Effective exchange interaction for terahertz spin waves in iron layers

The exchange stiffness is a central material parameter of all ferromagnetic materials. Its value controls the Curie temperature as well as the dynamic properties of spin waves to a large extent. Using ultrashort spin current pulses we excite perpendicular standing spin waves (PSSW) in ultrathin epitaxial iron layers at frequencies of up to 2.4 THz. Our analysis shows that for the PSSWs the observed exchange stiffness of iron is about 20% smaller compared to the established iron bulk value. In addition, we find an interface-related reduction of the effective exchange stiffness for layers with the thickness below 10 nm. To understand and discuss the possible mechanisms of the exchange stiffness reduction we develop an analytical one-dimensional model. In doing so we find that the interface induced reduction of the exchange stiffness is mode dependent.

Figure 1

(a) Schematic spin-wave dispersion of Fe in the Brillouin zone with indicated measurement ranges for neutron scattering (e.g., [1]) and our work. (b) Sample layout and optical pump probe measurement geometry.

Figure 2

(a) Time-resolved polar MOKE rotation and ellipticity data measured at several thicknesses of the Fe collector. (b) Sum of damped cosines fit (black line) to the MOKE rotation and ellipticity data exemplified for the 11.8-nm-thick collector; (c) FFT amplitude (normalized to the amplitude of first PSSW mode) of MOKE rotation and ellipticity data shown in (b). The FMR mode as well as two PSSW modes are visible in the spectrum obtained for the long-time range ($t_{\text{max}}=700\mathrm{ps}$, $\mathrm{\Delta }t=0.2\mathrm{ps}$) and another three PSSW modes are visible in the spectrum of the short-time-range measurement ($t_{\text{max}}=30\mathrm{ps}$, $\mathrm{\Delta }t=0.02\mathrm{ps}$).

Figure 3

(Lower graph) FMR and (upper graph) PSSW frequencies obtained by fits to the time-resolved-MOKE traces [Eq. (1)]. The error bars are given by the fit procedure. The solid lines are a fit to the Kittel formula [Eq. (2)] using an exponential thickness dependence of the stiffness [Eq. (5)] with $D^{\infty }=220$ meV Å${}^2$. Due to large error bars in the frequencies of the sixth and seventh mode these modes were not included in the fit.

Figure 4

Effective stiffness of PSSWs measured by time-resolved-MOKE (points). The diamonds are the results obtained for the first PSSW mode on the three additional samples using FMR (see Appendix pp3). The dashed line in the inset represents the exponential fit according to Eq. (5) corresponding to a bulk stiffness of $D^{\infty }=220$ meV Å${}^2$ and a critical thickness of $d_D^0=2.4\mathrm{nm}$ resulting from the fit by the Kittel formula [Eq. (2)]. The triangles show the literature values obtained by Prokop et al. [6] (triangle down) and Razdolski et al. [5], Supplementary Note 4 (triangle up). The gray thick line is a guide to the eye.

Figure 5

(a) Normalized magnetization as a function of thickness for a magnet without vacancies and three cases with a redistribution of magnetic moments in 1, 2, and 3 ML around the boundary. (b) Exchange parameters as a function of shell number, reduced to 80% for second layer and to 60% for first layer to interface, respectively.

Figure 6

Effective exchange stiffness as a function of thickness for a magnetic layer without vacancies (a), for vacancies in 1 ML (b), 2 ML (c), and 3 ML (d). (e) The measured values as shown in Fig. 4 normalized to $D^{\infty }=220$ meV Å${}^2$ (points) and for comparison analytical results for vacancies in 3 ML (lines).

Figure 7

Effective exchange stiffness as a function of thickness for several magnon modes in magnetic materials where exchange constants are reduced at the interface. (a) All eight parameters are reduced. (b) $J_1$ and $J_2$ are lowered. (c) $J_3\text{−}{J}_8$ are lowered. Note that no changes of Fe concentration are considered here. (d) Measured values as shown in Fig. 4 normalized to $D^{\infty }=220\mathrm{meV}{\AA }^2$ and for comparison analytical results for lowered $J_1$ and $J_2$ (lines).

Figure 8

(a) Effective out-of-plane anisotropy constant $K_{\perp }^{\mathrm{eff}}{d}_{\mathrm{Fe}}$ obtained by out-of-plane static MOKE measurements. (b) Effective fourfold anisotropy constant $K_4^{\mathrm{eff}}{d}_{\mathrm{Fe}}$ and (c) uniaxial anisotropy constant $K_u^{\mathrm{eff}}{d}_{\mathrm{Fe}}$ obtained by in-plane static MOKE measurements

Figure 9

Measured resonance fields for FMR (brown points) and first PSSW (blue points) modes and resulting fit (black lines) for 45 (a), 69 (b), and 87.5 nm (c) thick iron samples. The error bars for thickness and demagnetization field are only the result of the fit.

Figure 10

Resulting frequency spectrum for 30 ML (4.3 nm) iron with vacancies in 1 ML obtained by atomistic spin dynamics simulation. Amplitudes are normalized to peak amplitude and frequencies are normalized to bulk values.