Cu(C$_8$H$_6$N$_2$)Cl$_2$, a strong-rung spin-1/2 Heisenberg ladder compound,
is probed by means of electron spin resonance (ESR) spectroscopy in the
field-induced gapless phase above $H_{c1}$. The temperature dependence of the
ESR linewidth is analyzed in the quantum field theory framework, suggesting
that the anisotropy of magnetic interactions plays a crucial role, determining
the peculiar low-temperature ESR linewidth behavior. In particular, it is
argued that the uniform Dzyaloshinskii-Moriya interaction (which is allowed on
the bonds along the ladder legs) can be the source of this behavior in
Cu(C$_8$H$_6$N$_2$)Cl$_2$

Landee, C.

Ozerov, M.

Ponomaryov, A. N.

Povarov, K. Yu.

Wosnitza, J.

Xiao, F.

Zheludev, A.

Zviagina, L.

Zvyagin, A. A.

Zvyagin, S. A.

Čižmár, E.

English

arXiv

Ponomaryov, A.N.

Povarov, K.Y.

Cizmar, E.

Zvyagin, A.A.

Zvyagin, S.A.

NARCIS 

Electron spin resonance in a strong-rung spin-1/2 Heisenberg ladder

Cu(C8H6N2)Cl-2, a strong-rung spin-1/2 Heisenberg ladder compound, is probed by means of electron spin resonance (ESR) spectroscopy in the field-induced gapless phase above H-c1. The temperature dependence of the ESR linewidth is analyzed in the quantum field theory framework, suggesting that the anisotropy of magnetic interactions plays a crucial role, determining the peculiar low-temperature ESR linewidth behavior. In particular, it is argued that the uniform Dzyaloshinskii-Moriya interaction (which is allowed on the bonds along the ladder legs) can be the source of this behavior in Cu(C8H6N2)Cl-2

Ponomaryov, A.

Povarov, K.

Zvyagin, A.

Zvyagin, S.

MPG.PuRe

Contains fulltext :
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arXiv:1509.01022v3  [cond-mat.str-el]  18 Apr 2016Electron spin resonance in a strong-rung spin-1/2 Heisenberg ladder.A. N. Ponomaryov,1 M. Ozerov,1, ∗ L. Zviagina,1 J. Wosnitza,1, 2 K. Yu. Povarov,3F. Xiao,3, 4, † A. Zheludev,3 C. Landee,5 E. Čižmár,6 A. A. Zvyagin,7, 8 and S. A. Zvyagin11Dresden High Magnetic Field Laboratory (HLD-EMFL),Helmholtz-Zentrum Dresden-Rossendorf, D-01328 Dresden, Germany2Institut für Festkörperphysik, TU Dresden, D-01062 Dresden,Germany3Neutron Scattering and Magnetism, Laboratory for Solid State Physics, ETH Zürich, Switzerland4Clark University, Worcester, MA 01610, USA5Clark University, 950 Main Street, Worcester, MA 01610, USA6Institute of Physics, P.J. Šafárik University, Košice, Slovakia7Max Planck Institute for the Physics of Complex Systems, D-01187 Dresden, Germany8B. I. Verkin Institute for Low Temperature Physics and Engineering ofthe National Academy of Science of Ukraine, Kharkov 61103, Ukraine(Dated: June 20, 2018)Cu(C8H6N2)Cl2, a strong-rung spin-1/2 Heisenberg ladder compound, is probed by means ofelectron spin resonance (ESR) spectroscopy in the field-induced gapless phase above Hc1. Thetemperature dependence of the ESR linewidth is analyzed in the quantum field theory framework,suggesting that the anisotropy of magnetic interactions plays a crucial role, determining the peculiarlow-temperature ESR linewidth behavior. In particular, it is argued that the uniform Dzyaloshinskii-Moriya interaction (which is allowed on the bonds along the ladder legs) can be the source of thisbehavior in Cu(C8H6N2)Cl2.PACS numbers: 75.10.Pq, 75.10.Jm, 75.50.Ee, 76.30.-vQuantum spin ladders exhibit a variety of exoticstrongly correlated states continuing to attract a greatdeal of attention. One of the most remarkable dis-coveries on this family of low-dimensional spin systemsis superconductivity, observed in Sr0.4Ca13.6Cu24O41.84under pressure [1]. Recently, spin ladders have beenused to address a number of fundamental questions,related to field-induced phase transitions in quantummatter. In magnetic fields, spin ladders undergo atransition from a disordered gapped to gapless phase,where they can be mapped onto a system of interact-ing bosons with the density of states easily controlled bythe applied magnetic field. At low-enough temperatures,when three-dimensional interactions become important,the system undergoes a transition into the magneticallyordered state (often recalled as to the magnon Bose-Einstein condensation [2]), while above these tempera-tures (when one-dimensional interactions become domi-nant) the quantum spin ladders provide a remarkable re-alization of the Tomonaga-Luttinger liquid (TLL) state[3].Among other spin-Hamiltonian parameters, theanisotropy of spin-spin interactions (hereafter magneticanisotropy) is known to play a particularly significantrole [4, 5], being, for instance, of crucial importance forthe realization of the magnon Bose-Einstein condensa-tion in gapped quantum magnets (where the presence ofthe uniaxial U(1) symmetry, corresponding to the globalrotational symmetry of the bosonic field phase, is oneof the most essential conditions). Electron spin reso-nance (ESR) spectroscopy is traditionally recognized asone of the most sensitive tools to probe the magneticanisotropy in magnets. In particular, a theory of thelow-temperature ESR was developed for uniform spin-1/2 Heisenberg antiferromagnetic (AF) chains and spin-1/2 Heisenberg AF chains perturbed by a staggered mag-netization [6–8]. The theory allowed to explain the ef-fect of the anisotropy on the ESR linewidth and fieldshift in a number of spin-chain systems (see [8–10] andthe references therein). A new approach to determineanisotropy parameters from the low-temperature ESRfrequency shift in a spin-1/2 strong-rung ladder has beenrecently developed by Furuya et al. [11]. The theorywas applied to (C5H12N)2CuBr4 (also known as BCPBor (Hpip)2CuBr4), indicating a good agreement with theexperimental data [12]. It was shown also, that theanisotropy strongly affects the ESR linewidth in spin-1/2strong-leg ladders [13], in particular in the TLL phase[14].The strong-rung spin-1/2 ladder material,Cu(C8H6N2)Cl2 [hereafter Cu(Qnx)Cl2, where Qnxindicates quinoxaline, C8H6N2] with Jr/kB = 34.2 Kand Jl/kB = 18.7 K [15], where Jr and Jl are theexchange coupling constants along the rungs andlegs, respectively. This material is characterized byHc1 ≃ 14.3 T [15, 16], which can be easily reachedusing a superconducting magnet (the second criticalfield corresponding to the fully spin polarized phase isHc2 ≃ 52 T [17]). The relatively low Hc1 makes thiscompound a perfect model system for studying spectraland magnetic properties of strong-rung spin-1/2 laddersin the field-induced gapless phase, where the TLL regimecan be realized [18].Series of Cu(Qnx)Cl2 single crystals of typically 1 mm32acbaaca(a) (b)baFIG. 1: (color online) Two views of the crystal structure ofthe spin-1/2 spin-ladder compound Cu(Qnx)Cl2. The Cu,N, C, and Cl ions are shown in red, blue, gray, and green,respectively. The ladders run along the b-axis direction.size were synthesized at the ETH Zürich using slow diffu-sion in methanol solution, as described in Ref. [16]. Thecrystal structure of Cu(Qnx)Cl2 is shown schematicallyin Fig. 1. The compound belongs to the monoclinic spacegroup C2/m with unit-cell parameters a = 13.237 Å,b = 6.935 Å, c = 9.775 Å, and β = 107.88◦ (Z = 4),measured at room temperature [19, 20]. The spin-1/2Cu2+ ions (shown in red in Fig. 1) are bridged by quinox-aline molecules, forming chains along the two-fold rota-tion axis b. Two chains are coupled into a two-leg ladderover two Cl− ions in each rung.The samples first were characterized using the X-bandBruker Elexsys E500 ESR spectrometer operated at afrequency of 9.4 GHz. High-quality, twin-free sampleswere chosen for high-field experiments. At room temper-ature X-band ESR experiments revealed ga = 2.16(1),gb = 2.03(1), and gc = 2.09(1). High-field ESR ex-periments were performed using a spectrometer, sim-ilar to that described in Ref. [21]. The spectrome-ter was equipped with a 16 T superconducting magnet,transmission-type probe in the Faraday configuration,and VDI radiation sources (product of Virginia DiodesInc.). The magnetic field was applied along the b axis.Examples of ESR spectra for the frequency 446.7 GHzare shown in Fig. 2. The ESR spectra were fit using theLorentzian lineshape function. With decreasing temper-ature, starting at T ∼ Jl/kB the ESR line shifts towardssmaller magnetic fields (Fig. 3), indicating the enhance-ment of short-range spin correlations. These correlationsappear also to be be responsible for the pronounced ESRnarrowing revealed by us in Cu(Qnx)Cl2 (Fig. 4).As shown in Ref. [22], the Hamiltonian of a spin lad-der for H > Hc1 can be reduced to an effective spin-1/2Heisenberg AF chain Hamiltonian. To set the stage wecan describe the low-temperature behavior of the systemwith the Hamiltonian H using the bosonization (or con-formal field theory) description, namelyH = H0 +Hpert, (1)15.3 15.4 15.5 15.6 15.7 15.8 15.9 16.0 = 446.7 GHzH II b21 K19 K13 K11 KTransmitance (arb. units)Magnetic field (T)7 KFIG. 2: (color online) Examples of ESR spectra obtained at446.7 GHz.0 3 6 9 12 15 18 21 24 2715.6015.6115.6215.63  Resonance field (T)Temperature (K) = 446.7 GHzH II bFIG. 3: ESR field as function of temperature obtained at446.7 GHz.whereH0 =v2∫dx[Π2 + (∂xφ)2] (2)is the Hamiltonian of the free boson field φ and its con-jugated momentum Π. The velocity of low-lying spinexcitations, v, can be written asv =πJeff√1−∆22 arccos∆. (3)Here, Jeff is the effective exchange interaction in thechain (proportional to the exchange coupling Jl, [22])and ∆ is a parameter describing the uniaxial anisotropy.The perturbation can be written asHpert = λ∫dx cos(√8πKφ) , (4)30 3 6 9 12 15 18 21 24 270102030405060708090100  ESR linewidth (mT)Temperature (K) = 446.7 GHzH II bFIG. 4: ESR linewidth as function of temperature obtainedat 446.7 GHz (symbols). The line is the fit results using Eq.(11) (see the text for details).with the conformal field theory exponent, K, given byK =ππ − arccos∆. (5)The ESR intensity, I(ω,H, T ), in the standard geom-etry can be expressed in the framework of the bosoniza-tion approach [6] for H < Jl (we use the units in whichh̄ = gµB = kB = 1) asI(ω,H, T ) ∝ −χ(T,H)ωImH2 + ω2ω2 −H2 −Π(ω,H, T ) ,(6)where χ(T,H) is the static susceptibility of the chain,ω is the frequency of the ac field and vq = H (whereq is the wave vector). In the above expression the selfenergy Π(ω,H, T ) can be calculated in the framework ofthe perturbation theory with respect to λ. One can showthatΠ(ω, q) = 4πKv2λ2[F r(ω, q)− F r(0, 0)] , (7)where F r(ω, q) is the retarded propagator of the bosonfield given byF r(ω, q) ∝ −24K−2vπ2T 2(πTv)4KΓ2(1− 2K)××Γ(K − i(ω+vq)2π )Γ(K −i(ω−vq)2π )Γ(1−K − i(ω+vq)2π )Γ(1−K −i(ω−vq)2π ),where Γ(x) is the Gamma function. In the case of res-onance we have to use ω = H . Then, it is clear thatthe shift of the resonance position due to spin-spin in-teractions and anisotropy is given by the real part ofthe self energy, while the ESR linewidth is related to itsimaginary part. In particular, the linewidth, ∆B, can beexpressed as∆B ≈ λ2v2√πK22KT 4K−3(2πv)4K−2×Γ(12 −K)Γ2(K)Γ(1− 2K)Γ(1−K). (8)We can consider the weak anisotropy A = ∆Jeff ofthe system as a perturbation with respect to the isotropicantiferromagnetic case K = 1. Notice that only the real(not effective) magnetic anisotropy reveals itself in theESR experiments. Then, according to the above and ne-glecting corrections due to marginal perturbations, suchas logarithmic ones, one finds∆B ≈(AJeff)2T. (9)Such a linear dependence between ∆B and T should ex-ist up to T ≈ Jeff . Taking into account logarithmiccorrections, one obtains∆B(T ) ≈ A2 ln2( vT) Tv2. (10)The linear fit of the ESR linewidth behavior for 5 K< T < 21 K [23], using the equation∆Btot = ∆B0 +δ(∆B)δTT, (11)gives ∆BTI = 21.7(5) mT andδ(∆B)δT= 3.2(1) mT/K,for the temperature-independent and temperature-dependent contributions, respectively (Fig. 4). The first,temperature-independent, contribution can be of thevan Vleck origin. Accordingly to Eq. (9), the secondterm of Eq. (11) givesA ≈ 0.066Jeff . (12)Assuming Jeff = Jl = 18.7 K, one obtains A ≈ 1.2 K.One source of ESR line broadening is the exchangeanisotropy, which can be roughly estimated using theformula DE ≈ Jeff (∆g/g)2 (where ∆g is the deviationof the g factor from the free electron value and assum-ing Jeff = Jl) [24], giving DE/kB ∼ 0.1 K. On theother hand, it is worthwhile to mention that the uni-form Dzyaloshinskii-Moriya (DM) interaction is allowedon the bonds along the ladder legs in Cu(Qnx)Cl2 bythe symmetry. The rough estimate using the formulaDDM ≈ Jeff (∆g/g) [24] gives for the DM interactionDDM/kB ∼ 1.5 K. This value agrees well with the oneobtained using Eq. (12), A ≈ 1.2 K, suggesting that theuniform DM interaction can be the source of the ESRlinewidth dependence observed by us in Cu(Qnx)Cl2.In conclusion, we have presented systematic ESR stud-ies of the spin-1/2 strong-rung Heisenberg ladder com-pound Cu(C8H6N2)Cl2. 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