4 Spin correlations in the A-site spinel MnSc 2 S 4
4.1 Introduction to magnetic frustration on the diamond lattice
4.1.1 Classical approach
The essence of frustration on the diamond lattice can be captured by theJ1-J2model [151, 156, 157]:
H =J1X
〈i j〉
Si·Sj+J2 X
〈〈i j〉〉
Si·Sj. (4.1)
Its classical ground state can be solved using the Luttinger-Tisza method, which basically calculates the eigen-problem of the exchange coupling matrixJ(q) similar to that outlined in
H (r.l.u.)
Figure 4.1 –(a)TheA-sites of the spinel AB2X4constitute a diamond lattice. The first-, second-, and third-neighbour couplings are indicated withJ1,J2, andJ3. (110) planes are shaded blue.
(b)Spiral surface (grey) predicted by mean-field theory for the J1-J2model with the ratio
|J2/J1| =0.85. The red ring emphasizes a cut within the (hk0) plane (blue).
section 3.3.2, and then looks for possible equal-spin configurations with propagation vectors qat theJ(q) minimum positions. For the 2×2 coupling matrix defined on the primitive unit cell of the diamond lattice, the eigen-energies are calculated to be [156]:
E±(q)=16J2
Noticing thatΛ(q)≤1, we immediately arrive at an important conclusion of theJ1-J2model:
for frustration ratio|J2/J1| ≤1/8, the minimum ofE±(q) is always realized atq=0 of theE− branch; while for|J2/J1| >1/8, the minimum ofE±(q) can be found instead from the relation Λ(q)= |J1/8J2|, which forms a two-dimensional surface in reciprocal space. For the frustration ratio|J2/J1| =0.85, the spiral surface is shown in Fig. 4.1b. Since eachqposition over this surface represents a spiral state, such a surface is called the ‘spiral surface’. Furthermore, when all the possible spiral states are populated at the same time, the system then realizes the so-called spiral spin-liquid state [151].
The effects of an additional third-neighbour couplingJ3and of spin anisotropy have been studied theoretically in Ref. [156]. Under the perturbation of an anti-ferromagneticJ3coupling, the extensive degeneracy of theJ1-J2model is relieved, and the ground state will possess a single propagation vector. Specifically, in the regime of|J2/J1| >0.65, the spiral state with propagation vectorqx=qy andqz=0 will be selected as the ground state. With this fixedq position, the spin anisotropy will determine the direction of the spiral plane. Two factors have been found to contribute to the spin anisotropy with possibly opposite effects. One factor is the dipolar interaction, which prefers a spiral plane perpendicular to the propagation vector.
Another factor is the spin-orbit coupling, which might prefer a spiral plane parallel to the propagation vector.
Of special interest is the effect of thermal fluctuations. Through a self-consistent treatment of
4.1. Introduction to magnetic frustration on the diamond lattice
the spin-fluctuations, it is found that ground states withqpoints on the spiral surface exhibit different susceptibility to thermal excitations [151]. Excitations around some specificqpoints are softer compared to others and might lead to an order-by-thermal-disorder effect where a long-range ordering transition is completely driven by thermal fluctuations [7].
4.1.2 Quantum approach
Very recent work by Gang Chen studies the effect of frustration on the diamond lattice through a quantum approach [155]. Similar to that in the classical approach, the Hamiltonian can be written as:
whereDz>0 describes an additional easy-plane anisotropy that might be realized with 3d ions ofS=1 like Ni2+[158]. In the regime of dominant anisotropy, the ground state of the Hamiltonian (4.4) describes a trivial quantum paramagnetic state withSiz =0. Using the quantum rotor model discussed in section 1.3, excitations in this quantum paramagnetic state can be calculated to be:
ωi,q=£
(4Dz+2ξi,q)(∆(T)+ξi,q)¤12
, (4.5)
where∆(T) is defined in the mean-field ansatz similar to that in eqn. (1.12) andξi,q with i=1,2 are the two eigenvalues of the exchange matrix as theE±(q) of eqn. (4.2). The excitation gap in eqn. (4.5) decreases with reducedDzand marks the transition into a long-range ordered state when the gap is reduced to zero.
The frustration ratio|J2/J1|again plays an important role. For|J2/J1| ≤1/8, the minimum ofωi,qis realized at the single pointq=0. But for|J2/J1| >1/8, the minimum ofωi,qcan be found on a continuum surface in reciprocal space:
coskx
which is exactly the same as the spiral surface described in eqn. (4.3). Furthermore, calcu-lations with the linear spin wave theory revealsq-dependent quantum renormalization for the ground state energies. Thus an order-by-quantum-disorder effect might be realized on a diamond lattice of quantum spins [155].
4.1.3 Experimental studies of frustratedA-site spinels
Experimental studies of frustrated A-site spinels cover a wide range of exotic properties [50, 158–160]. Here we concentrate on the endeavors to realize the spiral spin-liquid state. Till now, a series of compounds have been explored, including the cobaltates Co3O4and CoRh2O4
Figure 4.2 – The neutron diffuse scattering map of CoAl2O4reveals broad peaks at the Brillouin zone centers [166].
[161]; the aluminates MAl2O4 withM = Mn, Co [162–164]; and the scandium thiospinel MnSc2S4[165]. Among them, Co3O4, CoRh2O4, and MnAl2O4have been found to exhibit a relatively small ratio ofΘCW/TN=3.7, 1.2, and 3.6, respectively, and lie deep in the Néel phase.
In CoAl2O4and MnSc2S4, the ratioΘCW/TNis found to be 22 and 10, respectively, and have been suggest as possible candidates to realize the spiral spin-liquid state.
The spin correlations in CoAl2O4have been studied using both neutron scattering techniques [166–169] and local probes as NMR andµSR [170, 171]. Fig. 4.2 shows the neutron diffuse scattering map of CoAl2O4atT=1.5 K measured by Oksana Zaharkoet al.[166]. Broad peaks are observed at the centers of the Brillouin zone, indicating the incipient Néel order with q=0. Further studies of the spin wave excitations by inelastic neutron scattering reveal the frustration ratioJ2/J1∼0.1 [166], which is near but still smaller than the threshold of 0.125 to realize the spiral spin-liquid state.
Magnetic susceptibility measurements of MnSc2S4down to 1.8 K reveal two long-range-order transitions at 2.3 and 1.9 K, which are∼10 times smaller than the Weiss temperature of 23 K [165, 172] and suggest strong frustration in MnSc2S4. Neutron diffraction experiments on powder samples of MnSc2S4[172,173] confirm a long-range ordered state withq=(0.75 0.75 0) forT <1.9 K and a short-range correlated state with|q| ∼1 forT>2.3 K. And co-existence of the long- and short-range correlations is observed in the intermediate region between 2.3 and 1.9 K [173]. Of special interest is the short-range correlated state, where the propagation vectorsqare close to the Brillouin zone boundary and might originate from the spiral surface predicted for the highly frustrated spiral spin-liquid state [151].
Despite so much effort, direct observation of the spiral surface in MnSc2S4requires single crys-tal samples and was still lacking before our study [172]. Besides the short-range correlations, the long-range order of MnSc2S4also needs further investigations. Previous refinement of the