Effect of an applied magnetic field on the relaxation time of single-chain magnets

Single-chain magnets (SCMs) are one-dimensional systems in which the relaxation of the magnetization becomes very slow at low temperature. This singular behavior is due to the vicinity of the critical point located at vanishing temperature $(T=0)$ and applied magnetic field $(H=0)$. In order to optimize the properties of these nano-objects, detailed studies of the observed critical behavior are necessary. However, previous works on the SCM relaxation have essentially analyzed experimental data in the absence of applied magnetic field. We discuss in this paper the effect of applying a magnetic field on two different examples of single-chain magnets. These samples have been previously described in the absence of magnetic field and considered as model systems since their chains are composed of a regular one-dimensional (1D) arrangement of anisotropic trimer units. These magnetic trinuclear motifs can be described at low temperature as effective spins coupled ferromagnetically. Theoretical results relevant to analyze our data in the presence of an applied magnetic field are first described. We also present a simple numerical approach to discuss finite-size effects relevant in SCM systems. Experimental data, including ac and dc data on powder samples and single crystals, are then presented. These results are analyzed and compared with the theoretical predictions deduced for the 1D Ising model. At low field, for $h\xi ⪡1$ (where $\xi$ is the correlation length normalized to the unit cell parameter and $h=\mu H∕k_{B}T$ is the dimensionless applied field) or for $hL⪡1$ when finite-size effects are relevant ($L$ being the chain length normalized to the unit cell parameter), we show that experimental data reproduce the critical behavior expected from the theory. Moreover, the obtained values of $\xi$ or $L$ are in excellent agreement with the estimation deduced from susceptibility data. At higher fields, for $h\xi ⪢1$ or $hL⪢1$, we show that the field dependence of the relaxation time is drastically different for the two samples. This difference is understood taking into account the field dependence of the relaxation time of the effective spins located inside a domain wall.

Figure 1

Definition of the different kinetic constants for the dynamic equations: (a) cluster growth and (b), (c) cluster nucleation.

Figure 2

(Color online) The eigenfrequencies (normalized to $1∕{\tau }_0$) of three slowest symmetric modes (◼, ▴, ●) as a function of $C$ for $K=3$ and $L=20$. The continuous line gives the result for the initial time of the infinite chain. The dotted line gives the result for the finite chain ($K=3$ and $L=20$) using the simplified Schwarz model (Ref. 19).

Figure 3

(Color online) Components of the eigenvectors of three slowest symmetric modes for $K=3$ and $L=20$: (a) $C=0.1$, (b) $C=0.25$, and (c) $C=0.6$. The normalization is such as ${\sum }_{p=1}^L{x}_p^2=1$.

Figure 4

(Color online) Weight of the different modes $\beta ={\sum }_{p=1}^L{x}_p$ for the three modes considered in Fig. 3 as a function of $C$.

Figure 5

(Color online) The eigenfrequencies (normalized to $1∕{\tau }_0$) of the three slowest symmetric modes as a function of $C$ for $K=3$ and $L=100$. The continuous line gives the result for the initial time of the infinite chain. The dotted line gives the result for the finite chain ($K=3$ and $L=100$) using the simplified Schwarz model (Ref. 19).

Figure 6

(Color online) Components of the eigenvectors of three slowest symmetric modes for $K=3$ and $L=100$: (a) $C=0.03$ and (b) $C=0.2$. The normalization is the same as for Fig. 3.

Figure 7

(Color online) Weight of the three modes considered in Fig. 6 as a function of $C$.

Figure 8

(Color online) Numerical estimation of $a_i$ normalized to the chain length as a function of $x=L∕\xi$. This parameter is defined from the field dependence of the normalized frequency, ${\nu }_{\mathit{nor}}\left(h\right)=1+a_i^2{h}^2$, valid at low field.

Figure 9

(Color online) Numerical estimation of $a_i$ normalized to $2\xi =e^{2K}$ as a function of $x=L∕\xi$. The continuous curve is the function $f$ introduced in Ref. 15 plotted as a function of $0.82x$.

Figure 10

(Color online) Semilogarithmic plot of $\lambda \left(T\right)$ (defined in the text) as a function of $1∕T$ (full dots). (a) $\mathrm{Mn}∕\mathrm{Fe}$ chain. Open dots and open squares give the value deduced from the analysis given in Sec. 4, respectively, for ac and dc relaxation data. (b) Semilogarithmic plot of $\lambda \left(T\right)$ for the $\mathrm{Mn}∕\mathrm{Ni}$ chain.

Figure 11

(Color online) Real part (top) and imaginary part (bottom) of the ac susceptibility at $1.8\mathrm{K}$ of the $\mathrm{Mn}∕\mathrm{Fe}$ chain for selected values of the applied magnetic field.

Figure 12

(Color online) Normalized characteristic frequency deduced from ac data for the $\mathrm{Mn}∕\mathrm{Fe}$ chain on a powder sample: full dots, $T=2.3\mathrm{K}$; full triangles, $T=2\mathrm{K}$; full squares, $T=1.8\mathrm{K}$. The continuous lines give the fit discussed in Sec. 4.

Figure 13

(Color online) Time dependence of the magnetization (normalized between $t=0$ and $t=\mathrm{\infty }$) obtained at $T=1.05\mathrm{K}$ from the dc measurement on single crystal of the $\mathrm{Mn}/\mathrm{Fe}$ chain when the field is applied in the easy direction.

Figure 14

(Color online) For the $\mathrm{Mn}∕\mathrm{Fe}$ chain: (a) Quadratic behavior obtained at low field: a linear plot is obtained plotting ${\nu }_{\mathit{nor}}\left(h^2\right)$. The deduced values of the effective length $\lambda$ are 20 and 30 at 1.05 and $1.25\mathrm{K}$, respectively. (b) Normalized relaxation frequency obtained from dc measurements on single crystals as a function of $h\lambda$: full dots, $T=1.05\mathrm{K}$; full squares, $T=1.25\mathrm{K}$. The continuous and dotted lines correspond to the numerical calculation for ($L=20$, $K=2.9$) and ($L=30$, $K=2.44$), respectively.

Figure 15

(Color online) Normalized characteristic frequency deduced from ac susceptibility data on a powder sample for the $\mathrm{Mn}∕\mathrm{Ni}$ chain: full dots, $T=3.8\mathrm{K}$; full squares, $T=4.4\mathrm{K}$.

Figure 16

(Color online) Normalized characteristic frequency deduced from dc measurements at $5\mathrm{K}$ on single crystal for the $\mathrm{Mn}∕\mathrm{Ni}$ chain when the field is applied in the easy direction. The dotted line gives the theoretical prediction for $L=80$, $K=2.8$ assuming a constant value of ${\tau }_0$. Inset: The derivative of the normalized frequency relative to $h^2$ is shown to emphasize the extrapolation close to 80 for $h=0$. Continuous lines are guides for the eyes.

Figure 17

(Color online) (a) Deduced field dependence of ${\tau }_0\left(h\right)∕{\tau }_0\left(0\right)$ for the $\mathrm{Mn}∕\mathrm{Fe}$ chain (full dots) and the $\mathrm{Mn}∕\mathrm{Ni}$ chain (full squares), normalizing the single crystal data obtained, respectively, at 1.25 and $5\mathrm{K}$ to the theoretical result. The continuous lines give the fits to a quadratic dependence $[{\tau }_0\left(h\right)∕{\tau }_0\left(0\right)=1+{(h∕h_0)}^2]$, with $h_0=0.385$ and 0.139 for the $\mathrm{Mn}∕\mathrm{Fe}$ and $\mathrm{Mn}∕\mathrm{Ni}$ chains, respectively. (b) Comparison of the averaged result obtained from the data shown in (a) for the $\mathrm{Mn}∕\mathrm{Fe}$ chain (full dots) to the experimental data obtained at $1.8\mathrm{K}$ for $\left(\mathrm{N}{\mathrm{Et}}_4\right)$ $\left[{\mathrm{Mn}}_2{\left(\mathrm{salmen}\right)}_2{\left(\mathrm{Me}\mathrm{O}\mathrm{H}\right)}_2\mathrm{Fe}{\left(\mathrm{C}\mathrm{N}\right)}_6\right]$ (full squares).

Figure 18

(Color online) (a) Calculated normalized relaxation frequency for a powder sample in the infinite chain regime for different values of $K$. (b) Zoom on the low field regime where a single curve is obtained as a function of $C_a{e}^{2K}$ [this curve is still valid up to $C_a{e}^{2K}=40$ for the largest value of $K$ shown in (a)]. The continuous line gives the empirical fit with ${\nu }_{\mathit{nor}}=2\sqrt{1+0.406{\left(C_a{e}^{2K}\right)}^{2.16}}-1$.