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The magnetic structure of AuMn
J. Phys. Chem. So/ids
Pergamon Press 1968. Vdl. 29, pp. 225-236.
THE MAGNETIC
Printed in Great Britain.
STRUCTURE
OF AuMn*
W. BINDLOSSt Department of Physics, University of California, Berkeley, California 94720 (Received
20 March 1967; in revisedform
12 September
1967)
Abstract-The results are reported of room-temperature magnetic torsion measurements on a single crystal of the t, phase of the tetragonal, antiferromagnetic compound AuMn. It is found that the magnetic torsion resulting from an external field is dependent on the displacement of domain walls. The domain walls are assumed to result from localized pinning of the magnetization at random sites throughout the crystal. The torsion measurements are consistent with a magnetic structure in which the Mn magnetic moments align ferromagnetically in a-a planes, with adjacent planes antiferromagnetically aligned. The sublattice magnetization is found to lie along an a-axis. A uniaxial anisotropy field HA, = 39,800 Oe favoring alignment of the spin direction in the a-a plane is calculated from classical dipolar interactions. The torsion measurements indicate that the in-plane anisotropy field H,, stabilizing the magnetization direction against rotation in the a-a plane is larger than 120 Oe. The roomtemperature susceptibility difference x1- x,,is found to be approximately 9.5 x IO-“. 1. INTRODUCTION
AuMn crystallizes in the ordered CsCl structure over a broad range of concentration around 50 at.% Au and, for concentrations close to stoichiometric AuMn, undergoes a martensitic transformation at about 500°K to a tetragonal phase (cl phase) with a c/a ratio of 0.97. At a temperature close to or equal to that of the cubic-tetragonal transition the crystal undergoes a magnetic transition to a two sublattice antiferromagnetic spin structure. The crystallographic and magnetic transitions have been investigated by neutron diffraction [ 11and X-ray [2] studies. The Mn magnetic moments align ferromagnetically in a-a planes, with adjacent planes antiferromagnetically aligned as shown in Fig. 1. For alloys with gold concentrations of less than 5 1 at.% a second transformation to a tetragonal phase (tz) occurs at a somewhat lower temperature. This phase has a c/a ratio of 1.03 and has a more complicated magnetic THE ALLOY
*Supported by the U.S. Atomic Energy through Contract AT( 1 I -l)-34 Project 47. UCB-34P47-5. tPresent address: Central Research E. 1. du Pont de Nemours and Company, Delaware.
structure [ 11. The tl * t2 transformation is inhibited for bulk samples with Au concentrations in excess of 51 at.%. In this paper the results are presented of room-temperature magnetic torsion measurements on a single crystal of the I, phase (51.5 at.% Au) of AuMn. It is found that the Mn spins align along an Mn
Mn
Commission Report Code Department, Wilmington, Fig. 1. The magnetic structure of the tl phase of AuMn. :2.s
226
W. BINDLOSS
a-axis as indicated in Fig. 1. Strictly speaking, the symmetry of the structure shown in Fig. 1 is orthorhombic, however no distortion of the tetragonal lattice has been detected by X-ray analysis [2]. The susceptibility [ 31 and the electrical resistivity[ 4,5J of the AuMn alloy have been measured over a wide range of temperature. The susceptibility measurements by Gainsoldati et a1.[3] indicate a magnetic moment in the neighborhood of 4.8 Bohr magnetons per Mn atom, while the neutron diffraction measurements by Bacon [ 11give a magnetic moment between 4.0 and 4.2 pB per Mn atom. Measurements of the electronic specific heat [6], which were made on the same crySta1 as is discussed here, showed a remarkably small density of electronic states at the Fermi energy, suggesting that a magnetic moment of close to 5 pB per Mn atom may be present. It was also found that single crystal resistivity measurements[5] made on a portion of the crystal discussed in this paper could be interpreted using a model which, for the usual picture of a localized Mn state [7], is consistent with a 5 pg state. The results reported here indicate that the magnetic torsion resulting from an external field is dependent on the displacement of domain walls. It is assumed that the domain walls are the result of localized pinning of the sublattice magnetization at random sites throughout the crystal. Thus it is possible that a significant portion of the magnetization may be constrained to lie in directions out of the a-a plane. Any substantial pinning of the magnetization out of the a-a plane could cause a reduction in the intensity of the magnetic reflections observed in neutron diffraction measurements. 2. SAMPLE
PREPARATION
The alloy discussed here was formed from manganese of 99.98 per cent purity and gold of 99.999 per cent purity, using crucibles machined from high-purity graphite. The initial alloy was made from stoichiometric (5Oat.%) amounts of Au and Mn in an atmosphere of
argon, using a crucible with a tightly-fitted cap to minimize losses of Mn through vaporization. Approximately 40 per cent of the surface of the initial ingot was found to be coated with a thin layer of manganese carbide. This carbide layer was removed, and no further carbide formation resulted from subsequent melting of the alloy in the graphite containers. The alloy was remelted several times, and then was cast into a ain. dia. ingot by pouring the melt from one crucible into another in an atmosphere of argon. This procedure was necessary in order to remove holes and inclusions in the ingot. Finally, the ingot was machined to fit into a 0.22 in. i.d. crystal mold with a tightly-fitted cap. A single crystal approximately 3 in. in length was grown in an argon atmosphere, using the modified Bridgman technique. The graphite mold was heated by a high frequency magnetic induction coil, which was raised at a uniform rate of approximately # in./hr during the growth of the crystal. The crystal was then heat treated for several hours at both 1000 and 600°C to insure homogenization and complete chemical ordering. Chemical analysis of several slices of the crystal indicated a composition of 51.5 4 0.2 at.% Au. This composition was constant within the indicated error over the length of the crystal, excluding the top and bottom 4 in. The crystallography of the cubic + tetragonal transition at 500°K has been studied by Smith and Gaunt[2], using both X-ray and optical techniques. The lattice transforms discontinuously to the tetragonal t1 structure by means of a martensitic shear transformation, resulting in a finely twinned product. The single crystal discussed here contained only two twin-related orientations of the tl phase. The two orientations shared a common a-axis, and were in an approximately 2: 1 volume ratio for any large region of the crystal. However, the volume occupied by a single orientation was found to be much larger than was observed in the polycrystalline ingots obtained prior to the growth of the final
THE
MAGNETIC
STRUCTURE
crystal. In some portions of the crystal regions of millimeter size were found in which only one crystallographic orientation was detectable by X-ray analysis. A number of oriented samples, ranging in size from 20 to 50mm3, were spark-cut from the crystal. From torsion measurements of the kind described in Section 3, it was found that the vol. ratio of the two twin-orientations in these samples varied widely from the theoretical 2: 1 ratio. The magnetic torsion measurements described in Section 3 were made on an irregularly shaped sample of about 3 mm3 vol., in which only one crystallographic orientation was detectable from X-ray and torsion measurements. 3. EXPERIMENTAL
RESULTS
Room temperature torsion measurements were made on the untwinned sample des-
-15,000
OF AuMn
227
cribed above using essentially the same apparatus described by Lee et d.[8]. The magnetic torque was measured as a function of field strength and orientation for fields up to 18 kOe. Figure 2 shows a typical set of torsion curves for field directions lying in the a-a plane of the crystal. The magnetic torque is plotted vs. the angle Q,, of the magnetic field with respect to an a-axis. The theoretical curve; obtained from Section 4 are also shown. Hysteresis was observed in these torsion measurements for field directions near a,, = 45” and Q0 = 135”. Typical hysteresis loops for field directions near Q0 = 45” are shown in Fig. 3. For field strengths in excess of 6 kOe the torsion measurements in the a-a plane show a fourfold symmetry, and the maximum torque is proportional to HU2. The slope of the torque curve, for a given field, is
-
I I
I
I
0
45
90
Magnet
I 135
I 180
angle QO, degrees
Fig. 2. Magnetic torque vs. field direction for field directions lying in the a-a plane. The maximum torque is proportional to H,2. The slope of the torque curves at Q0 = 0, a0 = W”, and UQ,= 180” is proportional to H,,‘; the slope at a0 = 45” and Q0 = 135” is proportional to-H,‘.
W. BINDLOSS
228
the torque curves at @,,= 45” is shown in Fig. 5, which is a plot of slope vs. field. The experimental points plotted in Fig. 5 were obtained from data such as is shown in Fig. 3, using the slope measured at the point of zero torque. The slope determined in this way is independent of the direction of rotation of the magnetic field and is proportional to Ho4 for fields in excess of 10 kOe. I
-10,000
II 40
42 Magnet
44 angle
a,,
46
48
,
degrees
Fig. 3. Hysteresis loops of torque curves directions near CJ,)= 45”. The field is rotated plane.
I
I
I
J 50
for field in the a-a
the same at @,, = 0”, a0 = !20”,and Q0 = 180”, and is also proportional to HoZ. The field dependence of the slope of the torque curves at @0 = 0” is shown in Fig. 4, where the slope divided by the field is plotted versus the field strength. The field dependence of the slope of
Magnetic
field
H,,
I
10 Magnetic
I
I
I
IS
20
30
field
H,,
kOe
Fig. 5. Field dependence of the slope of the torque curves at @,, = 45”. The field is rotated in the a-a plane.
kOe
Fig. 4. Field dependence of the slope of the torque a0 = 0. The field is rotated in the a-a plane.
curves
at
THE MAGNETIC
STRUCTURE
2000
_E Y 6
229
(111) plane of the crystal. The resulting torsion curves had twofold symmetry and were found to be consistent at fields above 8 kOe with the domain wall model discussed in Section 4. Torsion measurements were made on a number of twinned samples oriented so that the field was rotated in the a-a and a-c planes of the two crystallographic orientations. The results of these measurements could in all cases be interpreted as a superposition of the curves shown in Figs. 2 and 6. Thus the torsion measurements served to determine the volume ratio of the two orientations in a twinned sample. The torsion measurements on one of the twinned samples were extended to magnetic fields up to 30 kOe. No deviation was observed from the behavior described above.
A set of typical torsion measurements for field directions lying in the a-c plane are shown in Fig. 6. The torque is plotted vs. the angle 8, of the field with respect to the c-axis. Considerable hysteresis was observed in these measurements. The experimental points shown in Fig. 6 are for increasing field angle 13~For decreasing field angle the relative magnitudes of the peaks near f30= 0” and &,= 180” are reversed. The theoretical curve plotted in Fig. 6 includes a parameter which takes account of hysteresis. The relative magnitudes of the theoretical peaks would also be reversed for decreasing magnetic field angle. The slope of the torque curves at 8, = 0” and 8,, = 180” is proportional to H,,2, and is approximately Q that observed at @‘. = 0” in the a-a plane. For fields of less than 5 kOe the torsion curves for the a-a and a-c planes were dominated by hysteresis effects, and, for fields of less than 2 kOe were dependent on the prior history of the sample. Torsion measurements were also made with the field rotated in the
1
OF AuMn
4. TORSION THEORY
Assuming a molecular field model, tetragonal symmetry, and keeping terms in the anisotropy energy up to fourth order in the direction cosines of the sublattice magnetiza-
I I I I I I I I I .-. .i
1000
d, E 6 i
0
5 P
.a
2 -1000
-
. H, = 13 kOe 0 Ho=
0 .
IO kOs
0
-Theory - 2000
J 0
I
.
_
:
I
I
40
I
I
80 Magnet
angle
I 120
8,,
I
I
I
160
degrees
Fig. 6. Magnetic torque versus field direction for field directions lying in the c-a plane. The experimental points and the theoretical curve are for increasing field angle Bw For decreasing field angle the relative magnitudes of the peaks near B,,= 0 and &, = 180” are reversed. The slope of the torque curves at B,,= 0” and B0= 180”is proportional to Ho2.
230
W. BINDLOSS
tions M1 and Mz, the energy per unit volume has the form U=-(M1+M2)
.Ho+X(M,.M2)
+ K, (cos281+
cos%J
+ K; (cos%, +
cos40,)
-
K2 ( sin4B, cos 4@, + sin48, cos 4Q2). (1)
The quantity A is the intersublattice molecular field constant, K1, K2, and K; are anisotropy constants, and the angles &, and Q1 define the direction of the magnetization M,. Let us define the x direction as the easy direction in the a-a plane, so that in the absence of external fields the equilibrium positions are given by 13~= d2 = ~12, a1 = 0, and
Ml and M2 point
in the plus- and minus-z directions respectively, then the dipolar field is 13,270 Oe and is antiparallel to the Mn magnetic moment. Identifying the energy difference per Mn site between the two configurations with the quantity 2K,/N, where N is the number of Mn atoms per unit volume, we find HA, = 39,800Oe. Thus the observed spin configuration is consistent with the configuration of minimum dipolar energy. To examine the influence of external fields let us consider a configuration in which the external field is allowed to rotate in the x-y plane. Since the x-y plane is an easy plane we may take t$= d2= m/2. No evidence of rotation of the magnetizations out of the a-a plane was observed in the torsion measurements reported here. The external field causes the sublattice magnetizations to cant through the angle A, resulting in a net moment M, in the direction @ = $(a1 +Q2) -r as shown in Fig. 7. The canting angle A is of the order of Y
Fig. 7. Definition of the angles used in calculating the torque for field directions lying in the a-a plane.
Ho/HE, where HE is the exchange field. Thus for laboratory fields we may use the approximation Q = Q1 - ~12 = a2 - 3~12. The equilibrium position Cpis given by the relation
(AxHo2/16K2) sin 2(@,,-(B)
= sin4@
(2)
THE MAGNETIC
STRUCTURE
(3)
where Ax is the susceptibility difference x1-x,,. In the limit of small external fields (Ax~Y3/16K, -+ l), for which Ca/$ + 0, the torque is given by 7 = ( AxH,~/~) sin 2@+,.
231
Domain wall model
and the torque is given by T = (AxHo2/2) sin2($--@)
OF AuMn
(4)
In this limit both the maximum torque and the slope of the torque curve at a0 = 0 are proportional to Ho2. In large fields such that AxHo2/16K2 2 1 the spin direction is rotated from one axis to the other as the field is rotated in the x-y plane, and the maximum torque is given by TV = 8K, independent of field strength. The experimental torsion curves obtained for fields in excess of 6 kOe (Fig. 2) show that the maximum torque is proportional to Ho2 for fields up to 30 kOe. This field dependence indicates that the low field condition AxHo2/ 16K2 6 1 is satisfied. For field directions such that @, =o”, Q’o = 90”, and Q0 = 180” the torsion curves are in agreement with the low field formula (3), but with a 90” phase change for every 90” rotation of the external field. The phase of the torsion curves indicates that the sublattice magnetizations lie perpendicular to the field direction when the field is along an a-axis. Thus the magnetization direction is transferred from one a-axis to another as the field is rotated 90” in the plane. Since the low field condition is satisfied, we conclude that the transfer of the direction of the sublattice magnetizations proceeds by a mechanism other than rotation against the anisotropy field HA2. The susceptibility difference AX may be determined from the slope of the torque curves at a,, = 0. The data plotted in Fig. 4 are in agreement with the value Ax = 9.5 X lo-+. We also conclude that the easy direction for the sublattice magnetizations lies along an a-axis. Since the low field condition AxHo2/16K2 4 1 is satisfied for fields up to 30 kOe, the anisotropy field HA2 = 16K,/M must satisfy the condition HA2 P 120 Oe.
The transfer of the spin direction from one a-axis to another as the field is rotated in the a-a plane can be understood if we assume the presence of domain walls. We assume that in the absence of an external field the magnetization lies along one a-axis in some portions of the crystal and along the other a-axis in the remainder of the crystal. Then at the boundaries between these two regions there will be domain walls within which the spin direction rotates by 90”. In a sufficiently large field the domain walls will move until the entire volume of the sample contains the energetically favorable spin orientation. We will see that the field dependence of the slope of the torsion curves at a0 = 45” (dr/d@,, 0~ H04) indicates that the domain walls experience restoring forces tending to equalize the volume occupied by the two spin orientations. This behavior is assumed to result from pinning of the direction of the sublattice magnetization at random sites throughout the crystal. We shall consider here a simplified model in which the magnetization direction is assumed to be pinned along either easy axis. Let us consider, for example, two pinning sites separated by a distance L such that the magnetization direction is pinned at @ = 0 at one site and @ = 90” at the other site. Then between the two pinning sites the magnetization direction @ as a function of position will have the form shown in Fig. 8. The coordinate z measures the distance along a line between the two pinning sites. The rotation of the magnetization direction by 90” occurs primarily within a domain wall which lies perpendicular to the line between the pinning sites. We assume that the wall thickness d is small compared to the distance L between pinning sites. From the symmetry of this configuration it is clear that the lowest energy will be achieved if the wall is situated midway between the two pinning sites, so as to equalize the volume occupied by the two spin orientations. Thus we may assume a harmonic restoring force opposing a displacement 5 of
232
W. BINDLOSS Wall thickness d
0
L
L/2
Coordinate
z perpendicular
to plane of wall
Fig. 8. Spin direction as a function of position between two pinning sites. The sublattice magnetizations are assumed to be pinned such that Q = 0 at z = 0 and @ = 42 at z = L.
the wall away from its equilibrium position at z = L/2. To determine the magnetic torque in the presence of domain wall displacements we will treat the idealized model shown in Fig. 9. We consider a cube with side L, such that the magnetization is pinned at Q = 0 on one side of the cube (z = 0) and at Q, = 90” on the opposite side of the cube (z = L). The displacement of the wall away from the mid-point is denoted by 5, and the field H,, rotates in the a-a plane. Let the restoring force per unit area of the wall be given by F = -kg. Then, if Magnetization
we neglect the volume occupied the energy is given by
E = $L”kt2 - &L2AxHo2[ (L/2 + 5) COS~@~
+ (L/2--<) The wall displacement
5 is given by
(@=
(6)
where the quantity y = kL has the dimensions of energy/volume. The torque per unit volume .
ls
pinned
T = ( AxH,,~) (l/L) sin 2$. (@=7r/2/
at z = L
wall z
pinned
sin‘QI+J. (5)
5 = (AxHo2/2y) L cos 2Q0
-Domain
Magnetization
by the wall,
0) at z = 0
Fig. 9. Idealized model of spin pinning used in calculating the magnetic torque for field directions lying in the a-a plane. The sublattice magnetizations are assumed to be pinned such that @ = 0 on one side of the cube and @ = 5712on the opposite side.
(7)
‘ap!s al!soddo aql uo Z/S = Q pue aqm aql JO ap!s auo uo 0 = @ JEIJJqms pauurd aq 01 pawnsse am suoyxz!~au%xu axlvIqns aqL .aue[d IS-J all2 u! Bu!l([ suoyamp play IOJ anbIo1 ~!]auWm aql Buyu[nym u! pasn %u!uu!d u!ds JO [apow pazgeap1 ‘01 %!g
0 !*
-sip e gl!M pawr3ossE up y_xorn aIq!slaAa_u! (0 I ) z (Z/7 - 2) z7Y4 + oBzu!S%7z”HxVf = 5’ aql wp %uytunsst? Kq seInwlo3 1wya~oayj aq1 uogEp_l olu! s!salawQ aDnpo.xy llew ah .khaua 30 aql Icq uay8 s! lli21aua [E?JOJ ay,~ *anblol SSOIaIq~S.IaAa.Lyaq$ Icq pa!wduroD%? s! SIIEM atjl 03 uognqyuo~ ou saywu uo$a~ syl snqL u!w~op aql 30 ~uaura~eIds!p aq$ JXX# aw ‘P-7)z7(Z/z0HXV)= 2 AquaA? s;r006 = Q, -!pu! E +!!d u! u~oys sdool s!saJalslCy aqL ljD!ljM II! UO!sal aqJ qJ!M paJE!DOSST &aUa *suoyEA.Iasqo Iwuawyadxa I.@M$UaUIaaJi% aw ‘ZvH Q “H pUt? IvH 4 3H JW$j%U!UInSS+’ uy s! s!y,z. *a0 001~ < OH SpIay 103 pamas *(z/7 - 2)y- = d Aq uaA$i s! aDJo %uIJ03sal -qo aq Quo p1noqs 0 = QI 1~ saA.uw anblo) ayl ~wj~ OS ‘2 Lq paJouap MOU s! [IEM ayl 30 ay) 30 adoIs arg 30 acwapuadap z”H aql snyl uoysod agJ_ ‘0 1 ‘%+Ju! UMO~Sso UO~yL&!LJUO~ ‘a0 ()()[p < OH Aq UaAp s! 0 = ‘@ UayM Z/7 = 2 arg faAoqE passnDs!p w?ql se Iapow paz!pap! )uacuaDEIds!p wnur~xt?uI sl! 01 IIE?M ayl ahup auw au[l awns% ah ~~EwCI:,aql30 auald a-3 01 pa_ynbaJ pIag ayl .eur@la 0~91 = h pug agl u! pawoJ s! pIag ayl YD~M u! as23 ayl -lo3 aM qD!qM way ‘s ‘%A 30 BJEp ayl q,iM waw uoyo~ ~gau?iw_u ayl ssn3s;rp MOUj@ys aM -aa.& ur sy 1p-tsa.Isy~ ‘+OH01 @uoyodold s! *anblol olaz 30 $u!od ,&p = %$I1~ saAm3 anblol aqI 30 adoIs agl snql (8)
(6)
“
uwjqo aM (L) uoynba %!sn ‘olaz aq aDlo lau ayl wy~ uoyrpuo3 aql hq pauywalap s! 11~~ aql30 uop +od ayL *uo~~our30 uogDa_np aql 01 alisoddo can )!un lad do = d aDlo3 ‘e aDuauadxa II’M uo!loW u! jI’?M ‘1? snq.l $pld= Mp hq uaAp s! ea.re l!un 30 IIEMa 30 f2p wawaDyd
“@ti u!s [ht/z(zoHXV)]
= L
uyqo aM uaqL ‘(L) uogenba o)u! (9) uogenba ajnlgsqns aM .sp = OQ,01 asol:, suog -Da.np pIag ~03 anb.Iol arl, au!walap 0~ ‘0 = L put? 0 = fi u!wqo aM pu?i? luapi?A!nba IlIjWga8 -laua a32 suoywa!~o u!ds 0~1 aql .sp = O@ uoyal!p pIay ayl god *,M = OQ, 103 uS!s 30 @sJaAal e ql!~ lnq ‘(p) uoynba 0~ Iwyuap! SI (L) uoyznba liq uaA!% anblol ayl suo!lDanp play asaql JOT *72/i- = 2 aAw_jaM .()6 = O@ “03 f7Z/I = 2 alleq aN\ .()fj[ 2: O@pul? .() = OQ 103 uaqL ‘~~81= Oa pw ‘,M iT OQ, ‘0 = OQ, suogDa.np play ~03 Z/7 = (21 luatua~t?Ids!p aIq!ssod uInur~xEuI sl! 01 uaA!.Ip aq II!M I~?M aqi (I < X/zo~X~) spiay a8-w 43uapgns -w
W. BINDLOSS
234
and the torque
per unit volume is given by
7 = ( AxHo2/2) (z/L) sin 28,,.
(11)
Since significant hysteresis is observed experimentally for all fields up to 18 kOe, we must include the force per unit area F = &p associated with the irreversible loss of energy in the presence of wall motion. Setting the net force on the wall equal to zero we find z/L = THI-
(b~H,,~/y) sin2&,-+ 2P/r]
(12)
where the plus and minus signs apply to the cases of decreasing and increasing wall displacement respectively. Using p = 200 erg/cm3 we find 201-y = 0.24, which indicates that hysteresis effects should be very significant. When the field is directed along the c-axis (0, = 0)) the two spin orientations (Fig. 10) are energetically equivalent and the wall position z tends toward z = L/2. If the field direction is rotated away from the c-axis, the region within which 8 = 90” becomes energetically favorable and the wall position tends toward z = 0, in which case the torque tends toward zero. Thus in large fields the torque will be zero except for field directions in the neighborhood of B0= 0 and 8, = 180”. Theoretical torque curves obtained using equations (11) and (12) are compared with experiment in Fig. 6. The curves are drawn for the case of increasing 13~.The magnitude and field dependence of the slope of the torque curves at I$, = 0 (dr/d&, 0~Ho2) are found to be in good agreement with the theoretical result. We do not expect to obtain agreement between the theoretical and experimental values for either the maximum torque or the magnitude of the torque for field directions far away from the c-axis, because these quantities depend critically on the amount of irreversible energy loss associated with domain wall displacements. Domain wall restoring force We will now examine in detail the origins
of the domain wall restoring force for the configuration shown in Fig. 8. We define the
position of the wall as the point at which the spin direction a,(z) is equal to 45”. The displacement of the wall away from its equilibrium position is denoted by 4, and the wall thickness is defined by d = r/2 [ (dz/d@),=,,,], as shown in Fig. 8. Let us denote the energy per unit area of the wall, for zero wall displacement, by E,(L). This energy will decrease with increasing L, approaching a constant value as L + 03. Now consider a small displacement 5 of the wall away from its equilibrium position. This displacement will raise the wallenergy to a value =itE,(L-t25)
E,(L)
+B&-225).
(13)
Equation (13) is obtained by assuming that the range of exchange interactions is small compared to the wall thickness do. The first and second terms are the energies associated with the regions 0 < z < L/2+5 and L/2+ s < z < L respectively. We have previously defined the restoring force per unit area of the wall by F =-k{, so that E,(L) -E,(L) = $k(L)t2. Using a Taylor expansion of E,(L), we find k(L)
= 4d2E,(L)/dL2.
(14)
We will now obtain an expression for the energy E,,(L) per unit wall area following the theory of ferromagnetic domain walls [lo]. The exchange-energy density associated with the spin rotation has the form A(d@/d~)~. Denoting the anisotropy-energy density by g(a), the wall energy is given by E,(L)
= j;
[A(z)‘+,(@)]dz
(15)
where g(
(16)
If we consider the particular case in which the wall is perpendicular to the c-axis and include only exchange interactions between spins lying within neighboring a-a planes, then we find A = a(gpdhM)
(c/a”).
(17)
THE MAGNETIC
STRUCTURE
By minimizing equation (15) with respect to small variations of the function a(z) , subject to the conditions that @( 0) = 0 and @(L) = 7r/2, one obtains the Euler equation
_2A@Q+dgW -dz2
_
o
dz
’
(18)
Multiplying equation (18) by (d@/dz)dz, and integrating from zero to z, we find
=A($)z-B(L)
g(Q)
(19)
OF AuMn
To determine the constant B(L) rewrite equation (24) in the form
(20) In the limit as L --, q, B(L) + 0. Denoting the wall thickness in this limit by d,,, and using equation (19) one finds do = (r/4) (A/K,) 1’2.
(21)
Equation ( 19) implies that
dz=
(
A
112
>
g(@)+B(L)
(22) da
hence z(Q) = A”2 f;
[g(Q) +B(L)]-1’2d@
(23)
where the constant of integration is zero because z = 0 when Q>= 0. Noting that z(7r/2) = L, one finds L = Al’2 0n’2[g(m) +B(L)]-“2dQ. I
(24)
Substituting equations (19) and (22) ‘into equation (27), and making use of equation (24), one finds E,(L) = 2Al12J;” [g(a) +B(L)]“2d@ -LB(L).
(25)
Then, using equations (24) and (25), one finds dE,(L)/dL=-B(L). Thus the constant given by
k(L)
(26)
defined by (14) is
k(L) = -4dB(L)/dL.
(27)
let us
2L (K,/A ) l/2 = &j: [ sin28 + l2] -l/zdO = 1” [sin28+ e2]-1’2d& I
(28)
where 0 = 2+ and l2 = B (L)/4K,. This integration can be divided into two parts such that 2L(K,/A) 1’2= I[’ sin28 + E2]-1/2dfI 0 +Jgi'12
where
235
[
sin20 + l2] -1/2d& (29)
We are interested in the case where L is large compared to the wall thickness do. Since E + 0 in this case we can evaluate equation (29) in the limit that l2 + a2 4 1, finding 2L (K,/A ) 1/2= In (4/e). Thus we obtain B(L) = 64K, exp [-4L(K2/A)1’2].
(30)
The quantity y(L) = LK (L) has been determined from the data of Fig’s. (4) and (5). Using equations (2 l), (27) and (30), one finds y(L) = 2%rK, (L/d,) e-flLldo).
(31)
Using the experimental value y = 1640 erg/ cm3, we can determine the ratio L/d, for various values of K,. For various values of K, between lo4 erg/cm3 and lo6 erg/cm3 (240 Oe < HA2 < 24,000 Oe) we find that 3.1 < L/d, < 4.7. Thus for any reasonable value of the anisotropy energy K, we find Lid, = 4. Then l2 = 6 X 10e5, so that the approximations used in evaluating the integral (29) are justified. We can make a rough estimate of the distance L between pinning sites by using equation (21) for the wall thickness do, and equation (27) for the exchange-energy constant A. The constant A can be estimated from the susceptibility [3] by assuming that x( TN) = x, = l/h. Then for K, = IO4erg/cm3 we find do = 1670 A; for K, = 105erg/cm3 we find do = 520 A. Thus, since L/d, = 4, we conclude that the distance between pinning sites is of the order of 2000-8000 A. There may exist several different mechan-
236
W. BINDLOSS
isms which could account for the localized pinning of the sublattice magnetizations. One such mechanism would be the presence of local crystalline anisotropy fields due to lattice distortions in the immediate vicinity of a dislocation. A mean separation between dislocations of the order of 2000-8000 A is possible in a strained crystal[ I 11. This is a reasonable possibility for the case discussed here since considerable strain is introduced by the cubic to tetragonal transformation. 5. CONCLUSIONS We find that the torsion measurements presented here are in qualitative agreement with a theoretical model which includes the presence of domain walls. The measurements show that the sublattice magnetizations lie along an a-axis, and indicate that the anisotropy field H.,,2 is in excess of 120 Oe. This anisotropy field is sufficient to prevent the rotation of the magnetization in the a-a plane in external fields up to 30 kOe. The approximate value 9.5 X IO-" which we find for the room temperature susceptibility difference xL -x is seen, by comparison with the powder susceptibility(31, to be a reasonable value. We also find that the direction of the sublattice magnetization is shifted successively from one a-axis to another as the external field H,, is rotated in the a-a plane, and account for this behavior in terms of domain-wall displacements. The H, dependence of the torsion curves indicates that the domain walls experience restoring forces tending to equalize the volume occupied by the two spin orientations. This behavior is explained on the basis of a model in which the magnetization is pinned along one or the other a-axis, with equal probability, at random sites throughout the crystal. For this model a distance between pinning sites of the order of 2000-8000 A is found to be consistent with the experimental results. Thus local anisotropy fields associated with dislocations are a possible mechanism for the spin-pinning. The idealized model of the spin-pinning
treated here may be a considerable oversimplification of the true state of the crystal, since the magnetization direction may be pinned other than along an a-axis at a certain percentage of the pinning sites. Thus the numerical values obtained for the parameters associated with domain wall displacements should be considered only as crude estimates. As is noted in Section 1, any substantial pinning of the magnetization direction out of the a-a plane would also effect the intensity of magnetic reflections observed in neutron diffraction measurements. A(.~no~~~led~c~rn~fIls- My gratitude is extended to Professor A. M. Portis for the critical guidance he has provided throughout this work. I am indebted to Professor S. F. Ravitz for his many helpful suggestions during the sample preparation. I am particularly grateful to Dr. I.. B. Welsh for his assistance in the sample preparation and in the initial torsion measurements. Thanks are also due to Professor J. W. Garland for several helpful discussions. During the sample preparation this research benefited from the partial support of the National Science Foundation. REFERENCES
I. 2. 3.
4. 5. 6.
BACON G. E., Proc. phys. Sot. 79,938 (1961). SMITH J. H. and GAUNT P., Acta Mr~ull. 9, 819 (1961). GIANSOLDATI A., J. Phys. Radium. Paris 16. 342 (1955): GlANSOLDATl A., LINDE J. 0. and BOREI.IUSG.,J.Ph.vs.Chcm.So/id.~11,46(19S9). GIANSOLDATI A. and LINDE J. 0.. J. Phys. Rrrdium. Paris 16,341 (1955). BINDLOSS W.. Phy.;. Rev. In press. HO J. C. and BINDLOSS W.. Physics Lerlers 20. 459 (1966). The value for the Fermi level density of states is incorrectly reported. The correct value is 0445 el./eV-atom; thus the ratio of the measured
density of states to the free-electron density of states is I .49. 7. FRIEDEL J., Suppl. Nuouo Cim. 7, 287 (1958); BI.ANDIN A. and FRIEDEL J., J. Phps. Rudium. Paris 20. 160 (1959); ANDERSON P. W. Phps. Rec. P. A., Phys. Rev. 124. 1030 J. R. and MATTIS D.,,C.. (1965); KJOLLERSTROM D. J. and SCHRIEFFER J. R.,
124, 41 (1961); WOLFF (1961); SCHRIEFFER Phys. Rec. 140, A1412 B., SCALAPINO
Phys. Rec.
148,665(1966).
A. M. and WI’I”f G. L.. Phy.5. Rec. 132, 144 (1963). H., Phys. Rec. 47,947 (1935). 9. MEULl.ER C., Reo. mod. Phys. 21, 541 (1949): IO. KIT-TEL CHIKAZUMI S., Physics of Mqnerism. Wiley, New York ( 1964). C., Introduckm to Solid SIUIP Physics. Il. KIT-TEL Wiley, New York (1956).
8. LEE K.. PORTIS
Pergamon Press 1968. Vdl. 29, pp. 225-236.
THE MAGNETIC
Printed in Great Britain.
STRUCTURE
OF AuMn*
W. BINDLOSSt Department of Physics, University of California, Berkeley, California 94720 (Received
20 March 1967; in revisedform
12 September
1967)
Abstract-The results are reported of room-temperature magnetic torsion measurements on a single crystal of the t, phase of the tetragonal, antiferromagnetic compound AuMn. It is found that the magnetic torsion resulting from an external field is dependent on the displacement of domain walls. The domain walls are assumed to result from localized pinning of the magnetization at random sites throughout the crystal. The torsion measurements are consistent with a magnetic structure in which the Mn magnetic moments align ferromagnetically in a-a planes, with adjacent planes antiferromagnetically aligned. The sublattice magnetization is found to lie along an a-axis. A uniaxial anisotropy field HA, = 39,800 Oe favoring alignment of the spin direction in the a-a plane is calculated from classical dipolar interactions. The torsion measurements indicate that the in-plane anisotropy field H,, stabilizing the magnetization direction against rotation in the a-a plane is larger than 120 Oe. The roomtemperature susceptibility difference x1- x,,is found to be approximately 9.5 x IO-“. 1. INTRODUCTION
AuMn crystallizes in the ordered CsCl structure over a broad range of concentration around 50 at.% Au and, for concentrations close to stoichiometric AuMn, undergoes a martensitic transformation at about 500°K to a tetragonal phase (cl phase) with a c/a ratio of 0.97. At a temperature close to or equal to that of the cubic-tetragonal transition the crystal undergoes a magnetic transition to a two sublattice antiferromagnetic spin structure. The crystallographic and magnetic transitions have been investigated by neutron diffraction [ 11and X-ray [2] studies. The Mn magnetic moments align ferromagnetically in a-a planes, with adjacent planes antiferromagnetically aligned as shown in Fig. 1. For alloys with gold concentrations of less than 5 1 at.% a second transformation to a tetragonal phase (tz) occurs at a somewhat lower temperature. This phase has a c/a ratio of 1.03 and has a more complicated magnetic THE ALLOY
*Supported by the U.S. Atomic Energy through Contract AT( 1 I -l)-34 Project 47. UCB-34P47-5. tPresent address: Central Research E. 1. du Pont de Nemours and Company, Delaware.
structure [ 11. The tl * t2 transformation is inhibited for bulk samples with Au concentrations in excess of 51 at.%. In this paper the results are presented of room-temperature magnetic torsion measurements on a single crystal of the I, phase (51.5 at.% Au) of AuMn. It is found that the Mn spins align along an Mn
Mn
Commission Report Code Department, Wilmington, Fig. 1. The magnetic structure of the tl phase of AuMn. :2.s
226
W. BINDLOSS
a-axis as indicated in Fig. 1. Strictly speaking, the symmetry of the structure shown in Fig. 1 is orthorhombic, however no distortion of the tetragonal lattice has been detected by X-ray analysis [2]. The susceptibility [ 31 and the electrical resistivity[ 4,5J of the AuMn alloy have been measured over a wide range of temperature. The susceptibility measurements by Gainsoldati et a1.[3] indicate a magnetic moment in the neighborhood of 4.8 Bohr magnetons per Mn atom, while the neutron diffraction measurements by Bacon [ 11give a magnetic moment between 4.0 and 4.2 pB per Mn atom. Measurements of the electronic specific heat [6], which were made on the same crySta1 as is discussed here, showed a remarkably small density of electronic states at the Fermi energy, suggesting that a magnetic moment of close to 5 pB per Mn atom may be present. It was also found that single crystal resistivity measurements[5] made on a portion of the crystal discussed in this paper could be interpreted using a model which, for the usual picture of a localized Mn state [7], is consistent with a 5 pg state. The results reported here indicate that the magnetic torsion resulting from an external field is dependent on the displacement of domain walls. It is assumed that the domain walls are the result of localized pinning of the sublattice magnetization at random sites throughout the crystal. Thus it is possible that a significant portion of the magnetization may be constrained to lie in directions out of the a-a plane. Any substantial pinning of the magnetization out of the a-a plane could cause a reduction in the intensity of the magnetic reflections observed in neutron diffraction measurements. 2. SAMPLE
PREPARATION
The alloy discussed here was formed from manganese of 99.98 per cent purity and gold of 99.999 per cent purity, using crucibles machined from high-purity graphite. The initial alloy was made from stoichiometric (5Oat.%) amounts of Au and Mn in an atmosphere of
argon, using a crucible with a tightly-fitted cap to minimize losses of Mn through vaporization. Approximately 40 per cent of the surface of the initial ingot was found to be coated with a thin layer of manganese carbide. This carbide layer was removed, and no further carbide formation resulted from subsequent melting of the alloy in the graphite containers. The alloy was remelted several times, and then was cast into a ain. dia. ingot by pouring the melt from one crucible into another in an atmosphere of argon. This procedure was necessary in order to remove holes and inclusions in the ingot. Finally, the ingot was machined to fit into a 0.22 in. i.d. crystal mold with a tightly-fitted cap. A single crystal approximately 3 in. in length was grown in an argon atmosphere, using the modified Bridgman technique. The graphite mold was heated by a high frequency magnetic induction coil, which was raised at a uniform rate of approximately # in./hr during the growth of the crystal. The crystal was then heat treated for several hours at both 1000 and 600°C to insure homogenization and complete chemical ordering. Chemical analysis of several slices of the crystal indicated a composition of 51.5 4 0.2 at.% Au. This composition was constant within the indicated error over the length of the crystal, excluding the top and bottom 4 in. The crystallography of the cubic + tetragonal transition at 500°K has been studied by Smith and Gaunt[2], using both X-ray and optical techniques. The lattice transforms discontinuously to the tetragonal t1 structure by means of a martensitic shear transformation, resulting in a finely twinned product. The single crystal discussed here contained only two twin-related orientations of the tl phase. The two orientations shared a common a-axis, and were in an approximately 2: 1 volume ratio for any large region of the crystal. However, the volume occupied by a single orientation was found to be much larger than was observed in the polycrystalline ingots obtained prior to the growth of the final
THE
MAGNETIC
STRUCTURE
crystal. In some portions of the crystal regions of millimeter size were found in which only one crystallographic orientation was detectable by X-ray analysis. A number of oriented samples, ranging in size from 20 to 50mm3, were spark-cut from the crystal. From torsion measurements of the kind described in Section 3, it was found that the vol. ratio of the two twin-orientations in these samples varied widely from the theoretical 2: 1 ratio. The magnetic torsion measurements described in Section 3 were made on an irregularly shaped sample of about 3 mm3 vol., in which only one crystallographic orientation was detectable from X-ray and torsion measurements. 3. EXPERIMENTAL
RESULTS
Room temperature torsion measurements were made on the untwinned sample des-
-15,000
OF AuMn
227
cribed above using essentially the same apparatus described by Lee et d.[8]. The magnetic torque was measured as a function of field strength and orientation for fields up to 18 kOe. Figure 2 shows a typical set of torsion curves for field directions lying in the a-a plane of the crystal. The magnetic torque is plotted vs. the angle Q,, of the magnetic field with respect to an a-axis. The theoretical curve; obtained from Section 4 are also shown. Hysteresis was observed in these torsion measurements for field directions near a,, = 45” and Q0 = 135”. Typical hysteresis loops for field directions near Q0 = 45” are shown in Fig. 3. For field strengths in excess of 6 kOe the torsion measurements in the a-a plane show a fourfold symmetry, and the maximum torque is proportional to HU2. The slope of the torque curve, for a given field, is
-
I I
I
I
0
45
90
Magnet
I 135
I 180
angle QO, degrees
Fig. 2. Magnetic torque vs. field direction for field directions lying in the a-a plane. The maximum torque is proportional to H,2. The slope of the torque curves at Q0 = 0, a0 = W”, and UQ,= 180” is proportional to H,,‘; the slope at a0 = 45” and Q0 = 135” is proportional to-H,‘.
W. BINDLOSS
228
the torque curves at @,,= 45” is shown in Fig. 5, which is a plot of slope vs. field. The experimental points plotted in Fig. 5 were obtained from data such as is shown in Fig. 3, using the slope measured at the point of zero torque. The slope determined in this way is independent of the direction of rotation of the magnetic field and is proportional to Ho4 for fields in excess of 10 kOe. I
-10,000
II 40
42 Magnet
44 angle
a,,
46
48
,
degrees
Fig. 3. Hysteresis loops of torque curves directions near CJ,)= 45”. The field is rotated plane.
I
I
I
J 50
for field in the a-a
the same at @,, = 0”, a0 = !20”,and Q0 = 180”, and is also proportional to HoZ. The field dependence of the slope of the torque curves at @0 = 0” is shown in Fig. 4, where the slope divided by the field is plotted versus the field strength. The field dependence of the slope of
Magnetic
field
H,,
I
10 Magnetic
I
I
I
IS
20
30
field
H,,
kOe
Fig. 5. Field dependence of the slope of the torque curves at @,, = 45”. The field is rotated in the a-a plane.
kOe
Fig. 4. Field dependence of the slope of the torque a0 = 0. The field is rotated in the a-a plane.
curves
at
THE MAGNETIC
STRUCTURE
2000
_E Y 6
229
(111) plane of the crystal. The resulting torsion curves had twofold symmetry and were found to be consistent at fields above 8 kOe with the domain wall model discussed in Section 4. Torsion measurements were made on a number of twinned samples oriented so that the field was rotated in the a-a and a-c planes of the two crystallographic orientations. The results of these measurements could in all cases be interpreted as a superposition of the curves shown in Figs. 2 and 6. Thus the torsion measurements served to determine the volume ratio of the two orientations in a twinned sample. The torsion measurements on one of the twinned samples were extended to magnetic fields up to 30 kOe. No deviation was observed from the behavior described above.
A set of typical torsion measurements for field directions lying in the a-c plane are shown in Fig. 6. The torque is plotted vs. the angle 8, of the field with respect to the c-axis. Considerable hysteresis was observed in these measurements. The experimental points shown in Fig. 6 are for increasing field angle 13~For decreasing field angle the relative magnitudes of the peaks near f30= 0” and &,= 180” are reversed. The theoretical curve plotted in Fig. 6 includes a parameter which takes account of hysteresis. The relative magnitudes of the theoretical peaks would also be reversed for decreasing magnetic field angle. The slope of the torque curves at 8, = 0” and 8,, = 180” is proportional to H,,2, and is approximately Q that observed at @‘. = 0” in the a-a plane. For fields of less than 5 kOe the torsion curves for the a-a and a-c planes were dominated by hysteresis effects, and, for fields of less than 2 kOe were dependent on the prior history of the sample. Torsion measurements were also made with the field rotated in the
1
OF AuMn
4. TORSION THEORY
Assuming a molecular field model, tetragonal symmetry, and keeping terms in the anisotropy energy up to fourth order in the direction cosines of the sublattice magnetiza-
I I I I I I I I I .-. .i
1000
d, E 6 i
0
5 P
.a
2 -1000
-
. H, = 13 kOe 0 Ho=
0 .
IO kOs
0
-Theory - 2000
J 0
I
.
_
:
I
I
40
I
I
80 Magnet
angle
I 120
8,,
I
I
I
160
degrees
Fig. 6. Magnetic torque versus field direction for field directions lying in the c-a plane. The experimental points and the theoretical curve are for increasing field angle Bw For decreasing field angle the relative magnitudes of the peaks near B,,= 0 and &, = 180” are reversed. The slope of the torque curves at B,,= 0” and B0= 180”is proportional to Ho2.
230
W. BINDLOSS
tions M1 and Mz, the energy per unit volume has the form U=-(M1+M2)
.Ho+X(M,.M2)
+ K, (cos281+
cos%J
+ K; (cos%, +
cos40,)
-
K2 ( sin4B, cos 4@, + sin48, cos 4Q2). (1)
The quantity A is the intersublattice molecular field constant, K1, K2, and K; are anisotropy constants, and the angles &, and Q1 define the direction of the magnetization M,. Let us define the x direction as the easy direction in the a-a plane, so that in the absence of external fields the equilibrium positions are given by 13~= d2 = ~12, a1 = 0, and
Ml and M2 point
in the plus- and minus-z directions respectively, then the dipolar field is 13,270 Oe and is antiparallel to the Mn magnetic moment. Identifying the energy difference per Mn site between the two configurations with the quantity 2K,/N, where N is the number of Mn atoms per unit volume, we find HA, = 39,800Oe. Thus the observed spin configuration is consistent with the configuration of minimum dipolar energy. To examine the influence of external fields let us consider a configuration in which the external field is allowed to rotate in the x-y plane. Since the x-y plane is an easy plane we may take t$= d2= m/2. No evidence of rotation of the magnetizations out of the a-a plane was observed in the torsion measurements reported here. The external field causes the sublattice magnetizations to cant through the angle A, resulting in a net moment M, in the direction @ = $(a1 +Q2) -r as shown in Fig. 7. The canting angle A is of the order of Y
Fig. 7. Definition of the angles used in calculating the torque for field directions lying in the a-a plane.
Ho/HE, where HE is the exchange field. Thus for laboratory fields we may use the approximation Q = Q1 - ~12 = a2 - 3~12. The equilibrium position Cpis given by the relation
(AxHo2/16K2) sin 2(@,,-(B)
= sin4@
(2)
THE MAGNETIC
STRUCTURE
(3)
where Ax is the susceptibility difference x1-x,,. In the limit of small external fields (Ax~Y3/16K, -+ l), for which Ca/$ + 0, the torque is given by 7 = ( AxH,~/~) sin 2@+,.
231
Domain wall model
and the torque is given by T = (AxHo2/2) sin2($--@)
OF AuMn
(4)
In this limit both the maximum torque and the slope of the torque curve at a0 = 0 are proportional to Ho2. In large fields such that AxHo2/16K2 2 1 the spin direction is rotated from one axis to the other as the field is rotated in the x-y plane, and the maximum torque is given by TV = 8K, independent of field strength. The experimental torsion curves obtained for fields in excess of 6 kOe (Fig. 2) show that the maximum torque is proportional to Ho2 for fields up to 30 kOe. This field dependence indicates that the low field condition AxHo2/ 16K2 6 1 is satisfied. For field directions such that @, =o”, Q’o = 90”, and Q0 = 180” the torsion curves are in agreement with the low field formula (3), but with a 90” phase change for every 90” rotation of the external field. The phase of the torsion curves indicates that the sublattice magnetizations lie perpendicular to the field direction when the field is along an a-axis. Thus the magnetization direction is transferred from one a-axis to another as the field is rotated 90” in the plane. Since the low field condition is satisfied, we conclude that the transfer of the direction of the sublattice magnetizations proceeds by a mechanism other than rotation against the anisotropy field HA2. The susceptibility difference AX may be determined from the slope of the torque curves at a,, = 0. The data plotted in Fig. 4 are in agreement with the value Ax = 9.5 X lo-+. We also conclude that the easy direction for the sublattice magnetizations lies along an a-axis. Since the low field condition AxHo2/16K2 4 1 is satisfied for fields up to 30 kOe, the anisotropy field HA2 = 16K,/M must satisfy the condition HA2 P 120 Oe.
The transfer of the spin direction from one a-axis to another as the field is rotated in the a-a plane can be understood if we assume the presence of domain walls. We assume that in the absence of an external field the magnetization lies along one a-axis in some portions of the crystal and along the other a-axis in the remainder of the crystal. Then at the boundaries between these two regions there will be domain walls within which the spin direction rotates by 90”. In a sufficiently large field the domain walls will move until the entire volume of the sample contains the energetically favorable spin orientation. We will see that the field dependence of the slope of the torsion curves at a0 = 45” (dr/d@,, 0~ H04) indicates that the domain walls experience restoring forces tending to equalize the volume occupied by the two spin orientations. This behavior is assumed to result from pinning of the direction of the sublattice magnetization at random sites throughout the crystal. We shall consider here a simplified model in which the magnetization direction is assumed to be pinned along either easy axis. Let us consider, for example, two pinning sites separated by a distance L such that the magnetization direction is pinned at @ = 0 at one site and @ = 90” at the other site. Then between the two pinning sites the magnetization direction @ as a function of position will have the form shown in Fig. 8. The coordinate z measures the distance along a line between the two pinning sites. The rotation of the magnetization direction by 90” occurs primarily within a domain wall which lies perpendicular to the line between the pinning sites. We assume that the wall thickness d is small compared to the distance L between pinning sites. From the symmetry of this configuration it is clear that the lowest energy will be achieved if the wall is situated midway between the two pinning sites, so as to equalize the volume occupied by the two spin orientations. Thus we may assume a harmonic restoring force opposing a displacement 5 of
232
W. BINDLOSS Wall thickness d
0
L
L/2
Coordinate
z perpendicular
to plane of wall
Fig. 8. Spin direction as a function of position between two pinning sites. The sublattice magnetizations are assumed to be pinned such that Q = 0 at z = 0 and @ = 42 at z = L.
the wall away from its equilibrium position at z = L/2. To determine the magnetic torque in the presence of domain wall displacements we will treat the idealized model shown in Fig. 9. We consider a cube with side L, such that the magnetization is pinned at Q = 0 on one side of the cube (z = 0) and at Q, = 90” on the opposite side of the cube (z = L). The displacement of the wall away from the mid-point is denoted by 5, and the field H,, rotates in the a-a plane. Let the restoring force per unit area of the wall be given by F = -kg. Then, if Magnetization
we neglect the volume occupied the energy is given by
E = $L”kt2 - &L2AxHo2[ (L/2 + 5) COS~@~
+ (L/2--<) The wall displacement
5 is given by
(@=
(6)
where the quantity y = kL has the dimensions of energy/volume. The torque per unit volume .
ls
pinned
T = ( AxH,,~) (l/L) sin 2$. (@=7r/2/
at z = L
wall z
pinned
sin‘QI+J. (5)
5 = (AxHo2/2y) L cos 2Q0
-Domain
Magnetization
by the wall,
0) at z = 0
Fig. 9. Idealized model of spin pinning used in calculating the magnetic torque for field directions lying in the a-a plane. The sublattice magnetizations are assumed to be pinned such that @ = 0 on one side of the cube and @ = 5712on the opposite side.
(7)
‘ap!s al!soddo aql uo Z/S = Q pue aqm aql JO ap!s auo uo 0 = @ JEIJJqms pauurd aq 01 pawnsse am suoyxz!~au%xu axlvIqns aqL .aue[d IS-J all2 u! Bu!l([ suoyamp play IOJ anbIo1 ~!]auWm aql Buyu[nym u! pasn %u!uu!d u!ds JO [apow pazgeap1 ‘01 %!g
0 !*
-sip e gl!M pawr3ossE up y_xorn aIq!slaAa_u! (0 I ) z (Z/7 - 2) z7Y4 + oBzu!S%7z”HxVf = 5’ aql wp %uytunsst? Kq seInwlo3 1wya~oayj aq1 uogEp_l olu! s!salawQ aDnpo.xy llew ah .khaua 30 aql Icq uay8 s! lli21aua [E?JOJ ay,~ *anblol SSOIaIq~S.IaAa.Lyaq$ Icq pa!wduroD%? s! SIIEM atjl 03 uognqyuo~ ou saywu uo$a~ syl snqL u!w~op aql 30 ~uaura~eIds!p aq$ JXX# aw ‘P-7)z7(Z/z0HXV)= 2 AquaA? s;r006 = Q, -!pu! E +!!d u! u~oys sdool s!saJalslCy aqL ljD!ljM II! UO!sal aqJ qJ!M paJE!DOSST &aUa *suoyEA.Iasqo Iwuawyadxa I.@M$UaUIaaJi% aw ‘ZvH Q “H pUt? IvH 4 3H JW$j%U!UInSS+’ uy s! s!y,z. *a0 001~ < OH SpIay 103 pamas *(z/7 - 2)y- = d Aq uaA$i s! aDJo %uIJ03sal -qo aq Quo p1noqs 0 = QI 1~ saA.uw anblo) ayl ~wj~ OS ‘2 Lq paJouap MOU s! [IEM ayl 30 ay) 30 adoIs arg 30 acwapuadap z”H aql snyl uoysod agJ_ ‘0 1 ‘%+Ju! UMO~Sso UO~yL&!LJUO~ ‘a0 ()()[p < OH Aq UaAp s! 0 = ‘@ UayM Z/7 = 2 arg faAoqE passnDs!p w?ql se Iapow paz!pap! )uacuaDEIds!p wnur~xt?uI sl! 01 IIE?M ayl ahup auw au[l awns% ah ~~EwCI:,aql30 auald a-3 01 pa_ynbaJ pIag ayl .eur@la 0~91 = h pug agl u! pawoJ s! pIag ayl YD~M u! as23 ayl -lo3 aM qD!qM way ‘s ‘%A 30 BJEp ayl q,iM waw uoyo~ ~gau?iw_u ayl ssn3s;rp MOUj@ys aM -aa.& ur sy 1p-tsa.Isy~ ‘+OH01 @uoyodold s! *anblol olaz 30 $u!od ,&p = %$I1~ saAm3 anblol aqI 30 adoIs agl snql (8)
(6)
“
uwjqo aM (L) uoynba %!sn ‘olaz aq aDlo lau ayl wy~ uoyrpuo3 aql hq pauywalap s! 11~~ aql30 uop +od ayL *uo~~our30 uogDa_np aql 01 alisoddo can )!un lad do = d aDlo3 ‘e aDuauadxa II’M uo!loW u! jI’?M ‘1? snq.l $pld= Mp hq uaAp s! ea.re l!un 30 IIEMa 30 f2p wawaDyd
“@ti u!s [ht/z(zoHXV)]
= L
uyqo aM uaqL ‘(L) uogenba o)u! (9) uogenba ajnlgsqns aM .sp = OQ,01 asol:, suog -Da.np pIag ~03 anb.Iol arl, au!walap 0~ ‘0 = L put? 0 = fi u!wqo aM pu?i? luapi?A!nba IlIjWga8 -laua a32 suoywa!~o u!ds 0~1 aql .sp = O@ uoyal!p pIay ayl god *,M = OQ, 103 uS!s 30 @sJaAal e ql!~ lnq ‘(p) uoynba 0~ Iwyuap! SI (L) uoyznba liq uaA!% anblol ayl suo!lDanp play asaql JOT *72/i- = 2 aAw_jaM .()6 = O@ “03 f7Z/I = 2 alleq aN\ .()fj[ 2: O@pul? .() = OQ 103 uaqL ‘~~81= Oa pw ‘,M iT OQ, ‘0 = OQ, suogDa.np play ~03 Z/7 = (21 luatua~t?Ids!p aIq!ssod uInur~xEuI sl! 01 uaA!.Ip aq II!M I~?M aqi (I < X/zo~X~) spiay a8-w 43uapgns -w
W. BINDLOSS
234
and the torque
per unit volume is given by
7 = ( AxHo2/2) (z/L) sin 28,,.
(11)
Since significant hysteresis is observed experimentally for all fields up to 18 kOe, we must include the force per unit area F = &p associated with the irreversible loss of energy in the presence of wall motion. Setting the net force on the wall equal to zero we find z/L = THI-
(b~H,,~/y) sin2&,-+ 2P/r]
(12)
where the plus and minus signs apply to the cases of decreasing and increasing wall displacement respectively. Using p = 200 erg/cm3 we find 201-y = 0.24, which indicates that hysteresis effects should be very significant. When the field is directed along the c-axis (0, = 0)) the two spin orientations (Fig. 10) are energetically equivalent and the wall position z tends toward z = L/2. If the field direction is rotated away from the c-axis, the region within which 8 = 90” becomes energetically favorable and the wall position tends toward z = 0, in which case the torque tends toward zero. Thus in large fields the torque will be zero except for field directions in the neighborhood of B0= 0 and 8, = 180”. Theoretical torque curves obtained using equations (11) and (12) are compared with experiment in Fig. 6. The curves are drawn for the case of increasing 13~.The magnitude and field dependence of the slope of the torque curves at I$, = 0 (dr/d&, 0~Ho2) are found to be in good agreement with the theoretical result. We do not expect to obtain agreement between the theoretical and experimental values for either the maximum torque or the magnitude of the torque for field directions far away from the c-axis, because these quantities depend critically on the amount of irreversible energy loss associated with domain wall displacements. Domain wall restoring force We will now examine in detail the origins
of the domain wall restoring force for the configuration shown in Fig. 8. We define the
position of the wall as the point at which the spin direction a,(z) is equal to 45”. The displacement of the wall away from its equilibrium position is denoted by 4, and the wall thickness is defined by d = r/2 [ (dz/d@),=,,,], as shown in Fig. 8. Let us denote the energy per unit area of the wall, for zero wall displacement, by E,(L). This energy will decrease with increasing L, approaching a constant value as L + 03. Now consider a small displacement 5 of the wall away from its equilibrium position. This displacement will raise the wallenergy to a value =itE,(L-t25)
E,(L)
+B&-225).
(13)
Equation (13) is obtained by assuming that the range of exchange interactions is small compared to the wall thickness do. The first and second terms are the energies associated with the regions 0 < z < L/2+5 and L/2+ s < z < L respectively. We have previously defined the restoring force per unit area of the wall by F =-k{, so that E,(L) -E,(L) = $k(L)t2. Using a Taylor expansion of E,(L), we find k(L)
= 4d2E,(L)/dL2.
(14)
We will now obtain an expression for the energy E,,(L) per unit wall area following the theory of ferromagnetic domain walls [lo]. The exchange-energy density associated with the spin rotation has the form A(d@/d~)~. Denoting the anisotropy-energy density by g(a), the wall energy is given by E,(L)
= j;
[A(z)‘+,(@)]dz
(15)
where g(
(16)
If we consider the particular case in which the wall is perpendicular to the c-axis and include only exchange interactions between spins lying within neighboring a-a planes, then we find A = a(gpdhM)
(c/a”).
(17)
THE MAGNETIC
STRUCTURE
By minimizing equation (15) with respect to small variations of the function a(z) , subject to the conditions that @( 0) = 0 and @(L) = 7r/2, one obtains the Euler equation
_2A@Q+dgW -dz2
_
o
dz
’
(18)
Multiplying equation (18) by (d@/dz)dz, and integrating from zero to z, we find
=A($)z-B(L)
g(Q)
(19)
OF AuMn
To determine the constant B(L) rewrite equation (24) in the form
(20) In the limit as L --, q, B(L) + 0. Denoting the wall thickness in this limit by d,,, and using equation (19) one finds do = (r/4) (A/K,) 1’2.
(21)
Equation ( 19) implies that
dz=
(
A
112
>
g(@)+B(L)
(22) da
hence z(Q) = A”2 f;
[g(Q) +B(L)]-1’2d@
(23)
where the constant of integration is zero because z = 0 when Q>= 0. Noting that z(7r/2) = L, one finds L = Al’2 0n’2[g(m) +B(L)]-“2dQ. I
(24)
Substituting equations (19) and (22) ‘into equation (27), and making use of equation (24), one finds E,(L) = 2Al12J;” [g(a) +B(L)]“2d@ -LB(L).
(25)
Then, using equations (24) and (25), one finds dE,(L)/dL=-B(L). Thus the constant given by
k(L)
(26)
defined by (14) is
k(L) = -4dB(L)/dL.
(27)
let us
2L (K,/A ) l/2 = &j: [ sin28 + l2] -l/zdO = 1” [sin28+ e2]-1’2d& I
(28)
where 0 = 2+ and l2 = B (L)/4K,. This integration can be divided into two parts such that 2L(K,/A) 1’2= I[’ sin28 + E2]-1/2dfI 0 +Jgi'12
where
235
[
sin20 + l2] -1/2d& (29)
We are interested in the case where L is large compared to the wall thickness do. Since E + 0 in this case we can evaluate equation (29) in the limit that l2 + a2 4 1, finding 2L (K,/A ) 1/2= In (4/e). Thus we obtain B(L) = 64K, exp [-4L(K2/A)1’2].
(30)
The quantity y(L) = LK (L) has been determined from the data of Fig’s. (4) and (5). Using equations (2 l), (27) and (30), one finds y(L) = 2%rK, (L/d,) e-flLldo).
(31)
Using the experimental value y = 1640 erg/ cm3, we can determine the ratio L/d, for various values of K,. For various values of K, between lo4 erg/cm3 and lo6 erg/cm3 (240 Oe < HA2 < 24,000 Oe) we find that 3.1 < L/d, < 4.7. Thus for any reasonable value of the anisotropy energy K, we find Lid, = 4. Then l2 = 6 X 10e5, so that the approximations used in evaluating the integral (29) are justified. We can make a rough estimate of the distance L between pinning sites by using equation (21) for the wall thickness do, and equation (27) for the exchange-energy constant A. The constant A can be estimated from the susceptibility [3] by assuming that x( TN) = x, = l/h. Then for K, = IO4erg/cm3 we find do = 1670 A; for K, = 105erg/cm3 we find do = 520 A. Thus, since L/d, = 4, we conclude that the distance between pinning sites is of the order of 2000-8000 A. There may exist several different mechan-
236
W. BINDLOSS
isms which could account for the localized pinning of the sublattice magnetizations. One such mechanism would be the presence of local crystalline anisotropy fields due to lattice distortions in the immediate vicinity of a dislocation. A mean separation between dislocations of the order of 2000-8000 A is possible in a strained crystal[ I 11. This is a reasonable possibility for the case discussed here since considerable strain is introduced by the cubic to tetragonal transformation. 5. CONCLUSIONS We find that the torsion measurements presented here are in qualitative agreement with a theoretical model which includes the presence of domain walls. The measurements show that the sublattice magnetizations lie along an a-axis, and indicate that the anisotropy field H.,,2 is in excess of 120 Oe. This anisotropy field is sufficient to prevent the rotation of the magnetization in the a-a plane in external fields up to 30 kOe. The approximate value 9.5 X IO-" which we find for the room temperature susceptibility difference xL -x is seen, by comparison with the powder susceptibility(31, to be a reasonable value. We also find that the direction of the sublattice magnetization is shifted successively from one a-axis to another as the external field H,, is rotated in the a-a plane, and account for this behavior in terms of domain-wall displacements. The H, dependence of the torsion curves indicates that the domain walls experience restoring forces tending to equalize the volume occupied by the two spin orientations. This behavior is explained on the basis of a model in which the magnetization is pinned along one or the other a-axis, with equal probability, at random sites throughout the crystal. For this model a distance between pinning sites of the order of 2000-8000 A is found to be consistent with the experimental results. Thus local anisotropy fields associated with dislocations are a possible mechanism for the spin-pinning. The idealized model of the spin-pinning
treated here may be a considerable oversimplification of the true state of the crystal, since the magnetization direction may be pinned other than along an a-axis at a certain percentage of the pinning sites. Thus the numerical values obtained for the parameters associated with domain wall displacements should be considered only as crude estimates. As is noted in Section 1, any substantial pinning of the magnetization direction out of the a-a plane would also effect the intensity of magnetic reflections observed in neutron diffraction measurements. A(.~no~~~led~c~rn~fIls- My gratitude is extended to Professor A. M. Portis for the critical guidance he has provided throughout this work. I am indebted to Professor S. F. Ravitz for his many helpful suggestions during the sample preparation. I am particularly grateful to Dr. I.. B. Welsh for his assistance in the sample preparation and in the initial torsion measurements. Thanks are also due to Professor J. W. Garland for several helpful discussions. During the sample preparation this research benefited from the partial support of the National Science Foundation. REFERENCES
I. 2. 3.
4. 5. 6.
BACON G. E., Proc. phys. Sot. 79,938 (1961). SMITH J. H. and GAUNT P., Acta Mr~ull. 9, 819 (1961). GIANSOLDATI A., J. Phys. Radium. Paris 16. 342 (1955): GlANSOLDATl A., LINDE J. 0. and BOREI.IUSG.,J.Ph.vs.Chcm.So/id.~11,46(19S9). GIANSOLDATI A. and LINDE J. 0.. J. Phys. Rrrdium. Paris 16,341 (1955). BINDLOSS W.. Phy.;. Rev. In press. HO J. C. and BINDLOSS W.. Physics Lerlers 20. 459 (1966). The value for the Fermi level density of states is incorrectly reported. The correct value is 0445 el./eV-atom; thus the ratio of the measured
density of states to the free-electron density of states is I .49. 7. FRIEDEL J., Suppl. Nuouo Cim. 7, 287 (1958); BI.ANDIN A. and FRIEDEL J., J. Phps. Rudium. Paris 20. 160 (1959); ANDERSON P. W. Phps. Rec. P. A., Phys. Rev. 124. 1030 J. R. and MATTIS D.,,C.. (1965); KJOLLERSTROM D. J. and SCHRIEFFER J. R.,
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148,665(1966).
A. M. and WI’I”f G. L.. Phy.5. Rec. 132, 144 (1963). H., Phys. Rec. 47,947 (1935). 9. MEULl.ER C., Reo. mod. Phys. 21, 541 (1949): IO. KIT-TEL CHIKAZUMI S., Physics of Mqnerism. Wiley, New York ( 1964). C., Introduckm to Solid SIUIP Physics. Il. KIT-TEL Wiley, New York (1956).
8. LEE K.. PORTIS