- Email: [email protected]
Theory of helical spin structure in itinerant electron systems
Solid State Communications, Vol. 20, pp. 291—294, 1976.
Pergamon Press.
Printed in Great Britain
THEORY OF HELICAL SPIN STRUCTURE IN ITINERANT ELECTRON SYSTEMS T. Moriya Institute for Solid State Physics, University of Tokyo, Roppongi, Tokyo (Received 6 July 1976 by T. Nagamiya) Helical spin structure of a long period shown by an itinerant electron system and the effect of an external magnetic field on it are discussed from a general point of view by extending the recent self-consistent renormalization theory of spin fluctuations for weakly ferro- and antiferromagnetic metals. The magnetic properties of MnSi are discussed in the light of the present theory. IT HAS BEEN REPORTED recently”2 that a cubic intermetallic compound MnSi shows a helical spin structure with a long period (Q = 0.035 A-’) below the Curie temperature T~= 30°K.Under an external magnetic field smaller than a certain critical field H~,H~(T = 0) being 6.2 kOe, the spin structure becomes conical and for H> H~we have a paramagnetic phase with the spin density parallel to the external field. In the paramagnetic 3 as well as the region the observed relaxation magnetic properties nuclear spin-lattice rate4 seem to be interpreted in terms of the recent theory of weakly ferromagnetic metals, where the dominating long wave components of the spin fluctuations are renormalized in a self-consistent manner.5 This self-consistent renormalization theory of spin fluctuations5’6 has been quite successful in interpreting and predicting a number of important physical properties of weakly ferroand anti7 ferromagnetic itinerant electron systems. The purpose of the present note is to extend the above mentioned theory of wealdy ferro- and antiferromagnetic metals to a system with helical spin structure in a simple way and to discuss its magnetic properties. We start with the free energy expression as a function of the uniform and Q-components of the spin density: M 0 = dRM(R),
j
.
order terms in the magnetization components: F(MO MCQ M~)= F(0) +
M~+ (M~Q 2X 2X~ 0 + MS2Q) M 4 + ~lQ(McQ + M 2Q )2 0 H + ~ y (M0 4~3Q[~M 8 2 (2) + 2~ + + 2 02. MCQ) + 2QM~M~ 2 M~Q) 1 2 MSQ) I + 7.~ —M8Q) h + 2~sQ(McQ4Q(McQ Ms~)2} 1/ = (l/xoo) —I + a2L~FIaMO2X / 0 / + 2 F’ 2 ‘JiXo~) I ~ z~,aMCQX, 7 = 70 + (1/6) a4L1F/aM~~, (3) 70 = (l/16)[p(eo)] _3{ [p’(eo)/p(eo)]2 [p”(e 0)/3 p (60)] } —
—
~
—
—
—
where is the density of states,interaction 60 the Fermi energy at T = p0,(e) I the electron—electron constant, and ~F is the correction term to the Hartree—Fock free energy and is given in terms of the dynamical susceptibiity of the system under investigation.5 Here we consider only the temperature dependent part of 1~F,and assume that the temperature independent parts are absorbed in the corresponding coefficients of the 5 whose weak temperature Hartree—Fock expression, dependences are neglected here for brevity. The coefficients ~ (ii = 1, 2, 5) can be calculated in principle. When Q is small, however, we may use a long wave approximation, and in the long wave limit all of these coefficients go to unity.~For brevity we use this simplification: ~ = 1 (n = 1, 2,. 5) in what follows. We first consider the case of no applied magnetic field; H = 0. By minimizing the free energy equation (2) with H = M0 = 0 with respect to M~and M8Q we get the following two solutions corresponding to helical and sinusoidal spin structures, respectively: (1) Helical solution: .
MCQ
=
\/2J dR M(R) cos
(Q. R),
(1)
—~--
. . ,
. .
M~Q =
dR M(R) sin (Q. R)
where Q is the wave vector giving rise to the largest value for the wave vector-dependent susceptibility Xoq of a non-interacting electron system. Assuming that the systern is isotropic in the spin space, we have the following general expression for the free energy up to the fourth
291
,
292
THEORY OF HELICAL SPIN STRUCTURE
Vol. 20, No. 3
M~ = M~, M~ M8~= 0, 1 1 1 1 1 1 ——I, 77o. M~=M~+M~Q= (—vXQY’, Fh~ = —l/4’y4. Xo XQ = Xoo XOQ XQ = XOQ (2) Sinusoidal solution: (9) 2/37XQ~ Since Q is small for MnSi, the slope [(lIx~) M~Q1 M8Q, M~ F~ = M~+ M~Q= 1~ = l/6yX~. (5) (1/xoQ)Y’ oftheM—Hcurve should be very steep. We can deduce the values of~~ x~,XQ and 7o and then Clearly, the helical structure has a lower free energy and x0 from the observed magnetization curve as extrait tends continuously to a ferromagnetic state as Q ~ 0, polated to T = 0.~We tentatively estimate them as in contrast with the case of sinusoidal structure. Thus we follows: may generally expect a stable helical structure for the xQ(0) = 0.694 x l0-~emu/g, 3. isotropic with maximum of Xoqfield at small x(O) = 0.706 x iO~emu/g, ~ = 22.4 (emu/g) Now system we apply thethe external magnetic H to Q. the system. We can show that the spin structure becomes These values should not be taken seriously. (10) conical with the cone axis parallel to H; we have At finite temperatures we have to calculate the .
—
—
—
—
—
—
—
—
MCQ . M
3Q
= H. MCQ = H. M~Q = 0, M0 I M~I = I M6~I = MQ/~!~.
II H,
contribution of spin fluctuations in a self-consistent (6) manner as was done previously for ferro- and antiferromagnets. We first express the dynamical susceptibffity Minimizing the free energy with respect to M0 and MQ, in a matrix form as follows:
(
x+(q, ~) x~(q, q 2Q; x~(q+2Q,—q;~) x~(q+ 2Q,w)
x(q,c~)=
w)
—
x~~(q+Q,—q;w) xt(q+Q,_q_2Q;~)
—x~(q+Q,~)
—xt(q+Q,~)
x~(q+Q,—q;w)
—x~(q+Q,w)
—x~(q+Q,w)
x~(q+Q,—q—2Q;w)
)
x~(q,—q —Q; ce,) ~t(q, —q Q; w) —x~(q+ 2Q,—q—Q;c~) _~+t(q+ 2Q,—q—Q;~
(11) we get the following solutions for the conical and paramagnetic phases: (i) Conical phase: 2]112 MQ(T,H) = [— (l/7~) M0 Mo(T,H) = H[(l/xO)—(l/xQ)]’, forH
/i ~XO
i\
—— )MQ(T, 0).
where x~~(q, w), x0(q, w), are defined in a usual way and we have x~(q+ Q,—q;w) =
~
i
w)
J~t e~t( [~(q =
+ Q, t), S+(—q, 0)]), (12)
I Jdt eIWt < [p
0(q, t), p~(—q, 0)]), etc. XQ/ 0
(ii) Paramagnetic phase: MQ(T, H) = 0, 3 = H, forH>H~. (8) X~’Mo+7Mo This result gives a clear qualitative explanation of the magnetization curve of MnSi. The M—H curve is a straight line with a steep slope for H < H~.In this region the local magnetization is kept constant, i.e. ~ + M~ is independent of H. At H = H~theM—H curve breaks and for H>HC it has a drastically reduced but still large slope and a negative curvature. This significant increase ofM with H for H> 14, or large high field susceptibility, is a typical of wealdy ferromagnetic metals. We are property now left with the temperature dependences of various parameters given by equation (3). Let us first consider the case T = 0, where we have
This matrix is expressed in terms of the irreducible susceptibility ~(q, w) as ~(q,w)= [1 + X(q, w)— I~ 0(q,w)]’x0(q, w). (13) By using the interaction Hamiltonian as expressed by H1
=
~IN + ~I ~q [~ p(q)p(—q) — S(q). S(— q)],
we calculate L~Fwith the use an approximation valid 5 Weofget for weakly magnetic limit. L~F= — ~kB T ~ {— log det (I + X — IXo) ~ q
Vol. 20, No.3
THEORY OF HELICAL SPIN STRUCTURE
— Ix; + (q ~ — ix~(q, iw,~) (14) + ix~(q, kin) + ~ (q, ~ 2~F/~M~ ~where ~ = 2nlrkB T. In what follows we make further approximations of neglecting the non-diagonal matrix elements and the q- and w-dependences of X. In the limit of small magnetization, A is simply a scalar. The selfconsistency condition above T~is given after a somewhat lengthy calculation by ~,
6=X aQ
=
0Q/XQ
X(T,6) (kB T/XOQ) ~ ~
=
=
C
=
—
H/H~(0)= h,
mQ(t)
=
Mo(T)/MQ(0)
=
m
0, m~, -y(T)/’y(0) = i~. (20) 1~2[1—t4’3]~2, t< 1, (21) =
~~
J
(22) (23)
and for h > h~,or in the paramagnetic phase, we have (t4”3—l.’Q)mo+flm~= (1—VQ)h, fort~l, (15)
with
F1
t,
p /XOQ = mQ(t). 6 + 1 —f0(q + 2Q,iw~) For h
+ L~(q+Q,Q)/(q +2Q)2)1 6+1 _fo(q+2Q,iwn)jj =
=
and the critical boundary between the conical and paramagnetic phases is given by
2 ~ L~(q,Q)/q2 + Q) 6+ l_fo(q+Q,iwn46+1_fo(q,iwn)
aQ
T/Tc
We have
XOQaLIF/aMQ I(F 1 + Ri) f0(q, j’,~~) 6 + 1 f0(q, ~
4xoQ 6 + 1 +f 1 0(q + ~ P /XOQ ~ ___________________
+—1
and 6~obeying an approximate Curie—Weiss law above T~.We also neglect the small difference between and a2~FI8M~J, which actually seems to be negligible in MnSi where the initial slope of the M—H plot is constant below T~.These approximations enable us to obtain the following results as expressed in reduced units:
MQ(T)/MQ(O)
l—aQ+X(T,6),
293
[1 —VQ+I.LQ(t)]mo+flmo = (1 with VQ = XQ(0)/Xo(0), I.LQ(t) =
—VQ)h, fort~l, —
XQ(0)/XQ(T).
1x 0(Q, 0), fo(q, 2, w)R = x0(q, w)/x0(Q, 0),2, (p”/3p) + (p’/p) 1 = — (p”/p) + (p’/p) p = p(eo), p’ = p’(eo), etc.,
(24) To visualize the above results we show in Fig. 1 an example of a set ofM—H curves. We assume rather arbitrarily the following form for i~:10
L 1(q,Q)
q1q+Q1f’(e~) (16) (e~÷~ k)(k+q — Ek_Q) 5 if This reduces toThe the Curie previous result for isferromagnets we put Q -~0. temperature given by =
I
—~ k
—
Ilat~’~, 1 — a — aI.LQ(t),
-
—
+ X(Tc, 0)
=
0.
(17)
fort<1 for t> 1. (25)
We take vç~= 59/60, ,uQ(t) = 0.29(t — 1), a = 2/3. These values are chosen so that the calculated M—H
Below T~we know that the longitudinal and one of the transversal components of x(Q, 0) should diverge for 05
H = 0 in the present model. Also, as long as we deal with the second order term, we may neglect the MQdependence of A and set 6 = 0 in equation (15). Thus we get XOQ/XQ = X(T,0)A(Tc,0), (T
294
Vol. 20, No. 3
curves simulate to some extent the experimental results for MnSi.11 From Fig. 1 we see that the qualitative nature of the M—H curve for MaSi is fairly well reproduced. However, when we look into the experimental results more precisely, we see that the agreement is not quite satisfactory. Particularly, the experimental plot of M2 HIM (Arrott plot) does not make straight lines.9 This seems to suggest either the importance of higher
substance where the saturation moment is not quite small (0.4 fiB) and the region of q-space with strongly enhanced x~ is expected to be particularly large. Further detailed discussions on the experimental results of MnSi will be given later.
-~
6~
.
order terms such asM in the free energy owmg to the special band structure near the Fermi level, or the special importance of the short range order effect in this
Acknowledgements — The writer wishes to thank K. Makoshi for critical reading of the manuscript and H. Hasegawa, K. Ueda and H. Yasuoka for stimulating discussions.
REFERENCES 1. 2.
MOTOYA K., YASUOKA H. & NAKAMURA Y., Solid State Commun. 19, 529 (1976). ISHIKAWA Y., TAJIMA K., BLOCH D. & ROTH M., Solid State Commun. 19,525 (1976).
3.
WERNICK H., WERTHEIM G.K. & SHERWOOD R.C.,Mat. Res. BulL 7, 1431 (1972); LEVINSON L.M., LANDER G.H. & STEINTZ M.O., AlP Conf Proc. 10, 1138 (1970). YASUOKA H. (private communication).
4.
mt.
5.
MORIYA T. & KAWABATA A.,J. Phys. Soc. Japan 34, 639 (1973); ibid. 35, 669 (1973); Proc. Magnetism, IV 5. Moscow (1973).
6.
HASEGAWA H. & MORIYA T.,J. Phys. Soc. Japan 36, 1542 (1974).
7.
One remarkable example may be the nuclear spin-lattice relaxation rate. See MORIYA T. & UEDA K., Solid State Commun. 15, 169 (1974); KONTANI M., HIOKI T. & MASUDA Y., Solid State Commun. 18, 1251 (1976). This may be seen from the fact that in the long wave approximation the fourth order term of the free energy should be proportional to f dR[M(R)]4.
8. 9. 10. 11.
Conf on
BLOCH D., VOIRON J., JACCARINO V. & WERNICK J.H., Phys. Lett. 51A, 259 (1975). A leading term of ~ in equation (3) is proportional to A, though the full calculation of -y has not yet been performed. Qualitative nature of the M—H curves is insensitive to the value of a.
Pergamon Press.
Printed in Great Britain
THEORY OF HELICAL SPIN STRUCTURE IN ITINERANT ELECTRON SYSTEMS T. Moriya Institute for Solid State Physics, University of Tokyo, Roppongi, Tokyo (Received 6 July 1976 by T. Nagamiya) Helical spin structure of a long period shown by an itinerant electron system and the effect of an external magnetic field on it are discussed from a general point of view by extending the recent self-consistent renormalization theory of spin fluctuations for weakly ferro- and antiferromagnetic metals. The magnetic properties of MnSi are discussed in the light of the present theory. IT HAS BEEN REPORTED recently”2 that a cubic intermetallic compound MnSi shows a helical spin structure with a long period (Q = 0.035 A-’) below the Curie temperature T~= 30°K.Under an external magnetic field smaller than a certain critical field H~,H~(T = 0) being 6.2 kOe, the spin structure becomes conical and for H> H~we have a paramagnetic phase with the spin density parallel to the external field. In the paramagnetic 3 as well as the region the observed relaxation magnetic properties nuclear spin-lattice rate4 seem to be interpreted in terms of the recent theory of weakly ferromagnetic metals, where the dominating long wave components of the spin fluctuations are renormalized in a self-consistent manner.5 This self-consistent renormalization theory of spin fluctuations5’6 has been quite successful in interpreting and predicting a number of important physical properties of weakly ferroand anti7 ferromagnetic itinerant electron systems. The purpose of the present note is to extend the above mentioned theory of wealdy ferro- and antiferromagnetic metals to a system with helical spin structure in a simple way and to discuss its magnetic properties. We start with the free energy expression as a function of the uniform and Q-components of the spin density: M 0 = dRM(R),
j
.
order terms in the magnetization components: F(MO MCQ M~)= F(0) +
M~+ (M~Q 2X 2X~ 0 + MS2Q) M 4 + ~lQ(McQ + M 2Q )2 0 H + ~ y (M0 4~3Q[~M 8 2 (2) + 2~ + + 2 02. MCQ) + 2QM~M~ 2 M~Q) 1 2 MSQ) I + 7.~ —M8Q) h + 2~sQ(McQ4Q(McQ Ms~)2} 1/ = (l/xoo) —I + a2L~FIaMO2X / 0 / + 2 F’ 2 ‘JiXo~) I ~ z~,aMCQX, 7 = 70 + (1/6) a4L1F/aM~~, (3) 70 = (l/16)[p(eo)] _3{ [p’(eo)/p(eo)]2 [p”(e 0)/3 p (60)] } —
—
~
—
—
—
where is the density of states,interaction 60 the Fermi energy at T = p0,(e) I the electron—electron constant, and ~F is the correction term to the Hartree—Fock free energy and is given in terms of the dynamical susceptibiity of the system under investigation.5 Here we consider only the temperature dependent part of 1~F,and assume that the temperature independent parts are absorbed in the corresponding coefficients of the 5 whose weak temperature Hartree—Fock expression, dependences are neglected here for brevity. The coefficients ~ (ii = 1, 2, 5) can be calculated in principle. When Q is small, however, we may use a long wave approximation, and in the long wave limit all of these coefficients go to unity.~For brevity we use this simplification: ~ = 1 (n = 1, 2,. 5) in what follows. We first consider the case of no applied magnetic field; H = 0. By minimizing the free energy equation (2) with H = M0 = 0 with respect to M~and M8Q we get the following two solutions corresponding to helical and sinusoidal spin structures, respectively: (1) Helical solution: .
MCQ
=
\/2J dR M(R) cos
(Q. R),
(1)
—~--
. . ,
. .
M~Q =
dR M(R) sin (Q. R)
where Q is the wave vector giving rise to the largest value for the wave vector-dependent susceptibility Xoq of a non-interacting electron system. Assuming that the systern is isotropic in the spin space, we have the following general expression for the free energy up to the fourth
291
,
292
THEORY OF HELICAL SPIN STRUCTURE
Vol. 20, No. 3
M~ = M~, M~ M8~= 0, 1 1 1 1 1 1 ——I, 77o. M~=M~+M~Q= (—vXQY’, Fh~ = —l/4’y4. Xo XQ = Xoo XOQ XQ = XOQ (2) Sinusoidal solution: (9) 2/37XQ~ Since Q is small for MnSi, the slope [(lIx~) M~Q1 M8Q, M~ F~ = M~+ M~Q= 1~ = l/6yX~. (5) (1/xoQ)Y’ oftheM—Hcurve should be very steep. We can deduce the values of~~ x~,XQ and 7o and then Clearly, the helical structure has a lower free energy and x0 from the observed magnetization curve as extrait tends continuously to a ferromagnetic state as Q ~ 0, polated to T = 0.~We tentatively estimate them as in contrast with the case of sinusoidal structure. Thus we follows: may generally expect a stable helical structure for the xQ(0) = 0.694 x l0-~emu/g, 3. isotropic with maximum of Xoqfield at small x(O) = 0.706 x iO~emu/g, ~ = 22.4 (emu/g) Now system we apply thethe external magnetic H to Q. the system. We can show that the spin structure becomes These values should not be taken seriously. (10) conical with the cone axis parallel to H; we have At finite temperatures we have to calculate the .
—
—
—
—
—
—
—
—
MCQ . M
3Q
= H. MCQ = H. M~Q = 0, M0 I M~I = I M6~I = MQ/~!~.
II H,
contribution of spin fluctuations in a self-consistent (6) manner as was done previously for ferro- and antiferromagnets. We first express the dynamical susceptibffity Minimizing the free energy with respect to M0 and MQ, in a matrix form as follows:
(
x+(q, ~) x~(q, q 2Q; x~(q+2Q,—q;~) x~(q+ 2Q,w)
x(q,c~)=
w)
—
x~~(q+Q,—q;w) xt(q+Q,_q_2Q;~)
—x~(q+Q,~)
—xt(q+Q,~)
x~(q+Q,—q;w)
—x~(q+Q,w)
—x~(q+Q,w)
x~(q+Q,—q—2Q;w)
)
x~(q,—q —Q; ce,) ~t(q, —q Q; w) —x~(q+ 2Q,—q—Q;c~) _~+t(q+ 2Q,—q—Q;~
(11) we get the following solutions for the conical and paramagnetic phases: (i) Conical phase: 2]112 MQ(T,H) = [— (l/7~) M0 Mo(T,H) = H[(l/xO)—(l/xQ)]’, forH
/i ~XO
i\
—— )MQ(T, 0).
where x~~(q, w), x0(q, w), are defined in a usual way and we have x~(q+ Q,—q;w) =
~
i
w)
J~t e~t( [~(q =
+ Q, t), S+(—q, 0)]), (12)
I Jdt eIWt < [p
0(q, t), p~(—q, 0)]), etc. XQ/ 0
(ii) Paramagnetic phase: MQ(T, H) = 0, 3 = H, forH>H~. (8) X~’Mo+7Mo This result gives a clear qualitative explanation of the magnetization curve of MnSi. The M—H curve is a straight line with a steep slope for H < H~.In this region the local magnetization is kept constant, i.e. ~ + M~ is independent of H. At H = H~theM—H curve breaks and for H>HC it has a drastically reduced but still large slope and a negative curvature. This significant increase ofM with H for H> 14, or large high field susceptibility, is a typical of wealdy ferromagnetic metals. We are property now left with the temperature dependences of various parameters given by equation (3). Let us first consider the case T = 0, where we have
This matrix is expressed in terms of the irreducible susceptibility ~(q, w) as ~(q,w)= [1 + X(q, w)— I~ 0(q,w)]’x0(q, w). (13) By using the interaction Hamiltonian as expressed by H1
=
~IN + ~I ~q [~ p(q)p(—q) — S(q). S(— q)],
we calculate L~Fwith the use an approximation valid 5 Weofget for weakly magnetic limit. L~F= — ~kB T ~ {— log det (I + X — IXo) ~ q
Vol. 20, No.3
THEORY OF HELICAL SPIN STRUCTURE
— Ix; + (q ~ — ix~(q, iw,~) (14) + ix~(q, kin) + ~ (q, ~ 2~F/~M~ ~where ~ = 2nlrkB T. In what follows we make further approximations of neglecting the non-diagonal matrix elements and the q- and w-dependences of X. In the limit of small magnetization, A is simply a scalar. The selfconsistency condition above T~is given after a somewhat lengthy calculation by ~,
6=X aQ
=
0Q/XQ
X(T,6) (kB T/XOQ) ~ ~
=
=
C
=
—
H/H~(0)= h,
mQ(t)
=
Mo(T)/MQ(0)
=
m
0, m~, -y(T)/’y(0) = i~. (20) 1~2[1—t4’3]~2, t< 1, (21) =
~~
J
(22) (23)
and for h > h~,or in the paramagnetic phase, we have (t4”3—l.’Q)mo+flm~= (1—VQ)h, fort~l, (15)
with
F1
t,
p /XOQ = mQ(t). 6 + 1 —f0(q + 2Q,iw~) For h
+ L~(q+Q,Q)/(q +2Q)2)1 6+1 _fo(q+2Q,iwn)jj =
=
and the critical boundary between the conical and paramagnetic phases is given by
2 ~ L~(q,Q)/q2 + Q) 6+ l_fo(q+Q,iwn46+1_fo(q,iwn)
aQ
T/Tc
We have
XOQaLIF/aMQ I(F 1 + Ri) f0(q, j’,~~) 6 + 1 f0(q, ~
4xoQ 6 + 1 +f 1 0(q + ~ P /XOQ ~ ___________________
+—1
and 6~obeying an approximate Curie—Weiss law above T~.We also neglect the small difference between and a2~FI8M~J, which actually seems to be negligible in MnSi where the initial slope of the M—H plot is constant below T~.These approximations enable us to obtain the following results as expressed in reduced units:
MQ(T)/MQ(O)
l—aQ+X(T,6),
293
[1 —VQ+I.LQ(t)]mo+flmo = (1 with VQ = XQ(0)/Xo(0), I.LQ(t) =
—VQ)h, fort~l, —
XQ(0)/XQ(T).
1x 0(Q, 0), fo(q, 2, w)R = x0(q, w)/x0(Q, 0),2, (p”/3p) + (p’/p) 1 = — (p”/p) + (p’/p) p = p(eo), p’ = p’(eo), etc.,
(24) To visualize the above results we show in Fig. 1 an example of a set ofM—H curves. We assume rather arbitrarily the following form for i~:10
L 1(q,Q)
q1q+Q1f’(e~) (16) (e~÷~ k)(k+q — Ek_Q) 5 if This reduces toThe the Curie previous result for isferromagnets we put Q -~0. temperature given by =
I
—~ k
—
Ilat~’~, 1 — a — aI.LQ(t),
-
—
+ X(Tc, 0)
=
0.
(17)
fort<1 for t> 1. (25)
We take vç~= 59/60, ,uQ(t) = 0.29(t — 1), a = 2/3. These values are chosen so that the calculated M—H
Below T~we know that the longitudinal and one of the transversal components of x(Q, 0) should diverge for 05
H = 0 in the present model. Also, as long as we deal with the second order term, we may neglect the MQdependence of A and set 6 = 0 in equation (15). Thus we get XOQ/XQ = X(T,0)A(Tc,0), (T
294
Vol. 20, No. 3
curves simulate to some extent the experimental results for MnSi.11 From Fig. 1 we see that the qualitative nature of the M—H curve for MaSi is fairly well reproduced. However, when we look into the experimental results more precisely, we see that the agreement is not quite satisfactory. Particularly, the experimental plot of M2 HIM (Arrott plot) does not make straight lines.9 This seems to suggest either the importance of higher
substance where the saturation moment is not quite small (0.4 fiB) and the region of q-space with strongly enhanced x~ is expected to be particularly large. Further detailed discussions on the experimental results of MnSi will be given later.
-~
6~
.
order terms such asM in the free energy owmg to the special band structure near the Fermi level, or the special importance of the short range order effect in this
Acknowledgements — The writer wishes to thank K. Makoshi for critical reading of the manuscript and H. Hasegawa, K. Ueda and H. Yasuoka for stimulating discussions.
REFERENCES 1. 2.
MOTOYA K., YASUOKA H. & NAKAMURA Y., Solid State Commun. 19, 529 (1976). ISHIKAWA Y., TAJIMA K., BLOCH D. & ROTH M., Solid State Commun. 19,525 (1976).
3.
WERNICK H., WERTHEIM G.K. & SHERWOOD R.C.,Mat. Res. BulL 7, 1431 (1972); LEVINSON L.M., LANDER G.H. & STEINTZ M.O., AlP Conf Proc. 10, 1138 (1970). YASUOKA H. (private communication).
4.
mt.
5.
MORIYA T. & KAWABATA A.,J. Phys. Soc. Japan 34, 639 (1973); ibid. 35, 669 (1973); Proc. Magnetism, IV 5. Moscow (1973).
6.
HASEGAWA H. & MORIYA T.,J. Phys. Soc. Japan 36, 1542 (1974).
7.
One remarkable example may be the nuclear spin-lattice relaxation rate. See MORIYA T. & UEDA K., Solid State Commun. 15, 169 (1974); KONTANI M., HIOKI T. & MASUDA Y., Solid State Commun. 18, 1251 (1976). This may be seen from the fact that in the long wave approximation the fourth order term of the free energy should be proportional to f dR[M(R)]4.
8. 9. 10. 11.
Conf on
BLOCH D., VOIRON J., JACCARINO V. & WERNICK J.H., Phys. Lett. 51A, 259 (1975). A leading term of ~ in equation (3) is proportional to A, though the full calculation of -y has not yet been performed. Qualitative nature of the M—H curves is insensitive to the value of a.