source：nature
Abstract
Quasicrystals (QCs) possess a unique lattice structure without translational invariance, which is characterized by the rotational symmetry forbidden in periodic crystals such as the 5-fold rotation. Recent discovery of the ferromagnetic (FM) long-range order in the terbium-based QC has brought about breakthrough but the magnetic structure and dynamics remain unresolved. Here, we reveal the dynamical as well as static structure of the FM hedgehog state in the icosahedral QC. The FM hedgehog is shown to be characterized by the triple-Q state in the reciprocal-lattice qq space. Dynamical structure factor is shown to exhibit highly structured qq and energy dependences. We find a unique magnetic excitation mode along the 5-fold direction exhibiting the streak fine structure in the qq-energy plane, which is characteristic of the hedgehog in the icosahedral QC. Non-reciprocal magnetic excitations are shown to arise from the FM hedgehog order, which emerge in the vast extent of the qq-energy plane.
Quasicrystal (QC) has a unique lattice structure with rotational symmetry forbidden in periodic crystals1. Although progress has been made in unraveling their atomic structure2,3, the understanding of their electric properties remains a challenging and fascinating problem, because the Bloch theorem can no longer be applied.
The unresolved vital issue has been whether the magnetic long-range order is realized in the three-dimensional QC4,5,6,7,8,9,10,11,12,13,14,15,16. Recently, the ferromagnetic (FM) long-range order has been discovered experimentally in the QC Au-Ga-Tb17. Theoretically, the FM long-range order has actually been shown to be realized in the QC Au-SM-Tb (SM=Si, Al, Ge, Sn and Ga)18,19. Interestingly, the hedgehog state, where the magnetic moments at the Tb site located at each vertex of the icosahedron (IC) is directed outward (see Fig. 1A), has been shown to form a uniform long-range order as shown in Fig. 1B18. Moreover, the hedgehog state on the IC has been revealed to be characterized by the topological invariant, i.e., the topological charge of one, which exhibits emergent phenomena such as the topological Hall effect18.
Although the FM order has been detected in the QC, the detailed magnetic structure has not been resolved experimentally17. Theoretically, the configurations of the magnetic moments in real space have been identified but their magnetic structure factor in reciprocal space has not been clarified18,19. Furthermore, the dynamical property of the magnetism in the QC remains unresolved.
As for the dynamics in the QC, the lattice dynamics was studied by inelastic X-ray and neutron scattering measurements20,21. The dynamical structure factor was theoretically calculated in the spin 1/2 Heisenberg model on the Fibonacci chain for the FM ground state22 and in two-dimensional systems23. The dynamical structure factor was also calculated for antiferromagnetic spin 1/2 Heisenberg model on the two-dimensional octagonal tiling24. However, little has been known about the magnetic dynamics in the real three-dimensional QC theoretically nor experimentally.
In this report, we present the dynamical property of the uniform long-range order of the hedgehog state in the Tb-based QC. By calculating the magnetic structure factor, we show that the hedgehog is characterized as the triple-Q state. By analyzing the dynamical structure factor, we reveal unique energy and momentum dependences of the magnetic excitations. We find that the magnetic excitation mode along the 5-fold axis direction exhibiting streak fine structure with periodicity characterized by the wavelength of the diameter of the IC, which is considered to be characteristic of the hedgehog in the icosahedral QC. We also find the non-reciprocal magnetic excitation mode in the QC. We note that we take the unit of ℏ=1 hereafter where ℏ is reduced Planck constant.
(A) The hedgehog state in the IC. Each arrow illustrates the magnetic moment at Tb, which is directed to the pseudo 5-fold axis. (B) The hedgehog state in Cd5.7Yb-type QC. Green (brown) lines at the front (back) connect the vertices of the icosidodecahedron. Scale bar (5 Å) is shown in (B). (C) Local coordinate at the Tb site with the orthogonal unit vectors ee^1, ee^2, and ee^3 (see text).
Results
Lattice structure of QC
Let us start with the lattice structure of the QC. Although the FM long-range order has recently been identified by bulk measurements in the QC Au65Ga20Tb15, the detailed lattice structure has not been solved experimentally17. In general, the rare-earth atoms in the rare-earth-based icosahedral QC are considered to form the lattice structure of Yb in the Cd5.7Yb-type QC3. Figure 1B shows the main structure of the QC where the Tb-12 cluster, i.e., IC is located at each vertex of the icosidodecahedron with the total number of the vertices being 30. In the Cd5.7Yb-type QC, there exists a few other ICs as well as Tb sites located between the ICs. In this study, as a first step of analysis, we consider the Tb sites shown in Fig. 1B with the total lattice number N=12×30=360 to get insight into the magnetic dynamics in the QC. Here, we employ the real Tb configuration for the IC (see Fig. 1A) as well as the icosidodecahedron in the 1/1 approximant crystal (AC) Au70Si17Tb13 whose lattice structure was solved by the X-ray measurement25, as a typical case. The diameter of the IC is 10.56 Å. In Fig. 1B, the IC is located at 30 vertices of the τ3-times enlarged icosidodecahedron in the Tsai-type cluster of Au70Si17Tb13 with τ being the golden mean τ≡(1+5–√)/2.
Minimal model in rare earth-based QC
The Tb3+ ion with 4f8 configuration has the ground state of the crystalline electric field (CEF) with the total angular momentum J=6 according to the Hund’s rule. The quantization axis of the CEF is the vector passing through each Tb site from the center of the IC, which is the pseudo 5-fold axis (see Fig. 1A). The detailed analysis of the CEF has revealed that the magnetic anisotropy arising from the CEF plays a key role in realizing the unique magnetic state such as the hedgehog on the IC18,19. Then, we consider the minimal model for the magnetism in the Tb-based QC as
H=∑⟨i,j⟩JijSSi⋅SSj−D∑i(SSi⋅ee^3)2,
(1)
where Jij is the exchange interaction between the ith and jth Tb sites and SSi is the “spin” operator with Si=6. In the second term, the unit vector ee^3 indicates the direction of the magnetic anisotropy arising from the CEF, which can be controlled by the compositions of Au and SM in Au-SM-Tb18,19. This model is expected to be relevant to not only the Tb-based QC but also rare-earth based QCs. In this study, to discuss the hedgehog state, ee^3 is set to be the pseudo 5-fold axis direction. For the strong limit of the magnetic anisotropy, it has been shown that the uniform long-range order of the hedgehog state is realized in the QC for J2/J1>2 where J1(J2) is the nearest neighbor (N.N.) (next N.N.) exchange interactions (Supplementary information, Fig. S1). Each IC is characterized by the topological charge of one nTC=+1, which is distributed quasi-periodically in Fig. 1B18. The hedgehog is the source of emergent field, which is regarded as monopole with the charge nTC=+126,27.
In the hedgehog state, “spins” are non-collinearly aligned as shown in Fig. 1A. Hence it is convenient to introduce the local coordinate at each Tb site where the e^3 axis is set as the ordered “spin” direction as shown in Fig. 1C (see Methods section for detail). Then, by applying the Holstein-Primakoff transformation28 to H, the “spin” operators are transformed to the boson operators as S+i=2S−ni−−−−−−√ai, S−i=a†i2S−ni−−−−−−√ and SSi⋅ee^i3=S−ni with ni≡a†iai. Here, S−i(S+i) is the lowering (raising) “spin” operator and ai(a†i) is an annihilation (creation) operator of the boson at the ith Tb site. Here the quadratic terms of the boson operators are retained because the higher order terms are considered to be irrelevant at least for the ground state.
We employ J1=1.0 and J2=2.3 as a typical parameter for the Tb-based QC. Actually, J2/J1=2.3 has been experimentally identified in the model Eq. (1) for the large D limit applied to the 1/1 AC Au72Al14Tb1429. We confirmed that the hedgehog state shown in Fig. 1B with the N=360 sites under open boundary condition is realized as the ground state for D≥17.85 in Eq. (1), which gives the positive excitation energy ωi for i=1,⋯,N, as shown in Fig. 2A. The D dependence of the lowest excitation energy, i.e., the gap Δ≡ωN/(J1S) between the first-excited energy and the ground enegy is shown in Fig. 2B. In the spectrum, there exist several gaps, as remarkably seen in Fig. 2A as Δ1≡(ω90−ω91)/(J1S). As D increases, the energy gap Δ as well as Δ1 increases. Hereafter, we show the results for D=30 as the representative case. The lowest and highest energies of the excitation spectrum are Δ=ωN/(J1S)=15.47 and ω1/(J1S)=24.59, respectively.
The largest peak is located at QQ1≡(1.77,0,1.02) Å−1 as shown in Fig. 2C. Since the alignment of the magnetic moments in the hedgehog shown in Fig. 1A is invariant under the permutation of x, y, and z axis, the same results in Fs(qq) as Fig. 2C are obtained by replacing (qx,qy,qz) with (qy,qz,qx) and also with (qz,qx,qy). Indeed we confirmed that the largest peak in Fs(qq) appears at QQ2≡(0,1.02,1.77) and QQ3≡(1.02,1.77,0) in Fs(qq) (Supplementary information, Figs. S2A and S2B). Namely, Fs(QQ1)=Fs(QQ2)=Fs(QQ3) holds. Thus the hedgehog state is characterized by the triple-Q (QQ1,QQ2, and QQ3) state.
In Fig. 2C, the spots lie along the pseudo 5-fold axis indicated by the dashed line with an arrow named d∗ei. Here, d⃗ ∗ei (i=1,⋯,6) is the primitive vector of the six-dimensional reciprocal lattice space as the physical (external) space components as shown in Fig. 2C30. Hereafter, we express the pseudo-5 fold axis for the d⃗ ∗ei direction as the d∗ei line with an arrow. We note that the slope of the d∗ei line for i=2 in Fig. 2C is 1.736 reflecting the real configuration of the Tb sites in the IC25 shown in Fig. 1A, which is known to be τ in the regular IC30. The slope of the d∗e3, d∗e5, and d∗e4 lines is the sign-reversed value of the slope of the d∗e2, d∗e1, and d∗e6 lines within each qz-qx, qx-qy, and qy-qz plane, respectively (see Fig. 2D).
It is noted that Sxx(qq), Szz(qq), and Syy(qq) have the maximum at qq=QQ1, QQ2, and QQ3, respectively, where Sαβ(qq) is defined as Sαβ(qq)≡1N∑i,jeiqq⋅(rri−rrj)⟨SiαSjβ⟩ (α=x,y, and z).
Dynamical structure factors (A) Sxx(qq,ω) and (B) Sxx(qq,0) for qq along the d∗e2 line through QQ1 with qy=0. Inset illustrates the d∗e2 line through QQ1 inside the cube with a side length of 2×3.54 Å−1. (C) The ω dependence of Sxx(QQ1,ω). The dashed line in (B) is the guide for qq=QQ1.
The result of Sxx(qq,ω) for qq along the d∗e2 line in the qz-qx plane is shown in Fig. 3A. The spectra appear at ω/(J1S)=0 (see Fig. 3B) with strong intensity of ∼O(108) and also appear above the energy gap Δ with intensity of ∼O(104). The energy gap in the excitation spectra Δ reflects the magnetic anisotropy arising from the CEF. For ω/(J1S)>Δ, the large intensity appears at the energy ω90/(J1S)=22.90, where the highest peak appears at the Γ point. At ω=0, i.e., elastic energy, the maximum peak appears at qq=QQ1, as shown in Fig. 3B indicated by the dashed line. In the ω dependence of Sxx(qq,ω), spiky peak structures appear as shown in Fig. 3C for qq=QQ1. These results indicate that the peak Sxx(QQ1) is governed by the elastic contribution Sxx(QQ1,0), which is understandable from the sum rule with respect to ω as Sxx(QQ1)=12π∫dωSxx(QQ1,ω).
Dynamical structure factor Sxx(qq,ω) in the qz-qx plane with qy=3.436 Å−1 for (A) ω/(J1S)=22.90 and (B) 23.07. The dashed lines indicate the d∗e2 line and d∗e3 line. (C) Sxx(qq,ω) for qq along the d∗e2 line through qq0=(2.169,3.436,1.442) Å−1. Inset illustrates the d∗e2 line through qq0 inside the cube with a side length of 8.31×2 Å−1. (D) Top view of (C).
Full size image
This is in sharp contrast to the result recently reported in the uniform long-range order of the ferrimagnetic state in the icosahedral QC31. Namely, the high-intensity peak appears at the ordered vector qq=00 and the lowest CEF excitation energy ω/(|J1|S)=Δ, from which the high-intensity peaks are continuously formed in the dynamical structure factor, giving rise to the pseudo-magnon mode31.
Then we search the qq dependence of Sxx(qq,ω) for ω=ω90 where the large intensities appear as shown in Fig. 3A. Consequently, we identify that the maximum is located at qq0≡(2.169,3.436,1.442) Å−1. Around qq=qq0, we find that a series of the packet structures appears along the pseudo 5-fold axis direction, as shown in Fig. 4A where the d∗e2 line and d∗e3 line through qq=qq0 is illustrated by the dashed line in the qz-qx plane with qy=3.436 Å−1. The peak in the central packet gives the maximum Sxx(qq0,ω90)=1.027×105. A series of packet structures with sub-leading intensity is also aligned along the pseudo 5-fold axis direction. For slightly larger ω than ω90, the packets still appear along the d∗e2 line at slightly different positions as shown in Fig. 4B, which suggests the magnetic excitation propagating along the pseudo 5-fold direction.
Figure 4C shows Sxx(qq,ω) for qq along the d∗e2 line through qq=qq0. A series of the packet structures remarkably appears at the lower edge ω90 with strong intensity, which continuously forms the streak with fine structure down to the lower-ω region as also seen in the intensity plot in Fig. 4D.
Interestingly, we find that a series of the packet structures is the reflection of the bottom of the continuous mode periodic along the d∗e2 line in the qq-ω plane as shown in Fig. 4D. The period of the streak structure is evaluated as Δq∼0.6 Å−1 in the reciprocal space. From the relation of the wavenumber and wavelength Δq=2π/λ, the scale of the wavelength is estimated to be λ∼10 Å. It turns out that this corresponds to the diameter of the IC d=10.56 Å (see Fig. 1A). Since the hedgehog is the magnetic texture on the IC, the excitation gives rise to the dynamics whose intensity decreases with periodicity Δq∼2π/d with distance from qq0 in the reciprocal space of the QC. A series of the packet structure as well as the intensity streak in the qq-ω plane also appears along the d∗e3 direction (Supplementary information, Fig. S3). The emergence of the intensity streak with fine structure in the qq-ω plane indicates unique excitation mode along the 5-fold axis direction, which is considered to be characteristic of the hedgehog in the icosahedral QC.
To further clarify the general property of the dynamics of the hedgehog state, we show Sxx(qq,ω) at ω/(J1S)=23.18 for qq along the pink lines in the cube whose side is parallel to the 2-fold axis qα∈[0,2.56] Å−1 (α=x,y, and z) in the inset of Fig. 5A. Here we also plot Sxx(−qq,ω) along the green line in the inset of Fig. 5A. We see remarkable differences in the intensity for qq and −qq. In Fig. 5B, we plot |Sxx(qq,ω)−Sxx(−qq,ω)| for qq along the pink lines in the inset of Fig. 5A. The finite values indicate that Sxx(qq,ω)≠Sxx(−qq,ω). These results indicate that non-reciprocal magnetic excitation appears in the hedgehog state in the QC. This is, to our best knowledge, the first discovery of the non-reciprocal magnetic excitation in the topological magnetic long-range order in QC.
We confirmed that non-reciprocal magnetic excitation does not appear in the case of the collinear magnetic order in the QC (Supplementary information, Fig. S4). This implies that the noncollinear and noncoplanar magnetic structure on the IC of the hedgehog (Fig. 1A) is the origin of the nonreciprocal excitation. Recently, non-reciprocal magnetic excitation from the uniform ferrimagnetic order (characterized by the zero topological charge nTC=0) in the icosahedral QC has been shown to appear31. These results suggest that non-reciprocal excitation is common character of the noncollinear and noncoplanar alignment of the magnetic moments on the IC. As shown in Fig. 5B, emergence of many spiky peaks with fine structure as continuum are the consequence of the QC structure, which is in sharp contrast to the magnon branch in periodic crystals as the collective mode. This gives rise to the emergence of nonreciprocity as continuum in the vast extent of the qq-ω plane (see Fig. 5B), whose feature is unique to the QC.
Summary and discussion
We have revealed the dynamical as well as static property of the hedgehog state in the QC. The FM hedgehog state is shown to be characterized by the triple-Q state. The magnetic dynamical structure factor shows highly structured energy and momentum dependences unique to the QC. We have discovered the magnetic excitation mode along the pseudo 5-fold axis direction. A series of the packet structure in the dynamical structure factor is found to exist, which is shown to be the reflection of the periodic streak structure in the reciprocal lattice qq-energy ω plane. Non-collinear and non-coplanar magnetic alignment of the hedgehog state gives rise to non-reciprocal magnetic excitations which appear in the vast extent of the energy and momentum plane.
In the uniform long-range order of the ferrimagnetic state, the high-intensity peaks appear continuously from the ordered vector qq=0 and the lowest CEF excitation energy ω=Δ|J1|S, which are identified as the pseudo-magnon mode31. On the contrary, in the dynamical structure factor for the uniform hedgehog order, the high-intensity peaks do not appear at the ordered vector QQi for i=1, 2, and 3 beyond the CEF excitation energy. This implies that the peak in the static structure factor at the triple Q vector qq=QQi is governed by the elastic (ω=0) contribution of the dynamical structure factor for the uniform hedgehog order.
The streak structure with periodicity characterized by the wavelength corresponding to the diameter of the IC in the qq-ω plane is considered to be the unique character of the excitation from the uniform hedgehog order. To establish this point, the systematic analysis of the dynamical structure factor in the magnetically ordered states in the icosahedral QC is necessary, which is left for future studies.
The non-reciprocal magnetic excitation has also been found to emerge in the uniform ferrimagnetic order31. Hence, as noted above, non-reciprocity is considered to be general feature of the excitation from the non-collinear and non-coplanar magnetic texture on the IC.
Our results are useful not only for resolving the magnetic structure of the long-range order discovered recently in Tb-based icosahedral QC, but also for future neutron measurements of the magnetic dynamics in the QC. So far, the dynamical structure factor in the magnetically ordered phase in the QC has not been reported. It is expected that present study stimulates future experiments to detect the dynamical property in the QC and also in the approximant crystal.