Quantum fluctuations of the magnetization in high spin molecules - a muSR study.

Zaher Salman,Amit Keren,Philippe Mendels,Ariane Scuiller and Michel Verdaguer

Abstract:

Using zero field (ZF) and longitudinal field (LF) $\mu$SR we study the magnetic properties of high spin molecules (HSM) with spin S=15/2 and S=27/2. The LF-$\mu$SR at very low temperatures suggests that in both our samples dynamical field fluctuations are responsible for the muon relaxation. The relaxation rate $\lambda$ increases as the temperature decreases and then saturates below T<T_c indicating that the dynamics is of quantum nature. The fluctuation rate at $T\rightarrow 0$ of the different samples is compared.

High spin molecules (HSM) consist of magnetic ions coupled by ferromagnetic or antiferromagnetic interactions. These molecules crystallize in a lattice where neighboring molecules are magnetically separated, yielding, at low temperatures, noninteracting giant spins ${\bf S}$. Due to magneto-crystalline anisotropy, the dominant part of the Hamiltonian is an even function of ${\bf S}$, and when the temperature is much lower than some anisotropy barrier, the only possible relaxation mechanism is of quantum mechanical origins.

In this paper we report experiments on two compounds [Cr(CN)Ni(tetren)₆](ClO₄)₉ and [Cr(CN)Mn(trispicmeen)₆](ClO₄)₉ denoted here as CrNi₆ and CrMn₆ respectively. The CrNi₆ magnetic cores are situated on an ordered lattice, where the distance between two neighboring cores is 15.68 Å, while the distance between a Cr ion and a Ni ion in the same core is 5.28 Å. The CrMn₆ structure is not fully known but seems to be amorphous. The Hamiltonian of these systems at temperatures high above the anisotropy energy was found to agree with the form ${\cal H}=\sum_{{\rm M}}J_{{\rm Cr-M}}S_{{\rm Cr}}S_{{\rm M}}$ (where M is either Ni or Mn). In CrNi₆ [1], the Cr³⁺ ion ( $S=\frac{3}{2}$) interacts ferromagnetically with 6 Ni²⁺ ions (S=1) and creates a ground state of total spin $S=\frac{15}{2}$ [1]. Susceptibility measurements show that $J_{{\rm Cr-Ni}}\approx -24$ K, and the blocking temperature is $T_B \sim 4.1$ K [1]. In CrMn₆ the interaction between the Cr³⁺ ion ( $S=\frac{3}{2}$) and 6 Mn²⁺ ions ( $S=\frac{5}{2}$) is antiferromagnetic [2], and the total spin of the ground state is $S=\frac{27}{2}$. Similar susceptibility measurements [2] show $J_{{\rm Cr-Mn}}\approx 11.5$ K and a blocking temperature of $T_B \sim 5.3$ K. The different spin values of the samples allow us to investigate the effect of S on the fluctuation rate which we determined by $\mu$SR.

  

Figure 1: (a) The variation of the asymmetry in CrNi₆ as the temperature is changed. (b) The variation of the asymmetry in CrNi₆ as the field is changed.

Our experiments at high temperatures (weak relaxation) were performed in ISIS, and at low temperature (strong relaxation) in PSI. In Fig. 1(a) we present the LF dependence of the asymmetry A(t) in CrNi₆ at base temperature 50 mK. As can be seen the relaxation rate decreases as the field is increased. Two aspects of the data indicate that the muon polarization relaxes due to dynamical field fluctuations; the first is that no recovery ( $\lim_{t\rightarrow \infty }A(t)\equiv A_{\infty }=0$) is observed. Such recovery ( $A_{\infty }=A(0)/3$) appears in cases where the muon experiences static local field and zero external field [3]. The second is that the time scale of relaxation $[1/\lambda ]$ in ZF is 1 $\mu$sec; if this field were static it would have been of the order of $[B]\approx 10$ G (using $[B]=[\lambda ]/\gamma_{\mu }$ where $\gamma _{\mu }$ is the muon giromagnetic ratio). Such a field should have been completely decoupled ( $\lim_{t\rightarrow \infty}A(t)=A(0)$) with $\sim 100$ G LF or more [3]. However, even fields as high as 5000 G do not decouple the relaxation. Therefore, we conclude that even at 50 mK the CrNi₆ spins are dynamically fluctuating. Similar experiments and line of arguments indicate that CrMn₆ spins are also dynamically fluctuating at base temperature.

The temperature dependence of the asymmetry in CrNi₆ is presented in Fig. 1(b). As the temperature is decreased towards $T_c \approx 6$ K the relaxation rate increases. However, below 6 K there is no change in A(t) indicating that the relaxation rate reaches saturation at T_c. Such a saturation of the relaxation rate suggests that the fluctuations below T_c is of quantum nature. Again the same temperature dependence was observed in CrMn₆ (but with a different value of T_c).

  

Figure 2: The relaxation rate of the asymmetry as a function of temperature for different magnetic fields. The relaxation rate saturates at low temperatures.

We fit the asymmetry in ZF or LF using the form $A(t)\propto e^{-\left( \lambda t\right) ^{\beta }}$, where $\beta$ varies between samples but is a global parameter for a specific sample. In CrNi₆ and CrMn₆ we find the best $\beta$ to be 0.5 and 0.3 respectively. In Fig. 2 the relaxation rate for both CrNi₆ and CrMn₆ is plotted as a function of temperature for different LF values. One can see that as the temperature is decreased the relaxation rate is increased, and reaches saturation at a T_c which is different for different compounds. At high temperatures the relaxation is field independent, and becomes field dependent at lower temperatures.

The solid lines in Fig. 2 are fits of the relaxation rate $\lambda$ to a function of the form

$$\lambda (T,H)=\frac{1}{Q(H_{L})+C\exp \left( -\frac{U}{T}\right) }$$ (1)

where C and U are global parameters for all fields, $C=86 \pm 12,1.5 \pm 0.4$ $\mu$sec and $U=63 \pm 2, 73 \pm 7$ K for CrNi₆ and CrMn₆ respectively. This shows that the inverse relaxation rate (relaxation time) has a field dependent part which is temperature independent Q(H_L), and a temperature dependent part which is field independent $C\exp \left( -\frac{U}{T}\right)$. The value of Q(H_L) is the value of the relaxation time at low temperatures (saturation value). This parameter is found to be proportional to the LF squared, H_L², as shown in Fig. 3.

  

Figure 3: The saturated relaxation time Q as a function of H_L² for both CrMn₆ and CrNi₆. The solid lines are linear fits of Q vs. H_L².

The stretched exponential relaxation combined with the fact that $\lambda^{-1}$ depends linearly on H_L² could be explained by a single field-field correlation function [4], $\left\langle B_{\perp }(0)B_{\perp }(t)\right\rangle =\left\langle B_{\perp}^{2}\right\rangle e^{-\nu t}$ where $\nu$ is the field-field correlation rate, and $\left\langle B_{\perp}^{2}\right\rangle$ is the mean squared field (at a given muon site), combined with multiple muon occupation sites. The multiple sites introduce a distribution of $\left\langle B_{\perp}^{2}\right\rangle$. For long times $\nu t\gg 1$ (which is satisfied in our case) the relaxation rate $\lambda^{-1}$ reduces to [4]

$$\frac{1}{\lambda }=\frac{1}{\nu a^{2}}H_{L}^{2}+\frac{\nu }{\gamma _{\mu}^{2}a^{2}}$$ (2)

where a represents the range of possible $\left\langle B_{\perp}^{2}\right\rangle$. From Fig. 3 and Eq. (2), we find $a_{\rm Ni}=302\pm 5$ G and $a_{\rm Mn}=550\pm 30$ G for CrNi₆ and CrMn₆ respectively. The values of a in the two samples could not be directly compared since we have used different $\beta$ values. However, it is encouraging that the ratio $a_{\rm Ni}/a_{\rm Mn}$ differs from the ratio of the expectation values of $\sqrt{S^{2}}=\sqrt{\frac{15}{2}(\frac{15}{2}+1)/\frac{27}{2}(\frac{27}{2}+1)}$ in the two compounds by 4% only. This suggests that the muons occupy roughly the same sites in the two samples.

In addition, we calculated the fluctuations rates $\tau^{-1}=\frac{\nu}{2}=50\pm 5$ MHz and $60\pm 15$ MHz for CrNi₆ and CrMn₆ respectively. Quantum tunneling theories predict a strong dependence of the tunneling rate on the spin value, $\tau^{-1} = \frac{DS^2}{\pi S \hbar} \left(\frac{H_{\perp}S}{DS^2}\right)^{2S}$ [5] ($H_{\perp}$ is the transverse part of the Hamiltonian which induces tunneling and $T_B \sim DS^2$). However, according to this calculation the fluctuation rate yields $H_{\perp} \approx 4.3$ kG and 5.7 kG in CrNi₆ and CrMn₆, respectively. These fields are not consistent with the root mean squared field experienced by the muon. We thus conclude that the description of the quantum dynamics in this system could not be mapped into the double potential picture which is useful in other HSM such as Mn₁₂ [6] and Fe₈ [7].

These experiments were supported by the European Union through its TMR Program for Large Scale Facilities, the French Israeli cooperation program AFIRST and the Israeli ministry of science.