361
Progress of Theoretical Physics, Vol. 28, No. 2, August 1962
Fermi Surfaces and Spin Structures in Heavy Rare-Earth Metals
Kei YOSIDA and Atsuko W AT ABE
Institute for Solid State Physics
University of Tokyo, Azabu, Tokyo
(Received March 27, 1962)
The screw-like spin structures in heavy rare-earth metals found ·by neutron diffraction
experiments are investigated on the basis of the s-f exchange model. The calculation shows
that a screw structure with the screw axis along the c axis is stable for a hcp lattice with
three conduction electrons per atom and that the period of the screw becomes longer with
decrease of the atomic number. From this and other considerations it is concluded that
the spin structure has a 'close correlation with the Fermi surface for heavy rare-earth metals.
§ 1. Introduction
The people at Oak Ridge1 J-e) have made-neutron diffraction experiments on
single crystals of heavy rare-earth metals from Tb to Tm and found that all of
these metals show a screw-like structure with periodic nature below the transition temperatures. The spin structures of these metals are different in detail
but they are regarded essentially as a screw structure and it is modified by a strong
crystalline anisotropy energy of each spin of rare earth ions.
Elliott/) Kaplan8l and Miwa and Yosida9l have independently shown that the
exchange interaction between spins which itself makes a screw structure stable
and the anisotropy energy which is described by spherical harmonics, Y2 , Y4 , Y 6
and Y 66 , are enough to understand the various types of periodic structure which
appear in these rare-earth metals.
The energy of the spin system with a screw structure can simply be described in a classical approximation by
(1)
where N is the total number of atoms, S is the magnitude of the spin and J(Q)
represents the Fourier transform of the exchange integral J(R 1 -Ri) between
two spins and it is written as
(2)
The vector Q which characterizes the screw structure has a direction parallel
to the screw axis and its magnitude is expressed by
Q=a/c',
(3)
where a 1s the turn angle of the spins and c' is the distance between adjacent
362
K. Yosida and A. W atabe
planes perpendicular to the screw axis.
The fact that the screw structure is stable means that J (Q) , as a function
of Q, has a maximum at a certain non-zero value of Q, as can be seen from
(1). In the rare-earth metals under consideration the screw axis is along the
c axis and the spins on the same c plane are all parallel. The value of Q0
which makes J(Q) maximum is 0.286 in the units of 27r/c for Er and Tm at
their Neel temperatures at which a spiral ordering of the spins sets in and it
becomes 0.112 at Tb decreasing with decrease of the atomic number. This
change of Qo from element to element is shown by the solid curve in Fig. 1.
Generally, the value of Q0 decreases linearly with decreasing temperature and
then it becomes constant below a certain temperature except for Tm which has
a temperature-independent pitch. For
0,3
Ho, for example, the turn angle decreases linearly from a value of 50° at
the Neel temperature to 36° at 35° K
and then it remains constant. These
0,2
low temperature values of Q 0 are also
plotted in Fig. 1.
For rare-earth metals the mam
exchange interaction acting on the
spins has been considered to be the
indirect interaction through the intermediary of conduction electrons, beGd
'fb
Dy
Ho
Er
Tm
cause 4f electrons of an atom are well
Fig. 1. The values of Q0 for heavy rare-e;;~rth
isolated from those of its nearest
metals. The solid curve shows the value
neighbor
atoms. To investigate wheat the 'Nee! temperature and the dashed
ther
or
not
this s-f' exchange interacline indicates the value at low temperatures.
tion is really responsible for the spin
structure of heavy rare-earth metals
which has been described above is the main purpose of the present paper.
Recently, Woll and NetteP 0l calculated on the basis of the s-f exchange theory
the spin wave frequencies in the ferromagnetic state of Gd in order to see
the Kohn anomalies11l which appear in the dispersion relation and found
that the spin wave dispersion becomes negative along the c axis. This result
suggests. that a stable spin structure is of a screw type. Our calculation will be
along a similar line to theirs but we shall elucidate the origin of the screw
structure in rare-earth metals in more detail and further discuss the relation
between the Fermi surface and the spin structure.
§ 2.
Calculation of J(Q) for a hexagonal close-packed structure
The exchange interaction between a conduction electron and a 4f spin can
be written in the following conventional form:
Fermi Surfaces and Spin Structures in Heavy Rare-Earth Metals
363
(4)
-2J(ri-R,.)si-Sn-
J(ri-Rn) can be considered to be the exchange integral between the Wannier
function made from a linear combination of Bloch functions and the 4f wave
function. si and S,. are re~pectively the spins of the conduction electron and
the 4f ion. If we treat ( 4) as a perturbation to the kinetic energy of conduction electrons, we obtain the effective exchange interaction between 4f spins
from the second order perturbed energy as follows :
J(R,-Ri)=_~
116
ir
~~
{N-l"'i:,qJ(q)"f(q)eiq<Rt-Ril},
(5)
where N is the total number of unit cells, N. the total number of conduction
electrons and E 1 the Fermi energy. g (q) means the Fourier transform of g (r)
and f(q) is the function defined by
j(q) =1+ 1-x2 log]! l+x
2x
l-x
I'
(6)
x=q/2k1 .
The expression for the
derived by Kasuya12J and
is taken so as to include
the Fourier transform of
J(Q)
indirect exchange interaction given by (5) was first
discussed by Yosida. 13) In this expression a unit cell
four atoms. With the use of (5) in (2), we obtain
the exchange integral J(R 1 -Ri) as
=~l_ N. - 1 {"'E,xJ([K+Q[):f([K+Q[)ReF(K)
8 16 N
E1
- N-1 "'i:,qJ(q):f(q)}.
Here the summation with respect to K
extends over all reciprocal lattice
vectors and F(K) is the form factor
which is given by
z
F (K) = "'E,,. exp (iKR,.),
y
:x
Fig. 2.
(7)
A unit cell for a hcp lattice.
reciprocal lattice vector is expressed by
(8)
where the summation is taken over the
atoms in a unit cell.
Now we shall calculate J(Q) given
by (7) as a function of Q along the c
axis and investigate the variation of
J(Q) with Q. For Q along the a
qxis, J(Q) has its maximum at Q=O.
For convenience, we shall take a unit
cell as shown by Fig. 2. Then the
364
K. Yosida and A. Watabe
(9)
and the form factor F (K) becomes
F(K) =0,
for nx -f- ny =odd integer,
=4,
for nx -f- ny/3 -f- n, =2m,
=0,
for nx -f- ny/3 -f- n. =2m± 1,
v3 i, for
= 1 ± v3 i, for
= 3±
(10)
nx -f- ny/3 -f- n. =2m± 1/3,
nx+ ny/3+n.=2m±2/3,
where nx, ny, n, and m are integers.
In calculating J(Q), we first assume that g(q) is constant. This assumption makes each of the two sums in Eq, (7) diverge but J(Q) itself is convergent. J (Q) - J (0) is then proportional to
:Eidf(!K+QI) -f(K)}F(K).
Therefore, it turns out that the present calculation is equivalent to Woll and
Nettel's calculation. 10J The ratio of the two lattice parameters, c/ a, is about
1.58 for Tb and about 1.57 for Ho, Er and Tm. Therefore, we take a value
of 1.57 for c/ a. The number of free electrons is assumed to be 3 per atom.
Then the magnitude of the Fermi wave vector is given by (2rr/a) X 1.6152.
Table I shows the values of f(K) and F(K) for reciprocal lattice points near
the origin.
Table I. The values of f(x) and F(K) for reciprocal lattice
cfa=1.57.
p~ints
of a hcp lattice with
F(~)
f(x)
z
(000)
(110)
0
0.7149
(002)
(111)
0.7887
1.5118
4
0.8164
1.4682
3±v3i
2
12
12
2.0000
4
1
1.6135
1±v'3i
6
(112)
1.0645
0.7832
1±v3i
(200)
1.2382
0.5177
4
(113)
1.3822
0.3973
3±v'3i
12
(220)
1.4298
0.3674
1±y3i
6
6
When K/'2.k1 =x is less than unity, the Fermi sphere is cut by the Brillouin
zone boundary corresponding to that reciprocal lattice point and when x exceeds
unity, the Brillouin zone boundary is outside the Fermi surface. The function
f' (x) becomes minus infinity at x= 1. Therefore, f(x) changes rapidly with
x in the neighborhood of this singular point, but for the values of x which are
far from unity the change of f(x) is rather slow. Therefore, we calculate the
Fermi Surfaces and Spin Structures in Heavy Rare-Earth Metals
.365
quantities of f([K + Q[) - f(K) as a function of Q for three sets of reciprocal
lattice points, i.e. (002), (111) and (112). For other lattice points, . however,
we expand f([K+Q[) -f(K) with respect to Q 2 as follows:
'L:'x {f([K+Q[) -f(K)}F(K)
=a~ 2 +b~\
(11)
~=_Q.__
<>kf '
~
f';:) +f";:) }{1-6( ?r +5( ~·r}
(? r
+ 6-----f"-----'-~(x-'---) (? r{1- (~ r} +
Fv(x)
]F(K),
(13)
where the prime attached to the symbol of summation means to exclude the
above-mentioned three sets of lattice points. The calculated values of a and b
are as follows :
a= 17.0,
b=29.2.
(14)
For b the convergence of the lattice sum is very good, but for a it is not so
good. Here we have taken into account the summation up to the points (117)
with K/2k1 =2.8515. The sum of (11) with the values of a and b given by
(14) and the contribution from the three sets of lattice points of (002), (111)
and (112) gives rise to the curve described in Fig. 3 for 'L::x{f([K+Q[)
- f(K)} F (K) which is proportional to J (Q) - J (0) .
The curve in this figure shows a maximum at Q/2k1 =0.105 which corresponds to a turn angle of 48° and to a period of 7.5 c layers. Woll and
0.24,---------------------,
0.20
0.16
0.12
0.08
0.04
0
0.02
0.04
Fig. 3. L:xU(IK+QI) -f(K)}F(K) as a function of Q along
the c axis. The arrow indicates the point of the Kahn
anomaly.
K. Yosida and A. Watabe
366
Nettel's calculation seems to show that for c/a=l.633 the maximum occurs at
Q/2k1 = 0.11. This value corresponds just to a period of seven c layers. Considering the fact that we have assumed a spherical Fermi surface for the present
calculation, the agreement of this result with experiments on Ho, Er and Tm
is surprisingly good and strongly supports the idea that the exchange interaction
in heavy rare-earth metals originates from the s-f exchange interaction.
§ 3.
Relation between the Fermi surface and the spin structure
In this section we shall elucidate the reason why J(Q) has its maximum
at a non-zero value of Q. When x=K/2k1 is larger than unity, the sum of the
contribution f(IK+QI) from the lattice points with the' same value of Kincreases with Q, and for x less than unity, it decreases with Q in most cases.
However, this change of f(IK + Ql) with Q is not so rapid if the value of x is
at a distance from unity. Accordingly, the feature of the curve J(Q) is essentially determined by the contributions of the lattice points of (002), (111)
and (112). Among these three sets of lattice points it i~ the lattice points
(112) that make J(Q) increase with Q. Moreover, for this set of lattice points
K/2k1 is in the vicinity of unity (K/2k1 = 1.0645) and for the lattice points
(±1, ±1, -2), IK+QI/2k 1 decreases with Q and passes through unity at which
f(x) has an infinite derivative. This fact is reflected on the curve of J(Q)
shown in Fig. 3 which has also an infinite derivative at the point Q/2k1 =0.089
which satisfies
(15)
This is the condition for the Kohn anomaly. 11l Thus, we can presume that a
ferromagnetic screw structure is likely to occur if the Brillouin zone boundary
exists just outside the Fermi surface.
The next problem we should consider is that the value of Q0 is different
from element to element. One possible cause for this experimental fact is a
difference in c/ a. However, a small difference in c/ a observed experimentally
cannot account for the change of Q0 , as will be shown later. A most plausible
source which causes various values of Q 0 is supposed to be a difference in
nuclear charge. If the nuclear charge decreases, the 4f wave function will expand. This makes the effective range of J(q) shorter as one goes from Tm
to Tb. Therefore, we shall consider the change of J(Q) curve with decrease
?f the range of J(q). For simplicity, we approximate the contribution from
all the lattice points but (ll2) points by their expanded form. Then, we obtain
2::Idf(IK+QI) -f(K)}F(K)
= - af2 + 6 { f( v"'.xi~;+ 2x~;,-:-~ + ~2 )
+ ft/ xi12- 2xm, • ~ + .;
2) -
2f(xm)} ReF (Km),
(16)
Fermi Surfaces and Spin Structures in Heavy Rare-Earth Metals
367
where ~=Q/2k 1 , x 112 =Km/2k1 , Xm,./2k 1 .
The value of ~ which makes (16) extremum
lowing equation :
IS
a~/ 3 ReF (K112 ) = f' V(V 2x~12 + 2xm, ~z ~ +~ 2~2 )
X112
+ 2xm, z <; + <;
given by a solution of the fol-
(
Xn2,z+
_f'(Vxi12-2xl12,z~+~ 2 ) (x
,/ 2
e: _::2
v Xn2 - 2xn2, z '> + •
112,z
~)
-~)
·
(17)
The right-hand side of Eq. (17) first increases linearly with ~ and then becomes
infinite at the value of ~ which satisfies the condition for the Kohn anomaly.
From this point, it decreases. Therefore, for a small value of a, Eq. (17) has
one non-zero solution which is larger than the value ~J( for the Kohn anomaly.
This solution makes J (Q) maximum. The curve given by Fig. 3 corresponds
to this case.
If the range of !f (q) becomes shorter, we must cut the summation over K
at a smaller value of K. This makes a larger and the value of ~ satisfying ·
(17) decreases. If a exceeds a critical value, J(O) changes from minimum to
maximum and at the same time a new minimum of J (Q) appears at a smaller
value of ~ than ~x- However, the old maximum still survives at a point slightly
larger than ~x· This maximum point cannot become smaller than ~I(· Thus, the
decrease of Q 0 with decrease of the atomic number as shown in Fig. 1 can be
attributed to a decrease of the nuclear charge. The fact that Gd shows a
simple ferromagnetism can also be understood by supposing that the maximum
value of J (Q 0) is lower than the maximum at Q = 0.
4
According to neutron diffraction experiments, the value of Q0 decreases
with decreasing temperature. The reason for this temperature dependence of
the pitch of the screw is not yet clarified, but if we assume that this occurs
due to the pseudo-quadrupole interaction between spins which makes a parallel
spin alignment favorable, as suggested by Elliott/l the value of Q 0 decreases
as the quadrupole coupling increases with decreasing temperature, because this
quadrupole interaction effectively increases the value of a. Whatever causes
may reduce the value of Q 0 with decreasing temperature, Q 0 can never become
smaller than the lower limit given by the condition for the Kohn anomaly as
we have shown before. In accordance with this expectation, the experimental
values of Q 0 are constant at low temperatures. Even for Dy, there exists a
temperature range of about 10 degrees in which Q 0 is constant before it undergoes a transformation to the ferromagnetic state. The constant values of Q0 at
low temperatures for Tb to Tm are plotted by a dashed line in Fig. 1. Our
argument identifies these values with those of the Kohn anomaly (15) . If the
ratio c/ a is constant, the value of Q satisfying the Kohn anomaly should be
constant. In actuality, this value varies almost linearly with the atomic number.
368
K. Yosida and A. Watabe
Table II shows the variation of Q0 for the Kohn anomaly with the c/a ratio.
As can be seen from this table, a change of c/ a from 1.57 to 1.58 corresponding
to the difference in c/ a between Tm and Tb is not sufficient to account for
the difference in Q0 at low temperatures.
Table II. The values of Q112 for the Kohn anomaly for various values of cfa ratio.
cfa
2ktf (277:/a)
Q 112 c/277:
Qm/2kt
1.57
1.6152
0.089
0.227
1.58
1.6118
0.088
0.223
1.59
1.6084
0.086
0.220
1.60
1.6051
0.084
0.216
1.633
1.5941
0.079
0.206
It is most likely that the variation of the value of Q for the Kohn anomaly
is caused by an anisotropy of the Fermi surface. For a general Fermi surface
the Kohn anomaly occurs at a value of Q such that the vector of K 112 + Q is
bisected by the Fermi surface. Therefore, it may be considered that the constant value of Q 0 at low temperetures is determined by the diameter of the
Table III. The diameters of the Fermi sphere.
Gd
Tb
Dy
Ho
Er
Tm
1.045
1.034
1.021
1.006
0.996
0.984
Fermi spheroid. The diameters of the. Fermi spheroid along the direction of
K 112 + Q evaluated on this viewpoint are listed in Table III. In this table k 10 is
the radius of the Fermi surface as it is assumed to be spherical, and the value
for Gd is evaluated by the value of Q 0 extrapolated from those for other metals.
The variation of the Fermi diameter ranging from Gd to Tm is within six
percent and such a difference seems to be quite conceivable.
To the above considerations on the behavior of J(Q), we shall add here
the discussion on the value of J(Q) at Q=O. The value of J(O) is related to the
paramagnetic or asymptotic Curie temperature (} by
kd= 2(gJ-1) 2J(J+l)J(O) .
3
(18)
The evaluation of J(O) for heavy rare-earth metals on the basis of the s-f exchange model has been made by de Gennes. 14J He has shown that the value
of J(O) obtained from this model agrees to a reasonable extent with the experimental values. In the present formulation, J(O) is written as
Fermi Surfaces and Spin Structures in Heav:y Rare-Earth Metals
369
where fP means a suitable average of !f(q) 2 • Our calculation gives a value of
5.62 for the quantity in the curly bracket with the use of c/ a= 1.57 and
(}j(gJ-I) 2 J(J+I) is obtained as 1·65°K if we assume the free electron value
of E 1 =7.38 ev and !f=O.l ev. This value does not sensitively depend on the
range of !J(q). Experimental values for heavy rare-earth metals are within the
temperature range betw<oen 17° K and 20~ K.
§ 4.
Summary and discussion
On the basis of the s-f exchange model, we have investigated the behavior
of the Fourier transform J(Q) of the exchange integral for heavy rare-earth
metals and found that J(Q) has a maximum at a non-zero value of Q along the
c axis. This means that a screw-like spin structure is stable in these metals.
The value of Q for maximum J(Q) denoted by Q0 corresponds to a period of
about seven c layers. This value is in good agreement with the experimental
values for Ho, Er and Tm and provides a strong support to the s-f exchang.
model for heavy rare-earth metals.
The value of Q0 decreases with decreasing nuclear charge. This tendency
accounts for the experimental fact that the period of the screw-like modulation
of the spin moments gets longer as one goes from Tm to Tb. It was further
shown that Q0 :cannot become smaller than the value for the Kohn anomaly
corresponding to K 112 • Neutron diffraction experiments show that the t1:1rn angle
of the screw decreases linearly with lowering temperature from the Neel temperature and then it becomes· constant at low temperatures. Therefore, it may be
reasonable to identify this constant value of Q0 at low temperatures with the
value of Q at which the Kohn anomaly occurs. This identification enables us
to deduce the radius of the Fermi surface for each element of heavy rare-earth
metals.
The consideration presented here for the occurrence of the screw spin
structure may have some resemblance to the idea proposed by Slater15> and
Overhauser,I 6> although these authors have considered that the spin ordering of
the conduction electrons is stabilized by the interaction between themselves
instead of the s-f interaction. Each term in the first sum of Eq. (7) arises
from the change in the energy of the conduction electrons due to the periodic
Hartree-Fock potential with K + Q produced by the s-f exchange interaction in
the case that the 4f spins take a screw spin arrangement. This periodic potential
gives rise to a new energy gap across the plane of (K + Q) /2 in k-space. The
first sum of Eq. (7) can, therefore, be interpreted as the energy change of the
conduction electrons due to the formation of the new zone boundaries. Eq.
(7) shows that this change in energy is proportional to the function f(IK + Ql).
For a one-dimensional case this energy change is largest when this zone boundary
touches the Fermi· surface. However, for the present three-dimensional case it
remains finite and only shows the Kohn anomaly at IK+QI =2k1 . Therefore,
K. Yosida and A. Watabe
370
the largest energy gain due to the formation of the new zone boundaries takes
place at a slightly larger value of Q than that for the Kohn anomaly, as has
been shown in this paper. Recently, the long period superlattice in CuAu and
other alloys has been discussed by Sato and Toth17J on the basis of the idea
of Hume-Rothery and Jones. The present theory gives a new theoretical basis
to Hume-Rothery and Jones' idea.
Acknowledgments
We are greatly indebted to Dr. ]. Kondo and Dr. H. Miwa for· their
illuminating discussions and valuable co.mments and we are also indebted to
profitable conversations with Professor T. Moriya.
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