Shape of spin density wave versus temperature in $A$Fe${}_2$As${}_2$($A$ $=$ Ca, Ba, Eu): A Mössbauer study

Parent compounds AFe${}_2$As${}_2$ (A $=$ Ca, Ba, Eu) of the 122 family of the iron-based superconductors have been studied by ${}^{57}\mathrm{Fe}$ Mössbauer spectroscopy in the temperature range 4.2–∼300 K. Spin density waves (SDW) have been found with some confidence. They are either incommensurate with the lattice period or the ratio of the respective periods is far away from the ratio of small integers. SDW shape is very unconventional (i.e., differs from the sinusoidal shape). Magnetic order starts with lowered temperature as narrow sheets of the significant electron spin density separated by areas with very small spin density. Magnetic sheets are likely to be ordered in the alternate antiferromagnetic fashion as the material as a whole behaves similarly to the collinear antiferromagnet. A further lowering of temperature simply expands sheet thickness leading to the near triangular SDW. Finally, sheets fill the whole available space and the almost rectangular shape of the SDW is reached. The substantial maximum amplitude of SDW appears at the temperature just below the magnetic onset temperature, and this maximum amplitude increases slightly with lowering temperature. The square root from the mean squared hyperfine field behaves versus temperature according to the universality class $(1,2)$ (i.e., with the electronic spin space having dimensionality equal to unity and the real space having dimensionality equal to 2). The more or less pronounced tail above transition temperature due to the development of incoherent SDW is seen.

Figure 1

Mössbauer spectra in the “nonmagnetic” region.

Figure 2

Mössbauer spectra, SDW shapes, and resulting hyperfine field distributions for CaFe${}_2$As${}_2$ in the magnetically ordered region versus temperature. Amplitudes of the first two harmonics $h_1$ and $h_3$ are listed on corresponding plots of SDW shape. Symbol $\langle \left|B\right|\rangle$ stands for the average of the respective field distribution, while $B_{\max }$ denotes maximum value of SDW.

Figure 3

Mössbauer spectra, SDW shapes, and resulting hyperfine field distributions for BaFe${}_2$As${}_2$ in the magnetically ordered region versus temperature.

Figure 4

Mössbauer spectra, SDW shapes, and resulting hyperfine field distributions for EuFe${}_2$As${}_2$ in the magnetically ordered region versus temperature.

Figure 5

Comparison of the hyperfine field distribution resulting from SDW shape with the corresponding field distribution yielded by the HR method for selected spectra.

Figure 6

Plot of square root from the mean squared amplitude of SDW $\sqrt{\langle B^2\rangle }$ versus temperature. Solid lines represent total, coherent, and incoherent contributions as described in the text. Error bars for both field and temperature are smaller than the symbol size.