PRL 110, 157205 (2013)
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PHYSICAL REVIEW LETTERS
Unconventional Magnetic and Thermodynamic Properties of S ¼ 1=2 Spin Ladder with Ferromagnetic Legs H. Yamaguchi,1 K. Iwase,1 T. Ono,1 T. Shimokawa,2 H. Nakano,3 Y. Shimura,4 N. Kase,4 S. Kittaka,4 T. Sakakibara,4 T. Kawakami,5 and Y. Hosokoshi1 1
Department of Physical Science, Osaka Prefecture University, Osaka 599-8531, Japan Center for Collaborative Research and Technology Development, Kobe University, Kobe 657-8501, Japan 3 Graduate School of Material Science, University of Hyogo, Hyogo 678-1297, Japan 4 Institute for Solid State Physics, The University of Tokyo, Chiba 277-8581, Japan 5 Department of Chemistry, Osaka University, Osaka 560-0043, Japan (Received 23 November 2012; revised manuscript received 16 January 2013; published 10 April 2013)
2
We have succeeded in synthesizing single crystals of a new organic radical 3-Cl-4-F-V [3-(3-chloro-4fluorophenyl)-1,5-diphenylverdazyl]. Through the ab initio molecular orbital calculation and the analysis of the magnetic properties, this compound was confirmed to be the first experimental realization of an S ¼ 1=2 spin-ladder system with ferromagnetic leg interactions. The field-temperature phase diagram indicated that the ground state is situated very close to the quantum critical point. Furthermore, we found an unexpected field-induced successive phase transition, which possibly originates from the interplay of low dimensionality and frustration. DOI: 10.1103/PhysRevLett.110.157205
PACS numbers: 75.10.Jm
Spin-ladder systems have been investigated from both theoretical and experimental points of view in relation to field-induced quantum phase transition (QPT) and high-Tc superconductors [1,2]. Among these systems, S ¼ 1=2 antiferromagnetic (AFM) two-leg spin ladders, which have AFM rung and leg interactions, have been most extensively studied theoretically as a simple spin-ladder model. Since such spin ladders have an energy gap in the spin excitation spectrum, magnetic fields cause QPTs, and the field-induced gapless phase is expected to be a Tomonaga-Luttinger liquid (TLL), which is a quantum critical state with fractional S ¼ 1=2 spinon excitations. Recently, two types of copper complexes, ðC5 H12 NÞ2 CuBr4 and ðC7 H10 NÞ2 CuBr4 , abbreviated as BPCB and DIMPY, respectively, have been identified as very good realizations of S ¼ 1=2 AFM two-leg spin ladders [3,4]. Their field-induced TLL phases have been quantitatively investigated in detail by a variety of measurement techniques [5–8]. These experimental results have promoted the understanding of the field-induced QPT to the TLL state and stimulated renewed interest in studies on the quantum spin systems. On the contrary, spin ladders with ferromagnetic (FM) leg interactions, which consist of antiferromagnetically coupled FM chains, have not been realized experimentally. Their ground states and magnetic behavior have been discussed extensively from a theoretical point of view [9–14]. The phase diagram is parametrized by isotropic interleg AFM coupling, Jrung , and intraleg XXZ-type exchange anisotropy, , which is defined as H leg ¼ P Jleg hi;ji fSxi;j Sxiþ1;j þ Syi;j Syiþ1;j þ Szi;j Sziþ1;j g. For an isotropic case ¼ 1, the fully gapped rung-singlet (RS) state is stabilized. Vekua et al. argued that the TLL state appears 0031-9007=13=110(15)=157205(5)
on application of a magnetic field [12]. For an anisotropic case 0 < < 1, the TLL is established at zero field for the weak Jrung region, and an ordered stripe-FM phase appears only in finite field region. The stripe-FM phase is characterized by uniform magnetization in the direction of the applied field with opposite magnetization on the neighboring legs. The verification of these properties is quite significant for studies on complicated quantum behavior unsuspected in conventional spin systems. In this Letter, we report the first model compound of an S ¼ 1=2 two-leg spin ladder with FM leg interaction. We have succeeded in synthesizing a new verdazyl radical 3-Cl4-F-V [3-(3-chloro-4-fluorophenyl)-1,5-diphenylverdazyl] and solved its crystal structure. The ab initio molecular orbital (MO) calculation indicated the formation of an S ¼ 1=2 two-leg spin ladder with FM leg and AFM rung interactions. We performed magnetic susceptibility, magnetization curve, and specific heat measurements and successfully explained those experimental results as the expected spin-ladder model by using the quantum Monte Carlo (QMC) method. Furthermore, we observed a phase transition to a magnetic ordered state and an unconventional field-induced successive phase transition. The synthesis of 3-Cl-4-F-V, whose molecular structure is shown in Fig. 1(a), was performed by using a conventional procedure [15]. The crystal structure was determined by the intensity data collected using a single-crystal x-ray diffractometer (Rigaku, Mercury-7R) at 296 K. Magnetic susceptibility was measured with a commercial SQUID magnetometer (MPMS-XL, Quantum Design) in combination with 3 He refrigerator. The magnetization curve was measured by a capacitive Faraday magnetometer with a dilution refrigerator. The experimental results are
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N N R1 N N
3.54
asinβ
R4
Cl
(c)
F
3.50
b
(d)
b
csinβ
Jleg 3.69
Jrung
a
FIG. 1 (color online). (a) The molecular structure of 3-Cl-4-FV. Crystal structures of 3-Cl-4-F-V viewed along the (b) c and (c) a axis, respectively. The dotted lines indicate C-C and N-C short contacts. Hydrogen atoms are omitted for clarity. (d) Twoleg ladder formed by intermolecular interactions Jleg and Jrung .
corrected for the diamagnetic contribution of 3-Cl-4-F-V, which is calculated to be 2:34 104 emu mol1 by Pascal’s method. Specific heat was measured using a commercial calorimeter (PPMS, Quantum Design) by a thermal relaxation method above 2.0 K and using a handmade apparatus by a standard adiabatic heat-pulse method for 0:35 < T < 2:0 K. Experiments were performed using single crystals with typical dimensions of 3:0 0:5 0:5 mm3 . The obtained crystallographic parameters are as follows: b¼ monoclinic, space group P21 =n, a ¼ 5:046ð2Þ A, c ¼ 10:284ð5Þ A, ¼ 96:233ð5Þ , V ¼ 33:343ð15Þ A, 3 , Z ¼ 4, R ¼ 0:0480, and Rw ¼ 0:1242 for 1719:9ð14Þ A 2242 observed reflections [I > 2ðIÞ], 276 variables, and goodness of fit is 1.048 [16]. The MO calculation indicated that there exists about 63% of total spin density on the verdazyl ring including nitrogen atoms labeled as R1 [17]. While the R2 and R3 have about 15% of relatively large spin densities for each phenyl ring, the R4 has a spin density less than 7%. Therefore, attention is directed mainly toward the structural features related to R1 R3 rings. The uniform chain structure along the a axis is remarkable, as shown in Fig. 1(b). The molecules stack themselves with their molecular planes parallel by a translation, and C-C and ˚ are found between N-C short contacts less than 3.6 A each ring. Considering the -molecular overlap, a C-C ˚ between R3 rings, which is doubled short contact of 3.69 A by an inversion symmetry, as shown in Fig. 1(c), must connect two neighboring chains to form a two-leg ladder structure, as shown in Fig. 1(d). We performed the MO calculations to evaluate quantitatively the intermolecular ˚ magnetic interactions on the molecular pairs within 4.0 A
considering the spin-density distribution and overlapping of the molecular orbitals [17]. Consequently, we found that there are two types of dominate interactions, which correspond to the expected intra- and interchain interactions, respectively, and they are evaluated to be Jleg =kB ¼ 9:2 K and Jrung =kB ¼4:7 K, which are defined in Eq. (1). The evaluated interactions with the relation ¼ jJrung =Jleg j ¼ 0:51 form an S ¼ 1=2 two-leg spin ladder with FM leg interaction. Figure 2 shows the temperature dependence of magnetic susceptibility ( ¼ M=H) at 0.5 T for H k a, which is corrected for the diamagnetic contribution. We observed a broad peak at about 4.8 K. Above 100 K, it follows the Curie-Weiss law. The Curie constant and Weiss temperature are estimated to be 0:378 emu K=mol and W ¼ þ4:5 K, respectively, which indicates the existence of FM interactions. As described in the following experimental results of the magnetization curve, we evaluated paramagnetic impurities to be about 1.1% of all spins, which is expected to have negligible effects on the magnetic susceptibilities in the magnetic field above 0.5 T. The contribution of FM interactions appears definitely in the temperature dependence of T, which increases with decreasing temperatures down to about 15.5 K, as shown in the upper right inset of Fig. 2. Below 15.5 K, T decreases with decreasing temperatures, which indicates the existence of additional AFM interactions. We calculated the magnetic susceptibility based on the S ¼ 1=2 two-leg spin ladder with FM leg interactions by using the QMC method as a function of ¼ jJrung =Jleg j [19]. 0.06
0.48 0.46
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C N Cl F
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(b) R3
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Susceptibility χ (emu/mol)
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PHYSICAL REVIEW LETTERS
PRL 110, 157205 (2013)
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FIG. 2 (color online). Temperature dependence of magnetic susceptibility ( ¼ M=H) of 3-Cl-4-F-V at 0.5 T for H k a. The upper right inset shows temperature dependence of T. The dotted, solid, and broken lines represent the calculated results for ¼ jJrung =Jleg j ¼ 0:40, 0.55, and 0.70, respectively. The lower left inset shows magnetization curves at 1.8 and 0.1 K. The solid lines represent the calculated results for ¼ 0:55. For clarity, the values at 1.8 K have been shifted up by 0:2 B =f:u:
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20 10
hiji
where Si;j are the spin operators acting on site i of the leg j ¼ 1, 2 of the ladder, g the g factor, g ¼ 2:00, B the Bohr magneton, and H the external magnetic field. Although there is no distinct dependence of , the maximum value of T is very sensitive to the change in . We then obtained good agreement between the experiment and calculation by considering the parameters Jleg =kB ¼ 13:3 K and Jrung =kB ¼ 7:32 K ( ¼ 0:55), as shown in Fig. 2 and its inset. Considering general comparison between experimental and ab initio evaluation, these parameters can be regarded as consistent with those evaluated from the MO calculation. The ground state for this model is the RS with the excitation gap ¼ 1:06 K to the lowest triplet state [22]. The lower left inset of Fig. 2 shows magnetization curves at 1.8 and 0.1 K. We found a small paramagnetic contribution given by the Brillouin function in the lowfield region below about 0.5 T in the magnetization curve at 0.1 K, which is evaluated to be about 1.1% of all spins. The QMC calculation using the obtained parameters from the analysis of the magnetic susceptibility well reproduced the experimental results. In the case of 0.1 K, there is a slight deviation originating from the gapped behavior in the calculation in the vicinity of zero field and the saturation field Hsat , which was evaluated to be about 5.5 T. Although the experimental temperature of 0.1 K is low enough to observe the expected energy gap of 1.06 K, there is gapless behavior in the experimental result, which indicates the disappearance of the energy gap owing to the interladder interaction. Next, we describe the specific heat. The inset of Fig. 3 shows the experimental result of the specific heat Cp for 0, 2.0, and 3.0 T. The magnetic specific heats Cm are obtained by subtracting the lattice contribution assuming Debye’s T 3 law as 0:0108T 3 (J=mol K), which is applicable below about 10 K in ordinary radical compounds. We calculated the magnetic specific heats using the parameters obtained by our magnetic susceptibility analysis and reproduced the experimental results with broad peaks, as shown in Fig. 3. We found deviations in the low-temperature region and sharp peaks at TN ¼ 1:1, 1.5, and 1.7 K for 0, 2.0, and 3.0 T, respectively, which indicate phase transitions to an ordered phase. The small interladder exchange interactions suppress the energy gap and induce the ordered state. Since there must exist a sufficient development of magnetic short-range order accompanied by a large entropy loss above TN , we have not observed divergent behavior just above TN . In fact, the magnetic entropy Sm , obtained
experiment
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As is often the case with conventional radical compounds, we assume a Heisenberg spin Hamiltonian, which is expressed as X X X H ¼ Jleg Si;j Siþ1;j þ Jrung Si;1 Si;2 gB H Si;j ; hiji
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PHYSICAL REVIEW LETTERS
PRL 110, 157205 (2013)
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Cm T 2.5
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FIG. 3 (color online). The closed circles denote temperature dependence of magnetic specific heat of 3-Cl-4-F-V at 0, 2.0, and 3.0 T. The values at 2.0 and 3.0 T have been shifted up by 0.8 and 1:6 J=mol K, respectively. The open triangles are corresponding magnetic entropy at 0 T. The solid lines with open circles represent calculated results for ¼ 0:55. The solid line shows the fit of Cm / T 2:5 . The inset shows the total specific heats, and the values at 2.0 and 3.0 T have been shifted up by 5 and 10 J=mol K, respectively.
through the integration of Cm =T at 0 T, shows that about 90% of the total entropy Stotal m ¼ R ln2 is consumed above TN , as shown in Fig. 3. The slight deviations above about 5 K are considered to originate from deviation of the lattice contribution from Debye’s T 3 law. The magnetic specific heat below 0.8 K at 0 T is well fitted by a power law Cm ¼ AT , where A is a constant and is 2.51(4), as shown in Fig. 3. Therefore, we propose the following dispersion relation for the long-wavelength spin excitaqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tions: ! ¼ vp kp þ vz k2z , where kp ¼ k2x þ k2y and vp and vz are the spin wave velocity for the planar AFM and linear FM excitations, respectively. Figures 4(a) and 4(b) show the low-temperature region of the magnetic susceptibility and magnetic specific heat in various magnetic fields, respectively. We found a distinct sharp peak at TN ¼ 1:1 K in the temperature derivative of at 0.5 T, as shown in the inset of Fig. 4(a). At higher magnetic fields, we observed extrema of magnetic susceptibility, which appear almost symmetrically with respect to the curve at 3.0 T, as shown in Fig. 4(a), and there is no distinct peak in d=dT. In the case of the magnetic specific heat, the anomaly observed at zero field splits into two peaks above 1.5 T, as shown in Fig. 4(b). We plotted these specific temperatures in the T-H phase diagram, as shown in Fig. 5. While the temperatures from the specific heat indicate two phase boundaries, those from the magnetic susceptibilities appear only on the higher temperature boundary, as shown in Fig. 5. The transition point shifts toward the high-temperature region with an increasing
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PRL 110, 157205 (2013)
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FIG. 4 (color online). Temperature dependence of (a) magnetic susceptibility and (b) Cm =T of 3-Cl-4-F-V in various magnetic fields for H k a. The arrows indicate phase transition temperatures. For clarity, Cm =T for 1.0, 1.5, 2.0, 3.0, 4.0, and 4.5 T have been shifted up by 0.2, 0.4, 0.7, 1.3, 1.7, and 2:5 J=mol K2 , respectively. The inset shows the temperature derivative of magnetic susceptibility at 0.5 T.
magnetic field up to 3.0 T. This type of phase boundary shape is often associated with the field-induced phase transition in a gapped spin system. Since the expected small energy gap of 1.06 K is considered to disappear owing to the weak interladder interactions, the present system must be in the vicinity of the quantum critical point between the gapped RS and the gapless ordered phases. Considering the formation of the spin ladder, the obtained phase diagram reminds us of field-induced TLL or Bose-Einstein condensation of magnons. In a weakly coupled S ¼ 1=2 AFM Heisenberg two-leg ladder system, an additional phase transition is expected from the TLL to 2.0
disorder
H || a
1.8 1.6
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Temperature (K)
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Cm χ dχ /dT
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phase 2
0.0 0
1
2
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Hsat 6
Magnetic Field (T)
FIG. 5 (color online). Magnetic field versus temperature phase diagram of 3-Cl-4-F-V for H k a. The closed circles, closed triangles, and open triangle indicate the transition temperatures determined from the magnetic specific heat, the magnetic susceptibility, and its temperature derivative, respectively. The solid vertical line indicates the saturation field.
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3D ordered phase, with decreasing temperature [25]. If such a situation is assumed, phase 1 and phase 2 in Fig. 5 correspond to the TLL and 3D order, respectively. In several theoretical works on gapped spin systems, the extrema of magnetic susceptibility with a symmetrical magnetic field dependence are predicted to appear at the crossover temperature to the TLL phase [25–27]. Such behavior has been experimentally observed in the recent investigation on BPCB and DIMPY [3,7]. Conversely, magnetic susceptibility at the lower phase transition from the TLL to 3D ordered phase is expected to show only a small change in slope [25], which must appear as a peak in d=dT. In the case of the magnetic specific heat, anomalous behavior is suggested at the crossover temperature to the TLL phase by numerical studies on S ¼ 1=2 AFM twoleg ladder systems [26], and corresponding behavior is observed in BPCB [3]. Our experimental results clearly exhibit the field-induced extrema of magnetic susceptibility expected at the crossover to TLL as seen in BPCB and DIMPY, which differ from singular cusplike ones at BoseEinstein condensation of magnons accompanied by 3D ordering. However, we did not observe remarkable changes at the lower phase boundary between phase 1 and phase 2 either in the magnetic susceptibilities and their temperature derivatives, which is consistent with predictions. Furthermore, since both double peaks observed in our specific heat measurements indicate relatively sharp changes, as seen in second-order phase transition, the present case is inconsistent with the above TLL scenario. Considering the experimental result of the magnetic specific heat, we should take into account the possibility of a successive phase transition, where the partial components of magnetic moment order in a stepwise fashion, often induced by large magnetic anisotropy and/or noncollinear magnetic structure. The magnetic anisotropy in organic radical systems is generally quite small. In fact, for H ? a, we obtained almost the same phase diagram as that for H k a from the specific heat measurements, which proves a negligibly weak magnetic anisotropy. Conversely, we can suggest the possibility of noncollinear magnetic structure induced by frustration. The MO caluculation indicated that their are three kinds of possible small interladder interactions, whose absolute values are less than about 0.2 K. Although the signs of those small interactions are not reliable in ab initio MO calculation considering a strong dependence on the calculation method [28], these three interactions can form frustrated lattices by the combination of their signs, and then a noncollinear magnetic structure will appear. However, we do not have any explanation for the fact that the intermediate phase only appears by applying the magnetic field. In summary, we have succeeded in synthesizing a new verdazyl radical crystal of 3-Cl-4-F-V, and an S ¼ 1=2 two-leg spin-ladder lattice was deduced from the molecular packings. The ab initio MO calculation also indicated the formation of the spin ladder, which has FM leg and
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PHYSICAL REVIEW LETTERS
AFM rung interactions. We have successfully explained the magnetic and thermodynamic properties based on the S ¼ 1=2 two-leg spin ladder with FM leg interactions by using the quantum Monte Carlo method. We confirmed that 3-Cl-4-F-V is the first experimental realization of an S ¼ 1=2 two-leg spin ladder with FM legs. Considering the T-H phase diagram, the ground state is expected to be in the vicinity of the quantum critical point between the gapped RS and the gapless ordered phases. Furthermore, we have found an unexpected field-induced successive phase transition, which possibly originates from the interplay of low dimensionality and frustration. We hope that the present results will stimulate studies on spin ladder with FM interactions and unconventional behavior of the complicated quantum spin systems. We thank M. Hagiwara and T. Okubo for the valuable discussions. This research was partially supported by KAKENHI (No. 24740241). Part of this work was performed under the interuniversity cooperative research program of the joint-research program of ISSP, the University of Tokyo. Some computations were performed using the facilities of the Supercomputer Center, ISSP, The University of Tokyo. Our QMC calculations were carried out using the ALPS application [29].
[1] E. Dagotto and T. M. Rice, Science 271, 618 (1996). [2] M. Uehara, T. Nagata, J. Akimitsu, H. Takahashi, N. Moˆri, and K. Kinoshita, J. Phys. Soc. Jpn. 65, 2764 (1996). [3] C. Ru¨egg, K. Kiefer, B. Thielemann, D. F. McMorrow, V. Zapf, B. Normand, M. B. Zvonarev, P. Bouillot, C. Kollath, T. Giamarchi, S. Capponi, D. Poilblanc, D. Biner, and K. W. Kra¨mer, Phys. Rev. Lett. 101, 247202 (2008). [4] T. Hong, Y. H. Kim, C. Hotta, Y. Takano, G. Tremelling, M. M. Turnbull, C. P. Landee, H.-J. Kang, N. B. Christensen, K. Lefmann, K. P. Schmidt, G. S. Uhrig, and C. Broholm, Phys. Rev. Lett. 105, 137207 (2010). [5] M. Klanjsˇek, H. Mayaffre, C. Berthier, M. Horvatic´, B. Chiari, O. Piovesana, P. Bouillot, C. Kollath, E. Orignac, R. Citro, and T. Giamarchi, Phys. Rev. Lett. 101, 137207 (2008). [6] B. Thielemann, C. Ru¨egg, H. M. Rønnow, A. M. La¨uchli, J.-S. Caux, B. Normand, D. Biner, K. W. Kra¨mer, H.-U. Gu¨del, J. Stahn, K. Habicht, K. Kiefer, M. Boehm, D. F. McMorrow, and J. Mesot, Phys. Rev. Lett. 102, 107204 (2009).
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[7] K. Ninios, T. Hong, T. Manabe, C. Hotta, S. N. Herringer, M. M. Turnbull, C. P. Landee, Y. Takano, and H. B. Chan, Phys. Rev. Lett. 108, 097201 (2012). [8] D. Schmidiger, S. Mu¨hlbauer, S. N. Gvasaliya, T. Yankova, and A. Zheludev, Phys. Rev. B 84, 144421 (2011). [9] M. Roji and S. Miyashita, J. Phys. Soc. Jpn. 65, 883 (1996). [10] A. K. Kolezhuk and H.-J. Mikeska, Phys. Rev. B 53, R8848 (1996). [11] T. Vekua, G. I. Japaridze, and H.-J. Mikeska, Phys. Rev. B 67, 064419 (2003). [12] T. Vekua, G. I. Japaridze, and H.-J. Mikeska, Phys. Rev. B 70, 014425 (2004). [13] K. Hijii, A. Kitazawa, and K. Nomura, Phys. Rev. B 72, 014449 (2005). [14] G. I. Japaridze, A. Langari, and S. Mahdavifar, J. Phys. Condens. Matter 19, 076201 (2007). [15] R. Kuhn and H. Trischmann, Monatsh. Chem. 95, 457 (1964). [16] Crystallographic data have been deposited with Cambridge Crystallographic Data Center, Deposition No. CCDC 927278. [17] Ab initio MO calculations were performed using the UB3LYP method and 6-31G basis sets in the GAUSSIAN 09 program package. For the estimation of intermolecular magnetic interaction, we applied our evaluation scheme that has been studied previously [18]. [18] M. Shoji, K. Koizumi, Y. Kitagawa, T. Kawakami, S. Yamanaka, M. Okumura, and K. Yamaguchi, Chem. Phys. Lett. 432, 343 (2006). [19] Our QMC code is based on the continuous-imaginarytime-loop algorithm [20] and directed loop algorithm in the stochastic series expansion representation [21]. The calculation was performed for a system size of 96 under the periodic boundary condition. [20] H. G. Evertz, Adv. Phys. 52, 1 (2003). [21] A. W. Sandvik, Phys. Rev. B 59, R14157 (1999). [22] The calculation was performed under the open-boundary condition, and 900 basis states were kept. We employ the finite-system DMRG method [23,24]. [23] S. R. White, Phys. Rev. Lett. 69, 2863 (1992). [24] S. R. White, Phys. Rev. B 48, 10345 (1993). [25] X. Wang and L. Yu, Phys. Rev. Lett. 84, 5399 (2000). [26] S. Wessel, M. Olshanii, and S. Haas, Phys. Rev. Lett. 87, 206407 (2001). [27] Y. Maeda, C. Hotta, and M. Oshikawa, Phys. Rev. Lett. 99, 057205 (2007). [28] T. Kawakami, Y. Kitagawa, F. Matsuoka, Y. Yamashita, and K. Yamaguchi, Polyhedron 20, 1235 (2001). [29] A. F. Albuquerque et al., J. Magn. Magn. Mater. 310, 1187 (2007); see also http://alps.comp-phys.org.
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PHYSICAL REVIEW LETTERS
Unconventional Magnetic and Thermodynamic Properties of S ¼ 1=2 Spin Ladder with Ferromagnetic Legs H. Yamaguchi,1 K. Iwase,1 T. Ono,1 T. Shimokawa,2 H. Nakano,3 Y. Shimura,4 N. Kase,4 S. Kittaka,4 T. Sakakibara,4 T. Kawakami,5 and Y. Hosokoshi1 1
Department of Physical Science, Osaka Prefecture University, Osaka 599-8531, Japan Center for Collaborative Research and Technology Development, Kobe University, Kobe 657-8501, Japan 3 Graduate School of Material Science, University of Hyogo, Hyogo 678-1297, Japan 4 Institute for Solid State Physics, The University of Tokyo, Chiba 277-8581, Japan 5 Department of Chemistry, Osaka University, Osaka 560-0043, Japan (Received 23 November 2012; revised manuscript received 16 January 2013; published 10 April 2013)
2
We have succeeded in synthesizing single crystals of a new organic radical 3-Cl-4-F-V [3-(3-chloro-4fluorophenyl)-1,5-diphenylverdazyl]. Through the ab initio molecular orbital calculation and the analysis of the magnetic properties, this compound was confirmed to be the first experimental realization of an S ¼ 1=2 spin-ladder system with ferromagnetic leg interactions. The field-temperature phase diagram indicated that the ground state is situated very close to the quantum critical point. Furthermore, we found an unexpected field-induced successive phase transition, which possibly originates from the interplay of low dimensionality and frustration. DOI: 10.1103/PhysRevLett.110.157205
PACS numbers: 75.10.Jm
Spin-ladder systems have been investigated from both theoretical and experimental points of view in relation to field-induced quantum phase transition (QPT) and high-Tc superconductors [1,2]. Among these systems, S ¼ 1=2 antiferromagnetic (AFM) two-leg spin ladders, which have AFM rung and leg interactions, have been most extensively studied theoretically as a simple spin-ladder model. Since such spin ladders have an energy gap in the spin excitation spectrum, magnetic fields cause QPTs, and the field-induced gapless phase is expected to be a Tomonaga-Luttinger liquid (TLL), which is a quantum critical state with fractional S ¼ 1=2 spinon excitations. Recently, two types of copper complexes, ðC5 H12 NÞ2 CuBr4 and ðC7 H10 NÞ2 CuBr4 , abbreviated as BPCB and DIMPY, respectively, have been identified as very good realizations of S ¼ 1=2 AFM two-leg spin ladders [3,4]. Their field-induced TLL phases have been quantitatively investigated in detail by a variety of measurement techniques [5–8]. These experimental results have promoted the understanding of the field-induced QPT to the TLL state and stimulated renewed interest in studies on the quantum spin systems. On the contrary, spin ladders with ferromagnetic (FM) leg interactions, which consist of antiferromagnetically coupled FM chains, have not been realized experimentally. Their ground states and magnetic behavior have been discussed extensively from a theoretical point of view [9–14]. The phase diagram is parametrized by isotropic interleg AFM coupling, Jrung , and intraleg XXZ-type exchange anisotropy, , which is defined as H leg ¼ P Jleg hi;ji fSxi;j Sxiþ1;j þ Syi;j Syiþ1;j þ Szi;j Sziþ1;j g. For an isotropic case ¼ 1, the fully gapped rung-singlet (RS) state is stabilized. Vekua et al. argued that the TLL state appears 0031-9007=13=110(15)=157205(5)
on application of a magnetic field [12]. For an anisotropic case 0 < < 1, the TLL is established at zero field for the weak Jrung region, and an ordered stripe-FM phase appears only in finite field region. The stripe-FM phase is characterized by uniform magnetization in the direction of the applied field with opposite magnetization on the neighboring legs. The verification of these properties is quite significant for studies on complicated quantum behavior unsuspected in conventional spin systems. In this Letter, we report the first model compound of an S ¼ 1=2 two-leg spin ladder with FM leg interaction. We have succeeded in synthesizing a new verdazyl radical 3-Cl4-F-V [3-(3-chloro-4-fluorophenyl)-1,5-diphenylverdazyl] and solved its crystal structure. The ab initio molecular orbital (MO) calculation indicated the formation of an S ¼ 1=2 two-leg spin ladder with FM leg and AFM rung interactions. We performed magnetic susceptibility, magnetization curve, and specific heat measurements and successfully explained those experimental results as the expected spin-ladder model by using the quantum Monte Carlo (QMC) method. Furthermore, we observed a phase transition to a magnetic ordered state and an unconventional field-induced successive phase transition. The synthesis of 3-Cl-4-F-V, whose molecular structure is shown in Fig. 1(a), was performed by using a conventional procedure [15]. The crystal structure was determined by the intensity data collected using a single-crystal x-ray diffractometer (Rigaku, Mercury-7R) at 296 K. Magnetic susceptibility was measured with a commercial SQUID magnetometer (MPMS-XL, Quantum Design) in combination with 3 He refrigerator. The magnetization curve was measured by a capacitive Faraday magnetometer with a dilution refrigerator. The experimental results are
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N N R1 N N
3.54
asinβ
R4
Cl
(c)
F
3.50
b
(d)
b
csinβ
Jleg 3.69
Jrung
a
FIG. 1 (color online). (a) The molecular structure of 3-Cl-4-FV. Crystal structures of 3-Cl-4-F-V viewed along the (b) c and (c) a axis, respectively. The dotted lines indicate C-C and N-C short contacts. Hydrogen atoms are omitted for clarity. (d) Twoleg ladder formed by intermolecular interactions Jleg and Jrung .
corrected for the diamagnetic contribution of 3-Cl-4-F-V, which is calculated to be 2:34 104 emu mol1 by Pascal’s method. Specific heat was measured using a commercial calorimeter (PPMS, Quantum Design) by a thermal relaxation method above 2.0 K and using a handmade apparatus by a standard adiabatic heat-pulse method for 0:35 < T < 2:0 K. Experiments were performed using single crystals with typical dimensions of 3:0 0:5 0:5 mm3 . The obtained crystallographic parameters are as follows: b¼ monoclinic, space group P21 =n, a ¼ 5:046ð2Þ A, c ¼ 10:284ð5Þ A, ¼ 96:233ð5Þ , V ¼ 33:343ð15Þ A, 3 , Z ¼ 4, R ¼ 0:0480, and Rw ¼ 0:1242 for 1719:9ð14Þ A 2242 observed reflections [I > 2ðIÞ], 276 variables, and goodness of fit is 1.048 [16]. The MO calculation indicated that there exists about 63% of total spin density on the verdazyl ring including nitrogen atoms labeled as R1 [17]. While the R2 and R3 have about 15% of relatively large spin densities for each phenyl ring, the R4 has a spin density less than 7%. Therefore, attention is directed mainly toward the structural features related to R1 R3 rings. The uniform chain structure along the a axis is remarkable, as shown in Fig. 1(b). The molecules stack themselves with their molecular planes parallel by a translation, and C-C and ˚ are found between N-C short contacts less than 3.6 A each ring. Considering the -molecular overlap, a C-C ˚ between R3 rings, which is doubled short contact of 3.69 A by an inversion symmetry, as shown in Fig. 1(c), must connect two neighboring chains to form a two-leg ladder structure, as shown in Fig. 1(d). We performed the MO calculations to evaluate quantitatively the intermolecular ˚ magnetic interactions on the molecular pairs within 4.0 A
considering the spin-density distribution and overlapping of the molecular orbitals [17]. Consequently, we found that there are two types of dominate interactions, which correspond to the expected intra- and interchain interactions, respectively, and they are evaluated to be Jleg =kB ¼ 9:2 K and Jrung =kB ¼4:7 K, which are defined in Eq. (1). The evaluated interactions with the relation ¼ jJrung =Jleg j ¼ 0:51 form an S ¼ 1=2 two-leg spin ladder with FM leg interaction. Figure 2 shows the temperature dependence of magnetic susceptibility ( ¼ M=H) at 0.5 T for H k a, which is corrected for the diamagnetic contribution. We observed a broad peak at about 4.8 K. Above 100 K, it follows the Curie-Weiss law. The Curie constant and Weiss temperature are estimated to be 0:378 emu K=mol and W ¼ þ4:5 K, respectively, which indicates the existence of FM interactions. As described in the following experimental results of the magnetization curve, we evaluated paramagnetic impurities to be about 1.1% of all spins, which is expected to have negligible effects on the magnetic susceptibilities in the magnetic field above 0.5 T. The contribution of FM interactions appears definitely in the temperature dependence of T, which increases with decreasing temperatures down to about 15.5 K, as shown in the upper right inset of Fig. 2. Below 15.5 K, T decreases with decreasing temperatures, which indicates the existence of additional AFM interactions. We calculated the magnetic susceptibility based on the S ¼ 1=2 two-leg spin ladder with FM leg interactions by using the QMC method as a function of ¼ jJrung =Jleg j [19]. 0.06
0.48 0.46
0.05
χ Τ (emu K/mol)
C N Cl F
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0.44 0.42
0.5 T for H || a
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0.40 0.38
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T = 1.8 K experiment r = 0.55
1.0
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experiment γ = 0.40 γ = 0.55 γ = 0.70
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Magnetization (µB/f.u.)
(b) R3
R2
Susceptibility χ (emu/mol)
(a)
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PRL 110, 157205 (2013)
0.32
H || a
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6 8
10
2
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6 8
Temperature (K)
100
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0.4 T = 0.1 K
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0
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2
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6
Magnetic Field (T)
3
4 5 6 7
7
8
2 3 4 5 10 Temperature (K)
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100
2
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FIG. 2 (color online). Temperature dependence of magnetic susceptibility ( ¼ M=H) of 3-Cl-4-F-V at 0.5 T for H k a. The upper right inset shows temperature dependence of T. The dotted, solid, and broken lines represent the calculated results for ¼ jJrung =Jleg j ¼ 0:40, 0.55, and 0.70, respectively. The lower left inset shows magnetization curves at 1.8 and 0.1 K. The solid lines represent the calculated results for ¼ 0:55. For clarity, the values at 1.8 K have been shifted up by 0:2 B =f:u:
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4.5
(1)
γ = 0.55
3.0 T
30
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0
0
5
10
15
8
3.0 T
2.0 T 0T
10 9
20 10
hiji
where Si;j are the spin operators acting on site i of the leg j ¼ 1, 2 of the ladder, g the g factor, g ¼ 2:00, B the Bohr magneton, and H the external magnetic field. Although there is no distinct dependence of , the maximum value of T is very sensitive to the change in . We then obtained good agreement between the experiment and calculation by considering the parameters Jleg =kB ¼ 13:3 K and Jrung =kB ¼ 7:32 K ( ¼ 0:55), as shown in Fig. 2 and its inset. Considering general comparison between experimental and ab initio evaluation, these parameters can be regarded as consistent with those evaluated from the MO calculation. The ground state for this model is the RS with the excitation gap ¼ 1:06 K to the lowest triplet state [22]. The lower left inset of Fig. 2 shows magnetization curves at 1.8 and 0.1 K. We found a small paramagnetic contribution given by the Brillouin function in the lowfield region below about 0.5 T in the magnetization curve at 0.1 K, which is evaluated to be about 1.1% of all spins. The QMC calculation using the obtained parameters from the analysis of the magnetic susceptibility well reproduced the experimental results. In the case of 0.1 K, there is a slight deviation originating from the gapped behavior in the calculation in the vicinity of zero field and the saturation field Hsat , which was evaluated to be about 5.5 T. Although the experimental temperature of 0.1 K is low enough to observe the expected energy gap of 1.06 K, there is gapless behavior in the experimental result, which indicates the disappearance of the energy gap owing to the interladder interaction. Next, we describe the specific heat. The inset of Fig. 3 shows the experimental result of the specific heat Cp for 0, 2.0, and 3.0 T. The magnetic specific heats Cm are obtained by subtracting the lattice contribution assuming Debye’s T 3 law as 0:0108T 3 (J=mol K), which is applicable below about 10 K in ordinary radical compounds. We calculated the magnetic specific heats using the parameters obtained by our magnetic susceptibility analysis and reproduced the experimental results with broad peaks, as shown in Fig. 3. We found deviations in the low-temperature region and sharp peaks at TN ¼ 1:1, 1.5, and 1.7 K for 0, 2.0, and 3.0 T, respectively, which indicate phase transitions to an ordered phase. The small interladder exchange interactions suppress the energy gap and induce the ordered state. Since there must exist a sufficient development of magnetic short-range order accompanied by a large entropy loss above TN , we have not observed divergent behavior just above TN . In fact, the magnetic entropy Sm , obtained
experiment
C p ( J /mol K )
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Cm ( J /mol K)
hii
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40
7 2.0 T
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5
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Sm ( J /mol K)
As is often the case with conventional radical compounds, we assume a Heisenberg spin Hamiltonian, which is expressed as X X X H ¼ Jleg Si;j Siþ1;j þ Jrung Si;1 Si;2 gB H Si;j ; hiji
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3
0T 1.0
2
Cm T 2.5
0.5
1 0
0.0 3
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5
6 7 8 91
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FIG. 3 (color online). The closed circles denote temperature dependence of magnetic specific heat of 3-Cl-4-F-V at 0, 2.0, and 3.0 T. The values at 2.0 and 3.0 T have been shifted up by 0.8 and 1:6 J=mol K, respectively. The open triangles are corresponding magnetic entropy at 0 T. The solid lines with open circles represent calculated results for ¼ 0:55. The solid line shows the fit of Cm / T 2:5 . The inset shows the total specific heats, and the values at 2.0 and 3.0 T have been shifted up by 5 and 10 J=mol K, respectively.
through the integration of Cm =T at 0 T, shows that about 90% of the total entropy Stotal m ¼ R ln2 is consumed above TN , as shown in Fig. 3. The slight deviations above about 5 K are considered to originate from deviation of the lattice contribution from Debye’s T 3 law. The magnetic specific heat below 0.8 K at 0 T is well fitted by a power law Cm ¼ AT , where A is a constant and is 2.51(4), as shown in Fig. 3. Therefore, we propose the following dispersion relation for the long-wavelength spin excitaqffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi tions: ! ¼ vp kp þ vz k2z , where kp ¼ k2x þ k2y and vp and vz are the spin wave velocity for the planar AFM and linear FM excitations, respectively. Figures 4(a) and 4(b) show the low-temperature region of the magnetic susceptibility and magnetic specific heat in various magnetic fields, respectively. We found a distinct sharp peak at TN ¼ 1:1 K in the temperature derivative of at 0.5 T, as shown in the inset of Fig. 4(a). At higher magnetic fields, we observed extrema of magnetic susceptibility, which appear almost symmetrically with respect to the curve at 3.0 T, as shown in Fig. 4(a), and there is no distinct peak in d=dT. In the case of the magnetic specific heat, the anomaly observed at zero field splits into two peaks above 1.5 T, as shown in Fig. 4(b). We plotted these specific temperatures in the T-H phase diagram, as shown in Fig. 5. While the temperatures from the specific heat indicate two phase boundaries, those from the magnetic susceptibilities appear only on the higher temperature boundary, as shown in Fig. 5. The transition point shifts toward the high-temperature region with an increasing
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4.0
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(a)
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H || a
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-3
x10 12 10 8 6 4 2
d χ /dT (emu/mol K)
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Susceptibility χ (emu/mol)
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4 5 10 Temperature (K)
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FIG. 4 (color online). Temperature dependence of (a) magnetic susceptibility and (b) Cm =T of 3-Cl-4-F-V in various magnetic fields for H k a. The arrows indicate phase transition temperatures. For clarity, Cm =T for 1.0, 1.5, 2.0, 3.0, 4.0, and 4.5 T have been shifted up by 0.2, 0.4, 0.7, 1.3, 1.7, and 2:5 J=mol K2 , respectively. The inset shows the temperature derivative of magnetic susceptibility at 0.5 T.
magnetic field up to 3.0 T. This type of phase boundary shape is often associated with the field-induced phase transition in a gapped spin system. Since the expected small energy gap of 1.06 K is considered to disappear owing to the weak interladder interactions, the present system must be in the vicinity of the quantum critical point between the gapped RS and the gapless ordered phases. Considering the formation of the spin ladder, the obtained phase diagram reminds us of field-induced TLL or Bose-Einstein condensation of magnons. In a weakly coupled S ¼ 1=2 AFM Heisenberg two-leg ladder system, an additional phase transition is expected from the TLL to 2.0
disorder
H || a
1.8 1.6
phase 1
Temperature (K)
1.4 1.2 1.0 0.8 0.6 0.4
Cm χ dχ /dT
0.2
phase 2
0.0 0
1
2
3
4
5
Hsat 6
Magnetic Field (T)
FIG. 5 (color online). Magnetic field versus temperature phase diagram of 3-Cl-4-F-V for H k a. The closed circles, closed triangles, and open triangle indicate the transition temperatures determined from the magnetic specific heat, the magnetic susceptibility, and its temperature derivative, respectively. The solid vertical line indicates the saturation field.
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3D ordered phase, with decreasing temperature [25]. If such a situation is assumed, phase 1 and phase 2 in Fig. 5 correspond to the TLL and 3D order, respectively. In several theoretical works on gapped spin systems, the extrema of magnetic susceptibility with a symmetrical magnetic field dependence are predicted to appear at the crossover temperature to the TLL phase [25–27]. Such behavior has been experimentally observed in the recent investigation on BPCB and DIMPY [3,7]. Conversely, magnetic susceptibility at the lower phase transition from the TLL to 3D ordered phase is expected to show only a small change in slope [25], which must appear as a peak in d=dT. In the case of the magnetic specific heat, anomalous behavior is suggested at the crossover temperature to the TLL phase by numerical studies on S ¼ 1=2 AFM twoleg ladder systems [26], and corresponding behavior is observed in BPCB [3]. Our experimental results clearly exhibit the field-induced extrema of magnetic susceptibility expected at the crossover to TLL as seen in BPCB and DIMPY, which differ from singular cusplike ones at BoseEinstein condensation of magnons accompanied by 3D ordering. However, we did not observe remarkable changes at the lower phase boundary between phase 1 and phase 2 either in the magnetic susceptibilities and their temperature derivatives, which is consistent with predictions. Furthermore, since both double peaks observed in our specific heat measurements indicate relatively sharp changes, as seen in second-order phase transition, the present case is inconsistent with the above TLL scenario. Considering the experimental result of the magnetic specific heat, we should take into account the possibility of a successive phase transition, where the partial components of magnetic moment order in a stepwise fashion, often induced by large magnetic anisotropy and/or noncollinear magnetic structure. The magnetic anisotropy in organic radical systems is generally quite small. In fact, for H ? a, we obtained almost the same phase diagram as that for H k a from the specific heat measurements, which proves a negligibly weak magnetic anisotropy. Conversely, we can suggest the possibility of noncollinear magnetic structure induced by frustration. The MO caluculation indicated that their are three kinds of possible small interladder interactions, whose absolute values are less than about 0.2 K. Although the signs of those small interactions are not reliable in ab initio MO calculation considering a strong dependence on the calculation method [28], these three interactions can form frustrated lattices by the combination of their signs, and then a noncollinear magnetic structure will appear. However, we do not have any explanation for the fact that the intermediate phase only appears by applying the magnetic field. In summary, we have succeeded in synthesizing a new verdazyl radical crystal of 3-Cl-4-F-V, and an S ¼ 1=2 two-leg spin-ladder lattice was deduced from the molecular packings. The ab initio MO calculation also indicated the formation of the spin ladder, which has FM leg and
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PHYSICAL REVIEW LETTERS
AFM rung interactions. We have successfully explained the magnetic and thermodynamic properties based on the S ¼ 1=2 two-leg spin ladder with FM leg interactions by using the quantum Monte Carlo method. We confirmed that 3-Cl-4-F-V is the first experimental realization of an S ¼ 1=2 two-leg spin ladder with FM legs. Considering the T-H phase diagram, the ground state is expected to be in the vicinity of the quantum critical point between the gapped RS and the gapless ordered phases. Furthermore, we have found an unexpected field-induced successive phase transition, which possibly originates from the interplay of low dimensionality and frustration. We hope that the present results will stimulate studies on spin ladder with FM interactions and unconventional behavior of the complicated quantum spin systems. We thank M. Hagiwara and T. Okubo for the valuable discussions. This research was partially supported by KAKENHI (No. 24740241). Part of this work was performed under the interuniversity cooperative research program of the joint-research program of ISSP, the University of Tokyo. Some computations were performed using the facilities of the Supercomputer Center, ISSP, The University of Tokyo. Our QMC calculations were carried out using the ALPS application [29].
[1] E. Dagotto and T. M. Rice, Science 271, 618 (1996). [2] M. Uehara, T. Nagata, J. Akimitsu, H. Takahashi, N. Moˆri, and K. Kinoshita, J. Phys. Soc. Jpn. 65, 2764 (1996). [3] C. Ru¨egg, K. Kiefer, B. Thielemann, D. F. McMorrow, V. Zapf, B. Normand, M. B. Zvonarev, P. Bouillot, C. Kollath, T. Giamarchi, S. Capponi, D. Poilblanc, D. Biner, and K. W. Kra¨mer, Phys. Rev. Lett. 101, 247202 (2008). [4] T. Hong, Y. H. Kim, C. Hotta, Y. Takano, G. Tremelling, M. M. Turnbull, C. P. Landee, H.-J. Kang, N. B. Christensen, K. Lefmann, K. P. Schmidt, G. S. Uhrig, and C. Broholm, Phys. Rev. Lett. 105, 137207 (2010). [5] M. Klanjsˇek, H. Mayaffre, C. Berthier, M. Horvatic´, B. Chiari, O. Piovesana, P. Bouillot, C. Kollath, E. Orignac, R. Citro, and T. Giamarchi, Phys. Rev. Lett. 101, 137207 (2008). [6] B. Thielemann, C. Ru¨egg, H. M. Rønnow, A. M. La¨uchli, J.-S. Caux, B. Normand, D. Biner, K. W. Kra¨mer, H.-U. Gu¨del, J. Stahn, K. Habicht, K. Kiefer, M. Boehm, D. F. McMorrow, and J. Mesot, Phys. Rev. Lett. 102, 107204 (2009).
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[7] K. Ninios, T. Hong, T. Manabe, C. Hotta, S. N. Herringer, M. M. Turnbull, C. P. Landee, Y. Takano, and H. B. Chan, Phys. Rev. Lett. 108, 097201 (2012). [8] D. Schmidiger, S. Mu¨hlbauer, S. N. Gvasaliya, T. Yankova, and A. Zheludev, Phys. Rev. B 84, 144421 (2011). [9] M. Roji and S. Miyashita, J. Phys. Soc. Jpn. 65, 883 (1996). [10] A. K. Kolezhuk and H.-J. Mikeska, Phys. Rev. B 53, R8848 (1996). [11] T. Vekua, G. I. Japaridze, and H.-J. Mikeska, Phys. Rev. B 67, 064419 (2003). [12] T. Vekua, G. I. Japaridze, and H.-J. Mikeska, Phys. Rev. B 70, 014425 (2004). [13] K. Hijii, A. Kitazawa, and K. Nomura, Phys. Rev. B 72, 014449 (2005). [14] G. I. Japaridze, A. Langari, and S. Mahdavifar, J. Phys. Condens. Matter 19, 076201 (2007). [15] R. Kuhn and H. Trischmann, Monatsh. Chem. 95, 457 (1964). [16] Crystallographic data have been deposited with Cambridge Crystallographic Data Center, Deposition No. CCDC 927278. [17] Ab initio MO calculations were performed using the UB3LYP method and 6-31G basis sets in the GAUSSIAN 09 program package. For the estimation of intermolecular magnetic interaction, we applied our evaluation scheme that has been studied previously [18]. [18] M. Shoji, K. Koizumi, Y. Kitagawa, T. Kawakami, S. Yamanaka, M. Okumura, and K. Yamaguchi, Chem. Phys. Lett. 432, 343 (2006). [19] Our QMC code is based on the continuous-imaginarytime-loop algorithm [20] and directed loop algorithm in the stochastic series expansion representation [21]. The calculation was performed for a system size of 96 under the periodic boundary condition. [20] H. G. Evertz, Adv. Phys. 52, 1 (2003). [21] A. W. Sandvik, Phys. Rev. B 59, R14157 (1999). [22] The calculation was performed under the open-boundary condition, and 900 basis states were kept. We employ the finite-system DMRG method [23,24]. [23] S. R. White, Phys. Rev. Lett. 69, 2863 (1992). [24] S. R. White, Phys. Rev. B 48, 10345 (1993). [25] X. Wang and L. Yu, Phys. Rev. Lett. 84, 5399 (2000). [26] S. Wessel, M. Olshanii, and S. Haas, Phys. Rev. Lett. 87, 206407 (2001). [27] Y. Maeda, C. Hotta, and M. Oshikawa, Phys. Rev. Lett. 99, 057205 (2007). [28] T. Kawakami, Y. Kitagawa, F. Matsuoka, Y. Yamashita, and K. Yamaguchi, Polyhedron 20, 1235 (2001). [29] A. F. Albuquerque et al., J. Magn. Magn. Mater. 310, 1187 (2007); see also http://alps.comp-phys.org.
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