)Rz,
(x=2k°R~),
(23)
A. Narita, T. Kasuya / R K K Y interaction
36
where k f is the radius of the Fermi surface in the extended zone scheme (kf = ~ for 71 = 1.125, k f = Q for 7/= 0.75). A n and Bn depends in a complicated way on the potential. The values for A~ and B, obtained from the two wave approximation are shown in table 3. In this approximation, A, and B, for n > 2 and n < 0 are zero. Results for the constant matrix element are also shown. In fig. 12, the results for 7/= 0.75 obtained by the numerical calculations using eq. (16) and those obtained by the asymptotic form eq. (23) with table 3 are shown. It is seen that the asymptotic form is excellent in the whole region for the constant matrix element model (fig. 12a) and satisfactorily good even including the matrix element effect. The shift of the oscillation period with increasing field strength is simply due to the change of the Fermi wave vector 4. The increasing amplitude with increasing positive value of V is the matrix element effect seen in fig. 4b. As V increases, the Fermi surface in the second zone approaches the zone boundary, where C0(2, k) becomes large. The opposite happens for negative V, as seen in fig. 4c. The agreement for *1= 1.125 is less satisfactory, in particular, for small R,. This is because the interband contribution with exponential damping is more important for 7/= 1.125. Now we introduce the model to write J(Rz) in a similar form to the free electron model. J(Re)/C
0 = ( Is_f/l)2(kf/k°)4[sin(x
- (p) -- x c o s ( x
-- t ~ ) ] / x
(24)
4,
where x = 2krR z, / s - f is the modified s - f exchange coupling constant, q~ is the phase shift and k r is the effective radius of the Fermi surface. I S_r, q~ and k f are parameters. This model is referred to as the
Table 3 The values of A , and B, o b t a i n e d by the two wave a p p r o x i m a t i o n . The values in ( ) i n d i c a t e those in the c o n s t a n t m a t r i x element
VIE °
Ao
A1
Az
Bo
B1
B2
0.587 (0.945) 1.223 (0.945) 0.207 (0.733) 1.067 (0.733) 0.010 (0.310) 0.397 (0.310)
- 0.213 (0) 0.443 (0) - 0.161 (0) 0.829 (0) - 0.015 (0) 0,573 (0)
0.019 (0) 0.040 (0) 0.031 (0) 0.161 (0) 0.005 (0) 0.207 (0)
- 7.990 (0.094) 5.081 (0.094) - 9.278 (2.893) - 3.358 (2.893) - 1.660 (6.215) - 12.475 (6.215)
- 2.148 (0) 12.34 (0) 3.546 (0) 16.270 (0) 2.078 (0) - 5.765 (0)
0.652 (0) 2.070 (0) 0.023 (0) 6.827 (0) - 0.634 (0) 2.339 (0)
1.199 (0.964) 0.694 (0.964) 1.196 (0.845) 0.363 (0.845) 0.853 (0.593) 0.092 (0.593)
0.326 (0) - 0.188 (0) 0.692 (0) - 0.210 (0) 0.861 (0) - 0.093 (0)
0.022 (0) 0.013 (0) 0.100 (0) 0.030 (0) 0.217 (0) 0.024 (0)
- 0.889 (1.214) 4.095 (1.214) 0.619 (1.857) 5.705 (1.857) 5.704 (2.954) 3.639 (2.954)
- 4.150 (0) 1.150 (0) - 7.963 (0) - 0.775 (0) - 4.632 (0) - 2.547 (0)
- 0.547 (0) - 0.232 (0) - 2.356 (0) - 0.254 (0) - 3.792 (0) 0.359 (0)
1)~ =1.125 0.2 - 0.2 0.4 - 0.4 0.6 -0.6
2) n = 0.75 0.2 -
0.2 0.4
- 0.4 0.6 - 0.6
37
A. Narita, T. Kasuya / R K K Y interaction CONSTRNT M R T R I X ~1= 0 . 7 5 0
J (RzVc o
ELEMENT
v/E;
a
1.00
0.80
x10-2
0.60
i
&
0.2
Q
0.4
0
0.6
(3 x 104) -1
(6 x 10Z')
x 10-3
/
0.40 0.20
•.
O.
l/\..
•
-0.20
~0.40
0.2
-0.60
-1
X',o, ~.2
!o 7 1
I/R o 0.6
-0.80
0,4 0.6 ~ 2ktR z o
i 0
0.2
- I ,00 J(Rz)/Co 1.5 i 10 - 2 1.0
:0.75
b
10_z
(6x104)_ 1
/."~
,,!
10-3
(3xlOt') -1
V/E~ J(Rz~/C* 1.5r
~ : 0.75 ~ ~"
|
0.~
/
I
.~,
"
"I1" I/ ,"
-1 .C
-1.5
/ •
"~ - .......
i~ WE~' / oo
10-~
0.0 -0.2 0.4
..... :i. ...... 0.6 (3x10") i
.
,
(6x10")-'
I, Wti
H i/ ',,'e
0.2
it .o . ............. . . 0.4 0.6
10-3
1.0]-
- ff i.~11.\!
1, -0.5
10-2
~ = ...... =1 . . . .
II~ / '/~v / i / ; t..." I!/,' ; II
ii"=
~,
.c~lt~z~i . . . ~il
i \./
o| , ~ / , ~ .L!...~...! t/~['-...///I~":T"~'.[~:,."; [ ~If ~,1 ~/"~51 """"'~r/ I,,",.:.~. li :,1~,35~,~11;i~. \45
ll'
!~ o i "v"
_1.0L
L /
~' 2k~Rz
Fig. 12. The values for J ( R z ) / C o obtained by the asymptotic form (various lines) and by the numerical calculations (zx, ru, O) are shown as functions of 2 k ° R z for 7/= 0.75. Upward and downward arrows denote the positions specified by 2 k ° R z = 2~rh/71 (h = n for 1' and h = n + ½ for $, and n = 1-5), where the numerical calculations were carried out. (a) constant matrix element, (b) positive potential and (c) negative potential.
pseudo-free electron model in this paper. In the present case, kf is chosen to be equal to k f in eq. (23). J ( R z) in various cases are c o m p a r e d in fig. 13. It is seen clearly that J ( R , ) for the constant matrix element can be fitted by the pseudo-free electron model by choosing 1~_ f properly. Even when the matrix element effect is included, the m a i n terms are usually A 0 and B o in eq. (23) and effects of other terms are included in the phase shift ~ and the renormalized amplitude ls_ f. A typical example is the case for ~ = 1.125 and v = 0.4. W h e n the other terms are large, the pseudo-free electron model cannot fit the results. A typical example is the case for ~ = 1.125 and v = - 0 . 4 , where, due to a large B] term, a large oscillation with a long period is added. N o t e that 2kf = 1012k f - QI. The same situation exists for ~ = 0.75 and o = 0.4, but the period of the B 1 term is shorter than the former case, because now 2kf = 4.512k t - Q{.
38
A. Narita, T. Kasuya / R K K Y interaction
~= 1.125
J(Rz)/C o
................
1.5
k|
...... const,mat.ele ~-~v=0.4 ..... v= -0.4
V/E~'= v
!.".,.
1.0
10-2
10-3
/~
1
o.s
//'i il/'
~ I
O
:...... //'k "/,...%.
.....
lo -4
r,,
"' / "~\25t / ! ;j,'/""~:,:" \ , :"J', "".~.-..V~i
i'V] ii.,.;y
~.4
/
,
~:: !j ~.~ !(\1/
'/
/
/
i ~'-,~
'.
1.o i o - 2
,0-3
/!
,i~
io-~
(s,lofl-1
/
"
i i
-,.o
( 6 x 1 0 4 ) -1
0' //.7 "~
iI :i v
ill
~/ "~'.
~0~5
I -~.o / ..............,,, /
-2.C -1.5
......
/it ~' ,"
~ ~
--.--
°o,,,t.mot.~,, = .4 "O
. . . .
V= -0.4
\/
1 '~.,',! i /
,_~),~,:,
i!
!/
b
Fig. 13. Comparison of various J(R,.) for [VI/Er° = 0.4. The dotted line denotes J(R~.) given by eq. (24), with Is_ f = 1 and ~ = 0. for the constant matrix element (the dashed line), with the matrix element effect for positive V (the dot dashed line) and for negative V (the dot dot dashed line) are calculated by the asymptotic form of eq. (23) with the coefficients given in table 3. (a) ~1=1.125 and (b) ~ = 0.75.
J(R:)
8. J(Rl) T h e asymptotic form valid for large R~ was derived from eqs. (14) and (15), based on the method of the stationary phase by Roth et al. [11]. For the case under consideration, for example case B in ~/= 0.75,
J ( R ± ) / C = ba(O)2q~ F(qoR.) 8c~'(0)
bl(Q/2)2q4 F(qtR_L ) ~ b2(O/2)2q4 F(q2Rj_ ) 81(;'(Q/2) 1
+ b'(O'b2(Q/Z)[(q°+41%(O)(z(Q/2)]l/zq2)/214 [1-
8(~'(Q/Z) k(q°-qo + q2q2)2] F ( - ~
bl(O)bl(Q/2)[(q°+ql)/2]a[1-(q°-q']2]G(q°+q]R
R±/]
),
(25)
4 [ (~'(0)[(~'(Q/2)l] ,/2 where the n u m b e r of primes denotes the times of the differentiation with respect to k z, F(x) = (sin x x cos x)/x 4, G(x) = (cos x + x sin x)/x 4 and b,,(kz) = B,,k(O). The first term comes from scattering process a) from the belly(l) to the belly(l), in which (1) means the first zone, the second term b) from the neck(l) to the neck(l), the third term c) from the belly(2) to the belly(2), the fourth d) from the belly(l) to the belly(2) and its reversal, and the fifth e) from the belly(l) to the neck(l) and its reversal (see fig. 2c).
A. Narita, T. Kasuya / RKKY interaction
39
Note that there is no contribution due to the process from the neck(l) to the belly(2) and its reversal, since the product of the matrix element b ] ( Q / 2 ) b 2 ( Q / 2 ) is equal to zero. For case A in 77= 0.75 and 1.125, the terms related to the belly(2) vanish. For case B in ~/= 1.125, only process a) remains. When the potential V is positive, the second and fifth terms vanish, because b I ( Q / 2 ) = 0 and when the potential is negative, the third and the fourth terms vanish, becauseb2(Q/2 ) = 0. In eq. (25), the fourth and fifth terms are called the cross term. The behaviour of J(Rj_) for 7/= 0.75, 1 and 1.125 are shown as functions of 2k°R± in fig. 14. The case for T/= 1.125 is first considered. [VILE ° = 0.4 corresponds to case B and IVI/Ef° = 0.8 to case A. In all cases, the asymptotic form can reproduce the numerical results very well, even for the nearest neighbour distance, using the cubic lattice model. As was mentioned above, except for V / E ° = - 0 . 8 , only the first term of eq. (25) makes a contribution. Because the interband effect with the exponential damping form is small, in the present case as was mentioned before for J ( R z), the free electron model is applicable. The weak potential dependence of q0 leads to almost the same period as the free electron model. Change of amplitude is due to the matrix element near k z = 0, as is clearly seen in fig. 4. For V / E ° = - 0 . 8 , the first, second and fifth terms in eq. (25) make a contribution. In the whole region under consideration, the first term is the main term, but the second term becomes more important with increasing R~_, because of the smaller value of ql than q0- The effect of the second term is the phase shift type for a smaller R . region and the amplitude modulation type for larger R ± . The same argument is applicable for ~ = 1. For positive V, only the first term of eq. (25) contributes. The case of V / E f ° = - 0 . 4 is similar to the case of V / E f ° = - 0 . 8 for , / = 1.125. For V / E ° = - 0 . 8 , q] is about a half of q0 and thus the behaviour of J ( R . ) is well understood by these two wave models modulated by the interference term. For ~/= 0.75, the Fermi surface exists in the second zone for f V I / E ° = 0.4, but disappears for IVI/Ef° = 0.8. Therefore, the case for JVI/E ° = 0.8 is in the extension of ~ = 1.125 and 1. The behaviour for V / E ° = 0.8 is different from that of the free electron model, because the cylindrical feature of the Fermi surface is now important. For
V/E
J(R )/Co
0.60 I 0.40
+ -0.~
~.= 1.125 a
* A
-0.4 0.4
o
÷
0.8
x~°-3
x 1o -2
0.20 O. i
-
'
15
'
'
-0.20 -0.40
-0.60 -0.80
2:
-1.00
-1.20 -I
.40
+
> 2
R_L
40
A. Narita, T. Kasuya / R K K Y interaction
J(R-)/'Co
It= 1.000
+ .
b
-0.8 -0.4
0.4
a
0.60 I 0.40
o X(50) - I
0.8
x 1 0.3
0.20 O.
,,o
-0.20
-0.40 •8
-0.60
-0.80
V/Ef
0.8 0
0
-1.00 -1,20
-I
~
J(R.)/C, 1.5 -
"[ = 0.75 \
v/ET C
o.o
~% 1.0
\
0
~.--/~ --- 0,4 /" "~ '~. . . . . . . . . . . . 04 11A a-'0"8 . ---0.8
: .-.
'./
0.5
"i /
,
-0.5 ~ j "
2 k; R.L
>
.40
I
I I I
/
."
'
xlO-2 I -I.0----xlO-2 I u .......... xlo-1 I - x10-3 __.__x10-5 ..... x10-4 .... x10-1 ............. x10-2 -1.5 --"x10-4 " --"-- x10-2
I
/ I
~
"ll .:
l:*..
I
~ ~t I
d / ...."
~
I
/
j
"k// i . ..... '~
i
7"
xlO-4 xlO-4 .............xlO- ~, ----xlo -s .... x10-3 a
o >
2E~R.I"
Fig. 14. J ( R ± ) / C O are shown as functions of 2 k ° f R L . The results from numerical calculations ( + , *, zx, [3) for the cubic lattice model are compared with the asymptotic forms (solid lines in (a) and (b) and various lines of different types in (c)). (a) "q = 1.125, (b) ~1= 1.0 and (c) "1 = 0.75. For (c), note the different multiple factors on J ( R ± ) / C O for different values of V.
A. Narita, T. Kasuya/ RKKY interaction
41
V I E ° = - 0 . 8 , ql is nearly equal to q0 and thus the behaviour is similar to that of the free electron model, but the amplitude is very much enhanced. For V / E ° - 0.4, the contribution from the q2 term is added. Because of its large contribution, a complicated amplitude modulation type distortion is seen. For V / E ° = - 0 . 4 , t h e situation is similar to that for V / E ° = - 0 . 8 in , / = 1, because of the noncontribution from the q2 term.
9. Conclusions In the three-dimensional crystal, a one-dimensional simple periodic potential was introduced along the z-direction and its effect to the R K K Y interaction was investigated. This potential gives rise to two effects, that is; 1) the deviation of the Fermi surface from the sphere and 2) the matrix element effect due to the modification of the wave function of the conduction electron. A summary of the results obtained from the calculations carried out thus far are as follows.
9.1. J(Rz) 9.1.1. (Case A) No Fermi surfaces perpendicular to the z-axis 1) When both f-spins are located on the lattice points, the asymptotic form of the R K K Y interaction for a large distance shows exponential damping. This fundamental property does not depend upon the inclusion of the matrix element, but the detailed property is largely modified in quality by taking into account the matrix element. In the constant matrix element, the interaction is described very well by a simple single term with constant sign, because the only possible contribution is due to the interband process with finite energy gap IVI. The behaviour is well described by the analytic function given by eq. (17). When the matrix element effect is taken into account, the situation becomes much more complicated, because the intraband process is now not zero. Furthermore, in the interband process too, the contribution from the pair states with energy difference smaller than IVI can contribute. The sign of J ( R , ) depends on the sign of V. Because of competition among these different processes with different Rz-dependence, the sign of J ( R z) can change as a function of Rz. The sign change becomes evident as V decreases connecting to the R K K Y type continuously and also as the neck diameter q~ decreases connecting to case B. The matrix element effect is particularly important in this region. 2) When one of two interacting f-spins is located on the lattice point and the other on an interstitial position, the matrix element effect plays a crucial role. When the matrix element effect is taken into account accurately, the asymptotic form of J ( R , ) for a large distance indicates exponential damping similar to the previous case, including the matrix element effect. But in the constant matrix element, the asymptotic form for large R, shows a much slower damping, proportional to the inverse square of the distance. This is due to the fact that in the intraband contribution, for example, the energy difference between the pair states can be zero. By taking into account the matrix element effect correctly, the character of this R~-2 term changes into the exponential damping. Therefore, to obtain exponential damping, it is essential to take into account the matrix element effect correctly. In general, compared with the case 1), the R K K Y type effect appears more rapidly with decreasing V and q~. 9.1.2. (Case B) The Fermi surface has perpendicular planes to the z-axis 1) The asymptotic form, eq. (23), which is valid for a large distance, was derived up to the order of R~-a. Note that R z can be any value, including the interstitial point. This asymptotic form was shown to be applied practically from the nearest neighbour distance, except in the special case where the perpendicular part of the Fermi surface approaches the zone boundary extremely closely, because then the interband contribution considered in case A becomes relatively important.
42
,4. Narita, T. Kasuya / RKKY interaction
2) The formula called the pseudo-free electron model seen in the extended zone scheme is used together with that most cases, including the matrix element effect as fitted fairly well by the pseudo-free electron model, but with different frequencies become large.
was introduced by eq. (24), in which the single kf the phase shift and the amplitude. It was shown well as the constant matrix element case, can be some cases cannot be fitted, because other terms
9.2. J ( R j_) In this case, there is always a Fermi surface perpendicular to R ~ , the contribution of which is very similar to that of the free electron model. Therefore, the problem is the competition with other terms, the neck part, the Fermi surface in the second zone and the interband effect and the cross term a m o n g them. The general asymptotic form was derived and shown in eq. (25). The numerical results can be fitted very well by eq, (25), in most cases. W h e n the interband effect is large, it is possible to write J ( R ± ) in two terms, eq. (25) and the exponential d a m p i n g term due to the interband effect. Again the asymptotic form was checked to fit the pseudo-free electron model described above. W h e n the potential is small, it is possible to fit eq. (25) with the pseudo-free electron model, that is, the effects of other terms are approximately included in the phase shift and the effective amplitude. W h e n the potential becomes stronger, the behaviour of J ( R l ) becomes more complicated and thus cannot be fitted by the pseudo-free electron model. The results for the Heusler alloys should be considered, based on the present calculations. Comparison and interpretation of the results in actual materials will be carried out in separate papers.
Acknowledgement One of the authors (A.N.) would like to express his sincere thanks to Professor A. Yanase for helpful advice in the numerical calculations.
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