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Pairing of kinks observed in TMMC below TN
Journal of Magnetism and Magnetic Materials 104-107 (1992) 1077-1078 North-Holland
Pairing of kinks observed in T M M C below T N H. B e n n e r
a,
J.A. Hob'st a,b and J. L 6 w "
Institut fiir Festk6rperphysik, Technische Hochschule Darmstadt, W-6100 Darmstadt, Germany b Institute of Physics, Warsaw Technical University, 00-662 Warsaw, Poland By nuclear spin-lattice relaxation we have studied the soliton dynamics in the linear-chain antiferromagnet (CH3)4NMnCI3 (TMMC) below Try. The observed field and temperature dependences of T 1 clearly indicate a change of topology from "rr- to 2w-kinks accompanied by a doubling of their excitation energy. Nonlinear excitations in the form of kink solitons play an important role in understanding the low-temperature properties of quasi-one-dimensional (1D) magnets [1,2]. Until recently, solitons were considered as a specific property of one-dimensionality and only expected to occur above the three-dimensional (3D) ordering temperature. For weakly coupled chains like e.g. in T M M C , however, it was shown [3] that solitons should also occur in the 3D ordered phase: "rr-kink solitons on the same chain, moving independently above To, are bound into pairs below Tc. The pairing is effected through weak interchain interactions lifting the degeneracy of the ground state and giving rise to 3D long-range order. The transition results in a change of topology of the solitons and, moreover, in a doubling of their activation energy which should affect the soliton density in a dramatic way [3]. We have studied this effect by N M R in the 1D antiferromagnet T M M C above and below T N. The spin-lattice relaxation time T 1 of I4N was measured as a function of temperature and magnetic field within the ranges 34 k O e _ < H < 5 4 kOe and 1.6 K < T _ < 4 . 2 K. The ordering temperature of T M M C shows a m o n o t o n o u s increase from 0.85 K at zero magnetic field up to 3 K at about 100 kOe. The transition from the 1D to the 3D ordered phase was checked from the shape of the N M R spectrum: at TN(H) there occurs a characteristic splitting of lines. Above IN, in the 1D regime, T I data show a universal exponential decrease with respect to H / T [4,5]. Below TN, instead of a decrease a much steeper increase is observed which is also exponential in H~ T, but no longer universal for different fields (see fig. la). An adequate Hamiltonian for describing the low-temperature dynamics of T M M C is the following [3]:
9f = Y ' { - 2JSj,, "Sj,,+, + A ( S ; / ) 2 - glxBHS~.I}
,
10- l
(1)
The first part in braces represents the antiferromagnetic (afro) spin chains (J < 0) with an easy-plane anisotropy (A > 0) perpendicular to the chain direction £ and a transverse magnetic field 14£ (l refers to the site on the chain). Interchain interactions between neighbouring chains j and j ' , though mainly of dipolar origin, are described for simplicity by isotropic ferromagnetic coupling ( J ' > 0). For T M M C we have S = 5 / 2 , J / k n = - 6 . 7 K, A / k B = 0 . 3 K, g =2.01 and J ' / k B ranging from 0.1 to 1 m K [5,6].
I
.
.
.
.
'
,
' - -
I /
'
/_/
a)
T
O \Ox "1~'
1D
"~'.
3D
10-~
, , f~,
,
,
],
10-4 10
15
20
25
30
/,
TMIC 101°
b)
lo ~ 1° 8
[]
1D o n
o n o noo n
10'
40
35
1011
o o
10
ja-j',l
"
THHC
,
j,l
- ~-'~' J'Sj, I'S/, I.
Above TN, in absence of correlation between neighbouring chains, the last term of eq. (1) can be disregarded. Since the ground state of a single chain is represented by a spin-flop configuration, there exist two degenerate, but topologically inequivalent sublattice orientations in + y-direction. Transi-
I
15
~ /
S-
o o':'o° o ~ / I
I
20
(i
~(H)H/T.[kOe/K] I
25
,
I 30
i
I 35
I
40
Fig. 1. Field and temperature dependence of the spin-lattice relaxation time T l in TMMC. O p e n symbols: T >_ TN, full symbols: T < TN; H = 34.7 kOe (© t), 41.1 kOe (o i ) , 47.3 kOe (zx A) and 53.7 kOe (D II). (a) Universal behaviour of T 1 / H in the ID regime according to eq. (3). Dashed line yields the " I D slope" a . . . . p = (0.28_+ 0.02) K / k O e . Full lines correspond to slopes and distances predicted by eqs. (4) and (5) for the 3D regime; (b) universal behaviour of T t H 6 / T 3 in the 3D regime, including the renormalization due to "3D collisions", sO(H) denotes the ratio Ez~/2E,, , see eq. (4). Full line represents the theoretical results of eqs. (4)-(6) and yields a " 3 D slope" of a 2. . . . p = (0.57_+ 0.03) K / k O e .
0312-8853/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
H. Benner et al. / Pairing of kinks obsert,ed in TMMC below Z~,
1078
tions between them are described in terms of sine-Gordon-like solitons [1], which can be visualized by v-twists of the chain, reversing both sublattices when passing by. It was shown [4,5] that within the investigated regimes of H and T( > Try) the dominating contribution to 1/T~ results from soliton-induced sublattice flips, giving rise to large fluctuations of the local hyperfine field:
I/T, ~ A~S2/F.
(2)
A,~ denotes the corresponding hyperfine coefficient. The sublattice flipping rate I" = 4nvP,~ is proportional to the density n~ and average velocity P,~ of the v-kink solitons. In terms of field and temperature this rate can be expressed by [5]
1/T I~H
i exp(+os.~H/T).
(3)
e% represents a measure for the activation energy of a static v-kink E,~/k n =-a~H, which can be directly obtained from experiment. Below T N the weak (dipolar) couplings between neighbouring chains tend to align facing spins parallel to each other giving rise to 3D long-range order. The occurrence of two separate v-kinks on one of the chains would reverse the afm sublattices on a certain part of the chain and align the facing spins of neighbouring chains in an unfavourable antiparallel orientation. The corresponding increase of energy can be minimized if two kinks (or antikinks) move towards each other up to a certain minimum distance forming a + 2 v ( 2 v ) twist of the chain, since for topological reasons they cannot annihilate. A quantitative description of w-kink pairing has been given in terms of a double-sine-Gordon (DSG) equation [3] by mapping the effect of interchain interactions to a 1D staggered mean field. Analytic solutions of this equation which, in fact, resemble two coupled v-kinks and the respective thermodynamics have been treated in the literature [7]. The activation energy of such 2w-kinks exceeds that of a single v-kink by roughly a factor of 2:
E2~=2E~
l+x
+ x arsinh
, x
(glxBH)2
(4)
earlier results [4,5]. The deviation from the expected classical value cx,~,d = glx13/kB--0.34 K / k O e has been attributed to out-of-plane fluctuations and quantum effects. The data from the 3D regime, taken at four different fields, are no longer universal (mainly because of the field dependence of parameter x in eq. (4)). However, the respective slopes and distances are in very good agreement with eqs. (4) and (5) proving, in fact, the expected approximate doubling of activation energy. Surprisingly - and in contrast to the situation above T N [4] - the absolute values of 1 / T E data below 7"N are much larger than predicted [8]. The observed enhancement could be attributed to collisions between kinks and linear excitations. For the 1D case a renormalization of the soliton density due to collisions between kinks and magnons propagating in chain direction has been calculated and included in the standard expressions for n~ and n2= [I,7]. In presence of 3D order an additional effect should result from magnons propagating perpendicularly to the chains. We have shown [10] that such "3D collisions" result in a field and temperature dependent renormalization increasing the 2vkink density by a factor
( A ~ 2 ( H , T ) ~- 1+41~
)
(g#nH)4 4 S 2 j , I j [ ( k , f f ) 2"
(6)
At H = 40 kOe and T = 2 K this factor amounts to 1300 (!) which mainly accounts for the observed enhancement of 1 / T . In fig. lb the same data have been represented in a plot emphasizing the asymptotic behaviour in the 3D regime, eqs. (4) and (5), and including the H and T dependences of the renormalization factor (6). The data taken below 7'~ (full symbols) now follow a straight line and show completely universal behaviour. The best fit of the slope was obtained for .l' = ( 0 . 9 + 0 . 1 ) inK, which is consistent with the literature [6], and 2t~ . . . . p (0.57+_0.03) K / k O e , which almost exactly agrees with our prevkms 1D result. This project of SFB 185 "Nichtlineare Dynamik" was supported by the Deutsche Forschungsgemeinschaft.
References approaching 2E.~ = 2o%Hk B in the limit x ~ 0, i.e. for large magnetic fields. The corresponding relaxation rate is described by the following asymptotic expression [8]:
1 / T I ~ lq2T 1 e x p ( - E2.~/kBT).
(5)
The reverse exponential H~ T dependences of the spin-lattice relaxation time above and below T N - comparing eqs. (3) and (5) - are related to the different topology of v-kinks and 2"rr-kinks: Above T N (two degenerate ground states) the effect of sublattice flipping on I / T I is proportional to the incerse v-kink density n~, 1 ~ exp(+ E~,/knT). Below T N (single ground state) spin fluctuations only arise inside the moving 2w-kinks, so their effect on 1 / T 1 is directly proportional to n2,~ ~ e x p ( - E 2 . ~ / k ~ T ) . T 1 data from both above and below T N are compared in fig. la using a universal plot which should be appropriate only for the ID regime. Indeed, for T > TN(H) our data follow the expected universal behaviour of eq. 13), and a~ was evaluated from the slope to be 1/).28 + 0.02) K / k O e , in accordance with
[1] H.J. Mikeska, J. Phys. C 11 (1978) L29, C 13 (1980) 2913. [2] Nonlinearity in Condensed Matter, eds. A.R. Bishop et al. (Springer, Berlin, 1987). [3] J.A. Holyst and A. Sukiennicki, Phys. Rev. B 38 (1988) 6975. J.A. Hotyst, Z. Phys. B 74 (1989) 341. [4] J.P. Boucher and J.P. Renard, Phys. Rev. Lett. 45 (1980) 486. [5] ft. Benner, H. Seitz, J. Wiese and J.P. Boucher, J. Magn. Magn. Mater. 45 (1984) 354. [6] C. Dupas and J.P. Renard, Solid State Commun. 20 (1976) 581. M. Steiner, J. Villain and C.G. Windsor, Adv. Phys. 25 (1976) 87. [7] K.M. Leung, Phys. Rev. B 26 11982) 226, B 27 11983) 2877. [8] J.A. Holyst and H. Benner, Solid State Commun. 72 11989) 385. [9] J.F. Currie et al., Phys. Rev. B 22 11980) 477. [10] J.A. Hotyst and H. Benner, Solid State Commun. 79 11991) 703.
Pairing of kinks observed in T M M C below T N H. B e n n e r
a,
J.A. Hob'st a,b and J. L 6 w "
Institut fiir Festk6rperphysik, Technische Hochschule Darmstadt, W-6100 Darmstadt, Germany b Institute of Physics, Warsaw Technical University, 00-662 Warsaw, Poland By nuclear spin-lattice relaxation we have studied the soliton dynamics in the linear-chain antiferromagnet (CH3)4NMnCI3 (TMMC) below Try. The observed field and temperature dependences of T 1 clearly indicate a change of topology from "rr- to 2w-kinks accompanied by a doubling of their excitation energy. Nonlinear excitations in the form of kink solitons play an important role in understanding the low-temperature properties of quasi-one-dimensional (1D) magnets [1,2]. Until recently, solitons were considered as a specific property of one-dimensionality and only expected to occur above the three-dimensional (3D) ordering temperature. For weakly coupled chains like e.g. in T M M C , however, it was shown [3] that solitons should also occur in the 3D ordered phase: "rr-kink solitons on the same chain, moving independently above To, are bound into pairs below Tc. The pairing is effected through weak interchain interactions lifting the degeneracy of the ground state and giving rise to 3D long-range order. The transition results in a change of topology of the solitons and, moreover, in a doubling of their activation energy which should affect the soliton density in a dramatic way [3]. We have studied this effect by N M R in the 1D antiferromagnet T M M C above and below T N. The spin-lattice relaxation time T 1 of I4N was measured as a function of temperature and magnetic field within the ranges 34 k O e _ < H < 5 4 kOe and 1.6 K < T _ < 4 . 2 K. The ordering temperature of T M M C shows a m o n o t o n o u s increase from 0.85 K at zero magnetic field up to 3 K at about 100 kOe. The transition from the 1D to the 3D ordered phase was checked from the shape of the N M R spectrum: at TN(H) there occurs a characteristic splitting of lines. Above IN, in the 1D regime, T I data show a universal exponential decrease with respect to H / T [4,5]. Below TN, instead of a decrease a much steeper increase is observed which is also exponential in H~ T, but no longer universal for different fields (see fig. la). An adequate Hamiltonian for describing the low-temperature dynamics of T M M C is the following [3]:
9f = Y ' { - 2JSj,, "Sj,,+, + A ( S ; / ) 2 - glxBHS~.I}
,
10- l
(1)
The first part in braces represents the antiferromagnetic (afro) spin chains (J < 0) with an easy-plane anisotropy (A > 0) perpendicular to the chain direction £ and a transverse magnetic field 14£ (l refers to the site on the chain). Interchain interactions between neighbouring chains j and j ' , though mainly of dipolar origin, are described for simplicity by isotropic ferromagnetic coupling ( J ' > 0). For T M M C we have S = 5 / 2 , J / k n = - 6 . 7 K, A / k B = 0 . 3 K, g =2.01 and J ' / k B ranging from 0.1 to 1 m K [5,6].
I
.
.
.
.
'
,
' - -
I /
'
/_/
a)
T
O \Ox "1~'
1D
"~'.
3D
10-~
, , f~,
,
,
],
10-4 10
15
20
25
30
/,
TMIC 101°
b)
lo ~ 1° 8
[]
1D o n
o n o noo n
10'
40
35
1011
o o
10
ja-j',l
"
THHC
,
j,l
- ~-'~' J'Sj, I'S/, I.
Above TN, in absence of correlation between neighbouring chains, the last term of eq. (1) can be disregarded. Since the ground state of a single chain is represented by a spin-flop configuration, there exist two degenerate, but topologically inequivalent sublattice orientations in + y-direction. Transi-
I
15
~ /
S-
o o':'o° o ~ / I
I
20
(i
~(H)H/T.[kOe/K] I
25
,
I 30
i
I 35
I
40
Fig. 1. Field and temperature dependence of the spin-lattice relaxation time T l in TMMC. O p e n symbols: T >_ TN, full symbols: T < TN; H = 34.7 kOe (© t), 41.1 kOe (o i ) , 47.3 kOe (zx A) and 53.7 kOe (D II). (a) Universal behaviour of T 1 / H in the ID regime according to eq. (3). Dashed line yields the " I D slope" a . . . . p = (0.28_+ 0.02) K / k O e . Full lines correspond to slopes and distances predicted by eqs. (4) and (5) for the 3D regime; (b) universal behaviour of T t H 6 / T 3 in the 3D regime, including the renormalization due to "3D collisions", sO(H) denotes the ratio Ez~/2E,, , see eq. (4). Full line represents the theoretical results of eqs. (4)-(6) and yields a " 3 D slope" of a 2. . . . p = (0.57_+ 0.03) K / k O e .
0312-8853/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
H. Benner et al. / Pairing of kinks obsert,ed in TMMC below Z~,
1078
tions between them are described in terms of sine-Gordon-like solitons [1], which can be visualized by v-twists of the chain, reversing both sublattices when passing by. It was shown [4,5] that within the investigated regimes of H and T( > Try) the dominating contribution to 1/T~ results from soliton-induced sublattice flips, giving rise to large fluctuations of the local hyperfine field:
I/T, ~ A~S2/F.
(2)
A,~ denotes the corresponding hyperfine coefficient. The sublattice flipping rate I" = 4nvP,~ is proportional to the density n~ and average velocity P,~ of the v-kink solitons. In terms of field and temperature this rate can be expressed by [5]
1/T I~H
i exp(+os.~H/T).
(3)
e% represents a measure for the activation energy of a static v-kink E,~/k n =-a~H, which can be directly obtained from experiment. Below T N the weak (dipolar) couplings between neighbouring chains tend to align facing spins parallel to each other giving rise to 3D long-range order. The occurrence of two separate v-kinks on one of the chains would reverse the afm sublattices on a certain part of the chain and align the facing spins of neighbouring chains in an unfavourable antiparallel orientation. The corresponding increase of energy can be minimized if two kinks (or antikinks) move towards each other up to a certain minimum distance forming a + 2 v ( 2 v ) twist of the chain, since for topological reasons they cannot annihilate. A quantitative description of w-kink pairing has been given in terms of a double-sine-Gordon (DSG) equation [3] by mapping the effect of interchain interactions to a 1D staggered mean field. Analytic solutions of this equation which, in fact, resemble two coupled v-kinks and the respective thermodynamics have been treated in the literature [7]. The activation energy of such 2w-kinks exceeds that of a single v-kink by roughly a factor of 2:
E2~=2E~
l+x
+ x arsinh
, x
(glxBH)2
(4)
earlier results [4,5]. The deviation from the expected classical value cx,~,d = glx13/kB--0.34 K / k O e has been attributed to out-of-plane fluctuations and quantum effects. The data from the 3D regime, taken at four different fields, are no longer universal (mainly because of the field dependence of parameter x in eq. (4)). However, the respective slopes and distances are in very good agreement with eqs. (4) and (5) proving, in fact, the expected approximate doubling of activation energy. Surprisingly - and in contrast to the situation above T N [4] - the absolute values of 1 / T E data below 7"N are much larger than predicted [8]. The observed enhancement could be attributed to collisions between kinks and linear excitations. For the 1D case a renormalization of the soliton density due to collisions between kinks and magnons propagating in chain direction has been calculated and included in the standard expressions for n~ and n2= [I,7]. In presence of 3D order an additional effect should result from magnons propagating perpendicularly to the chains. We have shown [10] that such "3D collisions" result in a field and temperature dependent renormalization increasing the 2vkink density by a factor
( A ~ 2 ( H , T ) ~- 1+41~
)
(g#nH)4 4 S 2 j , I j [ ( k , f f ) 2"
(6)
At H = 40 kOe and T = 2 K this factor amounts to 1300 (!) which mainly accounts for the observed enhancement of 1 / T . In fig. lb the same data have been represented in a plot emphasizing the asymptotic behaviour in the 3D regime, eqs. (4) and (5), and including the H and T dependences of the renormalization factor (6). The data taken below 7'~ (full symbols) now follow a straight line and show completely universal behaviour. The best fit of the slope was obtained for .l' = ( 0 . 9 + 0 . 1 ) inK, which is consistent with the literature [6], and 2t~ . . . . p (0.57+_0.03) K / k O e , which almost exactly agrees with our prevkms 1D result. This project of SFB 185 "Nichtlineare Dynamik" was supported by the Deutsche Forschungsgemeinschaft.
References approaching 2E.~ = 2o%Hk B in the limit x ~ 0, i.e. for large magnetic fields. The corresponding relaxation rate is described by the following asymptotic expression [8]:
1 / T I ~ lq2T 1 e x p ( - E2.~/kBT).
(5)
The reverse exponential H~ T dependences of the spin-lattice relaxation time above and below T N - comparing eqs. (3) and (5) - are related to the different topology of v-kinks and 2"rr-kinks: Above T N (two degenerate ground states) the effect of sublattice flipping on I / T I is proportional to the incerse v-kink density n~, 1 ~ exp(+ E~,/knT). Below T N (single ground state) spin fluctuations only arise inside the moving 2w-kinks, so their effect on 1 / T 1 is directly proportional to n2,~ ~ e x p ( - E 2 . ~ / k ~ T ) . T 1 data from both above and below T N are compared in fig. la using a universal plot which should be appropriate only for the ID regime. Indeed, for T > TN(H) our data follow the expected universal behaviour of eq. 13), and a~ was evaluated from the slope to be 1/).28 + 0.02) K / k O e , in accordance with
[1] H.J. Mikeska, J. Phys. C 11 (1978) L29, C 13 (1980) 2913. [2] Nonlinearity in Condensed Matter, eds. A.R. Bishop et al. (Springer, Berlin, 1987). [3] J.A. Holyst and A. Sukiennicki, Phys. Rev. B 38 (1988) 6975. J.A. Hotyst, Z. Phys. B 74 (1989) 341. [4] J.P. Boucher and J.P. Renard, Phys. Rev. Lett. 45 (1980) 486. [5] ft. Benner, H. Seitz, J. Wiese and J.P. Boucher, J. Magn. Magn. Mater. 45 (1984) 354. [6] C. Dupas and J.P. Renard, Solid State Commun. 20 (1976) 581. M. Steiner, J. Villain and C.G. Windsor, Adv. Phys. 25 (1976) 87. [7] K.M. Leung, Phys. Rev. B 26 11982) 226, B 27 11983) 2877. [8] J.A. Holyst and H. Benner, Solid State Commun. 72 11989) 385. [9] J.F. Currie et al., Phys. Rev. B 22 11980) 477. [10] J.A. Hotyst and H. Benner, Solid State Commun. 79 11991) 703.