- Email: [email protected]
Magnetocrystalline symmetry, spin statics and dynamics in high-temperature superconductors
Physica C 156 ( 1988) 667-678 North-Holland, Amsterdam
M A G N E T O C R Y S T A I J J N E SYMMETRY, S P I N STATICS AND D Y N A M I C S I N H I G H TEMPERATURE SUPERCONDUCTORS V.G. BAR'YAKHTAR *, V.M. LOKTEV and D.A. YABLONSKII ** Institute for Theoretical Physics, Kiev-130, USSR
Received i 8 August 1988 Manuscript received in final form 29 September 1988
The symmetry analysis of the magnetic structures of La2CuO, and YBazCu306+6 (6 ~ 0.4) antiferromagneticoxides has been carried out. The spin configurations and spin-reorientation transition fields have been found. The frequencies of uniform spin oscillations and the external field dependence of their polarization properties have been calculated. It has been shown that the spin excitation branches may include exchange modes among which some dipole-activemodes can be present. The spectrum of the inelastic light scatteringaccompanied with the excitation of spin and libration (tilting) freedom degreeshas been discussed.
1. Introduction
The magnetism of high-temperature superconductors ( H T S C ) is widely recognized now as one of the most important and interesting properties. Moreover, many investigators (e.g. refs. [ I - 5 ] ) believe that it is the magnetic properties that play a key part in the pairing mechanism. Up till now the antiferromagnetic (AFM) spin ordering has been established experimentally in two classes of the HTSC. Thus non-alloyed La2CuO4 exhibits the N~el spin ordering S = ½ of Cu 2+ ions at TN,,~250 K [ 6 ]; the same AFM ordering was also revealed in YBa2Cu306 +~ at TN > 500 K for 6 = 0.0 and TN ~, 400 K for 6 = 0.15 [7 ]. In both cases the AFM is positively fixed only in dielectric phases of the HTSC. In a metallic state, however, the long-range N~'el ordering is disturbed. Nevertheless, highly developed fluctuations persist even at room temperature [ 8 ]. This is also confirmed by the data on the nonelastic light scattering accompanied by the creation of spin excitation pairs both in La2_xCuO4 (x # 0) [ 9 ] and in YBa2Cu306 +6 (0.0 < 6 < 0.9) [ 10 ]. The position of the line scattering indicates that magnetic correlations in HTSC are characterized by the high value of the exchange energy J ~ 103 c m - ~ (most probably attributed to the indirect CuZ+-Cu 2+ interaction in CuO2 planes characteristic of the copper-containing HTSC). As for the theoretical papers (see also refs. [ l I - 13 ] ) dealing with the HTSC magnetism to this or that extent, they mostly concentrate their attention on the substantiation of some "magnetic model" of superconductivity. The aim of this paper, however, is to study the magnetic (quite standard ) peculiarities of the HTSC magnetism by an example of lanthanum and yttrium systems. Thus we investigate a symmetrical connection between the HTSC crystalline structure and spin ordering type, find the critical fields of spin-orientational phase transitions, calculate the values of resonance frequencies as well as the polarization and intensity of their excitation by external magnetic ( A F M R ) and electric (AFER) fields.
* Institute of Metal Physics, Kiev, USSR. **Physical and Technical Institute, Donetsk, USSR. 0921-4534/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division )
V.G. Bar°yakhtar et al. I Magnetocrystalline symmetry, spin statics and dynamics
668
2. La2Cu04 crystal
2.1. Spin Hamiltonian structure The high-temperature modification of La2CuO4 belongs to the space group I 4 / m m m ( D ~ ) and is a tetragonal lattice with one "octahedric molecule" of CuO6 in a unit cell (fig. 1 ). At T~450 K [ 14] (the particular value of the temperature depends both on the content of the alloying impurities and on the oxygen deficiency) a phase transition into a low-symmetry rhombic phase with the space group C,~(D,~hs ) occurs (fig. 2). This transition is due to a slight tilting of the CuO6 octahedra extended along the C4-axis about one of the horizontal axes (either (110)[IOX or (I]0)[IOY) and results in the cell doubling. As is seen from fig. 2, the lattice spacings in the basal plane change since the octahedra of different planes preserve their translational invariance. The latter, however, can be disturbed after the phase transition into such an AFM state where the spin directions in the translationaUy equivalent sites do not coincide. So one can assume that the spin structure of La2CuO4 is described not with two but, at least, with four magnetic sublattices with magnetizations
ILj=lzBgSv~o, j = l , 2, 3 , 4 ,
(1)
where Sj is the spin of the copper ion from the jth sublattice,/tB is the Bohr magneton, g is the g-factor, Vo is the cell volume. We do not intend here to investigate all possible magnetic structures compatible with the D ~ group. We restrict ourselves to the discussion of the most probable ones on the basis of the experimental data. First of all, one should pay attention to the importance of the intralayer AFM exchange which makes it possible to
I Z {IC~
|,\ |~ /
3
\/ ~
/ I I
t \I\i
~
lt/I
x fl .(~o) Fig. 1. The unit cell of La2CuO, in the tetra-phase ( l al = I b I );
,
/ ~ /
/
/
x
t Fig. 2. The unit cell of La2CuO4 in o r t h ~ p h a s e ; ~ - 1 ~ 3 , 2 ~ 4 ; ( ~ ) helical axis.
V.GI Bar "yakhtar ~t at: / M a g n e t o c r y s t a l l i n e
symmetry,
spin statics a ~ dynamics
669
Table I Classification of irreducible vectors in the irreducible representations of the D l~ group. 1
2x
2iv
2,z
t2x
/2t~"
t2tz
+
+
+
+
+
+
+
+
+
--
+
--
+
--
+
--
+
+
--
--
+
+
--
+
--
--
+
+
--
_
+
+
+
+
+
--
+
--
--
+
--
+
L2v
Lsz
+
+
--
_
_
_
+
+
+
--
_
+
--
+
+
--
L,_x L,.z
Lsr
F
L lx
Fr
_
Fx
Liz
+
Fz
Ltv
L~ L3~
L2
Ls
introduce "intralayer" vectors of ferromagnetism mj and antiferromagnetism 1~(j= 1,2) according to the definitions
!'1 +~2 =2Momt;
Its +p4 =2Morn2;
m} +! 3 = 1;
1'1-~2=2Mo1~;
lts-la4=2Mo12;
Me= #ag s . Vo
(2)
Moreover, measurements [ 6 ] show that the CuO2-planes are antiferromagneticaUy ordered with the "'easy plane" anisotropy (i.e. when spins are located within the XY plane) *. Then the standard symmetry analysis (see table I) allows us to write the spin Hamiltonian in the form 1 2 2 2Me ,,'~=½H, E mJ +H'd,12 +Hv E (ljzrnjv-ljvmjz) j=l
j=l
2
+ H~ ( mlzl2r-m~vl2z + m2zl, r-m2rllz) + ½ ~ ( Hazl~z-HArl~v- 2Hmj ) ,
(3)
j=l
where He is the constant of the intralayer (intraplane) exchange field and H i ( ~ He) is the constant of interlayer (interplane) exchange field; Hv and H ~ are the similar Dzyaloshinskii fields; HAZ and liar are the "easy plane" and rhombic anisotropy fields: H is the external magnetic field; mj, ij and Me are determined in eqs. (2) and (1). While writing ~ we omit the interplan¢ anisotropy field, the anisotropy fields associated with the ferromagnetism vectors mj (ifH~KH,, then mj~lj) and so on and restrict ourselves to the most essential terms. Pay attention to the fact that in the rhombic phase decompensation of the interlayer exchange (H" # 0) takes place and antisymmetric inter- and intralayer exchange interactions appear. It has been already pointed out recently [ 15 ] that the Dzyaloshinskii interaction can, in principle, appear in La2CuO4. These peculiarities of the low-symmetry phase of La2CuO4 have quite a clear physical nature. In fact, the appearance of the interlayer exchange is entirely due to the fact that in the process of the tetragonal-orthorhombic transition the shift of atoms along the (110) and ( 1 i 0 ) axes of the initial lattice is not the same and as a result, the 1-3 distance differs from the 2-3 distance (fig. 1 ). For Hi the rotations ofoctahedra may not play an essential role (though, as will be shown below this will not always be the case). However, the fact that spins go beyond the planes is, to a large extent, caused just by the change in the orientation of the crystalline field acting on them. And it should be noted that in the orthorhombic phase the axes of this field do not coincide with the lattice axes. The ° This experimental fact would seem to be a bit surprising, because for a copper ion in an elongated octahedron, gm> g~, as a rule, and this should result in a "'light axis" structure. Dipole and exchange-relativistic interactions seem to play a major role in this situation. And although we avoid discussing this point (magnetic anisotropy sign ) below and focus on the structures favoured by experiment [ 6,7 ], the other ease can easily be treated in a similar manner.
670
V.G.Bar'yakhtaret al. / Magnetocrystallinesymmetry, spinstaticsand dynamics
combined action of the AFM exchange interaction and local (single-ion) field of ligands can be expressed in the form of operator (3) based on the symmetry considerations.
2.2. Spin-reorientational transitions The behaviour of a crystal in the external field HIIOZ was studied experimentally [ 15b ] and a jump of total magnetization in a field H~. 4-5 T was found, which was attributed to the reversal of the magnetization vector in one of the layers. We consider this transition in detail on the basis o f Hamiltonian (3). First of all we assume that the interlayer interaction has the same AFM sign as the intralayer one (otherwise it would be the case of a weak ferromagnetism). Provided that 1
HAy+ -~. (H,, +H'._.)2>0,
(4)
the ground state of the magnetic subsystem can be characterized by the following vectors
f,,=-f~
l; ,~,z=-,h2z= H~+n'.~ ~ 1, He
(5)
where ,,0,, denotes that the vectors belong to the ground state (fig. 3). It should be noted that ~olation of condition (4) l e ~ s to a change in the structure, so projections l~x= -12x become non-zero. In this c a ~ ms-O hut it is not confirmed experimentally and the ip'ound state La2CuO4 is d¢~cTibed by eqs. (5). It follows from eq. (4) that the Dzyaloshinsldi interaction can quite well l ~ ' f o ~ the part of the rhombic anisotropy. In the external field HHOZ the crystal magnetization changes so that H mjz=rhjx+~,
j=l,2.
(6)
It is seen that in accordance with eq. (6) the crystal magnetization M z = 2MoY2=,msz grows linearly up to a field Sg HZ= Hv + H ~ He
(7)
(in fact a similar expression is given in ref. [ 15b ] ). At that point the orientation of vector M2z occurs so that (see eq. 6) m , z = m2z= rh,z +H/H~. The crystal magnetization jump is
~z
= 4Mo
Hv + H~
Ho
The static susceptibility, however, preserves its value Z~ =4Mo/He (see eq. 6) after the transition which is confirmed by experiment [ 15b ]. In the course of this transition not only projection m2z changes its direction but projection/2 ~ as well (fig. 3). The similar spin-reorientational transitions are typical of the quasi-two-dimensional four-sublattice AFMs of the (NH3) 2(CH2) ,MnC14-type. In these AFMs, however, reorientation of one of the vectors is caused by the field directed not across but along the "easy" axis [ 16 ].
~:o
z
~
~<~
~ "
H~H~z.
-
Fi8. 3. Spin-reorientationof La2CuO4whenH{IOZ.
V.G.Bar'yakhtaret:al. / Magnetocrystallinesymmetry,spinStaticsand dynamics
671
Since this transition is the first-order transition induced by the external field it should be accompanied by the appearance of an intermediate state which is actually a domain structure composed of an initial and reversed phases. The field width of the intermediate state stability range is
AHz = 16xNz Morhlz (Nz is the demagnetizing factor along the OZ axis). A more standard situation arises in a field H[IO Y. Up to field H ~ = [HA rile (He + H~ ) 2] t / 2 (cf. eq. 7 ) the magnetization grows very slowly and the longitudinal susceptibility is expressed by
(n ~ + n ~ )2 Xu =Z± (nAz+2I~e)ne ~ Z . . Then the system undergoes the conventional spin-flop transition into the state l~x= -12x; mix=O; mjr=H/ He; in this case the width of the intermediate state region is AHy= 16x3VrMo(H~/H,).
2.3. Antiferromagnetic resonance The AFMR frequencies and the magnetic susceptibility tensor can be conveniently calculated using the method proposed in ref. [ 17 ]. Let us consider the irreducible vectors
F=½(ml+m2);
Ll=½(l,+12);
L2=½(ml--m2);
L3=½(l,-12),
(8)
whose classification by the irreducible representations of group D ~ is given in table I. In addition to the rotary elements the table includes the transition t = ½(a+b+c) since the unit cell of La2CuO4 is twice as big as the crystallographic one. Thus the group action of i is determined by commutation 1~3, 2 ~ 4 as well as iml = m2, ilt =12. As is seen from eqs. (5) and (8), in the absence of field, the ground state is characterized by the vectors/~2z=rhm, L~3y~ 1. Then using the equations of motion for vectors (8) [ 17] we find that the system described by operator (3) has four uniform oscillations, two of which belong to the acoustic type (A 1, A2) and two others to the exchange one (E 1, E2). The dynamic variables of the corresponding modes are
AI:{Fx;O;L2y;L3z};
A2:{Fz;LIy;O;L3x};
EI:{O, Ltx;L2z;L3y};
E2:{Fr;Ltz;L2x;O}.
(9)
The resonant frequencies and the components of the high frequency magnetic susceptibility tensor are determined by the equations AI: k-~--~: =(HAz+HAy) A2: \ ~~A2 -~/
=HArHe+(H~+H~)2;
~
.J'~"AZn,;
Xxx(09) =X~ ~ l _092,
~X2 . Xzz(09)=X± Q22_092,
(ll)
(~'~El~2
El: \~--~/ =(2H', + HAr)He + (H~ + H~ )2; X~a(09)=O ;
(~E2~ 2
E2: \ - ~ /
= (2H'~ + HAz + ttAy)He +4H~(H~ + H~ ); ;eYY(09)=;e±
(10)
(12)
[lz~g(H~ + H~ ) ]2 ~222 --09 2
~Zu~2_092' (13)
where Z~,n are determined above. It is quite probable that interlayer exchange interaction is less than the uniaxial anisotropy ttAZ. In this case there is rather an abnormal situation where ~E~ <~AJ. This fact must be by all means taken into consideration at treating the results of the resonance studies of HTSC. An external magnetic field changes the selection rules of the resonant mode excitation (9). The symmetry
V.G. Bar'yakhtar et al. / Magnetocrystalline symmetry, spin statics and dynamics
672
Table II Tensor ~(t~) components in a magnetic field. Mode
t1= 0
H~OX
HHO Y
HIIOZ
A1
XX
XX
XX, XZ, Z Z
XX, XY, Y Y
YY, YZ, Z Z
XX, XZ, Z Z
ZZ
XX
YY
ZZ
YY, YZ, Z Z
YY
XX, XY, Y Y
(Xxx(co)) A2
ZZ
(Zzz(O))) E1
-
E2
YY
(Xrr(w)) analysis makes it possible to predict the polarization properties of the high frequency magnetic susceptibility tensor ~(to) at various directions of an external constant magnetic field (see table II).
2.4. Magnetostructure interactions It can be shown that interlayer and antisymmetric exchange interactions are possible only in the rhombic phase. We remind here that the transition* from "tetragonal D~[ phase to the rhombic D ~ one is due to rotation of the extended oxygen octahedra, surrounding the copper ions, about the OX or OY axes chosen along the basic square diagonals (figs. 1 and 2). As we are interested in the connection of the spin moments with the change in the crystalline structure which is determinexi only by the directions of the octahedra main axes, we introduce vectors-directions taflj= 1..... 4) whose directions coincide with those of the respective main axes. It can be easily shown that transverse components tojx, tOjr of these vectors determine the deviation of the octahedra axes from the crystal tetragonal axis and from the two-dimensional E~-representation of point group Ddh which describes the local symmetry of Cu 2+ ions in the tetra-phase. And in this case the order parameter describing the structural phase transition D4~-.DIS (figs. 1 and 2) is represented by a two-component value {r/,; ~/2} in which ~1 m~'('OI } ' - - ('/')2 }" "Jff0)3)'--(X~4}';
~2 ~-~O.)lrt'-- O.J2X--(~)3X"~'O.)4X.
(14)
The transformation properties of values ( 8 ) and (14) are determined by tables III, IV and V. In this case table IV presents transformations of vectors (8) neglecting the vector origin (i.e. only commutations are taken into account). Thus under the action of rotations component L 2 y is transformed as coordinate Y and under the action of commutations it is transformed as is shown in tables III and IV. Knowing the transformations of magnetic vectors and order parameter of the structural transition one can write the Hamiltonian of the spinspin and spin-libron interactions provided that small swingings of the octahedra in the vicinity of the ori* The symmetry analysis of the structural transition irrespective of the magnetic subsystem was considered in ref. [ 18 ].
Table IV. Table Ill,
l
i~
~
i,t~
+
+
+
+
+ + +
-+
+ -
-+
L3 Li
-
L,
F
1/., ~h
1
4z
Table V.
F Li L2 L3
F L3 L2 L~
1
2x
2r
~h ~/2
th -~/2
-rh
4z
~/2
-~h
rh
V.G. Bar'yakhtar ~ ~ ~¢Magnetoerystattmex3,mmctry, xpin staticsand dynamics
6 73
entationally ordered position belong to the libron-type excitations. The exchange part of the Hamiltonian has the form
~x =~,F2 +6L,(L~ + L~ ) + 6 J . ~,+al ( L ~ + L ~ ) + a 2 ( L ~ +L~ +7,: ) (tSe, 6~, A t and A2 are the corresponding constants). It implies that (see eqs. 8 and 2 ) that the transition results not only in some renormalization of the intralayer exchange fields but also in the interlayer A~21tl2 (or A~llt 12 depending on the component determining the structural domain by which the tetragonal-orthorhombic transition is realized). In the latter case the constant of the exchange field H~ appears to be small due to the obvious proportionality H~ ~ r/2. In a similar way one can draw the connection between the deformations and the antisymmetric exchange-relativistic interaction. In the general case they have the form a~fSv= dn [ t/t ( FzLI r - F r L 1 z ) + tl2( F z L 3 x - F x L 3 z ) ] + ~,[ ~l ( L 2 z L 3 r - L 2 v L 3 z ) + tl2( L t x L 2 z - LizL2x) ] • This expression, in its turn, indicates that the Dzyaloshinskii fields in eq. (3) Hv, H~ ~ ~b, i.e. they are actually induced by rhombic deformation. The experiments on the magnetization and the magnitude [also proportional to r/j (see eq. (7) ] of field H z of the magnetic sublattiee reversal at HHOZ quite positively confirm that the values H~ and Hv, H ~ are very small.
2. 5. Inelastic light scattering As far as we know, the HTSC AFMR spectra have not been measured. However, there are some data on the HTSC spin excitations. Thus Lyons et al. studied the inelastic light scattering spectrum of La2CuO4 [ 9,10 ]. In addition to the main two-magnon band (the spectral position of its maximum made it possible to estimate the magnitude of the above constant J) this spectrum possesses rather an intensive line at the half-frequency which was attributed to the defect mode excitation. But due to distinct polarization of this line (its scattering tensor is diagonal in the basal plane) this interpretation cannot be regarded as a final one. On the basis of the magnon-libron interaction one can propose another interpretation, namely, the observed absorption is really a two-particle one but it also includes the spin excitation and a quantum of soft libration oscillation (tilting mode) of octahedra. (Judging upon the structural transition [ 19 ] the frequency of this oscillation at the edge of the Brillouin zone is ~ 102 cm-t ,,: ~2j(kB)~ 10 3 cm -t. ) Indeed, a simple calculation shows (see tables IIIV) that the operator of the magnon-libron light scattering with polarization Eo, propagating along the OZaxis is of the form
~c---[ (E,o)x : + (Eo~)r 2 ] ( ~ l L 2 z L 3 r + ~ 2 L , x L 2 z ) ,
(15)
in which, depending on a domain, either L3v or Ltx projections should be substituted by L3y and Ltx, and tb and L2z are the dynamic variables. As is seen from eq. ( 15 ), in this process the exchange mode of frequency (12), which is passive in the AFMR, is excited together with the libron. The obtained (diagonal) symmetry of the magnon-libron scattering process fully agrees with the observed one; it is easy to check that such a scattering involving the rotation of the light-wave polarization plane is impossible, and this is also confirmed by the experiment [ 9,10 ].
3. YP,a2Cu~O~,4 crystal
3.1. Structure and spin-Hamiltonian Though crystalline structures of YBa2Cu3Or+ 6 crystal (fig. 4) and the lanthanum system are different, they have one feature in common, namely they possess the CuO2 planes with Cu 2+ ions (here Cu2). At J~<0.4 the Cul ions (CuO chains) predominantly find themselves in the Cu + state (3d ~°, S = 0 ) and do not practically
674
V.G. Bar'yakhtar et al. / Magnetocrystalline symmetry, spin staZicsand dynamics
2z E
7q-'f o- Caz
0-0
//
I C
[
2¥
jjJ .... Fig. 4. The unit cell of YBa2Cu306.a (only magnetically ordered planes are shown ); i - 1~2, 3=4.
contribute to the magnetism of Y B a 2 C u 3 0 6 + a [7 ]. Then, from fig. 4, where cations Cu2 and anions 0 2- are shown, it is seen that the unit cell of this compound belongs to rhombic syngony Pm=m(D~h) and the magnetic cell also contains four spins which can be conveniently used for making up the same "intralayer" combinations according to eq. (2). The above arguments for the lanthanum ceramics allow us to present the phenomenological Hamiltonian of the yttrium system in the form of two summands: homogeneous 1
2
2
2"~o ~Ss=½He ~ m2~ + H'~ltl2 + ½ ~ (HAzl~z -- HAzl2x-- 2Hmj) j=t
(16)
j=l
and inhomogeneous 2Mo ~inho==I~,
(
012 12 01' ~ + I7', ( m , Ore2 1,-~ -- OZ] \ "~
-/'/vxj=,~ ( - 1 )
m2 -~~' ) - -
~ljx ox-lJz OX]-H~vj=,
) ~ljr oy-ljz
,
(17)
in which the meaning of all constants is clear from the definition, but still we note that in eq. (17) values//e and//~ are of the exchange and/'?~,x and/'/~v of the exchange-relativistic origin. Even simple comparison of the latter expressions with Hamiltonian (3) shows that in the yttrium compounds there is no Dzyaloshinskii interaction * and in the initial structure the interlayer exchange Hi is non-zero. On the other hand, absence * Strictly speaking, this conclusion applies to the lattice shown in fig, 4 which, among other elements of symmetry, exhibits the translation along the bHOY axis. If, as is reported in ref. [20], even at small J the YBa2Cu306.a system experiences doubling (or even linger increase) of the period along the same axis, there appears a symmetry possibility for an antisymmetry interaction similar to the case of La2CuO4, and only experimental study can give an answer as to an existence of the weak intraplane moment.
V.G. Bar'yakhtar et al, / Magnetocrystalline symmetry, spin statics and dynamics
675
of the inversion centre at the Cu 2+ site makes the non-uniform interaction Hamiltonian (17) non-zero which, in its turn, causes and describes the long-period helicoidal incommensurate structure under definite conditions imposed on the values of respective constants [ 21 ]. Regarding HAz (it is shown experimentally [7] ) and HAx (for definiteness) as positive it is easy to find the ground state of the magnetic subsystem described by operator (16). It is in compliance with projections [ix = - 12x= 1 at Hi > 0, or/~ x = / ~ x = 1 at Hi < 0; the other projections of antiferromagnetic and ferromagnetic vectors do not exist. In this case the spin-flop transition in the magnetic subsystem will take place at H[IOX in field H x = ~/HAx(He +H',). At HAx< 0 the O Y-axis becomes the "easy" axis and the physical picture remains the same (if X--. Y).
3.2. Antiferromagnetic and antiferroelectric resonances Similarly to the above case, it is convenient to calculate the resonance absorption frequencies and the susceptibility tensors using vectors (8). Their commutative symmetry is defined by the following elements simple rotations 2x.y.z, inversion i, translation i-- 1 ~ 2 , 3 ~ 4 and product t. i. Table Vl presents the total classification of the irreducible magnetic vectors over the crystallographic group representations of orthorhombic YBa2Cu306+6. It is seen from table Vl that in case of the antiferromagnetic interplane exchange the phase with /-:3x-- 1 serves as the ground state and the dynamic variables resulting from the equations of motion are
AI:{Fz;O;O;L3r};
EI:{O;L,z;L2r;O};
A2: {Fr; 0;0; L3r};
E2:{O, Ltr;L2z;O}.
(18)
Difference between the number of dynamic variables ( 18 ) and (9) is due to the non-collinear character of the magnetic structure of La2CuO4. Another and more important difference of YBa2Cu306+6 from La2CuO4 is the possibility to observe, in the former, the antiferrodielectric resonance (AFER) or excitation of spin vibrations by the light wave electric field. Indeed, as is seen from table VI, vectors Lt and L3 are odd with respect to inversion [22,23 ] then it follows that their respective vibrations (here E 1 and E2) are electric-dipole active and characterized by tensor 6 (m) of electric (not magnetic) high-frequency susceptibility. For H = 0 the AFMR
Table
VI
Classification 1
2x
of irreducible 2v
I
2z
vectors
i2x
in the irreducible
|2~.
i2z
i
representations
&~.
/2,
of the Pmmm
/2~.
t
t2 +
group. v
i'2r
hz
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
--
+
--
+
--
+
--
+
--
+
--
+
-
+
+
--
_
+
+
--
_
+
+
--
_
+
+
--
--
Fy
+
+
--
--
+
+
--
--
+
+
-
-
+
Fz
B
B
+
+
+
+
--
+
--
+
+
--
+
--
-
+
-
+
L2x
Y
+
+
+
+
--
_
+
+
L2r
X
+
--
_
+
-
+
+
--
+
--
+
-
+
-
+
+
+
+
+
-
+
+
-
+
+
-
+
+
+
+
--
+
--
+
+
--
+
+
+
+
+
--
+
+
--
+ +
+
+
+
+
--
+
--
+
+
--
_
+
+
--
+
+
--
+
+
+
+
+
+
+
+
+
--
+
--
+
+
+
--
--
+
+
+
+
-
.0.
--
+
+
+
+
--
-
+
+
+
+
+
--
+
--
+
+
--
_
+
--
--
+
,
--
+
--
+
+
L2z
--
Z
+
Lj
LIz LIX Ltr
L2 L3
L3r L3x L3z
676
V.G.Bar'yakhtaret al. / Magnetocrystallinesymmetry, spinstaticsand dynamics
and the AFER frequencies and corresponding components oftensors ;~(oJ) and ~(ro) are of the form (cf. eqs. 10 and 11) AI: \ ~ a g / ----HAx(He+H'e+HAx);
Xzz(to)=Xj./~21 _to2 ;
A2:\-~agag/QA2 =(HAz+HAx)(He+H'-+HAx);
(19)
Xrr(to)--X± g~22_to2,"
2
\} = (2H: + HAz--HAx) (H. + n'e +HAx); El: /[g2El \~--'ag/
E2: ,,)--~g/ = ( 2 H ; + H A x ) ( H e + H ; + H ^ x ) ;
axx(to) =4MoR 2
(20) [#Bg(H, + H i +HAx) ]2 /22 _to2 ;
a,~B(o)) = 0 .
(21)
(22)
The relative position of frequencies of the acoustic (19), (20) and the exchange (21), (22) modes is largely defined by the relation between the values of uniaxial anisotropy HAz and exchange Hi. Also, pay attention to the exchange enhancement of the value of electric susceptibility (21 ) [22,23]. Selection rules for the resonance excitation of frequencies (19)-(22) for H # 0 are summarized in table VII. At deriving expression (21) for a,,x(ra) we have taken into account that the interaction of the crystal magnetic subsystem with the electric field E,o is described by the operator 1
4Mo
X ~im = E oZ ( r~,FL2 + 7f2LIL3 ) + Eo, Rx( LtzL3x-LixL3z) + Eo,Y R r ( L t z L 3 y - L I yL3z)
,
(23)
obtained in terms of table VI. In eq. (23) only exchange (7~ and 7t2) and exchange-relativistic (Rx and R r ) antiferroelectric constants are preserved. The operator corresponds to a single-particle process if the replacement L3x-'*L3x is done, similar to the case of scattering (see. eq. 15). Due to the fact that in magnetically ordered ("goffered') CuO2 planes all atoms are in noncentresymmetric positions C2v, Cu2+ and 02- ions possess the electric dipole moment pjUOZ. As a whole, however, the crystal is nonpolarized, i.e. it is antiferroelectric (to some extent this concerns the La2CuO4 rhombic phase where, however, only the O 2- anions "sit" in positions with no inversion centre). Among transverse vibrations of,ej vectors there are odd ones whose frequencies may be in the same frequency range as the exchange modes. Their interaction may be strong enough due to the relatively high polarizability of the 02- ions. As a result, the intrinsic vibrations of the system are actualy the coupled exchange-phonon modes which may described based on the operator of the type of eq. (23). In the latter, however, the replacement E,o--,P (P is the crystal polarization) should be done. On the other hand, an essential dependence of the exchange mode frequencies and polarization properties (table VII) on the external magnetic field makes it, in principle, possible to evaluate their contribution to tensor ~(to). Table Vll Tensors,~(to) and ti (to) componentsin magneticfield. Mode
H= O
HIIOX
HIIOY
HnOZ
AI
ZZ (Zzz(tO))
YY, ZY, ZZ
XX, XZ, ZZ
ZZ
A2
YY
YY, YZ, Z Z
YY
XX, YX, Y Y
XX
XX, XZ, Z Z
XX, XY, Y Y
XX
YY
ZZ
(Zyy(to) ) E1
XX
(t*x.(to)) E2
V.G. Bar'yakhtar et al. / Magnetocrystallinesymmetry, spin statics and dynamics
677
Finally it should be noted that the yttrium ceramics also intensively scatters light [ I 0 ] and in its spectrum the lines can appear described by operator ( 15 ) and caused by the rotations of the CuO5 semioctahedra.
4. Conclusion The above results show how extraordinary the HTSC "'usual" magnetic properties are in the region of dielectric phases and in the states nearest to the superconductivity phases. We believe that the magnetic properties of newly synthesized bismuth and thallium compounds are not less interest provided that conditions for the magnetism manifestation are created there by means of special alloying or variation of the oxygen content, because these systems also possess the typical quasi-two-dimensional structure with goffered CuO2 planes [ 24 ] similar to those in the yttrium system. We have good reasons to think that the HTSC magnetization and conductance are indeed "intermixed" which is confirmed by a sufficiently large (almost two-fold) change of La2CuO4 resistance in the normal phase after the spin flop transition in the H[IOZ field [ 15b]. In the superconducting state the long-range AFM order is absent and a similar direct connection was not found so unambiguously. Still we may hope that a more profound understanding of the HTSC magnetism will promote a deeper comprehension of the essense of this connection and, possibly, permit to establish the very mechanism
providing so high values of Tc. When the above results were almost ready we obtained the preprint by Prof. R. Birgeneau et al. on the experimental study of the spin statics of La2CuO4 [ 15b ]. We express Prof. R. Birgeneau our deep gratitude.
Note added in proof When our paper was submitted the work by A. Borovik-Romanov et al. had been published [P'isma JETP 47 (1988) 600 ] where the magnetic structure of La2CuO4 is symmetrically analised and some results analogous to ours are obtained. Besides we have found the results of AFMR measuring in this crystal. JR. Collins et al., Phys. Rev. B 37 (1988) 5817]. The resonance line with frequency 9 cm - j , probably, corresponds to a homogeneous A2 mode whose energy (see ( 11 ) ) depends both on rhombic anisotropy and on the Dzyaloshinskii interaction. Similarly to ref. [ 15b ] the relativistic constant HAy (which is also proportional to the squared parameter of structural deformation, or ~ ) is also extremely small. Therefore, one really can consider that ~A2 ~ H~ + H ~ -- 9 c m - ~ ( ~ 90 kOe ) which is close to the value H ~ 80 kOe obtained by A. Borovik-Romanov et al. On the other hand, if one determines H : from (7) by the value of H = = 4-5 T, then we find H : ,~ 5 × 10- 3 meV; this agrees well enough with the value ( ~ 3 × l O- 3 meV) given by the authors of ref. [ 15b ]. Nevertheless it should be recognized that unambiguous interpretation of spin excitation spectra can be based only on the field or polarization properties of AFMR. It should also be interesting to note that the soft mode of the spinflop transition proves to be one of the mixed (acoustic-exchange) in magnetic field modes. Finally, there appeared a communication [H. Kadowaki et al., Phys. Rev. B 37 (1988) 7932] about the observation of magnetic helicoid in a 1-2-3 system whose existence, in principle, has been shown above (see eq. 17).
References [ 1] P.W. Anderson, Science235 (1987) 1196. [2] V.J. Emery,Phys. Rev. Left. 58 (1987) 2794. [3] J.E. Hirsch, Phys. Rev. Lett. 59 (1987) 228. [4 ] V.L. Pokrovskii,G.V. Uimin and D.V. Khveshchenko,JETP Lett. 46 (Suppl.) (1987) 136. [5] J.R. Sehrieffer,X.-G. Wen and S.-C. Zhang, Phys. Rev. Lett. 60 (1988) 944.
678
V.G. Bar'yakhtar et al. / Magnetocrystalline symmetry, spin statics and dynamics
[6] D. Vaknin, S.K. Sinha, D.E. Moncton et al., Phys. Rev. Lett. 58 (1987) 2802. [7] J.M. Tranquada, D.E. Cox, W. Kunnmann et al., Phys. Rev. Lett. 60 (1988) 156. [8] G. Shirane, Y. Endoh, R.J. Birgeneau et al., Phys. Rev. Lett. 59 (1987) 1613. [9] K.B. Lyons, P.A. Fleury, J.P. Remeika et al., Phys. Rev. B 37 ( 1988 ) 2353. [ 10] K.B. Lyons, P.A. Fleury, L.F. Schneemeyer et al., Phys. Rev. Lett. 60 (1988) 732. [ 11 ] A. Aharony, RJ. Birgeneau, A. Coniglio et al., Phys. Rev. Lett. 60 (1988) i 330. [ 12 ] Yu.B. Gaididei and V.M. Loktev, Phys. Status Solidi b 146 ( 1988 ) 154. [ 13] I.E. Dzyaloshinskii, A.M. Polyakov and P.B. Wiegmann, Phys. Left. A 127 (1988) 112. [ 14] E.M. Tatarenko and V.A. Tatarenko, preprint ITP-88-85P, K.iev (1988). [ 15 ] (a) L.J. de Jongh, Solid State Commun. 65 (1988) 961. (b) T. Thio, T.R. Thurston and N.W. Preyer, preprint MIT, Cambridge (1988). [ 16] A.I. Zvyasin et al., JETP 89 (1985) 2298. [ 17 ] V.G. Bar'yaktar et al., JETP 76 (1979) 1381. [ 18 ] N.M. Plakida and V.S. Shakhmatov, preprint JINR Pl 7-87-488, Dubna ( 1988 ). [ 19 ] R.J. Birgeneau, C.Y. Chen, D.R. Gabbe et al., PhYs. Rev. Lett. 59 (1987) 1329. [20 ] C. Chaiilout, M.A. Alario-Fnmco, J.J. Capponi et al., Solid State Commun. 65 ( 1988 ) 283. [21 ] V.G. Bar'yaktar et al., Soy. Solid State Phys. 28 (1986) 504. [22 ] D.A. Yablon~di and V.N. Krivoruchko, SOy. Low Temp. Phys. 14 (1988) 656. [23 ] V.N. Krivoruchko and D.A. Yablonskii, JETP 95 (1988). [24] M.A. Subramanian, C.C. Torardi, J.C. Calabrese et al., Science 239 ( 1988 ) 1015.
M A G N E T O C R Y S T A I J J N E SYMMETRY, S P I N STATICS AND D Y N A M I C S I N H I G H TEMPERATURE SUPERCONDUCTORS V.G. BAR'YAKHTAR *, V.M. LOKTEV and D.A. YABLONSKII ** Institute for Theoretical Physics, Kiev-130, USSR
Received i 8 August 1988 Manuscript received in final form 29 September 1988
The symmetry analysis of the magnetic structures of La2CuO, and YBazCu306+6 (6 ~ 0.4) antiferromagneticoxides has been carried out. The spin configurations and spin-reorientation transition fields have been found. The frequencies of uniform spin oscillations and the external field dependence of their polarization properties have been calculated. It has been shown that the spin excitation branches may include exchange modes among which some dipole-activemodes can be present. The spectrum of the inelastic light scatteringaccompanied with the excitation of spin and libration (tilting) freedom degreeshas been discussed.
1. Introduction
The magnetism of high-temperature superconductors ( H T S C ) is widely recognized now as one of the most important and interesting properties. Moreover, many investigators (e.g. refs. [ I - 5 ] ) believe that it is the magnetic properties that play a key part in the pairing mechanism. Up till now the antiferromagnetic (AFM) spin ordering has been established experimentally in two classes of the HTSC. Thus non-alloyed La2CuO4 exhibits the N~el spin ordering S = ½ of Cu 2+ ions at TN,,~250 K [ 6 ]; the same AFM ordering was also revealed in YBa2Cu306 +~ at TN > 500 K for 6 = 0.0 and TN ~, 400 K for 6 = 0.15 [7 ]. In both cases the AFM is positively fixed only in dielectric phases of the HTSC. In a metallic state, however, the long-range N~'el ordering is disturbed. Nevertheless, highly developed fluctuations persist even at room temperature [ 8 ]. This is also confirmed by the data on the nonelastic light scattering accompanied by the creation of spin excitation pairs both in La2_xCuO4 (x # 0) [ 9 ] and in YBa2Cu306 +6 (0.0 < 6 < 0.9) [ 10 ]. The position of the line scattering indicates that magnetic correlations in HTSC are characterized by the high value of the exchange energy J ~ 103 c m - ~ (most probably attributed to the indirect CuZ+-Cu 2+ interaction in CuO2 planes characteristic of the copper-containing HTSC). As for the theoretical papers (see also refs. [ l I - 13 ] ) dealing with the HTSC magnetism to this or that extent, they mostly concentrate their attention on the substantiation of some "magnetic model" of superconductivity. The aim of this paper, however, is to study the magnetic (quite standard ) peculiarities of the HTSC magnetism by an example of lanthanum and yttrium systems. Thus we investigate a symmetrical connection between the HTSC crystalline structure and spin ordering type, find the critical fields of spin-orientational phase transitions, calculate the values of resonance frequencies as well as the polarization and intensity of their excitation by external magnetic ( A F M R ) and electric (AFER) fields.
* Institute of Metal Physics, Kiev, USSR. **Physical and Technical Institute, Donetsk, USSR. 0921-4534/88/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division )
V.G. Bar°yakhtar et al. I Magnetocrystalline symmetry, spin statics and dynamics
668
2. La2Cu04 crystal
2.1. Spin Hamiltonian structure The high-temperature modification of La2CuO4 belongs to the space group I 4 / m m m ( D ~ ) and is a tetragonal lattice with one "octahedric molecule" of CuO6 in a unit cell (fig. 1 ). At T~450 K [ 14] (the particular value of the temperature depends both on the content of the alloying impurities and on the oxygen deficiency) a phase transition into a low-symmetry rhombic phase with the space group C,~(D,~hs ) occurs (fig. 2). This transition is due to a slight tilting of the CuO6 octahedra extended along the C4-axis about one of the horizontal axes (either (110)[IOX or (I]0)[IOY) and results in the cell doubling. As is seen from fig. 2, the lattice spacings in the basal plane change since the octahedra of different planes preserve their translational invariance. The latter, however, can be disturbed after the phase transition into such an AFM state where the spin directions in the translationaUy equivalent sites do not coincide. So one can assume that the spin structure of La2CuO4 is described not with two but, at least, with four magnetic sublattices with magnetizations
ILj=lzBgSv~o, j = l , 2, 3 , 4 ,
(1)
where Sj is the spin of the copper ion from the jth sublattice,/tB is the Bohr magneton, g is the g-factor, Vo is the cell volume. We do not intend here to investigate all possible magnetic structures compatible with the D ~ group. We restrict ourselves to the discussion of the most probable ones on the basis of the experimental data. First of all, one should pay attention to the importance of the intralayer AFM exchange which makes it possible to
I Z {IC~
|,\ |~ /
3
\/ ~
/ I I
t \I\i
~
lt/I
x fl .(~o) Fig. 1. The unit cell of La2CuO, in the tetra-phase ( l al = I b I );
,
/ ~ /
/
/
x
t Fig. 2. The unit cell of La2CuO4 in o r t h ~ p h a s e ; ~ - 1 ~ 3 , 2 ~ 4 ; ( ~ ) helical axis.
V.GI Bar "yakhtar ~t at: / M a g n e t o c r y s t a l l i n e
symmetry,
spin statics a ~ dynamics
669
Table I Classification of irreducible vectors in the irreducible representations of the D l~ group. 1
2x
2iv
2,z
t2x
/2t~"
t2tz
+
+
+
+
+
+
+
+
+
--
+
--
+
--
+
--
+
+
--
--
+
+
--
+
--
--
+
+
--
_
+
+
+
+
+
--
+
--
--
+
--
+
L2v
Lsz
+
+
--
_
_
_
+
+
+
--
_
+
--
+
+
--
L,_x L,.z
Lsr
F
L lx
Fr
_
Fx
Liz
+
Fz
Ltv
L~ L3~
L2
Ls
introduce "intralayer" vectors of ferromagnetism mj and antiferromagnetism 1~(j= 1,2) according to the definitions
!'1 +~2 =2Momt;
Its +p4 =2Morn2;
m} +! 3 = 1;
1'1-~2=2Mo1~;
lts-la4=2Mo12;
Me= #ag s . Vo
(2)
Moreover, measurements [ 6 ] show that the CuO2-planes are antiferromagneticaUy ordered with the "'easy plane" anisotropy (i.e. when spins are located within the XY plane) *. Then the standard symmetry analysis (see table I) allows us to write the spin Hamiltonian in the form 1 2 2 2Me ,,'~=½H, E mJ +H'd,12 +Hv E (ljzrnjv-ljvmjz) j=l
j=l
2
+ H~ ( mlzl2r-m~vl2z + m2zl, r-m2rllz) + ½ ~ ( Hazl~z-HArl~v- 2Hmj ) ,
(3)
j=l
where He is the constant of the intralayer (intraplane) exchange field and H i ( ~ He) is the constant of interlayer (interplane) exchange field; Hv and H ~ are the similar Dzyaloshinskii fields; HAZ and liar are the "easy plane" and rhombic anisotropy fields: H is the external magnetic field; mj, ij and Me are determined in eqs. (2) and (1). While writing ~ we omit the interplan¢ anisotropy field, the anisotropy fields associated with the ferromagnetism vectors mj (ifH~KH,, then mj~lj) and so on and restrict ourselves to the most essential terms. Pay attention to the fact that in the rhombic phase decompensation of the interlayer exchange (H" # 0) takes place and antisymmetric inter- and intralayer exchange interactions appear. It has been already pointed out recently [ 15 ] that the Dzyaloshinskii interaction can, in principle, appear in La2CuO4. These peculiarities of the low-symmetry phase of La2CuO4 have quite a clear physical nature. In fact, the appearance of the interlayer exchange is entirely due to the fact that in the process of the tetragonal-orthorhombic transition the shift of atoms along the (110) and ( 1 i 0 ) axes of the initial lattice is not the same and as a result, the 1-3 distance differs from the 2-3 distance (fig. 1 ). For Hi the rotations ofoctahedra may not play an essential role (though, as will be shown below this will not always be the case). However, the fact that spins go beyond the planes is, to a large extent, caused just by the change in the orientation of the crystalline field acting on them. And it should be noted that in the orthorhombic phase the axes of this field do not coincide with the lattice axes. The ° This experimental fact would seem to be a bit surprising, because for a copper ion in an elongated octahedron, gm> g~, as a rule, and this should result in a "'light axis" structure. Dipole and exchange-relativistic interactions seem to play a major role in this situation. And although we avoid discussing this point (magnetic anisotropy sign ) below and focus on the structures favoured by experiment [ 6,7 ], the other ease can easily be treated in a similar manner.
670
V.G.Bar'yakhtaret al. / Magnetocrystallinesymmetry, spinstaticsand dynamics
combined action of the AFM exchange interaction and local (single-ion) field of ligands can be expressed in the form of operator (3) based on the symmetry considerations.
2.2. Spin-reorientational transitions The behaviour of a crystal in the external field HIIOZ was studied experimentally [ 15b ] and a jump of total magnetization in a field H~. 4-5 T was found, which was attributed to the reversal of the magnetization vector in one of the layers. We consider this transition in detail on the basis o f Hamiltonian (3). First of all we assume that the interlayer interaction has the same AFM sign as the intralayer one (otherwise it would be the case of a weak ferromagnetism). Provided that 1
HAy+ -~. (H,, +H'._.)2>0,
(4)
the ground state of the magnetic subsystem can be characterized by the following vectors
f,,=-f~
l; ,~,z=-,h2z= H~+n'.~ ~ 1, He
(5)
where ,,0,, denotes that the vectors belong to the ground state (fig. 3). It should be noted that ~olation of condition (4) l e ~ s to a change in the structure, so projections l~x= -12x become non-zero. In this c a ~ ms-O hut it is not confirmed experimentally and the ip'ound state La2CuO4 is d¢~cTibed by eqs. (5). It follows from eq. (4) that the Dzyaloshinsldi interaction can quite well l ~ ' f o ~ the part of the rhombic anisotropy. In the external field HHOZ the crystal magnetization changes so that H mjz=rhjx+~,
j=l,2.
(6)
It is seen that in accordance with eq. (6) the crystal magnetization M z = 2MoY2=,msz grows linearly up to a field Sg HZ= Hv + H ~ He
(7)
(in fact a similar expression is given in ref. [ 15b ] ). At that point the orientation of vector M2z occurs so that (see eq. 6) m , z = m2z= rh,z +H/H~. The crystal magnetization jump is
~z
= 4Mo
Hv + H~
Ho
The static susceptibility, however, preserves its value Z~ =4Mo/He (see eq. 6) after the transition which is confirmed by experiment [ 15b ]. In the course of this transition not only projection m2z changes its direction but projection/2 ~ as well (fig. 3). The similar spin-reorientational transitions are typical of the quasi-two-dimensional four-sublattice AFMs of the (NH3) 2(CH2) ,MnC14-type. In these AFMs, however, reorientation of one of the vectors is caused by the field directed not across but along the "easy" axis [ 16 ].
~:o
z
~
~<~
~ "
H~H~z.
-
Fi8. 3. Spin-reorientationof La2CuO4whenH{IOZ.
V.G.Bar'yakhtaret:al. / Magnetocrystallinesymmetry,spinStaticsand dynamics
671
Since this transition is the first-order transition induced by the external field it should be accompanied by the appearance of an intermediate state which is actually a domain structure composed of an initial and reversed phases. The field width of the intermediate state stability range is
AHz = 16xNz Morhlz (Nz is the demagnetizing factor along the OZ axis). A more standard situation arises in a field H[IO Y. Up to field H ~ = [HA rile (He + H~ ) 2] t / 2 (cf. eq. 7 ) the magnetization grows very slowly and the longitudinal susceptibility is expressed by
(n ~ + n ~ )2 Xu =Z± (nAz+2I~e)ne ~ Z . . Then the system undergoes the conventional spin-flop transition into the state l~x= -12x; mix=O; mjr=H/ He; in this case the width of the intermediate state region is AHy= 16x3VrMo(H~/H,).
2.3. Antiferromagnetic resonance The AFMR frequencies and the magnetic susceptibility tensor can be conveniently calculated using the method proposed in ref. [ 17 ]. Let us consider the irreducible vectors
F=½(ml+m2);
Ll=½(l,+12);
L2=½(ml--m2);
L3=½(l,-12),
(8)
whose classification by the irreducible representations of group D ~ is given in table I. In addition to the rotary elements the table includes the transition t = ½(a+b+c) since the unit cell of La2CuO4 is twice as big as the crystallographic one. Thus the group action of i is determined by commutation 1~3, 2 ~ 4 as well as iml = m2, ilt =12. As is seen from eqs. (5) and (8), in the absence of field, the ground state is characterized by the vectors/~2z=rhm, L~3y~ 1. Then using the equations of motion for vectors (8) [ 17] we find that the system described by operator (3) has four uniform oscillations, two of which belong to the acoustic type (A 1, A2) and two others to the exchange one (E 1, E2). The dynamic variables of the corresponding modes are
AI:{Fx;O;L2y;L3z};
A2:{Fz;LIy;O;L3x};
EI:{O, Ltx;L2z;L3y};
E2:{Fr;Ltz;L2x;O}.
(9)
The resonant frequencies and the components of the high frequency magnetic susceptibility tensor are determined by the equations AI: k-~--~: =(HAz+HAy) A2: \ ~~A2 -~/
=HArHe+(H~+H~)2;
~
.J'~"AZn,;
Xxx(09) =X~ ~ l _092,
~X2 . Xzz(09)=X± Q22_092,
(ll)
(~'~El~2
El: \~--~/ =(2H', + HAr)He + (H~ + H~ )2; X~a(09)=O ;
(~E2~ 2
E2: \ - ~ /
= (2H'~ + HAz + ttAy)He +4H~(H~ + H~ ); ;eYY(09)=;e±
(10)
(12)
[lz~g(H~ + H~ ) ]2 ~222 --09 2
~Zu~2_092' (13)
where Z~,n are determined above. It is quite probable that interlayer exchange interaction is less than the uniaxial anisotropy ttAZ. In this case there is rather an abnormal situation where ~E~ <~AJ. This fact must be by all means taken into consideration at treating the results of the resonance studies of HTSC. An external magnetic field changes the selection rules of the resonant mode excitation (9). The symmetry
V.G. Bar'yakhtar et al. / Magnetocrystalline symmetry, spin statics and dynamics
672
Table II Tensor ~(t~) components in a magnetic field. Mode
t1= 0
H~OX
HHO Y
HIIOZ
A1
XX
XX
XX, XZ, Z Z
XX, XY, Y Y
YY, YZ, Z Z
XX, XZ, Z Z
ZZ
XX
YY
ZZ
YY, YZ, Z Z
YY
XX, XY, Y Y
(Xxx(co)) A2
ZZ
(Zzz(O))) E1
-
E2
YY
(Xrr(w)) analysis makes it possible to predict the polarization properties of the high frequency magnetic susceptibility tensor ~(to) at various directions of an external constant magnetic field (see table II).
2.4. Magnetostructure interactions It can be shown that interlayer and antisymmetric exchange interactions are possible only in the rhombic phase. We remind here that the transition* from "tetragonal D~[ phase to the rhombic D ~ one is due to rotation of the extended oxygen octahedra, surrounding the copper ions, about the OX or OY axes chosen along the basic square diagonals (figs. 1 and 2). As we are interested in the connection of the spin moments with the change in the crystalline structure which is determinexi only by the directions of the octahedra main axes, we introduce vectors-directions taflj= 1..... 4) whose directions coincide with those of the respective main axes. It can be easily shown that transverse components tojx, tOjr of these vectors determine the deviation of the octahedra axes from the crystal tetragonal axis and from the two-dimensional E~-representation of point group Ddh which describes the local symmetry of Cu 2+ ions in the tetra-phase. And in this case the order parameter describing the structural phase transition D4~-.DIS (figs. 1 and 2) is represented by a two-component value {r/,; ~/2} in which ~1 m~'('OI } ' - - ('/')2 }" "Jff0)3)'--(X~4}';
~2 ~-~O.)lrt'-- O.J2X--(~)3X"~'O.)4X.
(14)
The transformation properties of values ( 8 ) and (14) are determined by tables III, IV and V. In this case table IV presents transformations of vectors (8) neglecting the vector origin (i.e. only commutations are taken into account). Thus under the action of rotations component L 2 y is transformed as coordinate Y and under the action of commutations it is transformed as is shown in tables III and IV. Knowing the transformations of magnetic vectors and order parameter of the structural transition one can write the Hamiltonian of the spinspin and spin-libron interactions provided that small swingings of the octahedra in the vicinity of the ori* The symmetry analysis of the structural transition irrespective of the magnetic subsystem was considered in ref. [ 18 ].
Table IV. Table Ill,
l
i~
~
i,t~
+
+
+
+
+ + +
-+
+ -
-+
L3 Li
-
L,
F
1/., ~h
1
4z
Table V.
F Li L2 L3
F L3 L2 L~
1
2x
2r
~h ~/2
th -~/2
-rh
4z
~/2
-~h
rh
V.G. Bar'yakhtar ~ ~ ~¢Magnetoerystattmex3,mmctry, xpin staticsand dynamics
6 73
entationally ordered position belong to the libron-type excitations. The exchange part of the Hamiltonian has the form
~x =~,F2 +6L,(L~ + L~ ) + 6 J . ~,+al ( L ~ + L ~ ) + a 2 ( L ~ +L~ +7,: ) (tSe, 6~, A t and A2 are the corresponding constants). It implies that (see eqs. 8 and 2 ) that the transition results not only in some renormalization of the intralayer exchange fields but also in the interlayer A~21tl2 (or A~llt 12 depending on the component determining the structural domain by which the tetragonal-orthorhombic transition is realized). In the latter case the constant of the exchange field H~ appears to be small due to the obvious proportionality H~ ~ r/2. In a similar way one can draw the connection between the deformations and the antisymmetric exchange-relativistic interaction. In the general case they have the form a~fSv= dn [ t/t ( FzLI r - F r L 1 z ) + tl2( F z L 3 x - F x L 3 z ) ] + ~,[ ~l ( L 2 z L 3 r - L 2 v L 3 z ) + tl2( L t x L 2 z - LizL2x) ] • This expression, in its turn, indicates that the Dzyaloshinskii fields in eq. (3) Hv, H~ ~ ~b, i.e. they are actually induced by rhombic deformation. The experiments on the magnetization and the magnitude [also proportional to r/j (see eq. (7) ] of field H z of the magnetic sublattiee reversal at HHOZ quite positively confirm that the values H~ and Hv, H ~ are very small.
2. 5. Inelastic light scattering As far as we know, the HTSC AFMR spectra have not been measured. However, there are some data on the HTSC spin excitations. Thus Lyons et al. studied the inelastic light scattering spectrum of La2CuO4 [ 9,10 ]. In addition to the main two-magnon band (the spectral position of its maximum made it possible to estimate the magnitude of the above constant J) this spectrum possesses rather an intensive line at the half-frequency which was attributed to the defect mode excitation. But due to distinct polarization of this line (its scattering tensor is diagonal in the basal plane) this interpretation cannot be regarded as a final one. On the basis of the magnon-libron interaction one can propose another interpretation, namely, the observed absorption is really a two-particle one but it also includes the spin excitation and a quantum of soft libration oscillation (tilting mode) of octahedra. (Judging upon the structural transition [ 19 ] the frequency of this oscillation at the edge of the Brillouin zone is ~ 102 cm-t ,,: ~2j(kB)~ 10 3 cm -t. ) Indeed, a simple calculation shows (see tables IIIV) that the operator of the magnon-libron light scattering with polarization Eo, propagating along the OZaxis is of the form
~c---[ (E,o)x : + (Eo~)r 2 ] ( ~ l L 2 z L 3 r + ~ 2 L , x L 2 z ) ,
(15)
in which, depending on a domain, either L3v or Ltx projections should be substituted by L3y and Ltx, and tb and L2z are the dynamic variables. As is seen from eq. ( 15 ), in this process the exchange mode of frequency (12), which is passive in the AFMR, is excited together with the libron. The obtained (diagonal) symmetry of the magnon-libron scattering process fully agrees with the observed one; it is easy to check that such a scattering involving the rotation of the light-wave polarization plane is impossible, and this is also confirmed by the experiment [ 9,10 ].
3. YP,a2Cu~O~,4 crystal
3.1. Structure and spin-Hamiltonian Though crystalline structures of YBa2Cu3Or+ 6 crystal (fig. 4) and the lanthanum system are different, they have one feature in common, namely they possess the CuO2 planes with Cu 2+ ions (here Cu2). At J~<0.4 the Cul ions (CuO chains) predominantly find themselves in the Cu + state (3d ~°, S = 0 ) and do not practically
674
V.G. Bar'yakhtar et al. / Magnetocrystalline symmetry, spin staZicsand dynamics
2z E
7q-'f o- Caz
0-0
//
I C
[
2¥
jjJ .... Fig. 4. The unit cell of YBa2Cu306.a (only magnetically ordered planes are shown ); i - 1~2, 3=4.
contribute to the magnetism of Y B a 2 C u 3 0 6 + a [7 ]. Then, from fig. 4, where cations Cu2 and anions 0 2- are shown, it is seen that the unit cell of this compound belongs to rhombic syngony Pm=m(D~h) and the magnetic cell also contains four spins which can be conveniently used for making up the same "intralayer" combinations according to eq. (2). The above arguments for the lanthanum ceramics allow us to present the phenomenological Hamiltonian of the yttrium system in the form of two summands: homogeneous 1
2
2
2"~o ~Ss=½He ~ m2~ + H'~ltl2 + ½ ~ (HAzl~z -- HAzl2x-- 2Hmj) j=t
(16)
j=l
and inhomogeneous 2Mo ~inho==I~,
(
012 12 01' ~ + I7', ( m , Ore2 1,-~ -- OZ] \ "~
-/'/vxj=,~ ( - 1 )
m2 -~~' ) - -
~ljx ox-lJz OX]-H~vj=,
) ~ljr oy-ljz
,
(17)
in which the meaning of all constants is clear from the definition, but still we note that in eq. (17) values//e and//~ are of the exchange and/'?~,x and/'/~v of the exchange-relativistic origin. Even simple comparison of the latter expressions with Hamiltonian (3) shows that in the yttrium compounds there is no Dzyaloshinskii interaction * and in the initial structure the interlayer exchange Hi is non-zero. On the other hand, absence * Strictly speaking, this conclusion applies to the lattice shown in fig, 4 which, among other elements of symmetry, exhibits the translation along the bHOY axis. If, as is reported in ref. [20], even at small J the YBa2Cu306.a system experiences doubling (or even linger increase) of the period along the same axis, there appears a symmetry possibility for an antisymmetry interaction similar to the case of La2CuO4, and only experimental study can give an answer as to an existence of the weak intraplane moment.
V.G. Bar'yakhtar et al, / Magnetocrystalline symmetry, spin statics and dynamics
675
of the inversion centre at the Cu 2+ site makes the non-uniform interaction Hamiltonian (17) non-zero which, in its turn, causes and describes the long-period helicoidal incommensurate structure under definite conditions imposed on the values of respective constants [ 21 ]. Regarding HAz (it is shown experimentally [7] ) and HAx (for definiteness) as positive it is easy to find the ground state of the magnetic subsystem described by operator (16). It is in compliance with projections [ix = - 12x= 1 at Hi > 0, or/~ x = / ~ x = 1 at Hi < 0; the other projections of antiferromagnetic and ferromagnetic vectors do not exist. In this case the spin-flop transition in the magnetic subsystem will take place at H[IOX in field H x = ~/HAx(He +H',). At HAx< 0 the O Y-axis becomes the "easy" axis and the physical picture remains the same (if X--. Y).
3.2. Antiferromagnetic and antiferroelectric resonances Similarly to the above case, it is convenient to calculate the resonance absorption frequencies and the susceptibility tensors using vectors (8). Their commutative symmetry is defined by the following elements simple rotations 2x.y.z, inversion i, translation i-- 1 ~ 2 , 3 ~ 4 and product t. i. Table Vl presents the total classification of the irreducible magnetic vectors over the crystallographic group representations of orthorhombic YBa2Cu306+6. It is seen from table Vl that in case of the antiferromagnetic interplane exchange the phase with /-:3x-- 1 serves as the ground state and the dynamic variables resulting from the equations of motion are
AI:{Fz;O;O;L3r};
EI:{O;L,z;L2r;O};
A2: {Fr; 0;0; L3r};
E2:{O, Ltr;L2z;O}.
(18)
Difference between the number of dynamic variables ( 18 ) and (9) is due to the non-collinear character of the magnetic structure of La2CuO4. Another and more important difference of YBa2Cu306+6 from La2CuO4 is the possibility to observe, in the former, the antiferrodielectric resonance (AFER) or excitation of spin vibrations by the light wave electric field. Indeed, as is seen from table VI, vectors Lt and L3 are odd with respect to inversion [22,23 ] then it follows that their respective vibrations (here E 1 and E2) are electric-dipole active and characterized by tensor 6 (m) of electric (not magnetic) high-frequency susceptibility. For H = 0 the AFMR
Table
VI
Classification 1
2x
of irreducible 2v
I
2z
vectors
i2x
in the irreducible
|2~.
i2z
i
representations
&~.
/2,
of the Pmmm
/2~.
t
t2 +
group. v
i'2r
hz
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
+
--
+
--
+
--
+
--
+
--
+
--
+
-
+
+
--
_
+
+
--
_
+
+
--
_
+
+
--
--
Fy
+
+
--
--
+
+
--
--
+
+
-
-
+
Fz
B
B
+
+
+
+
--
+
--
+
+
--
+
--
-
+
-
+
L2x
Y
+
+
+
+
--
_
+
+
L2r
X
+
--
_
+
-
+
+
--
+
--
+
-
+
-
+
+
+
+
+
-
+
+
-
+
+
-
+
+
+
+
--
+
--
+
+
--
+
+
+
+
+
--
+
+
--
+ +
+
+
+
+
--
+
--
+
+
--
_
+
+
--
+
+
--
+
+
+
+
+
+
+
+
+
--
+
--
+
+
+
--
--
+
+
+
+
-
.0.
--
+
+
+
+
--
-
+
+
+
+
+
--
+
--
+
+
--
_
+
--
--
+
,
--
+
--
+
+
L2z
--
Z
+
Lj
LIz LIX Ltr
L2 L3
L3r L3x L3z
676
V.G.Bar'yakhtaret al. / Magnetocrystallinesymmetry, spinstaticsand dynamics
and the AFER frequencies and corresponding components oftensors ;~(oJ) and ~(ro) are of the form (cf. eqs. 10 and 11) AI: \ ~ a g / ----HAx(He+H'e+HAx);
Xzz(to)=Xj./~21 _to2 ;
A2:\-~agag/QA2 =(HAz+HAx)(He+H'-+HAx);
(19)
Xrr(to)--X± g~22_to2,"
2
\} = (2H: + HAz--HAx) (H. + n'e +HAx); El: /[g2El \~--'ag/
E2: ,,)--~g/ = ( 2 H ; + H A x ) ( H e + H ; + H ^ x ) ;
axx(to) =4MoR 2
(20) [#Bg(H, + H i +HAx) ]2 /22 _to2 ;
a,~B(o)) = 0 .
(21)
(22)
The relative position of frequencies of the acoustic (19), (20) and the exchange (21), (22) modes is largely defined by the relation between the values of uniaxial anisotropy HAz and exchange Hi. Also, pay attention to the exchange enhancement of the value of electric susceptibility (21 ) [22,23]. Selection rules for the resonance excitation of frequencies (19)-(22) for H # 0 are summarized in table VII. At deriving expression (21) for a,,x(ra) we have taken into account that the interaction of the crystal magnetic subsystem with the electric field E,o is described by the operator 1
4Mo
X ~im = E oZ ( r~,FL2 + 7f2LIL3 ) + Eo, Rx( LtzL3x-LixL3z) + Eo,Y R r ( L t z L 3 y - L I yL3z)
,
(23)
obtained in terms of table VI. In eq. (23) only exchange (7~ and 7t2) and exchange-relativistic (Rx and R r ) antiferroelectric constants are preserved. The operator corresponds to a single-particle process if the replacement L3x-'*L3x is done, similar to the case of scattering (see. eq. 15). Due to the fact that in magnetically ordered ("goffered') CuO2 planes all atoms are in noncentresymmetric positions C2v, Cu2+ and 02- ions possess the electric dipole moment pjUOZ. As a whole, however, the crystal is nonpolarized, i.e. it is antiferroelectric (to some extent this concerns the La2CuO4 rhombic phase where, however, only the O 2- anions "sit" in positions with no inversion centre). Among transverse vibrations of,ej vectors there are odd ones whose frequencies may be in the same frequency range as the exchange modes. Their interaction may be strong enough due to the relatively high polarizability of the 02- ions. As a result, the intrinsic vibrations of the system are actualy the coupled exchange-phonon modes which may described based on the operator of the type of eq. (23). In the latter, however, the replacement E,o--,P (P is the crystal polarization) should be done. On the other hand, an essential dependence of the exchange mode frequencies and polarization properties (table VII) on the external magnetic field makes it, in principle, possible to evaluate their contribution to tensor ~(to). Table Vll Tensors,~(to) and ti (to) componentsin magneticfield. Mode
H= O
HIIOX
HIIOY
HnOZ
AI
ZZ (Zzz(tO))
YY, ZY, ZZ
XX, XZ, ZZ
ZZ
A2
YY
YY, YZ, Z Z
YY
XX, YX, Y Y
XX
XX, XZ, Z Z
XX, XY, Y Y
XX
YY
ZZ
(Zyy(to) ) E1
XX
(t*x.(to)) E2
V.G. Bar'yakhtar et al. / Magnetocrystallinesymmetry, spin statics and dynamics
677
Finally it should be noted that the yttrium ceramics also intensively scatters light [ I 0 ] and in its spectrum the lines can appear described by operator ( 15 ) and caused by the rotations of the CuO5 semioctahedra.
4. Conclusion The above results show how extraordinary the HTSC "'usual" magnetic properties are in the region of dielectric phases and in the states nearest to the superconductivity phases. We believe that the magnetic properties of newly synthesized bismuth and thallium compounds are not less interest provided that conditions for the magnetism manifestation are created there by means of special alloying or variation of the oxygen content, because these systems also possess the typical quasi-two-dimensional structure with goffered CuO2 planes [ 24 ] similar to those in the yttrium system. We have good reasons to think that the HTSC magnetization and conductance are indeed "intermixed" which is confirmed by a sufficiently large (almost two-fold) change of La2CuO4 resistance in the normal phase after the spin flop transition in the H[IOZ field [ 15b]. In the superconducting state the long-range AFM order is absent and a similar direct connection was not found so unambiguously. Still we may hope that a more profound understanding of the HTSC magnetism will promote a deeper comprehension of the essense of this connection and, possibly, permit to establish the very mechanism
providing so high values of Tc. When the above results were almost ready we obtained the preprint by Prof. R. Birgeneau et al. on the experimental study of the spin statics of La2CuO4 [ 15b ]. We express Prof. R. Birgeneau our deep gratitude.
Note added in proof When our paper was submitted the work by A. Borovik-Romanov et al. had been published [P'isma JETP 47 (1988) 600 ] where the magnetic structure of La2CuO4 is symmetrically analised and some results analogous to ours are obtained. Besides we have found the results of AFMR measuring in this crystal. JR. Collins et al., Phys. Rev. B 37 (1988) 5817]. The resonance line with frequency 9 cm - j , probably, corresponds to a homogeneous A2 mode whose energy (see ( 11 ) ) depends both on rhombic anisotropy and on the Dzyaloshinskii interaction. Similarly to ref. [ 15b ] the relativistic constant HAy (which is also proportional to the squared parameter of structural deformation, or ~ ) is also extremely small. Therefore, one really can consider that ~A2 ~ H~ + H ~ -- 9 c m - ~ ( ~ 90 kOe ) which is close to the value H ~ 80 kOe obtained by A. Borovik-Romanov et al. On the other hand, if one determines H : from (7) by the value of H = = 4-5 T, then we find H : ,~ 5 × 10- 3 meV; this agrees well enough with the value ( ~ 3 × l O- 3 meV) given by the authors of ref. [ 15b ]. Nevertheless it should be recognized that unambiguous interpretation of spin excitation spectra can be based only on the field or polarization properties of AFMR. It should also be interesting to note that the soft mode of the spinflop transition proves to be one of the mixed (acoustic-exchange) in magnetic field modes. Finally, there appeared a communication [H. Kadowaki et al., Phys. Rev. B 37 (1988) 7932] about the observation of magnetic helicoid in a 1-2-3 system whose existence, in principle, has been shown above (see eq. 17).
References [ 1] P.W. Anderson, Science235 (1987) 1196. [2] V.J. Emery,Phys. Rev. Left. 58 (1987) 2794. [3] J.E. Hirsch, Phys. Rev. Lett. 59 (1987) 228. [4 ] V.L. Pokrovskii,G.V. Uimin and D.V. Khveshchenko,JETP Lett. 46 (Suppl.) (1987) 136. [5] J.R. Sehrieffer,X.-G. Wen and S.-C. Zhang, Phys. Rev. Lett. 60 (1988) 944.
678
V.G. Bar'yakhtar et al. / Magnetocrystalline symmetry, spin statics and dynamics
[6] D. Vaknin, S.K. Sinha, D.E. Moncton et al., Phys. Rev. Lett. 58 (1987) 2802. [7] J.M. Tranquada, D.E. Cox, W. Kunnmann et al., Phys. Rev. Lett. 60 (1988) 156. [8] G. Shirane, Y. Endoh, R.J. Birgeneau et al., Phys. Rev. Lett. 59 (1987) 1613. [9] K.B. Lyons, P.A. Fleury, J.P. Remeika et al., Phys. Rev. B 37 ( 1988 ) 2353. [ 10] K.B. Lyons, P.A. Fleury, L.F. Schneemeyer et al., Phys. Rev. Lett. 60 (1988) 732. [ 11 ] A. Aharony, RJ. Birgeneau, A. Coniglio et al., Phys. Rev. Lett. 60 (1988) i 330. [ 12 ] Yu.B. Gaididei and V.M. Loktev, Phys. Status Solidi b 146 ( 1988 ) 154. [ 13] I.E. Dzyaloshinskii, A.M. Polyakov and P.B. Wiegmann, Phys. Left. A 127 (1988) 112. [ 14] E.M. Tatarenko and V.A. Tatarenko, preprint ITP-88-85P, K.iev (1988). [ 15 ] (a) L.J. de Jongh, Solid State Commun. 65 (1988) 961. (b) T. Thio, T.R. Thurston and N.W. Preyer, preprint MIT, Cambridge (1988). [ 16] A.I. Zvyasin et al., JETP 89 (1985) 2298. [ 17 ] V.G. Bar'yaktar et al., JETP 76 (1979) 1381. [ 18 ] N.M. Plakida and V.S. Shakhmatov, preprint JINR Pl 7-87-488, Dubna ( 1988 ). [ 19 ] R.J. Birgeneau, C.Y. Chen, D.R. Gabbe et al., PhYs. Rev. Lett. 59 (1987) 1329. [20 ] C. Chaiilout, M.A. Alario-Fnmco, J.J. Capponi et al., Solid State Commun. 65 ( 1988 ) 283. [21 ] V.G. Bar'yaktar et al., Soy. Solid State Phys. 28 (1986) 504. [22 ] D.A. Yablon~di and V.N. Krivoruchko, SOy. Low Temp. Phys. 14 (1988) 656. [23 ] V.N. Krivoruchko and D.A. Yablonskii, JETP 95 (1988). [24] M.A. Subramanian, C.C. Torardi, J.C. Calabrese et al., Science 239 ( 1988 ) 1015.