Magnetization plateaus of dipolar spin ice on kagome lattice
Y. L. Xie, Y. L. Wang, Z. B. Yan, and J.-M. Liua)
Laboratory of Solid State Microstructures, Nanjing University, Nanjing 210093, China
(Presented 5 November 2013; received 23 September 2013; accepted 1 November 2013; published online 31 January 2014)
Unlike spin ice on pyrochlore lattice, the spin ice structure on kagome lattice retains net magnetic charge, indicating non-negligible dipolar interaction in modulating the spin ice states. While it is predicted that the dipolar spin ice on kagome lattice exhibits a ground state with magnetic charge order and冑3冑3 spin order, our work focuses on the magnetization plateau of this system. By employing the Wang-Landau algorithm, it is revealed that the lattice exhibits the fantastic three- step magnetization in response to magnetic fieldhalong the [10] and [01] directions, respectively.
For the h//[1 0] case, an additional 冑3/6Ms step, where Ms is the saturated magnetization, is observed in a specific temperature range, corresponding to a new state with charge order and short- range spin order.VC 2014 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4863808]
Frustrated magnetic systems have been under intensive investigations for the rich ground states.1–3The spin structure is determined by the specific lattice symmetry and competing interactions.4,5 The well-known spin ices, including pyro- chlore oxides Ho2Ti2O74 and Dy2Ti2O7,6 are typical spin frustrated materials. Here, rare-earth ions Ho3þ and Dy3þ with large magnetic moments (10lB) are sited on vertices of the corner-shared tetrahedrons, and the moments can be treated as the Ising spins pointing along the localh111iaxes due to the huge crystal field.6If only the nearest-neighbor fer- romagnetic interaction is considered, the ground state is the spin ice state with two spins pointing into the center of tetra- hedron and the other two pointing out. However, in the spin ice state, the long-range dipolar interaction becomes impor- tant in the elementary excitations of quasi-particles carrying equivalent magnetic charges, while the dynamics of magnetic monopoles can be different from the cases free of magnetic charges.7,8Therefore, special attention should be paid to the long-range dipolar interaction.
In fact, as an analogous system, two dimensional spin ice on kagome lattice has been interested.3,9–11 The Ising spins point to the center of the triangles. Thei-th spin has moment Si¼lri withri being the easy axis vector andl being the total spin, as seen in Fig.1(a). At the ice state, each triangle of the lattice has two spins pointing in and one pointing out, or one spin pointing in and two out, shown in Fig. 1(b).
Obviously, this lattice exhibits regularly arranged magnetic monopoles (or magnetic charges). To have better illustration of the magnetic charges, the so-called dumbbell scenario is introduced, as plotted in Fig.1(c), where the spins (magnetic dipoles) can be simply seen as two opposite magnetic charges qisited at the center of the two nearest triangles7
qi¼6q¼6 ffiffiffi3 p l 2rnn
; (1)
wherernnis the separation between the two spins. The total magnetic charges for one triangle is
Qa¼X
i2a
qi; (2)
whereQais the charge on thea-th site of the honeycomb lat- tice and symbol i labels one of the three vertices of the triangle.
Unlike the pyrochlore spin ice lattice, the spin ice on kagome lattice retains net magnetic charges under the ice ruleQa¼6q. The nonzero magnetic charges make the dipo- lar interaction particularly important in determining the ice state. The Hamiltonian of a kagome spin ice should include the nearest exchange interaction and the long-range dipolar interaction, written as
HS¼ JX
<i;j>
SiSjþD 2
X
i6¼j
SiSj
jrijj3 3ðSirijÞðSjrijÞ jrijj5
" #
;
(3) whereJandDare the exchange interaction factor and dipo- lar interaction coefficient, respectively;rij¼rirjis the spa- tial vector connecting spinsSiandSj.
This so-called dipolar kagome spin ice model, corre- sponding to the artificial spin ice on kagome lattice, has been carefully studied theoretically.12 The ice rule applies when
FIG. 1. Geometry of kagome spin ice. (a) The spin triangle with the spin coor- dinates, given the [1 0] and [0 1] directions. (b) The冑3冑3 magnetic order state and (c) the corresponding dumbbell model with magnetic charge order.
a)Author to whom correspondence should be addressed. Electronic mail:
liujm@nju.edu.cn
0021-8979/2014/115(17)/17E122/3/$30.00 115, 17E122-1 VC2014 AIP Publishing LLC
JOURNAL OF APPLIED PHYSICS115, 17E122 (2014)
the effective neighbors interaction Jeff¼J/2þ7D/4>0. It should be mentioned that the transition into the long-range spin order state or even the short-range charge order state can be difficult due to two reasons.13One is the anisotropic barrier which prevents the spins from free flipping. The other is the long-range dipolar interaction which may trap the lat- tice into local intermediate states. To overcome this diffi- culty, a scenario was proposed, in which the dynamics is realized by applying a rotating magnetic fieldhso that the anisotropic barrier can be effectively avoided.14At the same time, a competition with the dipolar interactions is induced, which may bring in additional phenomena deserved for exploration.
In this work, we start from an alternative approach and study the magnetizationMof the dipolar kagome spin ices in order to understand the effect of dipolar interaction in com- peting with the Zeeman energy. It is interesting to observe the multi-step magnetization in this dipolar frustrated lattice, while such behavior is often observed in spin frustrated systems.15
Considering the Zeeman energy induced by magnetic fieldh
HZ ¼ X
i
hSi: (4)
We have the lattice Hamiltonian as
H¼HSþHZ: (5) The three interaction terms compete with one and another. If conventional algorithm like the Metropolis algo- rithm is employed to track the ground state, the lattice may trap at some metastable states. An advanced approach to fig- ure out this difficulty is to employ the WLS algorithm.16,17 In general, we simulate the two-dimensional energy-magnet- ization (E-M) joint density of state (DOS) g(E, M).18 However, the Hamiltonian is so complex that it is impracti- cable to calculate g(E,M) directly. Alternatively, one may calculate a physical quantityufor the one-dimensional DOS g(E) by
huiT¼ X
E
uðEÞgðEÞexpðE=kBTÞ X
E
gðEÞexpðE=kBTÞ
; (6)
where kB is the Boltzmann constant. The order parameter huias a function of Tcan be calculated if functionu(E) is available.
In the simulation, we calculate the DOS g(E) using the standard WLS with a nonzeroh. Then we use the obtained g(E) to calculateM(E). The procedure is described as the fol- lowing. First, random walk is done following the importance sampling rule of the WLS:
pðE1!E2Þ ¼min 1;gðE1Þ gðE2Þ
; (7)
wherep(E1! E2) is the transition probability of a random walk from event with energy level E1 to that with energy
levelE2. When the random walk visits event with energyE, M is recorded. The random walk does not stop until 107M-data for every energy E are recorded. Finally, we obtain spectrumM(E) from regularly selecting 106data from theMdatabase withEand calculating their statistic average.
The spectrumM(h,T) is then obtained from Eq.(6).
The simulation is performed on a kagome lattice of 1212 with periodic boundary conditions. It is noted that the qualitative characters of the step-like magnetization at lowTare negligibly affected by the exchange interaction fac- tor J as long as Jis larger than 7/2D. For simplicity, we chooseJ¼0.5Dand setl¼1 for all of our calculations. The long-range dipolar interaction is summed up using the stand- ard Ewald summation scheme.19 In the WLS sequence, the modification factor f starts from exp(1.0) and ends with 1.00000015. The criterion for the histogram flatness is that the data scattering is less than 610% of the average histo- gram over the wholeEspace.
We perform extensive calculations on theM(h,T). The step-like magnetization is observed in the low-T range. We present our results on two cases:h//[1 0] and h//[0 1]. The [1 0] axis deviates from any of the three anisotropic axes but the [0 1] axis is parallel with one of them. TheM(T,h) pro- files for the two cases are plotted in Figs.2(a)and2(b). It is noted that the saturated M is Ms¼2/3, where the spins are forced along the local axes and all theM-data are normalized by Ms. Two characters are shown. First, in the low-T range (T/D<0.02), a three-step Mprofile appears upon increas- ing hfor both h//[1 0] and h//[0 1]. Second, it was reported that the three-step M profile becomes blurry gradually with increasing T/D over 0.02.20 It is indeed the case for h//[0 1], and the three-stepM profile is replaced by a trivial monotonous increasing ofMwithhuntil theMsis reached.
For a clearer clarification, the spin lattices correspond- ing to these steps are shown in Figs.2(c)–2(e), respectively.
Such a three-stepMprofile as the ground state is the conse- quence of the competition between the dipolar interaction and the Zeeman energy. We look at the magnetic transition from local magnetic order (charge order) to long-range mag- netic order atT¼Tmwhere the residual entropySdecreases
FIG. 2. SimulatedM(h,t) profiles: (a)h//[1 0] and (b)h//[0 1]. The ground states of spin lattice with (c)M¼0, (d)M¼1/2, and (e)M¼1. All theM- data are normalized byMs.
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down to zero, as indicated in Fig.4(b). For theh//[1 0], the evaluatedTm(h) is shown in Fig.3(a)where the three-stepM profile is also plotted. It is clear that there are two critical fields ath/D0.1 andh/D0.128, respectively.
In Fig. 3(b) are plotted, respectively, the three-step M profiles atT/D¼0.001 for bothh//[1 0] andh//[0 1], and the four-stepMprofile atT/D¼0.025 forh//[1 0]. TheM(h) pro- files at a series of differentTare plotted in Fig.3(c). Clearly, one sees the coexisting 1/2 step and冑3/6 step in this narrow T-range (roughly 0.009<T/D<0.049). This special inter- mediate state is again the outcome of the delicate competi- tion between the Zeeman energy and dipolar interaction in the presence of thermal fluctuations. In our simulations,Mx
andMy are the components ofM along the [1 0] and [0 1]
axes, respectively. TheT-dependences ofM,Mx, andMyare shown in Fig.4(a)withh/D¼0.112 along the [1 0] direction.
At the ground state (orT/D<0.01),M¼1/2, andMx¼冑3/4, My¼1/4. Clearly,Mxis induced byh. However, the nonzero Myis generated by the dipolar interactions.
One notes that this 冑3/6 step corresponds to a special spin configuration with magnetic charge order and partial magnetic order. This characteristic can be confirmed from the residual entropy calculation in Fig.4(b). The residual en- tropy of the magnetic charge order is 0.108.3However, the residual entropy for the冑3/6 step here is0.081, implying the existence of the partial magnetic order in addition to the magnetic charge order.
Finally, we would like to mention that the significant role of the long-range dipolar interaction associated with the so-called magnetic charge is not easy to access in most spin-frustrated materials where no net magnetic charge is available, while it is effectively illustrated in the present kagome spin ice. A number of oxide compounds have the kagome-like lattice21 and the frustrated behaviors are of special interests not only for fundamental significance but also for application potentials.
This work was supported by the National 973 Projects of China (Grant No. 2011CB922101), the Natural Science Foundation of China (Grant Nos. 11234005 and 51332006), and the Priority Academic Program Development of Jiangsu Higher Education Institutions, China.
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FIG. 3. (a) Simulated M(h) at T/D¼0.001 and Tm(h) for h//[1 0]. (b) Simulated three-stepMprofiles forh//[0 1] andh//[1 0] atT/D¼0.001, and the four-stepMprofile atT/D¼0.025 forh//[1 0]. (c) SimulatedM(h) curves at differentTvalues forh//[1 0].
FIG. 4. (a) SimulatedM(T) and its componentMxalong the [1 0] and com- ponentMyalong the [0 1]. (b) Simulated specific heatC(T) and entropyS(T) per spin for h//[1 0]. The dashed lines showS¼0.693 for paramagnetic state, S¼0.501 for the spin ice state, andS¼0.108 for the charge order state. TemperatureTmis defined as the point at whichS!0.h/D¼0.112.
17E122-3 Xieet al. J. Appl. Phys.115, 17E122 (2014)
