Effect of Pnictogen Height on Spin Waves in Iron Pnictides

We use inelastic neutron scattering to study spin waves in the antiferromagnetic ordered phase of iron pnictide NaFeAs throughout the Brillouin zone. Comparing with the well-studied $A{\mathrm{Fe}}_2{\mathrm{As}}_2$ ($A=\mathrm{Ca}$, Sr, Ba) family, spin waves in NaFeAs have considerably lower zone boundary energies and more isotropic effective in-plane magnetic exchange couplings. These results are consistent with calculations from a combined density functional theory and dynamical mean field theory and provide strong evidence that pnictogen height controls the strength of electron-electron correlations and consequently the effective bandwidth of magnetic excitations.

Figure 1

(a) Schematic diagram of the Fe spin ordering in NaFeAs with the effective magnetic nearest-neighbor, next-nearest-neighbor exchange couplings, $J_{1a}$, $J_{1b}$ and $J_2$. $a_o$ and $b_o$ mark the orthorhombic lattice structure of the system. (b) Temperature dependence of the AF Bragg peak intensity showing $T_N=45\mathrm{K}$. (c) Schematic diagram of the FeAs tetrahedron, showing increased iron pnictogen height ($h_{MAs}$) from ${\mathrm{BaFe}}_2{\mathrm{As}}_2$ to NaFeAs ($\sim 0.058\AA$). The in-plane Fe-Fe distance changes are much smaller ($\sim 0.011\AA$) from ${\mathrm{BaFe}}_2{\mathrm{As}}_2$ ($2.802\AA$) to NaFeAs ($\text{2.791\hspace{0.17em}\hspace{0.17em}}\AA$). Schematic diagrams illustrating the direct and indirect electron hoppings of (d) $d_{xz}$, (e) $d_{yz}$, and (f) $d_{xy}$ orbitals. (g) Calculated energy dependence of the local dynamic spin susceptibility ${\chi }^{\prime \prime }(E)$ per Fe for ${\mathrm{BaFe}}_2{\mathrm{As}}_2$ and NaFeAs as obtained from the combined DFT and DMFT method [31]. (h) Energy dependence of the measured local dynamic spin susceptibility for NaFeAs, and ${\mathrm{BaFe}}_2{\mathrm{As}}_2$ [46].

Figure 2

Constant energy slices of the spin waves as a function of increasing energy at 5 K and fits using the Heisenberg Hamiltonian and $\mathrm{DFT}+\mathrm{DMFT}$ method. Wave vector dependence of the spin waves for energy transfers of (a) $E=15\pm 3\mathrm{meV}$ [$E_i=80\mathrm{meV}$ and $\mathbf{q}=(H,K,1.6)$]; (b) $E=45\pm 3\mathrm{meV}$ [$E_i=80\mathrm{meV}$ and $\mathbf{q}=(H,K,5)$]; (c) $E=60\pm 10\mathrm{meV}$ [$E_i=150\mathrm{meV}$ and $\mathbf{q}=(H,K,4.5)$]; (d) $E=80\pm 10\mathrm{meV}$ [$E_i=150\mathrm{meV}$ and $\mathbf{q}=(H,K,6)$]; (e) $E=100\pm 10\mathrm{meV}$ [$E_i=250\mathrm{meV}$ and $\mathbf{q}=(H,K,5.5)$]. (f)–(j) Model calculation of identical slices by using the Heisenberg Hamiltonian with a new damping function convolved with the instrumental resolution [44]. (k)–(o) Calculations of identical energy slices from $\mathrm{DFT}+\mathrm{DMFT}$ method. The color bars represent the vanadium-normalized absolute spin wave intensity in units of $\mathrm{mbar}/(\mathrm{sr}\hspace{0.17em}\hspace{0.17em}\mathrm{MeV}\hspace{0.17em}\hspace{0.17em}\mathrm{Fe}\hspace{0.17em}\hspace{0.17em}\mathrm{atom})$.

Figure 3

Constant energy ($E$) and wave vector($\mathbf{q}$) dependence of the spin wave excitations at 5 K. Constant $E$ cuts at $E=15\pm 3$, $45\pm 3$, $60\pm 10$, $80\pm 10$, and $100\pm 10\mathrm{meV}$ along the $[H0]$ direction (a)–(e) and along the $[1K]$ direction (f)–(j). Constant $Q$ cuts at $Q=(1,0.4)$ (k) and $Q=(1,0.5)$ (l). The solid lines in (a)–(e), (f)–(j) and (k),(l) are global fits of our spin wave model discussed in the text.

Figure 4

Dynamic magnetic structure factor $S(q,E)$ for ${\mathrm{BaFe}}_2{\mathrm{As}}_2$ (top) and NaFeAs (bottom). The theoretical data are shown as colored plots while the black dots with vertical-line error bars are experimental data. Panels (a) and (d) show the total $S(q,E)$, (b) and (e) the diagonal component $S_{xy,xy}(q,E)$ from the $d_{xy}$ orbital which has the largest contribution to the total $S(q,\omega )$ around the zone boundary $(1,1,L)$ points, and (c) and (f) the diagonal component $S_{yz,yz}(q,E)$ from the $d_{yz}$ orbital which contributes much less than the $d_{xy}$ orbital to the total $S(q,E)$ around the zone boundary (1,1,L) points. This orbital difference is enhanced in NaFeAs compared to ${\mathrm{BaFe}}_2{\mathrm{As}}_2$ due to the increased As height in NaFeAs and the kinetic frustration mechanism.