Spin glass state and enhanced spiral phase in doped delafossite oxide CuCrO
2Z. R. Yan,1M. H. Qin,1,*S. Dong,2M. Zeng,1X. B. Lu,1X. S. Gao,1and J.-M. Liu3,†
1Institute for Advanced Materials and Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials, South China Normal University, Guangzhou 510006, China
2Department of Physics, Southeast University, Nanjing 211189, China
3Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, China (Received 17 May 2016; revised manuscript received 23 June 2016; published 8 July 2016)
In this paper, we study the doping effects on the magnetic states of CuCrO2based on the classical frustrated spin model [Linet al.,Phys. Rev. B 89, 220405(R)(2014)]. Several experimental observations can be well reproduced by the Monte Carlo simulations of the modified spin models. Our paper suggests that the disorder induced by V/Al doping cooperating with the frustration in the system may contribute to the emergence of the spin glass state. Furthermore, the hole doping by Mg2+substituting Cr3+enhances the quantum fluctuations and bond disorder which modulate the biquadratic exchanges and in turn results in the promotion of the spiral phase, consistent with the experimental report.
DOI:10.1103/PhysRevB.94.024410
I. INTRODUCTION
During the past decades, nontrivial magnetic phases ob- served in so-called type-II multiferroic materials, such as the delafossite oxide CuCrO2, have drawn extensive attention due to their interesting physics and potential applications [1–6].
Specifically, the magnetic Cr3+ions (having a quasiclassical S=3/2 spin) in CuCrO2form a triangular lattice on theab plane, and the compound exhibits an incommensurate proper- screw spiral state (ICY state) below the critical temperature TN. In this state, the spiral plane is perpendicular to the ab layer, and the three spins in each triangle sublattice form the so-called 120◦structure as shown in Fig.1(a). Interestingly, an electric polarization (P) along the spiral propagation vectork is induced through the variation of the hybridization between the Cr d orbitals and the O p orbitals caused by spin-orbit coupling [6,7].
Owing to the strong magnetoelectric coupling in this system, a significant dependence ofP on the magnetic-field Hhas been observed in experiments [8–10]. For example, the flop ofP from thexaxis to theyaxis has been reported due to the 90°rotation of the spin-spiral plane underHapplied along theydirection [11]. Theoretically, severalH-induced cycloidal spiral phases have been uncovered in the anisotropic classical spin model for CuCrO2 [12,13], well consistent with the experimentally reported electric polarization based on Arima’s mechanism for multiferroic behavior [7]. The competitions among magnetic frustration, Zeeman energy, and thermal fluctuations are suggested to be responsible for the changes in magnetoelectric properties in CuCrO2underH[13].
On the other hand, the magnetic states of CuCrO2can also be effectively modulated through impurity doping, and several interesting phenomena have been reported in experiments [14].
For example, a spin glass state induced by the V3+for Cr3+ substitution has been observed in the CuCr1−xVxO2series for x >0.18 [15,16]. Similarly, a short-range antiferromagnetic (AFM) excitation resulted from the enhancement of a spin
glass component has been observed in nonmagnetic Al-doped systems [17]. More interestingly, significant hole-doping effects have been uncovered in CuCr1−xMgxO2forx 0.03.
In detail, TN shifts toward high T with the substitution of nonmagnetic Mg2+for Cr3+, indicating the important role of the coupling between the itinerant hole and the localized spin [18–21].
The study on the doping effects on magnetic states in multi- ferroic oxides becomes very important from the following two viewpoints. On one hand, this study helps one to understand the multiferroic physics and to search for more attractive systems with improved magnetoelectric performance. On the other hand, despite the long history of research, spin glass phase transition is still a hot topic in statistical mechanics, whereas few results have been reported on triangular antiferromagnets [22]. Thus, the study of doped CuCrO2 is essential both in application potential and in basic physical research. However, several doping effects on magnetic phase transitions reported in experiments are still far from well understood in theory, including: (1) spin glass behaviors reported in CuCr1−xVxO2 and CuCr1−xAlxO2 and (2) the unconventional promotion of the ICY state by the hole doping for the Mg substitution.
Fortunately, the earlier spin model for CuCrO2which has been successfully used to explain the magnetic-field effects allows one to explore the doping effects based on a modified model [13]. For example, in our earlier work, both the lattice defects and the random exchange induced by the isovalent substitution of nonmagnetic Al3+ for Cr3+ are confirmed to be responsible for the decrease in TN with the increasing Al-doping magnitude [23,24]. In this paper, we study the modified spin models to further investigate the doping effects on the magnetic phase transitions in doped CuCrO2. Several experimental observations are well explained in our Monte Carlo simulations. In detail, it is suggested that: (1) the spin glass state in CuCr1−xVxO2is resulted from the competitions between the AFM Cr3+-Cr3+coupling and the ferromagnetic (FM) V3+-Cr3+coupling, and (2) the hole doping by Mg2+for Cr3+enhances the quantum fluctuations and bond disorder and modulates the biquadratic interactions and in turn enhances the ICY state.
(a)
(b)
FIG. 1. Spin structure and exchange interaction in (a) a clean system and (b) a doped system. The blue/red dotted lines denote the AFM/FM exchanges, and the red circle is the V3+/Cr4+cation. The three sublattices are labeledA,B, andC, and the wave vectorkand induced polarizationPare also shown with the red arrow.
The remainder of this paper is organized as follows: in Sec.IIthe model and the simulation method will be described.
SectionIIIis attributed to the simulation results and discussion, and the conclusion is presented in Sec.IV.
II. MODEL AND METHOD
For the case of V3+substituting for Cr3+, no additional hole is produced in the system, whereas the competing interactions between Cr3+and V3+are available [strong AFM Cr3+-Cr3+ coupling, weak AFM V3+-V3+ coupling, and FM Cr3+-V3+ coupling as depicted in Fig.1(b)] [16]. Taking into account this fact, we studied a modified spin model for V3+randomly doped CuCrO2, and the Hamiltonian can be written as
H =
i,j
JijijSiSj +
i
1 2AxSi,x2
−
i
1
2AzSi,z2 −
i
H·Si, (1) with
ij =
⎧⎨
⎩
1, Cr3+−Cr3+,
−0.4, V3+−Cr3+, 0.2, V3+−V3+.
(2) The first term is the exchange interaction between the nearest neighbors, and a spatial anisotropy with J/J = 0.7654 is considered as shown in Fig. 1(a). The couplings between different ions are modulated by the parameterij which are chosen to be consistent with the experimental report.
The second term is the in-plane hard-axis anisotropy with Ax =0.005J, the third term is the out-of-plane easy-axis anisotropy withAz=0.05J, and the last term is the Zeeman coupling. Here,J/J,AzandAxare chosen to be the same as those in earlier work [13], which well reproduce the spin state
under zeroH. For simplicity, the length ofS(Cr3+and V3+), J, and the Boltzmann constant are set to unity.
Several parameters are calculated in order to characterize the ICY state and spin glass state. For example, the vector chirality is calculated by [25]
χ= 2 3√
3 1 N
r
(SA×SB+SB×SC+SC×SA), (3)
whereN =L×Lis the amount of the total spins and the sum is over all the plaquettes of the system (ABCin Fig.1). The components parallel (χ) and perpendicular (χ⊥) toHor to the easyzaxis (H=0) are calculated. For the ICY state,χ⊥>0 andχ =0 are expected. For the spin glass transition, the order parameter generalized to wave vectorkis defined to be as fol- lows:qμν(k)=N−1iSμ(1)i Sν(2)i exp(ikRi), whereμandνare spin components, “(1)” and “(2)” denote two identical copies of the system with the same interactions, andRiis the position vector at theisite [26,27]. Then, the spin glass susceptibility is calculated byXSG(k)=N
μ,ν[|qμν(k)|2]avg, where· · · denotes the thermal average and [· · ·]avg is the average over disorder. In this paper, disorder averages are taken overNs samples with Ns ranging from 400 for L=6 to Ns=10 for L=24. Subsequently, the spin glass correlation length is determined from
ξL= 1
2 sin(kmin
2)
XSG(0) XSG(kmin)−1
1/2
, (4)
wherekmin=(2π/L)(1,0,0). Thus, the freezing temperature Tgis estimated from the crossing points ofξL/Lfor different L’s, according to the scaling lawξL/L=f[L1/ν(T −Tg)], whereνis the correlation length exponent.
On the other hand, itinerant holes are produced by Mg2+
for Cr3+substitutions, enhancing the quantum fluctuations and bond disorder. In earlier works, it has been clearly proved that thermal fluctuations and bond disorder can produce an effec- tive biquadratic exchange in the classical Heisenberg triangular antiferromagnet [28]. Furthermore, quantum fluctuations can also generate a similar term as uncovered by the perturbation theory in earlier work [29]. Thus, the biquadratic interactions are further taken into account in the model to study the Mg2+ doping effects,
HK=K1
i,j
(SiSj)2+K2
[i,k]
(SiSk)2. (5)
Here, the nearest-neighbor and the next-nearest-neighbor biquadratic interactions are considered. Furthermore, Cr4+
cations are also produced due to the additional holes, and a double-exchange FM interaction between Cr4+-Cr3+ may be available [21]. In this paper, we set ij = −1 for the Cr4+-Cr3+ interaction and neglect the lattice defects and the itinerancy of Cr4+due to the very small amount of Mg2+.
Our simulation is performed using the standard Metropolis algorithm and temperature-exchange method [30,31]. Unless stated elsewhere, the simulation is performed on a 24×24 lattice with periodic boundary conditions.
(a)
(b)
FIG. 2. The (a) calculatedχ⊥and (b) specific heatCas a function ofT for variousx’s.
III. SIMULATION RESULTS AND DISCUSSION A. Spin glass state induced by magnetic/nonmagnetic
impurity doping
First, we study the effects of the V doping on the magnetic properties in CuCrO2. Figure2(a)shows the simulatedχ⊥as a function ofT for variousx’s. In the clean limitx=0 whenT falls down to the transition point, χ⊥ increases, whereasχ and the spin coplanarity remain small (not shown here), figuring the development of the ICY state. With the increase in x, theχ⊥curve shifts toward the low-Tside, demonstrating the suppression of the ICY state. The transition pointTN(Tpeak, exactly) can be roughly estimated from the position of the peak in the calculated specific heatC as given in Fig.2(b).
It is clearly shown thatTpeakdecreases with the increasingx, consistent with the experimental report [16]. Furthermore, the value ofχ⊥is also significantly decreased, indicating that the ICY state is not dominated at lowT (forx >0.2, at least).
Interestingly, spin glass order at low T emerges due to the combination of disorder and frustration forx >0.2. For example, Fig. 3(a)shows the calculated ξL/L as a function of T for various L’s at x=0.5. From the common well- defined crossing point, we estimate the freezing temperature Tg =0.087±0.005. Actually, the finite temperature spin glass transition has been reported in a dilute Ising system on the triangular lattice [22]. In this paper, it is suggested that the V doping produces disorder and leads to spin glass behavior in the frustrated Heisenberg model with the uniaxial anisotropy [32].
Furthermore, we plotξL/Lin the scaling form in Fig.3(b)and estimate that the spin glass transition is with a critical exponent ν=1.25±0.03.
(a) (b)
FIG. 3. The (a) calculatedξL/Las a function ofT and (b) as a function ofL1/ν(T −Tg) for variousL’s atx=0.5.
As a short summary, the simulated phase diagram for V doping is presented in Fig.4(a)which qualitatively reproduces the experimental one. The conventional ICY state is suppressed with the increase inx, and a spin glass state is favored beyond x >0.2 due to the introduction of disorder. Furthermore, the spatial anisotropy may be changed withx in real materials.
Thus, the case of the spatial isotropic model (J=J) is also investigated, and the corresponding results are shown in Fig.4(b). It is noted that the frustration is further enhanced in the model for J=J in favor of spin glass magnetism, resulting in the enlargement of the spin glass phase in the phase diagram. However, the estimatedν’s for x <0.2 are rather abnormal (ν >2), indicating that the spin glass behavior is a little different from that for x >0.2. Furthermore, the calculated χ⊥ curves show that the ICY state can be well stabilized at lowT forx <0.2 (the corresponding results are not shown here), further demonstrating that the estimatedTg in thisxregion is not a genuine one and additional disorder is needed for the development of the spin glass state.
On the other hand, it has been experimentally reported that the spin defects produced by Al doping destabilize the ICY state accompanying the enhancement of the spin glass component [17]. This behavior is also reproduced in our simulations in which the nonmagnetic Al3+ impurity is simply considered as a lattice defect. The spin glass order can be observed at low T when x increases above 0.35 as shown in Fig. 5(a). Similarly, the case of J=J is also investigated, and a rough phase diagram on the (x,T)
(a) (b)
FIG. 4. The estimated phase diagram on the (x,T) plane for V doping with (a) spatial anisotropy and (b) isotropy. The correlation exponents are also depicted.
(a) (b)
FIG. 5. The estimated phase diagram on the (x,T) plane for Al doping with (a) spatial anisotropy and (b) isotropy. The correlation exponents are also depicted.
parameter plane is presented in Fig. 5(b). The spin glass state shows significant dependence on the spatially anisotropy, which may provide useful information in understanding the experimental observations.
In short, our work in this part undoubtedly demonstrates the important role of disorder caused by V/Al doping in the development of the spin glass state in CuCrO2, although the accurate determination of transition points may be not available due to the finite-size effects in the simulations.
B. Enhanced ICY state in a Mg-doped system
At first glance, the introductions of the lattice defects and FM Cr4+-Cr3+interaction by the substitution of nonmagnetic Mg2+ for Cr3+ will definitely destabilize the ICY state, contrary to the experimental observation. Thus, the nontrivial promotion of the ICY state in Mg-doped CuCrO2 indicates the essential role of the doped holes in modulating the magnetic phase transition. It is expected that the interaction
(a)
(b)
FIG. 6. The calculatedCas (a) a function ofT for various (K1, K2)’s and (b) the contour plot ofTpeak on the (K1, K2) parameter plane.
(a)
(b)
FIG. 7. The calculated specific heat C as a function of T for (a)x=0 atHy=0 and Hy=0.3 (K1=0,K2=0), and (b)x= 0.03 atHy =0 andHy =0.3 (K1=0.05,K2= −0.02).
between the doped hole and the localized spin may enhance the quantum fluctuations which could be described by the classical biquadratic interactions [shown in Eq. (5)] [25]. Here, we introduce the additional biquadratic interactions in the model to study the hole-doping effects. Furthermore, in one of the earlier theoretical works studying a similar triangular antiferromagnet, it has been proved that quantum fluctuations may produce an effective biquadratic exchange (negativeK1 andK2), whereas bond disorder may generate a positiveK1 interaction [28]. However, the exact values ofK1andK2are not available so far, and we systematically studied the effects ofK1andK2on the multiferroic phase transition in this paper.
Figure 6(a) shows the calculated specific heat curves for various (K1, K2)’s. As the magnitude of the negative K2 increases from zero (−K2<0.08), the transition to the ICY state shifts toward the high-T side. It is easily noted that the ICY state can be further stabilized by the negative K2 interactions due to the fact that the angle between the next-nearest neighbors is much less thanπ/2. The simulated results are summarized in Fig.6(b)which presents the detailed phase diagram on the (K1,K2) plane. Considering the small amount of Mg doping in experiments, the effective biquadratic exchanges are expected to be much weaker than the spin exchanges. Thus, it is strongly suggested that the cooperation of the enhanced quantum fluctuations and bond disorder may cover the negative effect of the FM Cr4+−Cr3+ interaction and enhance the ICY state, leading to the increase inTpeak, qualitatively consistent with experimental observation [19].
Furthermore, earlier experiments reveal thatTpeakforx = 0.03 shifts toward lowT by applying a 9-T magnetic field,
whereas that for x=0 is almost unaffected, demonstrating improved magnetoelectric properties in the Mg-doped system [19]. This interesting phenomenon is also captured in our simulations, and the calculated results are shown in Fig. 7 which well reproduces the experimental observations. In the clean limit x =0, the specific heat curves for Hy =0 and Hy=0.3 (magnetic field applied along they axis) coincide with each other [Fig. 7(a)], indicating that Hy =0.3 never affects the AFM transition behavior. Forx=0.03,TNshifts toward the low-T side, and the peak height ofCis noticeably decreased by applyingHy=0.3. Thus, it is strongly suggested that the FM Cr4+-Cr3+interaction and biquadratic exchanges induced by Mg2+doping are responsible for the magnetic-field effects onTpeakin the hole-doped CuCrO2.
IV. CONCLUSION
In conclusion, we have studied the doping effects on the magnetic states of CuCrO2by the Monte Carlo simulation of
the frustrated spin models. It is suggested that the disorder induced by V/Al doping and the frustration in the system may result in the emergence of the spin glass state, consistent with the experimental observations. Furthermore, the hole doping by Mg2+substituting Cr3+enhances the quantum fluctuations and bond disorder and in turn leads to the unconventional promotion of the AFM ICY state.
ACKNOWLEDGMENTS
This work was supported by the National Key Projects for Basic Research of China (Grant No. 2015CB921202), the National Key Research Programme of China (Grant No.
2016YFA0300101), the Natural Science Foundation of China (Grants No. 51332007, No. 51322206, and No. 11274094), the Science and Technology Planning Project of Guangdong Province (Grant No. 2015B090927006), and Special Funds for the Cultivation of Guangdong College Students’ Scientific and Technological Innovation (Grant No. pdjh2016b0138).
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