a r X i v :c o n d -m a t /0004403v 2 [c o n d -m a t .s t r -e l ] 21 J u l 2000Weak ferromagnetism of quasi-one-dimensional S =1/2
三人行必有我师翻译
antiferromagnet BaCu 2Ge 2O 7
I.Tsukada,1J.Takeya,1T.Masuda,2and K.Uchinokura 21Central Rearch Institute of Electric Power Industry,2-11-1Iwadokita,Komae-shi,Tokyo 201-8511,Japan 3Department of Advanced Materials Science,The University of Tokyo,7-3-1Hongo,Bunkyo-ku,Tokyo 113-8656,Japan (February 1,2008)Abstract Weak ferromagnetism of quasi-one-dimensional S =1/2antiferromagnet BaCu 2Ge 2O 7is studied by the magnetization measurement.The sponta-neous magnetization appears along the b axis below 8.8K.The local sym-metry between the intra-chain nearest neighbor spins allows the prence of Dzyaloshinskii-Moriya interaction,and the only possible spatial configuration of the weak ferromagnetic moment per spin uniquely determines the sign of the inter-chain interaction.A weak a -axis magnetic field can change the di-rection of the magnetization to the a -axis direction,which shows that the spin chain forms a weakly coupled weak-ferromagnetic chain system.75.10.Jm,75.25.+z,75.40.Cx,75.50.Ee
Typet using REVT E X
One-dimensional(1D)quantum antiferromagnet(AF)is of particular interest in both theory and experiment of quantum magnetism.It is widely accepted that a pure1D S=1/2 antiferromagnet has no long-range order at T=0K.1On the other hand,almost all the actual spin-chain systems show magnetic three-dimensional long-range order(3D-LRO)at their ground states due to afinite inter-chain interaction.While in general there are many kinds of magnetic3D-LRO states,the transition to a normal3D AF-LRO state has been studied mostly.2One remarkable exception is a weak ferromagnetic transition that was recently found in1D compound Sr0.73CuO2.3However,this compound cannot be treated as a uniform spin-chain system but rather is well explained as an alternating-chain sys-tem.In this paper,we report weak ferromagnetism(WF)of a uniform S=1/2spin chain in BaCu2Ge2O7,which is isomorphous to antiferromagnet BaCu2Si2O7.4In BaCu2Ge2O7, intra-chain Dzyaloshinskii-Moriya(DM)interaction and inter-chain Heinberg interaction cooperatively add a spontaneous magnetization in contrast to AF BaCu2Si2O7.cri
尺子英文
The single-crystal sample of BaCu2Ge2O7was grown by afloating-zone method,and was cut into a rectangular shape(a×b×c=1.7×1.6×3.1mm3).Since the a-and the c-axis lengths are almost the same,particular attention was paid to determine the crystal orientation by x-ray diffraction.We took the diffraction at randomly lected three different points at every surface,and found no trace of a twi
nned structure.However,the possibility of inclusion of misoriented grains inside the crystal was not eliminated.Magnetization measurement was carried out with a commercial SQUID magnetometer(MPMS,Quantum Design)in thefield range from0to5T.We also measured transver magnetization with this system.
Figure1shows the magnetic susceptibility measured at H=1000Oe.A broad peak is obrved around300K,and the data can befitted with a calculation by Bonner and Fisher(BF),5which indicates that the system is well described as an1D Heinberg AF.χb andχc are wellfitted assuming different g values,and J=540K is obtained as an intra-chain coupling showing a good agreement with that for the polycrystalline sample.4 As the temperature decreas to the WF transition temperature at T N=8.8K,deviation
from the BF curves becomes significant.Below T N,spontaneous magnetization appears as shown in the int of Fig.1.The magnetization along the b axis(M b)shows a typical evolution for ferromagnetic transition and is saturated at approximately9×10−2emu/g for T→0,which corresponds only to0.42%of what we may obrve when all the S=1/2 spins on Cu2+site are aligned toward the same direction.Thus we eliminate the possibility of normal ferromagnetic transition.Ferrimagnetism is impossible,becau BaCu2Ge2O7has no magnetic ions other than Cu2+that occupies an equivalent
site.Since the sample is highly insulating,itinerant or band ferromagnetism is impossible.
To e the anisotropic magnetization,thefield dependence of magnetization was mea-sured at the WF state.Figure2shows low-field magnetization along three principal axes.华浦日语
A characteristic steep increa of the magnetization is found along the a and c axes up to H =280Oe(H c).M a reaches more than1.6×10−1emu/g,while M b remains approximately 9×10−2emu/g.Once magneticfield exceeds H c,the slopes become similar to one another as shown in the int of Fig.2.The magnetization continues to increa along all the axes, which is a typical behavior of WF-LRO state.We did not obrve a spin-flop anomaly in thisfield region,which indicates the lack of a strong easy-axis anisotropy in this system.
Let usfirst discuss the crystal structure to reveal this WF-LRO state.BaCu2Ge2O7has a structure isomorphous to BaCu2Si2O7(e Fig.1of Ref.4)with slightly longer lattice parameters:a=7.028˚A,b=13.403˚A,and c=7.044˚A.6Two neighboring Cu2+ions are connected by one O2−ion and they form a chain along the c axis.However,the bond of Cu-O-Cu is not straight as shown in Fig.3(a).The angle of
to allow the DM interaction.According to the rule given by Moriya,10the Dzyaloshinskii vectors D i at sugarcane
bond i are approximately defined as schematically drawn in Fig.3(a).The positions of only two neighboring Cu2+ions and one intermediate O2−ion are taken into account.According to the room-temperature structural data,6D i is written as D i=D ((-1)i0.8332,(-1)i0.5462,0.0860).Here,the components perpendicular and parallel to the chain have different effects on the spin order;the former(D⊥i=(-1)i D(0.8332,0.5462,0)) changes its sign from one bond to the next,while the latter(D i=D(0,0,0.0860))keeps its sign along the chain.For the rest of the paper we shall ignore the effect of D i and simply consider only D⊥i from the following two reasons:1)the magnitude of D i is far smaller than that of D⊥i,and2)D i will stabilize a spiral order that is against the experimental obrvation.
We must know the arrangement of D vectors in the whole magnetic unit cell,becau the bulk spontaneous magnetization is cloly related to how D vectors are distributed.It is noted that D vectors and spin operators are both an axial vector when we consider the symmetry operations.For the a-axis directions,one chain is transformed to the neighboring one by the successive operations of a)reflection in the ac plane and b)rotation around the axis located at the middle of two adjacent Cu2+ions along the[101]direction and parallel to the b axis.Through the operations,the a component of D⊥i changes its sign twice,the b component is unchanged,and the order of the concerning spin operators are exchanged. As a result,D⊥i at the a-axis neighboring bonds points an
opposite direction.We can also transform a chain to the b-axis neighboring one by the operation of reflection in the ac plane and following parallel shift to the b direction.This process changes the sign of the a component only.In this way we determined all the D vectors in the magnetic unit cell as shown in Fig.3(b).
specificNext we discuss how the D vectors tilt spins to create the b-axis spontaneous mag-netization.For this purpo,we must determine the principal direction of the spins.The crystal symmetry is high enough that it must be along the a,b,or c direction.If we take only DM interaction into account,the fact that D vectors are not collinear along the b direc-
tion can differentiate the three cas becau there is afinite inter-chain interaction along the b axis.However,the recent theoretical studies proved that DM interaction is always accompanied by an additional term quadratic to D,which recovers O(3)symmetry of the spin-spin interaction.11This term is called as KSEA interaction,and has been established also experimentally.12Therefore we cannot predict the principal spin direction from DM and KSEA interactions even in the prence of inter-chain interaction.
On the other hand,experimental results indicates that the principal spin direction is along the c axis.T
越狱第三季剧情介绍his is explained as follows;1)Each chain must have weak ferromagnetic moment along the b direction,while D⊥i has the a and b components,2)if the principal direction is along the a or b direction,spins are canted only toward the c direction and we will have no b-axis magnetization,3)only when the principal direction is along the c axis,we will obrve weak ferromagnetic moment along both the a and b directions.Thus,we safely concluded that the c axis is the easy-axis direction in the actual system.Under this condition, spins on a single chain are confined in the plane perpendicular to D vectors(we shall call it as easy plane for convenience)as shown in Fig.3(c).It is convenient to define the WF moment per chain as the sum of the two sub-lattice magnetizations:M W F,k=(M1,k+M2,k)/2,where M i,k denotes the magnetic moment at the i-th sub-lattice in the k-th chain.Obviously the direction of the weak ferromagnetic moment is parallel to the easy-plane and simultaneously perpendicular to the c axis.Once we obtain the direction of M W F,k on each spin chain,the spatial configuration of M W F,k determines the bulk magnetization.Relative directions of M W F,k between the neighboring chain depend on the inter-chain interaction.In Fig.3(d), four different cas according to the signs of J a and J b are schematically drawn.Among the four cas,a spontaneous magnetization along the b axis is realized only when J a>0 and J b>0,and therefore BaCu2Ge2O7is considered to be the ca.Conquently,the spin configurations in the magnetic unit cell os determined as shown in Fig.3(e).
类似摩登家庭的美剧So far,we have assumed that the DM interaction is the dominant perturbation to the spin system and the inter-chain interaction gives only a condary effect.This assumption is probably valid by the following discussion.The magnitude of DM interaction can be
inkel
课后辅导班estimated from the magnetization data.Using the angle between the easy plane and the ac plane being57.3◦,the canting angle of each spin from the c-axis direction is estimated as θ=0.95◦.13Then the coar relation,tan2θ∼=D/J,gives D as18K.On the other hand, we can also estimate the inter-chain interaction as an averaged value according to the chain mean-field theory:14J=540K and T N=8.8K give the averaged inter-chain interaction as|J⊥|=2.9K,which is sufficiently smaller than D.Of cour,the direct estimation of J a and J b is necessary for further quantitative discussion,but|D|≫|J a|and|D|≫|J b|are probably valid in BaCu2Ge2O7.The above discussion means that BaCu2Ge2O7is a good system to study the magnetism of weak-ferromagnetic spin chains that are weakly coupled with each other,and the sign of the inter-chain interaction can easily change the magnetic ground state demonstrated as the difference between BaCu2Ge2O7and BaCu2Si2O7.
Finally we discuss the low-field anisotropic magnetization of BaCu2Ge2O7upon this model structure.M a steeply increas up to H cfirst and suddenly changes its slope.This is qualitatively explained by the rotation of WF moment as schematically shown in Fig.4(a). When thefield is applied
parallel to the a axis,M W F,1and M W F,2hold their directions,while M W F,3and M W F,4are rotated almost by180◦,which can produce a large magnetization along the a axis.We consider that the spins on the chains3and4are cooperatively rotated without changing their relative angle so much,becau both J b and the Zeeman energy are too weak to change the relative angle of the spins against the dominant DM interaction. After this rotation has beenfinished at280Oe,large magnetization appears along the a axis,and simultaneously net magnetization along the b axis should disappear.In order to confirm this rotation,we carried out the transver magnetization measurement,and found that the b-axis magnetization is actually suppresd with increasingfield along the a axis as shown in Fig.4(b).Since the geometry of the pick-up coil for our transver magnetization measurement was the cond differential type,finite M b gives an even-function respon to the direct signal.Actually we got such a signal around0Oe,and obrved the evolution of M b.However,as thefield approach280Oe,the direct signal changes its shape andfinally turns to an odd-function respon,which means that the M a induced by thefield dominates
