Static and dynamic properties of single-chain magnets with sharp and broad domain walls

We discuss time-quantified Monte Carlo simulations on classical spin chains with uniaxial anisotropy in relation to static calculations. Depending on the thickness of domain walls, controlled by the relative strength of the exchange and magnetic anisotropy energy, we found two distinct regimes in which both the static and dynamic behavior are different. For broad domain walls, the interplay between localized excitations and spin waves turns out to be crucial at finite temperature. As a consequence, a different protocol should be followed in the experimental characterization of slow-relaxing spin chains with broad domain walls with respect to the usual Ising limit.

Figure 1

Domain-wall energy (in $J$ units) as a function of the ratio $D/J$, adapted from Ref. 33: discrete-lattice calculation ${\varepsilon }_{dw}$ (solid line), continuum-limit solution $2\sqrt{2DJ}$ (dashed line). Crosses and circles correspond to the ratios used in transfer-matrix and time-quantified Monte Carlo calculations, respectively. The vertical line indicates the value ($D/J=2/3$) at which the transition from broad- to sharp-wall regime occurs. Insets: sketch of the spatial dependence of $S_i^z$ as a function of $i$ in the broad-wall (up left) and in the sharp-wall (down right) regime.

Figure 2

Logarithm of the correlation length obtained by TM calculations as function of the inverse temperature in units of the DW energy ${\varepsilon }_w$ [defined in Eq. (2)] for $J=1$ and different values of $D$. Symbols, broad domain walls, from above: $D=$ 0.1 (black squares), 0.2 (red circles), 0.3 (green up-triangles), 0.4 (blue diamonds), 0.5 (cyan down-triangles), 0.6 (magenta pentagons). Solid lines, sharp domain walls, from below: $D=$ 1 (black), 2 (red), 3 (green), 4 (blue). Dashed horizontal lines give the reference for typical lengths of realistic spin chains.

Figure 3

The ratio between renormalized constant $D\left(L\right)$ and the bare constant $D\left(0\right)$ is plotted as a function of some selected intermediate values of $L$, for different values of $D\left(0\right)=0.05$ (black squares), 0.1 (red circles), 0.2 (blue crosses) and different temperatures ${\varepsilon }_w\left(0\right)/T=$ 0.0625, 0.106, 0.149, 0.193, 0.236. The renormalization flux stops at $L=\tilde{\Lambda }\left(T\right)$.

Figure 4

Same data as in Fig. 3 but plotted vs $L/w\left(0\right)$.

Figure 5

Logarithm of the correlation length vs ${\varepsilon }_w\left(0\right)/T$: TM calculation for $D=0.05$ (solid red line); $\tilde{\Lambda }\left(T\right)$ multiplied by a constant to match the TM results at low $T$ (circles); analytic result for $D=0$ given in Ref. 12 (dashed line). The straight lines are guide to the eye to highlight the change of slope due to spin-wave renormalization. The dotted line corresponds to the renormalized DW width $\ln \left[\tilde{w}\left(T\right)\right]$. Inset: Logarithm of $\xi /w\left(0\right)$ vs ${\varepsilon }_{dw}\left(0\right)/T$ for $D=$ 0.1, 0.2, 0.3, 0.4, 0.5, and 0.6 (same data as in Fig. 2).

Figure 6

Relaxation curves at different temperatures for $D/J=0.2$. Inset: FC curve obtained by TM calculations (solid line); open circles are the values of $M_0$ obtained by fitting the relaxation curves with Eq. (26) at different temperatures.

Figure 7

Arrhenius plot of the relaxation times obtained at different temperatures and for three values of the anisotropy, $D/J=0.1$ (full circles), 0.2 (squares), and 0.3 (triangles), with open boundary conditions. The dashed line is guide to the eyes and has slope $1.1$. Empty circles: periodic boundary conditions for $D/J=0.1$.

Figure 8

Temperature dependence of the diffusion coefficient for $D=1$ and 2 (sharp-wall regime).

Figure 9

Temperature dependence of the diffusion coefficient for two values of the anisotropy $D=0.1$ and 0.2 (broad-wall regime). The inset shows the scaled curves.

Figure 10

The solid lines represent the energy of spin waves $\hslash \omega \left(q\right)$ (in $J$ units) as a function of the wave vector $q$. Horizontal short-dashed lines correspond to the energy of one DW in the sharp- and in the broad-wall regime: for the left panel $D/J=2$ while for right panel $D/J=0.2$. The horizontal dashed line represents an indicative temperature smaller than ${\varepsilon }_w$ in both cases (see the text).

Figure 11

Square root of the mean-square deviation of a random walker obtained using different boundary and initial conditions. The time $t$ is measured in random-walk steps (RWS).

Figure 12

Average dispersion of the DW trajectories at $T/J=0.14$ for two damping constants and applied fields. In (a) and (c) the applied field is $H=0$ and the damping constant $a=0.25$ and $a=4$, respectively. In (b) and (d) the applied field $H=0.01$ and the damping constant $a=0.25$ and $a=4$, respectively.

Figure 13

Domain-wall trajectories at $T/J=0.14$ for two damping constant and applied fields. In (a) and (c) the applied field is $H=0$ and the damping constant $a=0.25$ and $a=4$, respectively. In (b) and (d) the applied field $H=0.01$ and the damping constant $a=0.25$ and $a=4$, respectively.