Spin dynamics of a magnetic Weyl semimetal ${\mathrm{Sr}}_{1-x}{\mathrm{Mn}}_{1-y}{\mathrm{Sb}}_2$

Dirac matters provide a platform for exploring the interplay of their carriers with other quantum phenomena. ${\mathrm{Sr}}_{1-x}{\mathrm{Mn}}_{1-y}{\mathrm{Sb}}_2$ has been proposed to be a magnetic Weyl semimetal and provides an excellent platform to study the coupling between Weyl fermions and magnons. Here, we report comprehensive inelastic neutron scattering (INS) measurements on single crystals of ${\mathrm{Sr}}_{1-x}{\mathrm{Mn}}_{1-y}{\mathrm{Sb}}_2$, which have been well characterized by magnetization and magnetotransport measurements, both of which demonstrate that the material is a topologically nontrivial semimetal. The INS spectra clearly show a spin gap of $\sim 6$ meV. The dispersion in the magnetic Mn layer extends up to about 76 meV, while that between the layers has a narrow band width of 6 meV. We find that the linear spin-wave theory using a Heisenberg spin Hamiltonian can reproduce the experimental spectra with the following parameters: a nearest-neighbor ($S{J}_1\sim 28.0$ meV) and next-nearest-neighbor in-plane exchange interaction ($S{J}_2\sim 9.3$ meV), interlayer exchange coupling ($S{J}_c\sim -0.1$ meV), and spin anisotropy constant ($SD\sim -0.07$ meV). Despite the coexistence of Weyl fermions and magnons, we find no clear evidence that the magnetic dynamics are influenced by the Weyl fermions in ${\mathrm{Sr}}_{1-x}{\mathrm{Mn}}_{1-y}{\mathrm{Sb}}_2$, possibly because that the Weyl fermions and magnons reside in the Sb and Mn layers separately, and the interlayer coupling is weak due to the quasi-two-dimensional nature of the material, as also evident from the small $S{J}_c$ of $-0.1$ meV.

Figure 1

(a) Crystal and magnetic structure of ${\mathrm{Sr}}_{1-x}{\mathrm{Mn}}_{1-y}{\mathrm{Sb}}_2$ (space group No. 62, $Pnma$) with $a\approx b=4.4342$, $c=$ 23.1359 Å, and $\alpha =\beta =\gamma ={90}^{\circ }$. Note that we only show half of the unit cell along the $c$ axis. Arrows represent the spins. The exchange paths are labeled with $J_1$ (nearest-neighbor), $J_2$ (next-nearest-neighbor), and $J_c$ (interlayer). (b) Coaligned single crystals we used for the inelastic neutron scattering measurements. (c) Single-crystal x-ray diffraction pattern of the cleavage plane. (d) Backscattering Laue x-ray diffraction pattern with the beam along the [001] direction. (e) Temperature dependence of the magnetization measured with magnetic field $B=1$ T applied along the $c$ axis. (f) Temperature dependence of the integrated intensities of the magnetic Bragg peak (1, 0, 0). The solid line in (f) is a guideline to the eye. Arrows in (e) and (f) indicate the Néel temperature $T_N$.

Figure 2

(a) The in-plane longitudinal (${\rho }_{xx}$) and transverse (Hall) (${\rho }_{xy}$) resistivity as a function of magnetic field applied along the $c$ axis measured at 2 K. (b) Magnetic-field dependence of the in-plane conductivity (${\sigma }_{xx}$). The solid line indicates the smooth background. The inset is the oscillatory component ($\mathrm{\Delta }{\sigma }_{xx}$) by subtracting the smooth background. (c) Magnetic-field dependence of the magnetization measured at 2 K, with the solid line being the fit of the smooth background. (d) Oscillatory component of the magnetization ($\mathrm{\Delta }M$) by subtracting the background at various temperatures. (e) Temperature dependence of the oscillatory amplitude, normalized by $\mathrm{\Delta }M$ at 2 K. The solid line through data is the fit as described in the main text, from which we obtain an effective mass of $0.1m_0$. (f) $\mathrm{\Delta }M$ at 2 K plotted as a function of $1/B$. The inset shows the fast Fourier transform spectra of $\mathrm{\Delta }M$ which show two oscillating frequencies. The solid line is the fit to $\mathrm{\Delta }M$ with a two-frequency Lifshitz-Kosevich formula. From the fit, we obtain the Berry phase of $0.3\pi$ and $\pi$ corresponding to the frequency of 73.2 and 57.2 T, respectively.

Figure 3

Magnetic dispersions of ${\mathrm{Sr}}_{1-x}{\mathrm{Mn}}_{1-y}{\mathrm{Sb}}_2$ measured at 6 K together with the theoretical calculation results. (a, b) Dispersions along the [100] and [110] directions, respectively, obtained on 4SEASONS with $E_i=100.1$ meV. (c, d) Dispersions along the [100] and [001] directions obtained on MERLIN with $E_i=16$ meV. Panel (c) focuses on the low-energy part of (a) to show the spin gap. In panels (a), (b), and (d), solid lines are the calculated dispersions. The calculated dispersions with intensities are shown on the right-hand side of each panel to be compared with the experimental data on the left-hand side. Data in (a) and (c) were integrated with $K=[-0.2,0.2]$ rlu, and all available $L\mathrm{s}$; data in (b) with a thickness of 0.1 rlu about each momentum along the [$-110$] direction; data in (d) with $H=[0.8,1.2]$ rlu, and $K=[-0.2,0.2]$ rlu. The dashed line in (c) illustrates an energy scan at (100). The scan profile is plotted on the right-hand side with the energy and intensity being the $y$ and $x$ axis, respectively. The arrow indicates the gap value. In (d), the low-energy part below about 4.5 meV is due to the Bragg peaks and some spurious signals, and the broad experimental data can be reproduced by spin-wave calculations convoluting the instrumental resolution [21].

Figure 4

Contour maps of the constant-energy cuts measured with $E_i=100.1$ meV at 6 K. The experimental data are shown on the left-hand side of each panel, while the theoretical results are shown on the right hand side. The energy range of the integration is shown on the top of each panel. All the data are from 4SEASONS.