Relaxation and Landau-Zener experiments down to 100 mK in ferritin

Temperature-independent magnetic viscosity in ferritin has been observed from 2 K down to 100 mK, proving that quantum tunneling plays the main role in these particles at low temperature. Magnetic relaxation has also been studied using the Landau-Zener method making the system crossing zero resonant field at different rates, alpha=dH/dt, ranging from 10^{-5} to 10^{-3} T/s, and at different temperatures, from 150 mK up to the blocking temperature. We propose a new Tln(Delta H_{eff}/tau_0 alpha) scaling law for the Landau-Zener probability in a system distributed in volumes, where Delta H_{eff} is the effective width of the zero field resonance.

Over the past decade there have been experimentally observed a large number of quantum phenomena in the dynamics of the magnetic moment of mesoscopic systems. Monodomain magnetic particles of nanometer size contain thousands of magnetic atoms strongly interconnected by exchange interaction. As a result of the exchange interaction, all the atomic spins align parallel or antiparallel between them, resulting in a ferro, or ferri-antiferromagnetic ordering respectively. One particular system has had much attention since Awschalom and co-workers announced the observation of a resonance near 1 MHz that interpreted in terms of quantum coherence of the magnetic moment [1,2]: the system is composed by antiferromagnetic particles which grow inside the cage of the horse spleen ferritin proteins [3].
Next experimental studies of the dynamics of the magnetization of ferritin particles, carried out at kelvin regime, showed different phenomena interpreted as quantum tunneling of the ferritin magnetic moment [3][4][5][6][7][8][9][10]. These phenomena can be differentiated as follows: The first was the temperature-independent relaxation of the magnetization below 2.3 K observed in the magnetic viscosity measurements [4]. Secondly, a non-monotonic behaviour of the blocking temperature, T B , on the magnetic field [2,5,6,[8][9][10], contrasting with the monotonic classical expectation. Thirdly, a maximum at zero field in the magnetic field derivative of the magnetization extracted from the hysteresis cycles over 4 K [2,[5][6][7][8]. And finally, a rapid increase of the magnetic viscosity as the magnetic field approaches to zero at temperatures higher than 3 K [6,8,9]. More recently, there have been done 57 Fe Mössbauer spectroscopy measurements on an artificial ferritin sample down to 50 mK showing nonincoherent tunnel fluctuations around 10 8 Hz [11]. Attending to the recent observation of resonant quantum tunneling of the spin observed in molecular clusters like Mn 12 -Acetate [12,13], one can conclude that all these phenomena observed in ferritin particles in the kelvin range can be attributed to thermally activated resonant quantum tunneling of the magnetic moment at zero field. The observation of tunneling only at zero field lies in the fact that ferritin has a broad distribution of energy barriers due to the distribution of volumes of the particles (f (U = KV ), where K is the anisotropy constant), and their anisotropy axes are randomly oriented. In the other hand, the temperature-independent viscosity below the crossover temperature, T c = 2.3 K, indicates that, below this temperature, the quantum magnetic relaxation of ferritin occurs through the lowest states.
In this paper we present new magnetic data which extend the quantum relaxation measurements to the millikelvin regime. At the same time, in order to estimate the value of the quantum splittings of ferritin particles, we have done measurements of the change of magnetization when the system crosses zero magnetic field at different rates of the field sweep and we analyze the results in terms of the Landau-Zener probability associated to the magnitude of the splitting playing the main role in the quantum relaxation. This is the same method used by Wernsdorfer et al. to determine the quantum splittings of Fe 8 molecular clusters [14].
Ferritin is an iron storage protein. It has a spherical cage of about 8 nm in diameter in whose interior grows mineral ferrihydrite combined with a phosphate. Its core is equivalent to a small antiferromagnetic particle. The size of the core in natural ferritin ranges from 3 to 7.5 nm. The fully packed ferritin contains 4500 F e 3+ ions. A small magnetic moment of the particle arises from the non-compensation of collinear spin sublattices due to the finite size and irregular shape of the core. The spin of the sublattice, S, is of the order of 5000, while the non-compensated spin, s, is below 100. This number corresponds to 15 non-compensated F e 3+ ions extracted from magnetic susceptibility measurements [4], in agreement with the theoretical expectation for the square root of the number of ions at the surface of the particle, N S ∼ (4500) 2/3 , which gives (4500) 1/3 ∼ 16. This non-compensated spin looks in one of two directions along the anisotropy axis of the particle. In our experiments we have used a Fluka Biochemical diluted natural ferritin sample. The distribution of volumes of the sample, f (V )dV , is plotted in the inset of figure 1 (extracted from reference [9]). The center volume is V 0 ∼ 150 nm 3 . The low moment of the antiferromagnetic particles makes the interactions between different particles to be negligible, as one can see in the inverse susceptibility in the superparamagnetic regime that extrapolates to zero at T →0 [6].
Low temperature magnetic relaxation measurements were done in an Oxford Instruments temperature and then a magnetic field of 1 T is applied during 10 minutes. After that, the field is switched off and the magnetization is measured during 3 hours. In order to avoid remanent fields in the superconducting magnet and to obtain the relaxation measurements as close as possible to zero field a demagnetizing cycle is immediately applied after switching off the field. The demagnetizing cycle was previously tested in a pure Pb diamagnetic sample.
This method makes the field along the major part of the relaxation to be zero with a precision of ±1 Oe. The measurements were done at different temperatures ranging from 100 mK up to 1,5 K in the dilution cryostat and repeated in the same manner in a Quantum Design MPMS magnetometer at temperatures up to 25 K. The logarithmic on time dependence of the magnetic relaxation is clearly observed over the whole measure. In a sample with a distribution of energy barriers, the quantitative magnitude which measures the relaxation time is the magnetic viscosity defined as [15] where M eq (H, T ) is the equilbrium magnetization of the system at fixed temperature and field, which is M eq (H, T ) = 0 in our case, and M ini (H, T ) is the initial magnetization.
In our experiments M ini (H, T ) was taken from the extrapolation at small time of each magnetic relaxation curve. It is known that after switching off the field the system rapidly runs to a critical state in a time much more shorter than the times involved in the slow relaxation process occurring after the system reaches this critical state and relaxes to the final equilibrium state [15]. The observed dependence of the magnetic viscosity with temperature is shown in figure 1. The viscosity shows a maximum at T B ∼ 10 K. This is the blocking temperature, defined as which for viscosity measurements, with a characteristic measuring time, t m , of hours, corresponds to the unfreezing of the magnetic moment of a particle of volume V which changes its orientation jumping over the energy barrier. In a sample distributed in size (see inset obtain T B ∼ 10 K, in good agreement with the experimental result. As the temperature decreases, the magnetic viscosity goes to zero, as expected for thermal relaxation in a system with barriers distribution. However, below ∼ 2 K the viscosity becomes independent on temperature down to 100 mK. This temperature, at which the system crosses from thermal to quantum relaxation regime is called crossover temperature, T c [4]. The new data showed in this paper extend the observation of the plateau of the magnetic viscosity down to a few millikelvin. This takes high relevance assuming the fact that below T c the system relaxes exclusively through the lowest levels of the magnetic structure by quantum tunneling. This temperature does not depend on the volume of the particles. The expression expected from theory which determines this temperature for antiferromagnetic monodomain particles is [16]. Taking ǫ an ∼ 0.1 K (anisotropy energy per spin) and ǫ ex ∼ 10 3 K (exchange energy per atom) [6], we obtain T c ∼ 2 K in good agreement with the experimental value. (2) we obtain that a significant number of particles smaller than 5 nm 3 is needed to obtain T B < 0.1 K. The ac-susceptibility measurements (window time of 10 −3 s) [9] and Mössbauer espectroscopy (10 −8 s) [11,17] show that there are not significant particles of this size behaving superparamagnetically at low temperature. The second explanation, quantum superparamagnetism, explains better the 1/T behaviour at low temperature. This is the quantum behaviour of the particles for which T B is smaller than T c . That is, these particles do not feel the anisotropy barrier because they can rotate their magnetic moments by quantum tunneling even if the temperature is not enough to jump across the barrier.
The quantum tunneling rate is determined by the WKB exponent, B ∼ KV /K B T c , in the following manner, Γ ∼ exp(−B). This means that the particles having smallest size have the higher probability to tunnel across the barrier.
The most direct way to measure the quantum tunneling splitting, ∆, is by using the Landau-Zener model [18], which gives the tunnel probability, P , when a resonance is crossed at a given sweeping rate, α: where h is the Plank's constant and S is the spin of a particle. Due to the distribution of volumes in ferritin there are a distribution of spin values, S(V ), and a distribution of quantum splittings, ∆(V ). Also, the random orientation of the anisotropy axis of the particles in the sample introduces a distribution of sweeping rates, α(θ), on the angle between the applied field and the anisotropy axis of each particle. This makes that different particles have different tunnel probability at a given sweeping rate depending in both volume and orientation respect to the applied magnetic field. Taking into account the mentioned conditions, we can express the change of magnetization of the whole sample as the zero resonant field is crossed from H i to H f at a given α in terms of the Landau-Zener probability as follows: where M i , M f and M eq are the initial, final and equilibrium magnetizations, respectively.
The integral over θ has been chosen to take into account the random orientation of the anisotropy axes of the particles respect to the applied field. The form of α(θ) for one particle is then cos(θ)α.
Our experiments where done in the following manner: First, a saturating magnetic field was applied at the measure temperature. Then, the field was changed to H i = 250 Oe at the highest sweeping rate and the magnetization was measured giving M i . Immediately, the field was changed to H f = -250 Oe at a given α, measuring M f after the process was finished. The procedure was repeated at different sweeping rates, ranging from 10 −5 T/s up to 10 −3 T/s and at different temperatures, from 100 mK up to the blocking temperature. The results are shown in figure 3. In order to make the nomenclature shorter we will use P ∆M (probability to change the magnetization) instead the expression given in eq. (4). One can see that, at a given temperature, P ∆M increases when α decreases. That is, as the zero field resonance is crossed slower the probability to change the magnetization of the sample is higher. With the same dependence in α, the probability becomes higher for higher temperatures. The bahaviour of P ∆M on 1/α is perfectly logarithmic. This dependence reminds the behaviour where Oe. It is observed that the data collapse into a master curve for temperatures higher than ∼5 K. The value of the effective resonance width, ∆H ef f = 5 Oe, is two orders of magnitude smaller than the width of the zero resonance observed in the magnetic hysteresis loops at the same temperatures, ∆H ∼ 1000 Oe [2,[5][6][7][8] and associated to thermally assisted resonant quantum tunneling [6,8,9]. The same phenomena was previously observed in molecular clusters [12,13]. In principle, the width of this resonance is associated to the quantum splitting of the blocking level, m B . This is the level through which the quantum tunneling occurs at a given temperature. In ferritin the width of the resonance is associated to the distribution of quantum splittings of the blocking levels due to the different volumes of the particles of the sample. This fact, together with the random orientation of the anisotropy axes respect to the applied magnetic field, makes the width of the zero field resonance to be several orders of magnitude higher than the width of the quantum splitting of one of the particles of the sample. However, the scaling law proposed here takes into account the effect of an average particle of the sample. Due to this, the physical meaning of ∆H ef f extracted from the master curve can be attributed to the width of the zero field resonance for an average particle of the sample. That is, we may associate ∆H ef f with the quantum splitting of the effective blocking level, ∆ ef f , of the distribution of particles in ferritin in the following manner: ∆ ef f ∼ gµ B S∆H ef f . Using S ∼ 50, we obtain ∆ ef f ∼ 700 MHz.
Taking into account the uncertainties associated to the random orientation of the anisotropy axes of the particles it seems clear that the obtained value of the quantum splitting of the effective blocking level agrees with the ∼1 MHz resonance found by Awschalom et al. [1] and attributed to the quantum splitting of the ground state of ferritin particles.
In conclusion, we have obtained temperature-independent magnetic relaxation from both magnetic viscosity measurements and Landau-Zener viscosity down to 100 mK. We have   Oe. The inset amplifies the low temperature zone. 1/T quantum superparamagnetism behaviour is observed below 0.5 K.