Dynamics at $T\rightarrow 0$ in the half-integer isotropic high spin molecules

Zaher Salman¹, Amit Keren¹, Philippe Mendels², Valerie Marvaud ³, Ariane Scuiller ³, Michel Verdaguer³, James S. Lord⁴, and Chris Baines⁵

¹Technion-Israel Institute of Technology, Physics Department, Haifa 32000, Israel
²Laboratoire de Physique des Solides, Bâtiment 510, UMR 8502 CNRS Université Paris Sud, 91405 Orsay, France
³ Laboratoire de Chimie des Métaux de Transition, URA CNRS 419 Université Pierre et Marie Curie, 75252 Paris, France
⁴ISIS Facility, Rutherford Appleton Laboratory, Chilton Didcot, Oxfordshire OX11 0QX, U.K.
⁵ Paul Scherrer Institute, CH 5232 Villegen PSI, Switzerland

Abstract:

We investigate the dynamical spin-spin auto-correlation function of the isotropic high spin molecules CrCu₆ (S=9/2), CrNi₆ (S=15/2) and CrMn₆ (S=27/2 ), using magnetization, $\mu$SR and NMR measurements. We find that the field autocorrelation time $\tau$ of the molecule's spin at zero and low fields is nearly temperature independent as $T\rightarrow 50$ mK. The high temperatures $\tau$ is very different between the molecules. Surprisingly, it is identical ($\sim$ 10 nsec) at low temperature. This suggests that $\tau$ is governed by hyperfine interactions.

High spin molecules (HSM) consist of clusters of metal ions, they are ordered in a crystal lattice, and coupled by Heisenberg ferromagnetic or antiferromagnetic interactions with coupling constant J, only between spins $\vec{S}_{i}$ in the molecule. At low temperatures k_BT<J these spins lie parallel or anti-parallel to each other, and the molecule is in its ground spin state, where $\vec{S}=\sum_{i}\vec{S}_{i}$ is high, with quantum number S, and 2S+1 degeneracy. At even lower temperatures, $k_{B}T\ll J$ the degeneracy can be removed by additional anisotropic interactions such as the uniaxial term DS_z², or rhombic term E[S_x²-S_y²] etc. Experiments on the two most famous high spin molecules Mn₁₂ [1] and Fe₈ [2,3] show that at low T the main interaction is, indeed, the uniaxial anisotropy, where up and down spin states $S_{z}=\pm S=\pm 10$ are degenerate, and tunneling is induced between these states by an additional term in the Hamiltonian that does not commute with S_z. This quantum tunneling of the magnetization (QTM) phenomenon has received considerable attention in recent years, focusing mainly on the additional terms in the Hamiltonian which are responsible for the tunneling [3,4,5]. Nevertheless, the theoretical picture is far from being clear and existing models are controversial, and often contradict each other [6]. In some cases, even qualitative understanding of the observed experimental data is absent [7].


In this paper we investigate three simple HSM systems, which are isotropic (D,E=0). In these systems no tunneling is observed due to the absence of the uniaxial term DS_z². However, spin dynamics is observed even at very low temperatures (T=50 mK). Therefore, the additional terms in the spin Hamiltonian could be probed directly in these systems. Such a study can highlight the role of phonons [6,8], dipolar interactions, and nuclear fluctuations [5] in a simple setup. In addition, it could serve as a large S limit for isotropic models usually applied only for the S=1/2 case [6], or it could serve as a $D\rightarrow 0$ test case for anisotropic HSM models [9].


We present an experimental investigation using three types of measurements: magnetization, spin lattice relaxation (T₁^-1) of muon spin ($\mu$SR), and of proton spin (NMR). Our molecules are [Cr{(CN)Cu(tren)}₆ ](ClO₄)₂₁ [10], [Cr{(CN)Ni(tetren)}₆ ](ClO₄ )₉ [11,12] and [Cr{(CN)Mn(tetren)}₆](ClO₄)₉ [10,13] which we label as CrCu₆, CrNi₆ and CrMn₆ respectively. In these molecules a Cr(III) ion is surrounded by six cyanide ions, each bonded to a Cu(II), Ni(II) or Mn(II) ion. The coordination sphere of Cr and Cu/Ni/Mn can be described as a slightly distorted octahedral. The objective in this work is to find the spin-spin correlation time $\tau (T)$ in the three systems, and to compare them. Our main findings are: (I) $\tau$ is nearly T independent at low temperatures, and (II) at very low temperature $\tau$ does not depend on S or J in this isotropic case.

Figure 1: The magnetization of CrCu₆, CrNi₆ and CrMn₆ as a function of the external applied field at T=2 K, respectively. The solid lines are the S=9/2, S=15/2 and S=27/2 Brillouin functions (see text).

In Fig. 1 we show the magnetization M per molecule in units of $\mu _{B}$ as a function of applied field H for the three molecules. The data are taken at T=2 K. In all cases M increases as the applied field is increased. The magnetization reaches a saturation value of $\frac{9}{2}$ $g \mu_{B}$, $\frac{15}{2}$ $g \mu_{B}$ and $\frac{27}{2}$ $g \mu_{B}$ in CrCu₆, CrNi₆ and CrMn₆ respectively. These saturation values are consistent with 6 Cu (S=1/2), Ni (S=1) or Mn (S=5/2) ions ferromagnetically (or anti-ferromagnetically in the case of CrMn₆) coupled only to a Cr ion of spin 3/2. The solid lines are Brillouin functions of the respective S. In Refs [10,11,12,13] the susceptibility was fitted to the one expected from the Heisenberg Hamiltonian and the values $J_{{\rm CrCu}_{6}}=77$ K, $J_{{\rm CrNi}_{6}}=24$ K and $J_{{\rm CrMn}_{6}}=-11$ K where found. Therefore, the highest spin value of each molecule is well seperated from other spin states (a few tens of degrees K). High field ESR measurements (on CrNi₆) [14] and susceptibility measurements (on CrCu₆, CrNi₆ and CrMn₆) found no evidence for anisotropy, namely, $D\simeq 0$. This is consistent with the octahedral character of the molecules.


In our $\mu$SR experiments we measure the polarization P(t,H) of a positive muon spin implanted in the sample, as a function of time t and magnetic field H, where P(0,H)=1. The field is applied in the direction of the initial muon polarization. The positive muon decays to a positron which is emitted in the direction of the muon spin, and the polarization as a function of time is reconstructed by the detection of the emitted positrons.


The measurements in all molecules are done at temperatures ranging from 25 mK up to 300 K, and in fields ranging between zero and 20 kG. These experiments were performed at both ISIS and PSI, exploiting the long time window in the first facility for slow relaxation (high T), and the high time resolution in the second facility for fast relaxation (low T).

Figure 2: The spin polarization as a function of time. (a) At zero field and different temperatures. (b) At field H=20 kG and different temperatures. The solid lines are fits of the data to square root exponential functions.

In Fig. 2(a) and (b) we present the muon spin polarization as a function of time and for different temperatures in CrNi₆ in zero field, and in H=20 kG, respectively. In zero field, the relaxation rate increases, as the temperature is decreased, and saturates at $\sim 5$ K. The increase at high temperatures is caused by thermally activated tarnsitions between excited spin states. However, at low temperatures, only the ground spin state is populated, and only transitions within the degenerate ground state are possible. In contrast, at H=20 kG, and temperatures lower than $\sim 17$ K, the relaxation rate decreases as the temperature is decreased, and does not saturate.


In Ref. [15] we demonstrated that the magnetic field experienced by the muon in all molecules is dynamically fluctuating even at T=50 mK. We therefore analyze our data using spin lattice relaxation theory. In this theory, the polarization of a local probe (muon or nucleus), in the fast fluctuation limit, is given by

$$P(H,t)=\left( P_{0}-P_{\infty }\right) \exp \left[ -t/T_{1}\right] +P_{\infty }$$ (1)

where P₀ is the initial polarization, $P_{\infty }$ is the equilibrium polarization [16],

T₁(H) = A+BH²,

$$\frac{1}{\Delta ^{2}\tau },$$ A = (2)

$$\frac{\gamma ^{2}\tau }{\Delta ^{2}},$$ B =

and $\gamma$ is the probe gyromagnetic ratio. The correlation time $\tau$ and mean square of the transverse field distribution at the probe site in frequency units $\Delta ^{2}$ are defined by

$$\gamma ^{2}\left\langle {\bf B}_{\bot }(t){\bf B\bot }(0)\right\rangle =\Delta ^{2}\exp \left( -t/\tau \right) .$$ (3)

The fast fluctuation limit is obeyed when $\tau \Delta \ll 1$.


In $\mu$SR $P_{\infty }=0$, and the muon could occupy many different sites in the sample, because the molecules are fairly large ($\sim 15$ Å diameter) and are embedded in an organic surrounding. As a result one must average over $\Delta$. Using the distribution [16]

$$\rho (\Delta )=\sqrt{\frac{1}{2\pi }}\frac{\Delta ^{\ast }}... ...\left[ \frac{\Delta ^{\ast }}{\Delta }\right] ^{2}\right) ,$$ (4)

and allowing for a constant background (B_g) due to muons stopping outside the sample one obtains

$$P(t)=P_{0}\exp \left( -\sqrt{t/T_{1}^{\mu }}\right) +B_{g},$$ (5)

where $1/T_{1}^{\mu }$ is the muon spin relaxation rate. This form is in agreement with the experimental results (see below). In addition, Eq. 2 still holds, while in Eq. 2 and , $\Delta$ is replaced by $\Delta ^{\ast }$.


The solid lines in Fig. 2 are fits of the data to Eq. 5 where P₀ is a global parameter. The parameter B_g is free within 10% of its mean value since the high fields affect the positron trajectory in a manner that is reflected in B_g. The fit is satisfactory in all cases apart from the highest H and lowest T in CrNi₆. The relaxation rate $1/T_{1}^{\mu }$ in the different compounds, obtained from the fits is presented in Fig. 3 as a function of temperature for different values of H. As pointed out above, at low fields $1/T_{1}^{\mu }$ increases with decreasing temperatures, and saturates at low temperatures. In addition the value of $1/T_{1}^{\mu }$ increases as the spin of the compound is higher, as expected from Eqs. 2- and 3, and the saturation temperature increases as the coupling constant J increases. This is in strong contrast to Mn₁₂ [17] and Fe₈ [18] where in zero field $1/T_{1}^{\mu }$ increases continuously upon cooling until the correlation time $\tau$ becomes so long that the molecule appears static in the muon dynamical window, and no T₁ saturation is observed at low temperatures.

Figure 3: 1/T₁ as a function of temperatures for different external fields, measured by $\mu$SR and by NMR (after scaling) in (a) CrCu₆ (b) CrNi₆ and (b) CrMn₆.

In Fig. 4 we plot the average relaxation time $T_{1}^{\mu }(H)$ at $T\rightarrow 0$ as a function of H² for all compounds, for fields up to 2 kG (note the axis break). We find that $T_{1}^{\mu }$ obeys Eqs. 2-. This implies that the muon spin relaxation is indeed due to dynamical field fluctuations, and that at low T the field autocorrelation can be described by a single correlation time as long as the applied field is not too strong. At high fields (> 2 kG) we find deviations (not shown) from the linear relation between T₁ and H². The deviation might be due to the impact of the field on the spin dynamics, i.e. the correlation function given by Eq. 3. From the linear fits in Fig. 4, and taking $(\Delta ^{\ast })^{2}=\gamma _{\mu }(AB)^{-1/2}$ from Eq. 2 and , where $\gamma _{\mu }=85.162$  MHz/kG, we find that for CrCu₆ $\Delta_{0}^{\ast }=4.9\pm 0.9$ MHz ($57 \pm 10$ G), for CrNi₆ $\Delta _{0}^{\ast }=26\pm 2$ MHz ($305\pm 25$ G), and for CrMn₆ $\Delta _{0}^{\ast }=38\pm 2$ MHz ($446 \pm 24$ G); the subscript 0 stands for $T\rightarrow 0$. Using $\tau =(B/A\gamma _{\mu }^{2})^{1/2}$ from the same equation we find $\tau _{0}=7\pm 1$ nsec for CrCu₆, $\tau _{0}=10\pm 1$ nsec for CrNi₆ and $\tau _{0}=11\pm 1$ nsec for CrMn₆. These values of $\Delta _{0}^{\ast }$ and $\tau _{0}$ are self consistent with the fast fluctuation limit. Most striking is the fact that all $\tau _{0}$ values are nearly equal.


Although our data support a picture where the muon spin relaxes due to dynamically fluctuating magnetic fields, they leave open the interpretation of these fluctuations. Are they due to the fluctuations of the molecular spins or a result of muon diffusion, muonium formation, etc.? In order to address this question we performed proton NMR T₁ measurements. We find that in all applied fields smaller than 20 kG the proton T₁ is shorter than the experimental window around the peak in $1/T_{1}^{\mu }$. Only in a field of $\sim 20$ kG were we able to perform the experiment at all temperatures. We measure T₁ using a saturation$-t-\pi /2-\pi$ pulse sequence. The proton polarization recovery follows Eq. 1 with P₀=0 from which we obtained T₁^H.

Figure 4: The saturation relaxation time as a function of H² for CrCu₆ , CrNi₆ and CrMn₆. The solid lines are linear fits of the data.

Since the proton gyromagnetic ratio $\gamma _{H}=42.57$ MHz/kG is very different from that of the muon, we scale the NMR results and plot them together with the $\mu$SR results in Fig. 3, for CrNi₆ (b) and CrMn₆ (c). The scaling factor C used in Fig. 3 is 0.6 for CrNi₆ and 8.8 for CrMn₆. It was chosen so that the scaled NMR relaxation rates will agree with the $\mu$SR rates at high T. After this scaling, we find a good agreement between the $\mu$SR and NMR data at all temperatures. In fields lower than 20 kG, where the NMR T₁ was measurable only at high T, we obtained the same agreement between the two techniques.


The scaling factor provides information on the ratio of the field experienced by a muon and by a proton. At high temperatures where T₁ shows no field dependence one can assume that $A\gg BH^{2}$ in Eqs. 2-, and therefore $1/T_{1}=\Delta ^{2}\tau$. Using the definition of $\Delta ^{2}$ given in Eq. 3 we can write $C\equiv \left( \gamma _{H}/\gamma _{\mu }\right) ^{2}\left( T_{1}^{H}/T_{1}^{\... ...}^{2}\right\rangle ^{\mu }/\left\langle {\bf B}_{\perp }^{2}\right\rangle ^{H}$ where $\left\langle {\bf B}_{\perp }^{2}\right\rangle ^{H}$ is the RMS of the transverse field at the proton site, and $\left\langle {\bf B}_{\perp }^{2}\right\rangle ^{\mu }$ is the RMS of the field at the muon site in its general sense given by Eq. 4. The proximity of C to 1 especially in the CrNi₆ is very encouraging and suggests that the muon sites are close to a proton in this system. Thus, we prove that in both techniques we are measuring the probe's spin lattice relaxation time due to the molecular spin fluctuations.


Finally, we would like to obtain the correlation time at all temperatures. This could be calculated from $T_{1}^{\mu }$ combined with magnetization measurements, at zero (or very low) fields. In zero order approximation we assume that $\left\langle {\bf B}_{\perp }^{2}\right\rangle$ is proportional to $\left\langle {\bf S}^{2}\right\rangle$ which is different from S(S+1) since at temperatures $k_{B}T\sim J$, states other than the ground state S can be populated. Therefore in zero field

$$\frac{1}{T_{1}}(T)=\frac{\Delta_{0}^{\ast 2}\left\langle S^{2}\right\rangle (T)\tau(T)}{S(S+1)}.$$ (6)

Taking

$$\left\langle S^{2}\right\rangle (T)=\frac{3k_{B}T\chi (T)}{N(g\mu _{B})^{2}},$$ (7)

where N is the number of molecules, $\mu _{B}$ is Bohr magneton, k_B is the Boltzman factor, and g=2, we find

$$\tau (T)=\frac{(g\mu _{B})^{2}}{3k_{B}}\frac{S(S+1)}{T_{1}(T)T\chi (T)\Delta _{0}^{\ast 2}}.$$ (8)

In the insets of Fig. 5 we present $\left\langle S^{2}\right\rangle (T)$, obtained from $H\rightarrow 0$ DC-susceptibility measurements and Eq. 7, as a function of temperature for CrCu₆, CrNi₆ and CrMn₆. In Fig. 5 we present $\tau (T)$ as calculated using Eq. 8 for the different compounds. The T dependence of the correlation time $\tau (T)$, unlike the muon spin lattice relaxation rate, reflects the dynamics of the molecular spin without the T dependence of the field at the muon site. At $T\sim 100$ K there is more than an order of magnitude difference in $\tau$ between the different molecules. As the temperature is lowered the correlation time in all compounds increases as the temperature is decreased, but reaches a common saturation value of $\sim 10$ nsec (within experimental error) at $T\sim 10$ K. At this temperature only the ground state S is populated. In other words, when the HSM are formed they all have the same correlation time at low T.

Figure 5: The correlation time $\tau$ as given by Eq. 8 as a function of temperature for CrCu₆, CrNi₆ and CrMn₆. The inset shows $\left\langle S^{2}\right\rangle (T)$ as obtained from Eq. 7

At very low temperatures one can interpret the correlation time $\tau$ in terms of a broadening of the spin levels, due to interactions between a molecular spins and other molecular spins, or the enviroment. This interaction is considered as a purturbation, the strength of which should be of the order of $h\tau ^{-1}$. Therefore, our results have two major indications. First, the broadening in the $T\rightarrow 0$ limit is not due to phonons since these die out exponentially with temperature at 0.05-1 K. Second, the broadening cannot be explained by interactions which are quadratic in S or have higher S dependence. This rules out the dipolar interaction between neighboring molecules since in the three compounds the nearest neighbor distance is $\sim 15$ Å. Similarly, crystal field terms which are allowed by the octahedral symmetry (S⁴ or higher [19]) are unlikely.


The only mechanisem suggested to date for level broadening of HSM, which is weakly (linear) S dependent is the hyperfine interaction between nuclear and electronic spins. This mechanism can account for the finite spin lattice relaxation rate at very low temperatures [20]. The hyperfine interactions in unisotropic high spin molecules were studied recently [22,23], and their effect on QTM is becoming clearer [24,25]. We believe that this interaction also governs the fluctuations of the isotropic molecules at very low temperatures (T<3 K). However, at high temperature (T>10 K) the fluctuations are governed by spin-phonon interactions [21].


We would like to thank the ISIS and PSI muon facilities for their kind hospitality and B. Barbara for helpful discussion. 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.