- Email: [email protected]
Magnetic field dependent thermodynamics of Fe8 single crystal in the thermally activated regime
Journal of Magnetism and Magnetic Materials 242–245 (2002) 915–920
Invited paper
Magnetic field dependent thermodynamics of Fe8 single crystal in the thermally activated regime G. Gaudina, P. Gandita,*, J. Chaussya, R. Sessolib a
CRTBT-CNRS, 25 av. des Martyrs, BP166, F - 38042 Grenoble Cedex 9, France b Dep. of Chemistry, Univ. Firenze, Via Maragliano 77, 550144 Firenze, Italy
Abstract Quantum tunneling of the magnetization in the thermally activated regime has been studied on Fe8 single crystals by frequency dependent specific heat measurements as a function of applied magnetic field. Curves are very well described by a microscopic theory based on phonon assisted spin tunneling. However, an additional peak appears that can be due to transitions in higher multiplets. r 2002 Elsevier Science B.V. All rights reserved. Keywords: MagnetizationFquantum tunneling; Specific heatFlow temperature; Spin–phonon interactions; Magnetic molecular materials
1. Introduction Magnetic quantum tunneling of large spins is one of the phenomena that have aroused much interest recently both for fundamental science and for future applications, for instance in quantum computing [1]. Magnetic molecular clusters are among the most promising candidates for observing it. In fact, they permit macroscopic observation of single particle properties, avoiding complications due to size and orientation distributions existing in traditional small magnetic particle systems. Fe8 clusters ð½ðtacnÞ6 Fe8 O2 ðOHÞ12 8þ Þ with a spin ground state of S ¼ 10 [2] are among the most promising candidates. Below 360 mK, the magnetization relaxes through a pure tunneling process [3]. At higher temperature, the tunneling becomes thermally assisted. This regime has been widely studied for another molecular cluster, Mn12 ; both experimentally [4–10] and theoretically [11–15] but very few references mention measurements in that regime for Fe8 : Specific heat measurements that have been performed at low temperature on both Mn12 and Fe8 [17,18] give a great deal of information such as the contribution of *Corresponding author. Tel.: +33-4-76-889069; Fax: +33-476-875060. E-mail address: [email protected] (P. Gandit).
hyperfine and dipolar fields. However the specific heat is a thermodynamic quantity defined at the equilibrium and the tunneling of a spin is a dynamic process. The study of the interplay between the spin system and the crystal lattice, e.g. clarifying how the phonons influence the macrospin tunneling, can only be done through dynamic measurements such as the frequency dependent specific heat measurements that will be presented. An original nanocalorimeter [22] capable of measuring with very high resolution the specific heat of very small samples (mg) will be presented. It permits the qualitative and quantitative study of the quantum tunneling of the magnetization in the thermally activated regime through measurements performed on Fe8 single crystals.
2. Frequency dependent specific heat The idea of measuring the ‘‘specific heat spectrum’’ of a system began with the development of the ac-steady state method [19]. In that method, an alternative current of frequency n=2 passes in a heater, producing a smallamplitude sinusoidal heat flux to the sample. The frequency dependent specific heat is calculated with the amplitude of the temperature oscillation at frequency n measured by a thermometer. This frequency dependent specific heat is a linear susceptibility describing the
0304-8853/01/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 1 ) 0 1 3 2 3 - 3
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response of the system to arbitrarily small perturbations away from equilibrium and can be related to the temporal fluctuations of the enthalpy [20]. In the limit n-0; the conventional heat capacity is measured and by scanning a wide range of frequencies, the ‘‘specific heat spectrum’’ of a system can be obtained. However the frequency range experimentally accessible is defined by the quasi-adiabatic conditions of the sample holder and is narrow. Spectroscopy measurements of the specific heat that cover several decades of frequencies have been performed using other methods (for a review see [21] and references therein). Another advantage of the AC-method is its ability to detect very small changes in heat capacity and the possibility of measuring heat capacities of the same order of magnitude as the so-called addenda: whole sample holder (sample holder, thermometer, heater) and grease to glue the sample. The sample is placed on a sample holder linked to the baseline by a thermal link of conductance Kb : This sample holder is coupled to a thermometer and heater (for simplicity we consider those thermal links as ideal) (Fig. 1). The principle of the method is that the sample ensemble (sample and sample holder) oscillates in phase and that the AC-power doesn’t escape to the bath through the thermal link during a period. These conditions are achieved if CS =KS ; tint 51=o5ðCS þ CSH Þ=Kb where o ¼ 2pn is the angular frequency, CS and CSH the specific heats of the sample and the whole sample holder and tint represents the internal thermal time constant of the sample holder and the sample. In that case, the amplitude of the temperature modulation is directly related to the total heat capacity C ¼ CS þ CSH in the following way:
where P0 is the AC-heating power at frequency n; dT the amplitude of the temperature oscillation at the same frequency, t1 ¼ ðCS þ CSHq Þ=K ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b the thermal relaxation time to the bath and t2 ¼ t2int þ ðCS =KS Þ2 the internal time constant of the whole (sample + sample holder). The main difficulty in developing an AC steady state device is actually to achieve these conditions.
3. Experimental set-up 3.1. Apparatus
ð1Þ
The apparatus has already been extensively described [22]. It consists in a silicon membrane (2–10 mm thick) linked to the silicon frame by 12 bridges which limit the thermal conductance to the bath, as shown in Fig. 2. This membrane is in silicon which has the advantage of having a high thermal conductivity and a low specific heat at low temperature. This is particularly interesting in specific heat measurements where measurements on the addenda and the sample are carried out at the same time. Other advantages are its good crystalline surface which allows high quality films to be grown on it and its robustness which allows crystals to be stuck to it. A thin film heater in CuNi (or in Cu for temperatures lower than 1 K) and a thin film thermometer in NbN are directly deposited on the membrane. The connecting wires are in NbTi superconductor to limit the thermal leakage to the bath. A thin film of Pt is used to achieve a good interface with the thermometer. This thermometer is deposited by DC-magnetron sputtering in a nitrogen atmosphere. The control of the nitrogen flow through the chamber determines the nitrogen content of the NbN and thus, the slope dR=dT of the thermometer. Different thermometers are deposited depending on
Fig. 1. Schematic diagram of sample coupled to a sample holder by a thermal conductance KS : The sample holder is coupled to a bath (thermal conductance Kb ) and to a thermometer and a heater by ideal thermal links.
Fig. 2. Photograph of the sample-holder. The serpentine is the CuNi heater and the bar is the NbN thermometer. The membrane is suspended by 12 bridges, some of which carry the electrical leads.
C¼
P0 P0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ; dTo 1 2 dTo 1 þ þ ðot Þ 2 ðot1 Þ2
G. Gaudin et al. / Journal of Magnetism and Magnetic Materials 242–245 (2002) 915–920
Fig. 3. Adiabatic plateau of the sample holder and single crystal of Fe8 at three different temperatures.
the temperature range explored. They are calibrated with a commercial Ge thermometer fixed on the copper holder. The advantage of using NbN is that it is only weakly affected by magnetic fields.
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Fig. 4. Heat capacity of a 1:7mg Fe8 single crystal as a function of an applied magnetic field for different temperatures and fixed frequency. The inset shows the result of the calculation for T ¼ 1:87 K:
The heat capacity of the whole sample holder is typically 3 nJ/K at 4.2 K and 0.5 nJ/K at 1.5 K. The resolution is DC=C ¼ 103 : The apparatus can be used in a He-pumped calorimeter from 1.5 to 20 K with a magnetic field up to 5 T and in a dilution refrigerator from 25 mK to 2 K with a magnetic field up to 0.7 T. The frequency range defined by the quasi-adiabatic conditions can be determined experimentally. According to Eq. (1), dTo versus the frequency can be plotted to observe below which frequencies the heat escapes through the thermal link to the bath within the measuring period, and above which frequencies the substrate cannot follow the rapid heat modulation. Between both cut-off frequencies, dTo remains constant so that the measured heat capacity does not depend on frequency, i.e., steady state conditions are achieved and the sample is adiabatically measured. We can notice that this frequency range, the adiabatic plateau, is not very sensitive to temperature down to 1.5 K (Fig. 3).
curves do not depend on the history of the sample and are perfectly symmetric. This allows us to verify the validity of anomalies in CðHÞ curves which have to be symmetric for H > 0 and o0: However, if the thermalization time is too short, the CðHÞ curves are asymmetric and discontinuities at discrete field values are observed when the field is applied antiparallel to the initial magnetization. The discontinuities strongly depend on the history of the sample. Fig. 4 shows C½H measurements at different temperatures for the same frequency 16.64 Hz. A symmetric pattern of peaks appears at field values of 70:265 T; 70:385 T; 70:53 T: The shape of these anomalies does not depend on the amplitude of the temperature oscillation in the studied temperature range (2 mKrDTr20 mK). For a given frequency, the height of the peaks increases when the temperature increases, then they disappear in a broad maximum. This broad maximum shifts to higher field values when the temperature is raised. If the temperature is kept constant and the frequency is changed, the description is quite similar: increasing (decreasing) the frequency at fixed temperature has the same consequence as decreasing (increasing) the temperature at fixed frequency.
4. Results
5. Interpretation
Two different samples of Fe8 (mass 1.7 and 0:4 mg) have been measured. These crystals are pasted with Apiezon grease on the back side of the membrane, just behind the thermometer. This position allows us to know the real temperature of the sample. At every magnetic field step (from 1:2 to 1.2 T here), thermalization of the sample is achieved before recording the heat capacity and the sample temperature values. The
The Fe8 system can be described by the following Hamiltonian X Si Hi H ¼ DSz2 þ EðSx2 Sy2 Þ þ gmB
3.2. Specifications
i 4 4 þ CðSþ þ S Þ
ð2Þ
where E ¼ 0:046 K; D ¼ 0:275 K and C ¼ 8:56:106 K [2,23], gE2 is the gyromagnetic factor and S ¼ 10 is the
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cluster spin. The magnetic levels can be sketched by a parabolic energy barrier separating two wells. The magnetization relaxation time t is the typical time for a spin to go from one well to the other. It has to be compared to 1=o to interpret the C½H curves at different temperatures. If t is much greater than the period of measurement, i.e. if otb1; the spins have not enough time to reverse and remain in the same potential well. This regime will be called the ‘‘unilateral’’ regime hereafter. The specific heat of this regime is the sum of the thermodynamic specific heats of the two wells independent of each other. On the other hand, if ot51; the spin states in both wells are in thermal equilibrium. We will call this regime the ‘‘bilateral’’ regime and its specific heat is the thermodynamic specific heat calculated with all the energy levels of both wells. Between these two regimes, the spins are blocked in one of the two wells, except when magnetic levels on both sides of the barrier cross together (we will call resonance fields, the fields which add a Zeeman energy which align magnetic levels on both sides of the barrier). The tunneling of a spin through the barrier is possible and the relaxation time suddenly decreases. More magnetic levels are then accessible for the spin and there is a peak in the specific heat. According to this qualitative explanation, we can compare the position of the peaks with the resonance fields. The peaks at 0:265 and 0:53 T correspond to the first and second resonances with an angle of 401 between the easy axis of the crystal and the applied magnetic field. This value of angle agrees with the estimation based on an SEM image. Between these two peaks another peak (so-called peak II) exists in the specific heat measurements that cannot correspond to a resonance field. Measurements of other Fe8 crystals with a different angle between the magneto crystalline
Fig. 5. Heat capacity of a 0:4mg Fe8 single crystal as a function of applied magnetic field. Cuni and Cbil are calculated specific heat in the unilateral and bilateral regimes respectively. The inset shows tðHÞ determined by the specific heat measurements and Eq. (4) for T ¼ 2:15 K:
easy axis and the applied magnetic field show the same behaviour (cf. Fig. 5). The angle between the easy axis of that crystal and the applied magnetic field is 151 in agreement with an SEM image. The magnetic steps for that curve are smaller than those of Fig. 4 and we can see that peak II has a complex structure. This structure is symmetric about the magnetic field. 5.1. Thermodynamic study The bilateral regime corresponds to the thermodynamic regime (t-0) because the system has enough time to reach its equilibrium. The specific heat is thus given by P @U d i Ei expðbEi Þ P C¼ ; ð3Þ ¼ kB @T P dT i expðbEi Þ where b ¼ 1=kB T; i represents the magnetic levels (S ¼ 10 so 21 levels here) and Ei are the energies of the system obtained by diagonalysing (2). We access this regime at fixed frequency by increasing the temperature. Fig. 6 shows the specific heat measured in this regime and that calculated with Eq. (3). We use the same angle as the one determined with the position of the peaks. The only free parameters are the mass of the sample and the specific heat of the phonons. The mass is 1:7 mg and is constant for all the simulations in the present work. We obtain a generalized Schottky curve. The maximum can be qualitatively explained as the energy difference mB DH between the level m ¼ þ10 and 10 so that mB DH ¼ kB T where T is the temperature of the sample. This is only a qualitative picture and the experimental curves are quantitatively described only by considering all the levels. In that regime, our measurements enable us to access the thermodynamic specific heat and to check the parameters of the Hamiltonian determined by EPR and neutron measurements [2,23].
Fig. 6. Heat capacity of a 1:7mg Fe8 single crystal as a function of an applied magnetic field in the bilateral regime. The line represents the heat capacity calculated with Eq. (2).
G. Gaudin et al. / Journal of Magnetism and Magnetic Materials 242–245 (2002) 915–920
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5.2. Dynamic study The unilateral specific heat is calculated by considering that there are no possible transitions between the two wells. That corresponds to the limit t-N: This specific heat decreases when jHj increases because the distance between the levels in each well increases. The specific heat measurements we performed correspond to a finite t: The experimental curves lie between the calculated curves corresponding to the unilateral and bilateral regimes (cf. Fig. 5). If we consider a unique relaxation time t from one well to the other, and C defined within the linear response theory, the measured magnetic specific heat can be related to t and the specific heats Cbil and Cuni in the bilateral and unilateral regime through [10] sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 þ Cbil Cuni ; jCðoÞj ¼ Cuni 2 1 þ o t2
Fig. 7. Heat emission of an Fe8 crystal with 401 between its easy axis and the applied magnetic field. The temperature of the baseline is 700 mK and different sweeping rates have been used. A schematic view of the phenomena is shown in the inset.
ð4Þ
where C is the magnetic specific heat, corrected for the phonon contribution and the addenda. This contribution can be found by noting that Cbil ðH ¼ 0Þ ¼ Cuni ðH ¼ 0Þ ¼ Cmagn ðo; H ¼ 0Þ whatever o: When H ¼ 0; the specific heat is independent of the frequency because considering the two wells with or without any transition between them does not change the specific heat. The magnetic relaxation time t½H can also be calculated from the measurements except for H ¼ 0 (cf. Fig. 5). For all other fields, t decreases globally when jHj increases because the barrier is classically lowered. Note that t suddenly decreases at the resonance field. The spin has also another possible way to cross the energy barrier. This is the signature of quantum tunneling of the magnetization through this barrier. Thus, our measurements enable us to access the dynamics of this quantum phenomena through the determination of the relaxation time. This time t can be compared with the results of magnetic measurements. It can also allow us to study the interplay between the spin system and the lattice of the crystal. Numerical calculations have been made [24], based on the master equation. Exact eigenstates of the Hamiltonian have been considered as for Mn12 in Ref. [12]. Transitions between these states are given by the spin-phonon coupling allowed by the D2 symmetry of the Fe8 crystal. Resonance widths have been broadened by a gaussian distribution of dipolar field (0.03 T in width, in agreement with [25]). We obtain the frequency dependent specific heat that can be compared to experimental measurements. An example of this comparison is given in the insert of Fig. 4. The agreement is excellent except for the additional peak that can not be described by this model because it is not included in the Hamiltonian (2).
A complete description of the model, the results and the physical discussion will be given elsewhere [24]. In order to know if peak II is due to a distribution of energy barriers inside Fe8 or to the presence of two crystals with different orientations in our sample, we performed heat emission measurements. The temperature of the sample is monitored while the magnetic field is swept. The temperature of the baseline is kept constant and the sample is in its demagnetized state, then the magnetic field is swept at different rates. Fig. 7 shows the heat emission recorded for a baseline temperature of 700 mK. The anomaly around H ¼ 50 mT is due to the sample holder. Heat emissions at resonance field are clearly evidenced. They are due to the relaxation of the spin to the fundamental after tunneling through the barrier as depicted in the inset. The temperature of the sample increases with the field rate but no avalanche effect, as observed in Mn12 [9,16], can be seen. The important point is that there is no indication of peak II. This peak does not appear to be due to a defect of the crystal. Moreover, other peaks appear by comparing numerical calculations and experimental results [24]. These anomalies are probably due to transitions in the S ¼ 9 multiplet that have to be taken into account in the description of the thermally activated regime even for temperatures around 2 K.
6. Conclusions Our nanocalorimeter has proved to be a powerful tool for performing specific heat measurements on single crystals of some mg with high resolution. It allows us to measure the thermodynamic specific heat and so to check the Hamiltonian parameters. Measurements of the frequency dependent specific heat permit the study
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of the dynamics of the system, not only due to magnetic, electrical or structural changes but to all modes of the system. C½H measurements have been performed on Fe8 single crystals. They exhibit anomalies at resonance fields that correspond to the tunneling of the magnetization. t½H can be determined from these measurements. Comparison between calculated C½H and experimental results outlines the role of S ¼ 9 multiplet in the thermally activated regime even at low temperature.
Acknowledgements We gratefully acknowledge J. Villain and W. Wernsdorfer for fruitful discussions and Th. Fournier for his technical assistance.
References [1] M.N. Leuenberger, D. Loss, Nature 410 (2001) 789. [2] A.-L. Barra, P. Debrunner, D. Gatteschi, Ch.E. Schulz, R. Sessoli, Europhys. Lett. 35 (1996) 133. [3] C. Sangregorio, T. Ohm, C. Paulsen, R. Sessoli, D. Gatteschi, Phys. Rev. Lett. 78 (1997) 4645. [4] C. Paulsen, J.-G. Park, Quantum Tunneling of the Magnetization, in: L. Gunther, B. Barbara (Eds.), NATO ASI Series E, Vol. 301, Kuwer, Dordrecht, 1995. [5] M.A. Novak, R. Sessoli, Quantum Tunneling of the Magnetization, in: L. Gunther, B. Barbara (Eds.), NATO ASI Series E, Vol. 301, Kuwer, Dordrecht, 1995. [6] L. Thomas, F. Lionti, R. Ballou, D. Gatteschi, R. Sessoli, B. Barbara, Letters to Nature, 383 1996 (145). [7] J.R. Friedman, M.P. Sarachik, J. Tejada, R. Ziolo, Phys. Rev. Lett. 76 (1996) 3830.
[8] J. M. Hern!andez, X.X. Zhang, F. Luis, J. Bartholom!e, J. Tejada, R. Ziolo, Europhys. Lett. 35 (1996) 301. [9] F. Fominaya, J. Villain, P. Gandit, J. Chaussy, A. Caneschi, Phys. Rev. Lett. 79 (1997) 1126. [10] F. Fominaya, J. Villain, T. Fournier, P. Gandit, J. Chaussy, A. Fort, A. Caneschi, Phys. Rev. B 59 (1999) 519. [11] F. Hartmann-Boutron, P. Politi, J. Villain, Int. J. Mod. Phys. 10 (1996) 2577. [12] F. Luis, J. Bartolom!e, J.F. Fern!andez, Phys. Rev. B 57 (1998) 505; J.F. Fern!andez, F. Luis, J. Bartolom!e, Phys. Rev. Lett. 80 (1998) 5659. [13] D.A. Garanin, E.M. Chudnovsky, Phys. Rev. B 56 (1997) 11102. [14] A. Fort, A. Rettori, J. Villain, D. Gatteschi, R. Sessoli, Phys. Rev. Lett. 80 (1998) 612. [15] M.N. Leuenberger, D. Loss, Phys. Rev. B 61 (2000) 1286. [16] E. del Barco, J.M. Hernandez, M. Sales, J. Tejada, H. Rakoto, J.M. Broto, E.M. Chudnovsky, Phys. Rev. B 60 (1999) 11898. [17] A.M. Gomes, M.A. Novak, R. Sessoli, A. Caneschi, D. Gatteschi, Phys. Rev. B 57 (1998) 5021. [18] A.M. Gomes, M.A. Novak, W.C. Nunes, R.E. Rapp, Condens Matter (1999) 9912224; A.M. Gomes, M.A. Novak, W.C. Nunes, R.E. Rapp, J. Magn. Magn. Mater. 226–230 (2001) 2015. [19] P.F. Sullivan, G. Seidel, Phys. Rev. 173 (1968) 679. [20] J.K. Nielsen, J.C. Dyre, Phys. Rev. B 54 (1996) 15754. [21] E. Gmelin, Thermochim. Acta 304/305 (1997) 1. [22] F. Fominaya, T. Fournier, P. Gandit, J. Chaussy, Rev. Sci. Instrum. 68 (1997) 4191. [23] R. Caciuffo, G. Amoretti, A. Murani, Phys. Rev. Lett. 81 (1998) 4744. [24] G. Gaudin, Ph. Gandit, J. Chaussy, R. Sessoli, to be published. [25] W. Wernsdorfer, T. Ohm, C. Sangregorio, R. Sessoli, D. Mailly, C. Paulsen, Phys. Rev. Lett. 82 (1999) 3903.
Invited paper
Magnetic field dependent thermodynamics of Fe8 single crystal in the thermally activated regime G. Gaudina, P. Gandita,*, J. Chaussya, R. Sessolib a
CRTBT-CNRS, 25 av. des Martyrs, BP166, F - 38042 Grenoble Cedex 9, France b Dep. of Chemistry, Univ. Firenze, Via Maragliano 77, 550144 Firenze, Italy
Abstract Quantum tunneling of the magnetization in the thermally activated regime has been studied on Fe8 single crystals by frequency dependent specific heat measurements as a function of applied magnetic field. Curves are very well described by a microscopic theory based on phonon assisted spin tunneling. However, an additional peak appears that can be due to transitions in higher multiplets. r 2002 Elsevier Science B.V. All rights reserved. Keywords: MagnetizationFquantum tunneling; Specific heatFlow temperature; Spin–phonon interactions; Magnetic molecular materials
1. Introduction Magnetic quantum tunneling of large spins is one of the phenomena that have aroused much interest recently both for fundamental science and for future applications, for instance in quantum computing [1]. Magnetic molecular clusters are among the most promising candidates for observing it. In fact, they permit macroscopic observation of single particle properties, avoiding complications due to size and orientation distributions existing in traditional small magnetic particle systems. Fe8 clusters ð½ðtacnÞ6 Fe8 O2 ðOHÞ12 8þ Þ with a spin ground state of S ¼ 10 [2] are among the most promising candidates. Below 360 mK, the magnetization relaxes through a pure tunneling process [3]. At higher temperature, the tunneling becomes thermally assisted. This regime has been widely studied for another molecular cluster, Mn12 ; both experimentally [4–10] and theoretically [11–15] but very few references mention measurements in that regime for Fe8 : Specific heat measurements that have been performed at low temperature on both Mn12 and Fe8 [17,18] give a great deal of information such as the contribution of *Corresponding author. Tel.: +33-4-76-889069; Fax: +33-476-875060. E-mail address: [email protected] (P. Gandit).
hyperfine and dipolar fields. However the specific heat is a thermodynamic quantity defined at the equilibrium and the tunneling of a spin is a dynamic process. The study of the interplay between the spin system and the crystal lattice, e.g. clarifying how the phonons influence the macrospin tunneling, can only be done through dynamic measurements such as the frequency dependent specific heat measurements that will be presented. An original nanocalorimeter [22] capable of measuring with very high resolution the specific heat of very small samples (mg) will be presented. It permits the qualitative and quantitative study of the quantum tunneling of the magnetization in the thermally activated regime through measurements performed on Fe8 single crystals.
2. Frequency dependent specific heat The idea of measuring the ‘‘specific heat spectrum’’ of a system began with the development of the ac-steady state method [19]. In that method, an alternative current of frequency n=2 passes in a heater, producing a smallamplitude sinusoidal heat flux to the sample. The frequency dependent specific heat is calculated with the amplitude of the temperature oscillation at frequency n measured by a thermometer. This frequency dependent specific heat is a linear susceptibility describing the
0304-8853/01/$ - see front matter r 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 0 4 - 8 8 5 3 ( 0 1 ) 0 1 3 2 3 - 3
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G. Gaudin et al. / Journal of Magnetism and Magnetic Materials 242–245 (2002) 915–920
response of the system to arbitrarily small perturbations away from equilibrium and can be related to the temporal fluctuations of the enthalpy [20]. In the limit n-0; the conventional heat capacity is measured and by scanning a wide range of frequencies, the ‘‘specific heat spectrum’’ of a system can be obtained. However the frequency range experimentally accessible is defined by the quasi-adiabatic conditions of the sample holder and is narrow. Spectroscopy measurements of the specific heat that cover several decades of frequencies have been performed using other methods (for a review see [21] and references therein). Another advantage of the AC-method is its ability to detect very small changes in heat capacity and the possibility of measuring heat capacities of the same order of magnitude as the so-called addenda: whole sample holder (sample holder, thermometer, heater) and grease to glue the sample. The sample is placed on a sample holder linked to the baseline by a thermal link of conductance Kb : This sample holder is coupled to a thermometer and heater (for simplicity we consider those thermal links as ideal) (Fig. 1). The principle of the method is that the sample ensemble (sample and sample holder) oscillates in phase and that the AC-power doesn’t escape to the bath through the thermal link during a period. These conditions are achieved if CS =KS ; tint 51=o5ðCS þ CSH Þ=Kb where o ¼ 2pn is the angular frequency, CS and CSH the specific heats of the sample and the whole sample holder and tint represents the internal thermal time constant of the sample holder and the sample. In that case, the amplitude of the temperature modulation is directly related to the total heat capacity C ¼ CS þ CSH in the following way:
where P0 is the AC-heating power at frequency n; dT the amplitude of the temperature oscillation at the same frequency, t1 ¼ ðCS þ CSHq Þ=K ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi b the thermal relaxation time to the bath and t2 ¼ t2int þ ðCS =KS Þ2 the internal time constant of the whole (sample + sample holder). The main difficulty in developing an AC steady state device is actually to achieve these conditions.
3. Experimental set-up 3.1. Apparatus
ð1Þ
The apparatus has already been extensively described [22]. It consists in a silicon membrane (2–10 mm thick) linked to the silicon frame by 12 bridges which limit the thermal conductance to the bath, as shown in Fig. 2. This membrane is in silicon which has the advantage of having a high thermal conductivity and a low specific heat at low temperature. This is particularly interesting in specific heat measurements where measurements on the addenda and the sample are carried out at the same time. Other advantages are its good crystalline surface which allows high quality films to be grown on it and its robustness which allows crystals to be stuck to it. A thin film heater in CuNi (or in Cu for temperatures lower than 1 K) and a thin film thermometer in NbN are directly deposited on the membrane. The connecting wires are in NbTi superconductor to limit the thermal leakage to the bath. A thin film of Pt is used to achieve a good interface with the thermometer. This thermometer is deposited by DC-magnetron sputtering in a nitrogen atmosphere. The control of the nitrogen flow through the chamber determines the nitrogen content of the NbN and thus, the slope dR=dT of the thermometer. Different thermometers are deposited depending on
Fig. 1. Schematic diagram of sample coupled to a sample holder by a thermal conductance KS : The sample holder is coupled to a bath (thermal conductance Kb ) and to a thermometer and a heater by ideal thermal links.
Fig. 2. Photograph of the sample-holder. The serpentine is the CuNi heater and the bar is the NbN thermometer. The membrane is suspended by 12 bridges, some of which carry the electrical leads.
C¼
P0 P0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ ; dTo 1 2 dTo 1 þ þ ðot Þ 2 ðot1 Þ2
G. Gaudin et al. / Journal of Magnetism and Magnetic Materials 242–245 (2002) 915–920
Fig. 3. Adiabatic plateau of the sample holder and single crystal of Fe8 at three different temperatures.
the temperature range explored. They are calibrated with a commercial Ge thermometer fixed on the copper holder. The advantage of using NbN is that it is only weakly affected by magnetic fields.
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Fig. 4. Heat capacity of a 1:7mg Fe8 single crystal as a function of an applied magnetic field for different temperatures and fixed frequency. The inset shows the result of the calculation for T ¼ 1:87 K:
The heat capacity of the whole sample holder is typically 3 nJ/K at 4.2 K and 0.5 nJ/K at 1.5 K. The resolution is DC=C ¼ 103 : The apparatus can be used in a He-pumped calorimeter from 1.5 to 20 K with a magnetic field up to 5 T and in a dilution refrigerator from 25 mK to 2 K with a magnetic field up to 0.7 T. The frequency range defined by the quasi-adiabatic conditions can be determined experimentally. According to Eq. (1), dTo versus the frequency can be plotted to observe below which frequencies the heat escapes through the thermal link to the bath within the measuring period, and above which frequencies the substrate cannot follow the rapid heat modulation. Between both cut-off frequencies, dTo remains constant so that the measured heat capacity does not depend on frequency, i.e., steady state conditions are achieved and the sample is adiabatically measured. We can notice that this frequency range, the adiabatic plateau, is not very sensitive to temperature down to 1.5 K (Fig. 3).
curves do not depend on the history of the sample and are perfectly symmetric. This allows us to verify the validity of anomalies in CðHÞ curves which have to be symmetric for H > 0 and o0: However, if the thermalization time is too short, the CðHÞ curves are asymmetric and discontinuities at discrete field values are observed when the field is applied antiparallel to the initial magnetization. The discontinuities strongly depend on the history of the sample. Fig. 4 shows C½H measurements at different temperatures for the same frequency 16.64 Hz. A symmetric pattern of peaks appears at field values of 70:265 T; 70:385 T; 70:53 T: The shape of these anomalies does not depend on the amplitude of the temperature oscillation in the studied temperature range (2 mKrDTr20 mK). For a given frequency, the height of the peaks increases when the temperature increases, then they disappear in a broad maximum. This broad maximum shifts to higher field values when the temperature is raised. If the temperature is kept constant and the frequency is changed, the description is quite similar: increasing (decreasing) the frequency at fixed temperature has the same consequence as decreasing (increasing) the temperature at fixed frequency.
4. Results
5. Interpretation
Two different samples of Fe8 (mass 1.7 and 0:4 mg) have been measured. These crystals are pasted with Apiezon grease on the back side of the membrane, just behind the thermometer. This position allows us to know the real temperature of the sample. At every magnetic field step (from 1:2 to 1.2 T here), thermalization of the sample is achieved before recording the heat capacity and the sample temperature values. The
The Fe8 system can be described by the following Hamiltonian X Si Hi H ¼ DSz2 þ EðSx2 Sy2 Þ þ gmB
3.2. Specifications
i 4 4 þ CðSþ þ S Þ
ð2Þ
where E ¼ 0:046 K; D ¼ 0:275 K and C ¼ 8:56:106 K [2,23], gE2 is the gyromagnetic factor and S ¼ 10 is the
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cluster spin. The magnetic levels can be sketched by a parabolic energy barrier separating two wells. The magnetization relaxation time t is the typical time for a spin to go from one well to the other. It has to be compared to 1=o to interpret the C½H curves at different temperatures. If t is much greater than the period of measurement, i.e. if otb1; the spins have not enough time to reverse and remain in the same potential well. This regime will be called the ‘‘unilateral’’ regime hereafter. The specific heat of this regime is the sum of the thermodynamic specific heats of the two wells independent of each other. On the other hand, if ot51; the spin states in both wells are in thermal equilibrium. We will call this regime the ‘‘bilateral’’ regime and its specific heat is the thermodynamic specific heat calculated with all the energy levels of both wells. Between these two regimes, the spins are blocked in one of the two wells, except when magnetic levels on both sides of the barrier cross together (we will call resonance fields, the fields which add a Zeeman energy which align magnetic levels on both sides of the barrier). The tunneling of a spin through the barrier is possible and the relaxation time suddenly decreases. More magnetic levels are then accessible for the spin and there is a peak in the specific heat. According to this qualitative explanation, we can compare the position of the peaks with the resonance fields. The peaks at 0:265 and 0:53 T correspond to the first and second resonances with an angle of 401 between the easy axis of the crystal and the applied magnetic field. This value of angle agrees with the estimation based on an SEM image. Between these two peaks another peak (so-called peak II) exists in the specific heat measurements that cannot correspond to a resonance field. Measurements of other Fe8 crystals with a different angle between the magneto crystalline
Fig. 5. Heat capacity of a 0:4mg Fe8 single crystal as a function of applied magnetic field. Cuni and Cbil are calculated specific heat in the unilateral and bilateral regimes respectively. The inset shows tðHÞ determined by the specific heat measurements and Eq. (4) for T ¼ 2:15 K:
easy axis and the applied magnetic field show the same behaviour (cf. Fig. 5). The angle between the easy axis of that crystal and the applied magnetic field is 151 in agreement with an SEM image. The magnetic steps for that curve are smaller than those of Fig. 4 and we can see that peak II has a complex structure. This structure is symmetric about the magnetic field. 5.1. Thermodynamic study The bilateral regime corresponds to the thermodynamic regime (t-0) because the system has enough time to reach its equilibrium. The specific heat is thus given by P @U d i Ei expðbEi Þ P C¼ ; ð3Þ ¼ kB @T P dT i expðbEi Þ where b ¼ 1=kB T; i represents the magnetic levels (S ¼ 10 so 21 levels here) and Ei are the energies of the system obtained by diagonalysing (2). We access this regime at fixed frequency by increasing the temperature. Fig. 6 shows the specific heat measured in this regime and that calculated with Eq. (3). We use the same angle as the one determined with the position of the peaks. The only free parameters are the mass of the sample and the specific heat of the phonons. The mass is 1:7 mg and is constant for all the simulations in the present work. We obtain a generalized Schottky curve. The maximum can be qualitatively explained as the energy difference mB DH between the level m ¼ þ10 and 10 so that mB DH ¼ kB T where T is the temperature of the sample. This is only a qualitative picture and the experimental curves are quantitatively described only by considering all the levels. In that regime, our measurements enable us to access the thermodynamic specific heat and to check the parameters of the Hamiltonian determined by EPR and neutron measurements [2,23].
Fig. 6. Heat capacity of a 1:7mg Fe8 single crystal as a function of an applied magnetic field in the bilateral regime. The line represents the heat capacity calculated with Eq. (2).
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5.2. Dynamic study The unilateral specific heat is calculated by considering that there are no possible transitions between the two wells. That corresponds to the limit t-N: This specific heat decreases when jHj increases because the distance between the levels in each well increases. The specific heat measurements we performed correspond to a finite t: The experimental curves lie between the calculated curves corresponding to the unilateral and bilateral regimes (cf. Fig. 5). If we consider a unique relaxation time t from one well to the other, and C defined within the linear response theory, the measured magnetic specific heat can be related to t and the specific heats Cbil and Cuni in the bilateral and unilateral regime through [10] sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 þ Cbil Cuni ; jCðoÞj ¼ Cuni 2 1 þ o t2
Fig. 7. Heat emission of an Fe8 crystal with 401 between its easy axis and the applied magnetic field. The temperature of the baseline is 700 mK and different sweeping rates have been used. A schematic view of the phenomena is shown in the inset.
ð4Þ
where C is the magnetic specific heat, corrected for the phonon contribution and the addenda. This contribution can be found by noting that Cbil ðH ¼ 0Þ ¼ Cuni ðH ¼ 0Þ ¼ Cmagn ðo; H ¼ 0Þ whatever o: When H ¼ 0; the specific heat is independent of the frequency because considering the two wells with or without any transition between them does not change the specific heat. The magnetic relaxation time t½H can also be calculated from the measurements except for H ¼ 0 (cf. Fig. 5). For all other fields, t decreases globally when jHj increases because the barrier is classically lowered. Note that t suddenly decreases at the resonance field. The spin has also another possible way to cross the energy barrier. This is the signature of quantum tunneling of the magnetization through this barrier. Thus, our measurements enable us to access the dynamics of this quantum phenomena through the determination of the relaxation time. This time t can be compared with the results of magnetic measurements. It can also allow us to study the interplay between the spin system and the lattice of the crystal. Numerical calculations have been made [24], based on the master equation. Exact eigenstates of the Hamiltonian have been considered as for Mn12 in Ref. [12]. Transitions between these states are given by the spin-phonon coupling allowed by the D2 symmetry of the Fe8 crystal. Resonance widths have been broadened by a gaussian distribution of dipolar field (0.03 T in width, in agreement with [25]). We obtain the frequency dependent specific heat that can be compared to experimental measurements. An example of this comparison is given in the insert of Fig. 4. The agreement is excellent except for the additional peak that can not be described by this model because it is not included in the Hamiltonian (2).
A complete description of the model, the results and the physical discussion will be given elsewhere [24]. In order to know if peak II is due to a distribution of energy barriers inside Fe8 or to the presence of two crystals with different orientations in our sample, we performed heat emission measurements. The temperature of the sample is monitored while the magnetic field is swept. The temperature of the baseline is kept constant and the sample is in its demagnetized state, then the magnetic field is swept at different rates. Fig. 7 shows the heat emission recorded for a baseline temperature of 700 mK. The anomaly around H ¼ 50 mT is due to the sample holder. Heat emissions at resonance field are clearly evidenced. They are due to the relaxation of the spin to the fundamental after tunneling through the barrier as depicted in the inset. The temperature of the sample increases with the field rate but no avalanche effect, as observed in Mn12 [9,16], can be seen. The important point is that there is no indication of peak II. This peak does not appear to be due to a defect of the crystal. Moreover, other peaks appear by comparing numerical calculations and experimental results [24]. These anomalies are probably due to transitions in the S ¼ 9 multiplet that have to be taken into account in the description of the thermally activated regime even for temperatures around 2 K.
6. Conclusions Our nanocalorimeter has proved to be a powerful tool for performing specific heat measurements on single crystals of some mg with high resolution. It allows us to measure the thermodynamic specific heat and so to check the Hamiltonian parameters. Measurements of the frequency dependent specific heat permit the study
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of the dynamics of the system, not only due to magnetic, electrical or structural changes but to all modes of the system. C½H measurements have been performed on Fe8 single crystals. They exhibit anomalies at resonance fields that correspond to the tunneling of the magnetization. t½H can be determined from these measurements. Comparison between calculated C½H and experimental results outlines the role of S ¼ 9 multiplet in the thermally activated regime even at low temperature.
Acknowledgements We gratefully acknowledge J. Villain and W. Wernsdorfer for fruitful discussions and Th. Fournier for his technical assistance.
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