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Influence of interchain coupling on the one-dimensional magnon Raman spectrum of CsCoBr3
~/
593—597. Solid State Communications, Vol.36,in pp. Pergamon Press Ltd. 1980. Printed Great Britain.
INFLUENCE OF %WTERCHMN COUPLING ON THE ONE-DIMENSIONAL MACNON RAMAN SPECTRUM OF CsCoBr 3 LW. Johnstone, D.J. Lockwood and N.W.C. Dharma—wardana Division of Physics, National Research Council, Ottawa, Canada K1A 0R6 (Received 30 July, 1980 by M.F. Collins
Light scattering from magnons In CsCoBr3 has been measured for temperatures well below the upper antiferromagnetic 3D ordering temperature of T~— 28 K. These experiments reveal 1 similar multiple to those magnon found in CsCoClof energies in the range 90 to 170 cm features 3 but previously unobserved for the bromide. Frominent features in the spectrum and their temperature dependence are described in terms of a recent theory by Shiba. Other, weaker features are explained by a simple extension of 1~whosf the theory Intensity to include drops fluctuations. sharply prior the band ~ower orderingat transition at TN1 — 10 K. This band is Ato new is 3D observed 178 cm assigned to magnon—phonon combination scattering.
1.
Introduction
3have been difficult Until now, the spin-wave Raman spectra to of understand terms of current theories of IsingCsCoCl3 and in CsCoBr3~ Hc!senbcrg chains, which are themselves the subject of much debate.”8 Both materials simulate i-D Ising antiferromagnetic chains but display 3—0 magnetic order at sufficiently low temperatures. In CsCoBr 3, the large intrachain nearest—neighbour exchange interaction, J, gives 100 For T ‘ 30 spin K the Raman spectrum rise cm~. to a transverse excitation band near appears as an asymmetric peak superimposed on a
explain the spin excitation spectrum of CsCoC1 3 with considerable success. As a further test of processes we apply Shiba’s magnon theory excitation to new lowthis approach to nonlinear tceperature data for the bromide, where the interchain ccupling is approximately twice as strong as in the chloride. 2.
7’8 spin The Hamiltonian for a one-dimensional Ising like Is written as riZZantiferromagnct xx yy U 0 — 2JLSS.S.+i+c(S.S.+i+S.S.+i) . (1) j ‘ ~ .‘ .‘ ~
broader band of excitations. On cooling, the spcctrum Is remarkable for the appearance of a large number of discrete lines whose intensities are critically dependent on the crystal temperature. Below 10 K, the spin arrangement in CsCoBr3 comprises that of a triangular (a-b plane) lslng lattice with ipterchain antiferromagnetic nearest—neighbour coupling, J~, very weak ferro— magnetic next-nearest—neighbour interchairi interactions3 J3, and the strong Ising coupling along the chains (o axis). Between 14 K and 1 < become 14 K, these 28.3 K, one-third of the chains paramagnetic, two magnetic phases while are for thought 10 K < T~ to coexist. Above Th all the chains are disordered but retain significant spin correlations along the o axis for temperatures beyond 100 K. Estimates of the strengths of the various magnetic Inter— actions J, J~and can be made from susceptibllity data9 and the Ned temperatures T~and T~1. Thus J 56 cm~, J~/J 0.013 and J~/J -0.03. The fine structure in the low temperature
3
Since CsCoBr3 exhibits 3-0 magnetic order at sufficiently low teriperatures the effec; of the relatively weak interchain coupling, u~i~), must be included so that the model Hamiltnnian has the form:
r
) ~
)J
HIf~’~+2J’ A
j,t
~
I’
+c’(S’~AS~ +sY~s~ ~ p
,
(2)
where A,p ~ and j,t ~ denote summations over n.n chain pairs and n.n intrachain spin pairs respectively. Since c’ u £ u 0.1, the c’J’ term (i.e. the Interchain x-y term) may be Ignored to a good approximation. The remaining Ising interchain interaction in (2) Is treated within a molecular fi-Id approximation (e.g. Scalapino, Imry and Pincus~0) to give a single-chain effective Hamlitonlan: if
—
H 0
CT
T~) Raman spectrum has been variously 3 Recently, ascribed to magnon coetinatlon states the or pertur— multipie batlon theory bound approach states.~ of ishimara and Shiba7 for a single chain has been extended by Shiba8 to include interchain coupling and thereby
Theory
—
~ h.S~
J
~
where h; (-i)’h and h ~ 0. The resulting energy ~bvei scheme Is constructed from the Ndel state IN> which is favoured by the. staggered molecular field h, together with 593
(3)
ONE—DIMENSIONAL MACNON RAMAN SPECTRUM OF CsCoBr
594
3
admixtures of the low lying subs~aceof excited states. These consist 2of— 0 two and blocks Sz — of 1 excitastates tions. Thus forto the S corresponding ik•r — 1: — ~ iS~IN> J ik~r
Js~s
s~ !N’
/~7ii~e
*5
ik~r J+3 j+14 /2/N ~ j e JS~S..4.1S2S s~ ~N>
i.e.
1’2i1’ I
=
j+l j+2
~
1, 2, etc.
0:
12
—
/~7ii ~ i
e
~~.
0. 8 the magnitude h for ench of the two low temperature phasesof may be Following Shiba, easily determined by considering the nett molecular field cxperienced by one chain due to its neighbours (see figure 1). For T < T~1, the a chains arc surrounded by six antiferromagneti— cally coupled neighbours so that the nett field is 6J’. This gives rise to the A series of excitations. In contrast, the nett field
and =
magnon—photon coupling such that their intensities relatevectors elgen to the and response eigenvalues .cS1~ S~>,we of (5)cantouse obtain the the relative strength of the lines, which drop off in intensity as we go to hiaher magnon states. Thus if we now use A to identify the diac~onalizedlinear combination states where the principle contribution comes from ~i~i.•l of (4a), then
(14a)
*3
—
j
Vol. 36, No. 7
ik~r ~S~S IN> j j+l
e ik.ri+. ~ j+l s~ j+2 51N’
(4b)
experienced by the 8 and y chains is zero, giviny rise to an excitation continuum. T~1 < I
j i.e. *2i~ i
1, 2, etc.
Note that the x-y interaction (the c term) has converted the Ising spin—flips (S+ S+S j’ j j+l’
respect to the other chains with equal probability. Hence the molecular fields experienced by the a and B chains are 6.1’, 4J’, 2J’ and 0, with probabilities of 1/8, 3/8, 3/8 and 1/8, respectively (for y chains, h — 0). Thus, in this phase we expect three series of
S~.
2,etc.) into l~dependent propagating modes which correspond to the 5 and soliton the dirner picturegas(moving picture of HubberdU bioch wall) 0 Villain(see also Fowler and Puga12). In the limit h •~ 0, c ~ 0 these states have energies E 2.1 E2I 2J with respect to the Ned state ~ When the molecular field is restored the degenerate energies split to give the series E2~_1 — 2J+2h(2il), and similarly the —x-y 2J+2h(2i). interaction Finally, couples when ~ toe *i+2 is nonzero to mix the Sz — 1 subspace. At the zone centre (~ 0) this requires the diagonalizacion of the tridiagonal matrix
discrete lines comprising the A, 61 and 62 series together with a continuum. 3. ExperIment and results Large single crystals (6OxlO mm2) of CsCoBr3 ware grown by R. Ritchie (Physics Department, University of Canterbury, N.Z.) and B. Briat (Laboratoire d’Optique Physique, Paris, 13 France) using the Bridgeman technique. Raman spectra50were using mW ofrecorded 501.7 nm under argoncomputer laser excitation control and a spectral resolution of 2 cmt. Suitably cleaved samples were mounted in a Thor S500 variable temperature cryostat where the sample is immersed in a helium exchange gas. The
<* 2~_lIH/2Jl*2I,_l>{l+jc2~6i,l÷(2i_l)~} for i c
for
~
—
±
Note that the Sz — 0 subspace mixes with IN> to give a perturbed ground state and a 2) but states do notextending mix with the S2 — from 1 cluster of excited upwards subspace 2.1 + 4h +which 0(e gives a set of energy levels
19>
extending upwards from 2J+2h + O(c ). However, if the Sz — 0 excitations are related to the 2k2 for small ~ <~ ~k’ response then they would have and hence are importance Intensities ofoftheno order of c at the zone centre. Upon diagonalizatlon of (5) the ei~en values Ai/2J may be obtained to order c , in terms of the input parameters h/J and c. Different values of h describe different series of lines in the spectrum. If the 5z — excitations can be described by an effective
sample be maintained to within ±0.2K, temperature but because could of local laser heating
(‘~.3 K) the actual crystal laser filament could only
temperature within the be estimated. Spectra were recorded in the 90° scattering geometry X(. .)Y, where X, y, z correspond to the crystal a, b, c axes, respectively. Typical spectra are presented in figure 2 118.1 166.114 cm~ for T and 6 and K. are The due peakstoatRaman 72.8,scattering 89.9, from the E 1 , E29, E2g and Alg zone-centre phonons, re~pectively.~With the exception of the Elg mode these bands arc forbidden in x(Zx)Y tion effects of and polarization but crystal appear birefringence through thc depolarizastrained cryostat windows. Other features present in the range 90 to 170 cm~ are (ZX) polarized and are due to zone centre magnon scattering analogous to that observed for CsCoCl 1 but extending over a wider range of energies. Frequencies of distinct features are 3, listed in table 1. On warming to 14 K, new bands appear and all bands grow markedly in intensity
Vol. 36, No. 7
ONE—DIMENSIONAL MACNON RAMAN SPECTRUM OF CsCoBr
593
3
P
P
P
C
~1v~J”J~ 100
70
30
FrequenCy, Figure 1
90
cm~
The hexagonal arrangement of cobalt ions in the oh plane of CsCoBr3 showing (a) the fully ordered magnetic structure for T < )II involving up(+) or down(-) spins, and (b) the partly ordered structure for i~l < < where the spin on atom y is now disordered Cu).
except those at 105.2 and 138.4 cm’, which decrease. The behaviour of the weak l6O-cm~ No shifts arebands observable within this bandfrequency is masked by new appearing nearby. temperature ranga. Another temperature dependent feature occurs at 177.7 cm~ in both (ZZ) and (lx) polarization with the intensity ratio lZZ/IZX — 2.5. This band drops even more rapidly in intensity with increasing temperature than the magnon band at 105.2 cm . 4.
60
Comparison with theory
At low temperatures the theory predicts a sequence of sharp excitations (the A series) superimposed on a broad continuum. These bands are predicted to reduce in intensity by a factor l/q in the intermediate phase T 1 < T C T 1ij 1~. From our spectra at 6 K (T
in table 1, are in excellent agreement with is also indeedconfirm an A series peak. experiment1 and that the weak The band relative at i6o cmintensities of these bands are also In good agreement with theory as can be seen In table I. In the intermediate phase, T~ < T c T~, the B series bands dominate the magnon spectrum. Their frequencies are accurately reproduced by theory for h — 0.077 (cf. 4J’/J — 0.073) for the 61 series, and 1 0.045 (cf. 2J’/.) — 0.037) for the ~a series, with .1 and c as given above. However, because of overlap of bands within Bi and 62 series it is difficult to make definite assignments in some cases e.g. the bands near 125 cni’, While this overlap also precludes the extraction of accurate intensity data, there Is a general qualitative agreement between theory and experimtnt apart from the band near 143 cm’. The theory of Shiba as applied here does band at 97.1 any cm~. However, features are the 62 not predict magnetic scattering below observed at 92.6 cm~ and ‘~.87cm~ on either side of the E 1. Although these features29 have lowatintensity phonon 89.9 cm they are reproducible and they maintain their intensity with increasing temperature as shown in figure2. An approach to understanding these low~ intensity features in the spectrum may be made
ONE—DIMENSIONAL MACNON RAZ4AN SPECTRUM OF CsCoBr
596
3
/
S
%
I
~
‘a
‘
~
‘
-,
‘
/
‘
h
‘
a
—~
+ ~Y
/ ~ ,
,
,
Figure 2
,
(bi
(a)
4J’
+
~ 2.1’ +
.1’ (3~8+t~y) .1’ (3i~8-Ay)
h — 0 + 3J’ (~8-~y) make the effective molecular fields differ The effect of these small corrections is to somewhat from the 6:4:2 ratios, as was already seen in our analysis of the A, B1, B2 series. These differences can be used to estimate values forM, ~8 and and thus the uncertainty in l~y the “hl0.002/J’I = 0” molecular field is
,
~
/
—
Vol. 36, No. 7
of the order of J’. This physical picture can thus explain the extra features ,~fthe spectrum in terms of weak transitions corresponding to h .1’, given in table 10 may I as bethe constructed C series. by A considering linear response of the chains more detailedthetheory
Temperature dependence of the 70 to 190 cm~ region of the Raman spectrum polarization with a spectral of CsCoBr3 recorded in x(zx)Y resolution of 2.cm1.
Table 1 Comparison between theory and experiment for the frequencies (in cm~) and relative intensities (in brackets) of peaks in the magnon Raman spectrum of CsCoBr 1 and c 0.13. The molecular field h — hid.3. For the calculations .1 — 50.5 cm A series
B
0.11) Theory
—
Expt.
1 series — 0.077) Expt. Theory
107 (1.00)
101.3
102 (1.00)
97.1
136 (0.12)
125.6
125 (0.22)
112.1
112 (0.62)
160
159 (0.01)
142.7
142 (0.03)
123.5
125 (0.15)
-
86 (1.00) 93 (o.74) 98 (0.56)
158
159 (0.002)
133.4
135
(o.o4)
—
103 (0.47)
149
145 (0.004)
-
108 (0.36)
(0.08)
—
~(l + t~B)
—
—~(l +
—
f(i 4
—
±~(i + ~y)
~)
hy)
—
6J’ + 3.1’ 0 0
96
for T
for
T,~
<
<
i~ <
(AB+~.y)
+ 3d’ (~y—&~) + 3.1’ (L~B-M)
and similarly for ~he 4.1’, 2.1’ and 0 fields which appear for TN < I ~ T 1~. Thus
(1.00)
87 92.6
to evaluate and so on, but is not warianted for the present spectra.
5.
temperature dependent. In fact Ay is expected to go to zero at the transition temperature. Then the molecular fields felt by the a, B and y atoms are for T < Th —
Theory
138.4 (0.25)
where the small correction terms !~a, tiB, ~y are
h a h6 h
Expt.
C series 0.01?) ExpI. Theory
(i~—
1g5.2 (1.00)
as follows. Within the mean—field scheme used here let us write
a
B~series
(i~—0.065)
Spin excitations in CsCoCl3
In view of the satisfactory picture of tie CsCoBr3 spectrum obtained within the analysis of the previous Sections, havewe re-examined the data for CsCoC13. Herewe agaIn use the well defined and unambiguous A series to fix a suitable choice of .1, c, and J’ within the neighbourhood of the values (.1—52 cmt, 8 C These 0.13, values then given used to the 61 and B J~/J = are 0.0056) in predict the literature. 2 series. Since J~in CsCoCI3 is considerably less than that of the bromide we do not expect the C-series type of features to be so important here. The results are given in table 2 and hence the spectrum is consistent with the parameters (.1, .1’) to be expected from susceptibility data.~’ 1 6. Combination band at 178 cm~ As mentioned previously, the intensity of the 178-cm1 band has a marked temperature dependence, and has very low intensity by T1~,
ONE—DIMENSIONAL MACNON RAMAN SPECTRUM OF CsCoBr
Vol. 36, No. 7
3
597
Table 2 1 for frequencies (in cm1) of Comparison peaks in between the magnon the Raman experimental spectrumvalues of CsCoC1 3 and the theoretical values computed using .1 49.2 cm~ and c — 0.13. The predicted relative intensities are given in brackets beside the frequencies.
A series 0.037) Expt. Theory
B1 series
(i~—
(i~— 0.026) Expt.
Theory
90.5
91.3 (1.00)
88.8
106.6
105.5 (0.48)
100.1
116.3
116.5
(0.21)
125.6 (0.07)
—
(0.59)
86.5 94.3
109.0
108.8 (0.35)
101.0
101.2 (0.48)
116.3
116.3 (0.18)
105.0
107.1
—
122.9 (0.09)
112.9 116.3
112.5 (0.24) 117.2 (0.16)
which would explain why this band has not been observed previously. The extremely rapid decrease in intensity when the temperature is increased from 6 to 14 K indicates that the band is associated In some way with the magnetic ordering giving rise to the A series of bands, However, no A-series magnetic scattering is expscted of this intensity at 178 cm~ and the Intensity decreases too rapidly. The band is assigned to E19 phonon-magnon combination scattering for the following reasons. 1 and Firstly, the strongest magnon the intenseA series E19 phonon at at 72.8105.2 cm cm1 can combine to give a combination mode at 178 cm”~. Secondly, the Elq phonon Intensity is spin— 3 and therefore the combination band will have a stronger temperature dependence dependent
88.4 99.8
B2 series — 0.017) Expt. Theory
(1.00)
85.5
(1.00)
914.3 (0.68)
(0.32)
than either parent alone in agreement with experiment. Thirdly, the combination has E19 x E19 — A19 + A29 + E2~symmetry in the D6h point group and therefore is al1owed in (ZZ) polarization. The fact that the band is also observed in (ZX) polarization can only be explained for such a combination by invoking the crystal moanetic space group symmetry, Cm’c2~ for I < ~ For then the combination has 2A1 + 2Bi symmetry in the C2,, point group and it is now active in both diagonal and (ZX) polarizations exclusively. Acknowledgements - We wish to thank Dr. G.D.Jones and Dr. B. Briat for providing the crystals of CsCoBr of his 3 work. and Dr. H. Shiba for providing preprints
References 1.
2.
3. 4.
5. 6.
7.
h,l. Bretling, Il. Lehmann, T.P. Srinivasan and R. Weber, Solid State Conwnun. 24, 267 (1977). — I.W. Johnstonc and D.J. Lockwood, Solid State Commun. 32, 285 (1979). 1.14. Johnstonc and L. Dubicki, J. Phys. C: Solid State Phys. 13, (1980) in press. J. des Cloizeaux a’ H. Gaudin, J. Math. Phys. 7, 1384 (1966). .3. Villain, Physica 798, 1 (1975). N. Fowler, Phys. RevT”Bl7, 2989 (1976). H. ishimura and H. Shiba, Prog. Theor. Phys. 63, 743 (1980).
8.
9. 10. 11. 12. 13. 14.
H. Shiba, .Prog. Theor. Phys. to be published. %4.B. Yelon, D.E. Cox and H. Eibschutz, Phys. Rev. 612, 5007 (1975). D.J. Scalapino, V. Imry and P. Pincus, Phys. Rev. Bll, 2042 (1975). .1. Hubbard, Phys. Rev. B17, 494 (1978). H. Fowler and H. Puga, Pi~s. Rev. Bl8, 421 (1978). N.L. Rowell, D.J. Lockwood and P. Grant, .1. Raman Spectroscopy in press. N. Achiwa, .1. Phys. Soc. Japan 27, 561 (1969).
593—597. Solid State Communications, Vol.36,in pp. Pergamon Press Ltd. 1980. Printed Great Britain.
INFLUENCE OF %WTERCHMN COUPLING ON THE ONE-DIMENSIONAL MACNON RAMAN SPECTRUM OF CsCoBr 3 LW. Johnstone, D.J. Lockwood and N.W.C. Dharma—wardana Division of Physics, National Research Council, Ottawa, Canada K1A 0R6 (Received 30 July, 1980 by M.F. Collins
Light scattering from magnons In CsCoBr3 has been measured for temperatures well below the upper antiferromagnetic 3D ordering temperature of T~— 28 K. These experiments reveal 1 similar multiple to those magnon found in CsCoClof energies in the range 90 to 170 cm features 3 but previously unobserved for the bromide. Frominent features in the spectrum and their temperature dependence are described in terms of a recent theory by Shiba. Other, weaker features are explained by a simple extension of 1~whosf the theory Intensity to include drops fluctuations. sharply prior the band ~ower orderingat transition at TN1 — 10 K. This band is Ato new is 3D observed 178 cm assigned to magnon—phonon combination scattering.
1.
Introduction
3have been difficult Until now, the spin-wave Raman spectra to of understand terms of current theories of IsingCsCoCl3 and in CsCoBr3~ Hc!senbcrg chains, which are themselves the subject of much debate.”8 Both materials simulate i-D Ising antiferromagnetic chains but display 3—0 magnetic order at sufficiently low temperatures. In CsCoBr 3, the large intrachain nearest—neighbour exchange interaction, J, gives 100 For T ‘ 30 spin K the Raman spectrum rise cm~. to a transverse excitation band near appears as an asymmetric peak superimposed on a
explain the spin excitation spectrum of CsCoC1 3 with considerable success. As a further test of processes we apply Shiba’s magnon theory excitation to new lowthis approach to nonlinear tceperature data for the bromide, where the interchain ccupling is approximately twice as strong as in the chloride. 2.
7’8 spin The Hamiltonian for a one-dimensional Ising like Is written as riZZantiferromagnct xx yy U 0 — 2JLSS.S.+i+c(S.S.+i+S.S.+i) . (1) j ‘ ~ .‘ .‘ ~
broader band of excitations. On cooling, the spcctrum Is remarkable for the appearance of a large number of discrete lines whose intensities are critically dependent on the crystal temperature. Below 10 K, the spin arrangement in CsCoBr3 comprises that of a triangular (a-b plane) lslng lattice with ipterchain antiferromagnetic nearest—neighbour coupling, J~, very weak ferro— magnetic next-nearest—neighbour interchairi interactions3 J3, and the strong Ising coupling along the chains (o axis). Between 14 K and 1 < become 14 K, these 28.3 K, one-third of the chains paramagnetic, two magnetic phases while are for thought 10 K < T~ to coexist. Above Th all the chains are disordered but retain significant spin correlations along the o axis for temperatures beyond 100 K. Estimates of the strengths of the various magnetic Inter— actions J, J~and can be made from susceptibllity data9 and the Ned temperatures T~and T~1. Thus J 56 cm~, J~/J 0.013 and J~/J -0.03. The fine structure in the low temperature
3
Since CsCoBr3 exhibits 3-0 magnetic order at sufficiently low teriperatures the effec; of the relatively weak interchain coupling, u~i~), must be included so that the model Hamiltnnian has the form:
r
) ~
)J
HIf~’~+2J’ A
j,t
~
I’
+c’(S’~AS~ +sY~s~ ~ p
,
(2)
where A,p ~ and j,t ~ denote summations over n.n chain pairs and n.n intrachain spin pairs respectively. Since c’ u £ u 0.1, the c’J’ term (i.e. the Interchain x-y term) may be Ignored to a good approximation. The remaining Ising interchain interaction in (2) Is treated within a molecular fi-Id approximation (e.g. Scalapino, Imry and Pincus~0) to give a single-chain effective Hamlitonlan: if
—
H 0
CT
T~) Raman spectrum has been variously 3 Recently, ascribed to magnon coetinatlon states the or pertur— multipie batlon theory bound approach states.~ of ishimara and Shiba7 for a single chain has been extended by Shiba8 to include interchain coupling and thereby
Theory
—
~ h.S~
J
~
where h; (-i)’h and h ~ 0. The resulting energy ~bvei scheme Is constructed from the Ndel state IN> which is favoured by the. staggered molecular field h, together with 593
(3)
ONE—DIMENSIONAL MACNON RAMAN SPECTRUM OF CsCoBr
594
3
admixtures of the low lying subs~aceof excited states. These consist 2of— 0 two and blocks Sz — of 1 excitastates tions. Thus forto the S corresponding ik•r — 1: — ~ iS~IN> J ik~r
Js~s
s~ !N’
/~7ii~e
*5
ik~r J+3 j+14 /2/N ~ j e JS~S..4.1S2S s~ ~N>
i.e.
1’2i1’ I
=
j+l j+2
~
1, 2, etc.
0:
12
—
/~7ii ~ i
e
0. 8 the magnitude h for ench of the two low temperature phasesof may be Following Shiba, easily determined by considering the nett molecular field cxperienced by one chain due to its neighbours (see figure 1). For T < T~1, the a chains arc surrounded by six antiferromagneti— cally coupled neighbours so that the nett field is 6J’. This gives rise to the A series of excitations. In contrast, the nett field
and =
magnon—photon coupling such that their intensities relatevectors elgen to the and response eigenvalues .cS1~ S~>,we of (5)cantouse obtain the the relative strength of the lines, which drop off in intensity as we go to hiaher magnon states. Thus if we now use A to identify the diac~onalizedlinear combination states where the principle contribution comes from ~i~i.•l of (4a), then
(14a)
*3
—
j
Vol. 36, No. 7
ik~r ~S~S IN> j j+l
e ik.ri+. ~ j+l s~ j+2 51N’
(4b)
experienced by the 8 and y chains is zero, giviny rise to an excitation continuum. T~1 < I
j i.e. *2i~ i
1, 2, etc.
Note that the x-y interaction (the c term) has converted the Ising spin—flips (S+ S+S j’ j j+l’
respect to the other chains with equal probability. Hence the molecular fields experienced by the a and B chains are 6.1’, 4J’, 2J’ and 0, with probabilities of 1/8, 3/8, 3/8 and 1/8, respectively (for y chains, h — 0). Thus, in this phase we expect three series of
S~.
2,etc.) into l~dependent propagating modes which correspond to the 5 and soliton the dirner picturegas(moving picture of HubberdU bioch wall) 0 Villain(see also Fowler and Puga12). In the limit h •~ 0, c ~ 0 these states have energies E 2.1 E2I 2J with respect to the Ned state ~ When the molecular field is restored the degenerate energies split to give the series E2~_1 — 2J+2h(2il), and similarly the —x-y 2J+2h(2i). interaction Finally, couples when ~ toe *i+2 is nonzero to mix the Sz — 1 subspace. At the zone centre (~ 0) this requires the diagonalizacion of the tridiagonal matrix
discrete lines comprising the A, 61 and 62 series together with a continuum. 3. ExperIment and results Large single crystals (6OxlO mm2) of CsCoBr3 ware grown by R. Ritchie (Physics Department, University of Canterbury, N.Z.) and B. Briat (Laboratoire d’Optique Physique, Paris, 13 France) using the Bridgeman technique. Raman spectra50were using mW ofrecorded 501.7 nm under argoncomputer laser excitation control and a spectral resolution of 2 cmt. Suitably cleaved samples were mounted in a Thor S500 variable temperature cryostat where the sample is immersed in a helium exchange gas. The
<* 2~_lIH/2Jl*2I,_l>{l+jc2~6i,l÷(2i_l)~} for i c
for
~
—
±
Note that the Sz — 0 subspace mixes with IN> to give a perturbed ground state and a 2) but states do notextending mix with the S2 — from 1 cluster of excited upwards subspace 2.1 + 4h +which 0(e gives a set of energy levels
19>
extending upwards from 2J+2h + O(c ). However, if the Sz — 0 excitations are related to the 2k2 for small ~ <~ ~k’ response then they would have and hence are importance Intensities ofoftheno order of c at the zone centre. Upon diagonalizatlon of (5) the ei~en values Ai/2J may be obtained to order c , in terms of the input parameters h/J and c. Different values of h describe different series of lines in the spectrum. If the 5z — excitations can be described by an effective
sample be maintained to within ±0.2K, temperature but because could of local laser heating
(‘~.3 K) the actual crystal laser filament could only
temperature within the be estimated. Spectra were recorded in the 90° scattering geometry X(. .)Y, where X, y, z correspond to the crystal a, b, c axes, respectively. Typical spectra are presented in figure 2 118.1 166.114 cm~ for T and 6 and K. are The due peakstoatRaman 72.8,scattering 89.9, from the E 1 , E29, E2g and Alg zone-centre phonons, re~pectively.~With the exception of the Elg mode these bands arc forbidden in x(Zx)Y tion effects of and polarization but crystal appear birefringence through thc depolarizastrained cryostat windows. Other features present in the range 90 to 170 cm~ are (ZX) polarized and are due to zone centre magnon scattering analogous to that observed for CsCoCl 1 but extending over a wider range of energies. Frequencies of distinct features are 3, listed in table 1. On warming to 14 K, new bands appear and all bands grow markedly in intensity
Vol. 36, No. 7
ONE—DIMENSIONAL MACNON RAMAN SPECTRUM OF CsCoBr
593
3
P
P
P
C
~1v~J”J~ 100
70
30
FrequenCy, Figure 1
90
cm~
The hexagonal arrangement of cobalt ions in the oh plane of CsCoBr3 showing (a) the fully ordered magnetic structure for T < )II involving up(+) or down(-) spins, and (b) the partly ordered structure for i~l < < where the spin on atom y is now disordered Cu).
except those at 105.2 and 138.4 cm’, which decrease. The behaviour of the weak l6O-cm~ No shifts arebands observable within this bandfrequency is masked by new appearing nearby. temperature ranga. Another temperature dependent feature occurs at 177.7 cm~ in both (ZZ) and (lx) polarization with the intensity ratio lZZ/IZX — 2.5. This band drops even more rapidly in intensity with increasing temperature than the magnon band at 105.2 cm . 4.
60
Comparison with theory
At low temperatures the theory predicts a sequence of sharp excitations (the A series) superimposed on a broad continuum. These bands are predicted to reduce in intensity by a factor l/q in the intermediate phase T 1 < T C T 1ij 1~. From our spectra at 6 K (T
in table 1, are in excellent agreement with is also indeedconfirm an A series peak. experiment1 and that the weak The band relative at i6o cmintensities of these bands are also In good agreement with theory as can be seen In table I. In the intermediate phase, T~ < T c T~, the B series bands dominate the magnon spectrum. Their frequencies are accurately reproduced by theory for h — 0.077 (cf. 4J’/J — 0.073) for the 61 series, and 1 0.045 (cf. 2J’/.) — 0.037) for the ~a series, with .1 and c as given above. However, because of overlap of bands within Bi and 62 series it is difficult to make definite assignments in some cases e.g. the bands near 125 cni’, While this overlap also precludes the extraction of accurate intensity data, there Is a general qualitative agreement between theory and experimtnt apart from the band near 143 cm’. The theory of Shiba as applied here does band at 97.1 any cm~. However, features are the 62 not predict magnetic scattering below observed at 92.6 cm~ and ‘~.87cm~ on either side of the E 1. Although these features29 have lowatintensity phonon 89.9 cm they are reproducible and they maintain their intensity with increasing temperature as shown in figure2. An approach to understanding these low~ intensity features in the spectrum may be made
ONE—DIMENSIONAL MACNON RAZ4AN SPECTRUM OF CsCoBr
596
3
/
S
%
I
~
‘a
‘
~
‘
-,
‘
/
‘
h
‘
a
—~
+ ~Y
/ ~ ,
,
,
Figure 2
,
(bi
(a)
4J’
+
~ 2.1’ +
.1’ (3~8+t~y) .1’ (3i~8-Ay)
h — 0 + 3J’ (~8-~y) make the effective molecular fields differ The effect of these small corrections is to somewhat from the 6:4:2 ratios, as was already seen in our analysis of the A, B1, B2 series. These differences can be used to estimate values forM, ~8 and and thus the uncertainty in l~y the “hl0.002/J’I = 0” molecular field is
,
~
/
—
Vol. 36, No. 7
of the order of J’. This physical picture can thus explain the extra features ,~fthe spectrum in terms of weak transitions corresponding to h .1’, given in table 10 may I as bethe constructed C series. by A considering linear response of the chains more detailedthetheory
Temperature dependence of the 70 to 190 cm~ region of the Raman spectrum polarization with a spectral of CsCoBr3 recorded in x(zx)Y resolution of 2.cm1.
Table 1 Comparison between theory and experiment for the frequencies (in cm~) and relative intensities (in brackets) of peaks in the magnon Raman spectrum of CsCoBr 1 and c 0.13. The molecular field h — hid.3. For the calculations .1 — 50.5 cm A series
B
0.11) Theory
—
Expt.
1 series — 0.077) Expt. Theory
107 (1.00)
101.3
102 (1.00)
97.1
136 (0.12)
125.6
125 (0.22)
112.1
112 (0.62)
160
159 (0.01)
142.7
142 (0.03)
123.5
125 (0.15)
-
86 (1.00) 93 (o.74) 98 (0.56)
158
159 (0.002)
133.4
135
(o.o4)
—
103 (0.47)
149
145 (0.004)
-
108 (0.36)
(0.08)
—
~(l + t~B)
—
—~(l +
—
f(i 4
—
±~(i + ~y)
~)
hy)
—
6J’ + 3.1’ 0 0
96
for T
for
T,~
<
<
i~ <
(AB+~.y)
+ 3d’ (~y—&~) + 3.1’ (L~B-M)
and similarly for ~he 4.1’, 2.1’ and 0 fields which appear for TN < I ~ T 1~. Thus
(1.00)
87 92.6
to evaluate
5.
temperature dependent. In fact Ay is expected to go to zero at the transition temperature. Then the molecular fields felt by the a, B and y atoms are for T < Th —
Theory
138.4 (0.25)
where the small correction terms !~a, tiB, ~y are
h a h6 h
Expt.
C series 0.01?) ExpI. Theory
(i~—
1g5.2 (1.00)
as follows. Within the mean—field scheme used here let us write
a
B~series
(i~—0.065)
Spin excitations in CsCoCl3
In view of the satisfactory picture of tie CsCoBr3 spectrum obtained within the analysis of the previous Sections, havewe re-examined the data for CsCoC13. Herewe agaIn use the well defined and unambiguous A series to fix a suitable choice of .1, c, and J’ within the neighbourhood of the values (.1—52 cmt, 8 C These 0.13, values then given used to the 61 and B J~/J = are 0.0056) in predict the literature. 2 series. Since J~in CsCoCI3 is considerably less than that of the bromide we do not expect the C-series type of features to be so important here. The results are given in table 2 and hence the spectrum is consistent with the parameters (.1, .1’) to be expected from susceptibility data.~’ 1 6. Combination band at 178 cm~ As mentioned previously, the intensity of the 178-cm1 band has a marked temperature dependence, and has very low intensity by T1~,
ONE—DIMENSIONAL MACNON RAMAN SPECTRUM OF CsCoBr
Vol. 36, No. 7
3
597
Table 2 1 for frequencies (in cm1) of Comparison peaks in between the magnon the Raman experimental spectrumvalues of CsCoC1 3 and the theoretical values computed using .1 49.2 cm~ and c — 0.13. The predicted relative intensities are given in brackets beside the frequencies.
A series 0.037) Expt. Theory
B1 series
(i~—
(i~— 0.026) Expt.
Theory
90.5
91.3 (1.00)
88.8
106.6
105.5 (0.48)
100.1
116.3
116.5
(0.21)
125.6 (0.07)
—
(0.59)
86.5 94.3
109.0
108.8 (0.35)
101.0
101.2 (0.48)
116.3
116.3 (0.18)
105.0
107.1
—
122.9 (0.09)
112.9 116.3
112.5 (0.24) 117.2 (0.16)
which would explain why this band has not been observed previously. The extremely rapid decrease in intensity when the temperature is increased from 6 to 14 K indicates that the band is associated In some way with the magnetic ordering giving rise to the A series of bands, However, no A-series magnetic scattering is expscted of this intensity at 178 cm~ and the Intensity decreases too rapidly. The band is assigned to E19 phonon-magnon combination scattering for the following reasons. 1 and Firstly, the strongest magnon the intenseA series E19 phonon at at 72.8105.2 cm cm1 can combine to give a combination mode at 178 cm”~. Secondly, the Elq phonon Intensity is spin— 3 and therefore the combination band will have a stronger temperature dependence dependent
88.4 99.8
B2 series — 0.017) Expt. Theory
(1.00)
85.5
(1.00)
914.3 (0.68)
(0.32)
than either parent alone in agreement with experiment. Thirdly, the combination has E19 x E19 — A19 + A29 + E2~symmetry in the D6h point group and therefore is al1owed in (ZZ) polarization. The fact that the band is also observed in (ZX) polarization can only be explained for such a combination by invoking the crystal moanetic space group symmetry, Cm’c2~ for I < ~ For then the combination has 2A1 + 2Bi symmetry in the C2,, point group and it is now active in both diagonal and (ZX) polarizations exclusively. Acknowledgements - We wish to thank Dr. G.D.Jones and Dr. B. Briat for providing the crystals of CsCoBr of his 3 work. and Dr. H. Shiba for providing preprints
References 1.
2.
3. 4.
5. 6.
7.
h,l. Bretling, Il. Lehmann, T.P. Srinivasan and R. Weber, Solid State Conwnun. 24, 267 (1977). — I.W. Johnstonc and D.J. Lockwood, Solid State Commun. 32, 285 (1979). 1.14. Johnstonc and L. Dubicki, J. Phys. C: Solid State Phys. 13, (1980) in press. J. des Cloizeaux a’ H. Gaudin, J. Math. Phys. 7, 1384 (1966). .3. Villain, Physica 798, 1 (1975). N. Fowler, Phys. RevT”Bl7, 2989 (1976). H. ishimura and H. Shiba, Prog. Theor. Phys. 63, 743 (1980).
8.
9. 10. 11. 12. 13. 14.
H. Shiba, .Prog. Theor. Phys. to be published. %4.B. Yelon, D.E. Cox and H. Eibschutz, Phys. Rev. 612, 5007 (1975). D.J. Scalapino, V. Imry and P. Pincus, Phys. Rev. Bll, 2042 (1975). .1. Hubbard, Phys. Rev. B17, 494 (1978). H. Fowler and H. Puga, Pi~s. Rev. Bl8, 421 (1978). N.L. Rowell, D.J. Lockwood and P. Grant, .1. Raman Spectroscopy in press. N. Achiwa, .1. Phys. Soc. Japan 27, 561 (1969).