Temperature changes of the Fe_{8} molecular magnet during its spin reversal process
TTemperature changes of the Fe molecular magnet during its spin reversal process Maayan Yaari and Amit Keren Department of Physics, Technion - Israel Institute of Technology, Haifa, 32000, Israel (Dated: November 11, 2018)Tunneling of the spins in the Fe molecular magnet from a metastable ground state to an excitedstate is accompanied by a decay of these spins to the global ground state, and an increase of thecrystal temperature. We measured this temperature using two thermometers, one strongly coupledand the other weakly coupled to the thermal bath. We found that the temperature increases to nogreater than 2 . crystals. The Fe single molecular magnet is an exciting systemto study since its dynamics are fully quantum mechanicalbelow a temperature of 400 mK . This molecule hasa spin of S = 10, and accounting for the crystal-fieldtogether with the spin-orbit interaction, it is governedby the spin Hamiltonian : H = DS z + H ⊥ + gµ B (cid:126)S · (cid:126)H (1)where the dominating S z term with D = − .
295 K givesrise to an anisotropy barrier . The H ⊥ term is respon-sible for the mixing of spin states and tunneling betweenthem. The Zeeman term removes the degeneracy be-tween S z = ± m and allows the spins of all moleculesto align at sufficiently low temperatures. Upon sweep-ing of the magnetic field from H to − H , the samplesmagnetization versus field curve exhibits a staircase hys-teresis loop . This is attributed to quantum tunnelingbetween magnetization states, which is only allowed fordiscrete ’matching fields’ corresponding to level cross-ings . The matching fields for transitions between thestates m to m (cid:48) are given by: H n = Dn/gµ B (cid:39) . n (2)where n = m + m (cid:48) .Due to H ⊥ the level crossing is in fact an avoided cross-ing with a tunnel splitting ∆ mm (cid:48) between the m and m (cid:48) levels. According to the Landau, Zener and Stuckelberg[LZS] solution of the time dependent Schr¨odingerequation for a multistate system, the probability for tran-sition between two states, when the external field is in thevicinity of a matching field is given by: P mm (cid:48) = 1 − exp ( − π ∆ mm (cid:48) gµ B ( m − m (cid:48) ) α B ) , (3)where for an isolated spin Correspondence should be addressed to A.K. (email:[email protected]) α B = α H = µ dH z dt . (4)However, upon sweeping the field at low temperatures,the transitions occur between a metastable spin state (say m = − m = 10) or an ex-cited state (e.g. m = 9). We name these transitionsaccording to their n value. For n ≥ → n = 0, namely ±
10 to ∓
10. Therefore, toproperly account for the tunneling probability of generalmolecular magnets embedded in a crystal and Fe in par-ticular, it is essential to determine how hot the crystalgets after a tunneling event that is followed by an energyrelease. This is the main objective of the experimentreported here.Our experiment is done using a sorption pumped HeOxford Instruments refrigerator. For each sample twoRuO resistance thermometers (thermistors) are glued tothe sample with super glue. One of the thermistors is an-chored to the refrigerator cold-finger with a copper beryl-lium spring; we refer to it as the cold thermistor since itis strongly coupled to the cold finger. The other ther-mistor is anchored to a teflon bar using a similar spring.The teflon bar, in turn, is attached to the cold refriger-ator finger; this is the hot thermistor since it is weaklycoupled to the cold finger and is expected to warm upmore upon energy release. Before the energy release bythe molecular magnets both thermistors are at the sametemperature. The springs are essential since the facetsof the Fe crystal are not perpendicular to the z direc-tion of the molecules. The springs also allow for thermalshrinking of the apparatus upon cooling without break-ing the crystal. The RuO thermistors are not sensitiveto magnetic fields. The apparatus is depicted in Fig. 1.The thermistors resistivity is measured using the fourwire method; two wires for current and two for voltage. a r X i v : . [ c ond - m a t . m e s - h a ll ] J un However, in order to check if heat leaks through thesecryogenic wires we modified the wiring between differentmeasurements. Sometimes we connected all four wires di-rectly to the thermistor, and sometimes we used only twowires, which were split into four outside of the cryostat.We found that the wiring method has no impact on theconclusions of our work. We calibrated the thermistorsagainst the built-in thermometer of the He refrigeratorwhile slowly cooling it to base temperature.
FIG. 1: The experimental setup which is mounted on a coldfinger and connected to the He refrigerator. This setup isin the center of a magnet with the field pointing along thecrystal z direction. We measured six crystals, which are quite differentfrom one another in shape. Consequently their demag-netization factors and internal fields are not identical.This leads to variations in their behavior. Nevertheless,each crystal shows reproducible data when repeating thesweeps and when reversing the sweep direction. Here wepresent data from two crystals with fundamentally dif-ferent behavior.In Fig. 2 we show the sample temperature as recordedby both thermistors while sweeping the magnetic fieldfrom positive to negative at four different sweep rates.As the sweep begins the cold thermistor immediately be-comes hotter despite its better thermal coupling to therefrigerator. This is due to eddy currents in the coppercold finger and spring. The hot thermistor is hardly af-fected by the sweep at first. As the field approaches zero,there is a temperature rise. This is believed to be dueto the superconducting transition of lead in the solder-ing material as the field crosses the lead H c . Once thefield crosses over to the negative side, the quantum na-ture of the molecule becomes obvious especially, at thehighest sweep rate ( α H = 8 .
33 mT/sec) and for the hotthermistor. In this case we see a clear broad tempera-ture increase that starts at µ H = − . n = 1), and a spike in the temperature at µ H = − .
42 T ( n = 2)with a tail towards higher fields. No tunneling events arenoticed in the cold thermistor for the highest sweep rate.At a lower sweep rate of ( α H = 6 .
66 mT/sec) we as-sociate the spike at µ H = − .
28 T with the n = 1transition. However due to the slow response of the hotthermistor it appears at a slightly higher field. In thiscase the cold thermistor begins to show some tempera-ture increase at µ H = − . n = 2).As we lower the sweep rate further to 5 mT/sec andthen to 3 .
33 mT/sec the response of the cold thermis-tor at µ H = − . n = 2 transition. It is notclear to us why in α H = 6 .
66 mT/sec the largest tem-perature increase is at n = 1 and in the other three casesit is at n = 2. In any case, at the two lowest sweeprates a clear temperature increase is detected in boththermistors. Both spikes are associated with the n = 2transition, but there is a field (hence time) delay betweenthem. The delay in terms of field difference is indepen-dent of the sweep rate, but of course there is a delayin time as α H is varied. This phenomenon is intriguingand we lack an explanation for it. Finally, for all sweeprates the temperature does not exceed 2 K in either ofthe thermistors. - 1 . 0 - 0 . 8 - 0 . 6 - 0 . 4 - 0 . 2 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 a H =5 m T / s e cS a m p l e 1 Temperature (K)
C o l d H o t a H =8.33 m T / s e c a H =6.66 m T / s e c a H =3.33 m T / s e c m H ( T )
FIG. 2: Temperature versus magnetic field for sample 1, usinga four-point probe measurement to determine the resistivityof the thermistor. Each panel shows results for a differentmagnetic field sweep rate α H . Figure 3 represents a special sample where the onlytunneling event detected is at n = 3 for all four sweeprates. In this case both thermistors warm up equally,exactly at the same field (or time). This suggests thatall the spins in the sample flip together. For this sample,after the tunneling, the spins decay from the | m | = 7excited state to the ground state. Each spin releases anenergy of 15 K. This is approximately three times biggerthan the energy release for n = 1 and approximately 1.5times bigger than for n = 2. Yet, the temperature of thesample barely reaches 2 K. In fact, among all the sam-ples we measured the temperature never reached 2 . n -states are detectedupon sweeping of an external field, the temperature doesnot exceed 2 . - 1 . 0 - 0 . 8 - 0 . 6 - 0 . 4 - 0 . 2 0 . 0 0 . 2 0 . 4 0 . 6 0 . 8 1 . 0 m H ( T )
C o l d H o t a H =6.66 m T / s e c a H =5 m T / s e c a H =3.33 m T / s e c Temperature (K)
S a m p l e 2 a H =8.33 m T / s e c FIG. 3: Temperature versus magnetic field for sample 2, us-ing a 2-point probe measurement to determine the resistivityof the thermistor. Each panel shows results for a differentmagnetic field sweep rate α H . It is interesting to compare our finding with the flametemperature derived from magnetic deflagration theoryfor Fe . The theory of deflagration relates the prop-agation velocity of the spin reversal front v f to the heatconductivity κ , the barrier height U , and flame temper- ature T f . The relation is: v f ( H, T f ) = (cid:115) κ ( T f ) τ exp (cid:18) − U ( H )2 k B T f (cid:19) . (5)In Fe , only three transitions are observed. Therefore theeffective barrier height is taken as the energy differencebetween m = −
10 to m = − n = 1 transition where deflagration was found, namely U = 10 K. Additionally, v ∼ m/s and κ ∼ × − m /s are known from previous measurements on partic-ular samples that happen to show deflagration . Thisgives T f = 2 . m = −
10) and the first and second excitedstates ( m = − , −
8) for the n = 1 and n = 2 transitionsare ≈ ≈
10 K, i.e. greater than 2 . n = 1tunneling event. Moreover, not all these excited spinsflip. Therefore, the LZS theory should work well for these n > a H (mT/sec) D M / M (cid:3) (cid:2) (cid:1)(cid:2) H x xx (cid:1) = += M/M0 m H [T ] a H ( m T / s ) = 1 . 6 64 . 1 68 . 3 31 2 . 5 FIG. 4: The magnetic field sweep rate plotted as a function ofthe first magnetization jump n = 1 of Fe . The solid line is afit to Eq. 7 derived in the text. The inset is raw hysteresis loopdata taken from Ref. from which the magnetization jumpsare derived. For the n = 1 jump we suggest the following explana-tion for the discrepancy between the LZS theory and theexperimental result. The magnetic induction experiencedby the spins in the z direction is given by B = µ ( H + M ),where H is the magnetic field and M is the uniform sam-ple magnetization. As we sweep H through a transition, B changes according to dBdt = α H + α M where now only α H = µ dHdt and α M = µ dMdH dHdt . We approximate dMdH by f (∆ M/M )( M / ∆ H ) where ∆ H is the field widthduring which the transition is taking place, ∆ M is themagnetization jump, M is the saturation magnetization,and f is on the order of unity. The reason for introducingthe factor f is that the local variation in the magnetiza-tion could be larger than the global one estimated from∆ M/ ∆ H . ∆ H is independent of sweep rates, thereforewe absorb M / ∆ H into f and write dBdt = α B = α H (1 + f ∆ MM ) . (6)For the first transition ∆ M = 2 M P where ∆ M and P stand for the magnetization jump and transition prob-ability at n = 1, respectively. Substituting Eq. 6 intoEq. 3 and solving for α H one finds the relation α H = − π ∆ gµ B (19)(1 + f ∆ M M ) ln(1 + ∆ M M ) . (7)In Fig. 4 we present the sweep rates as a function ofthe normalized magnetization jumps for n = 1. Theinset shows raw data taken from Leviant work . Dueto the limited number of data points the fit parametersare determined with large error bars. We therefore onlydemonstrate here that there is a set of parameters whichcapture the data points reasonably well, and estimatethe value of the tunnel splitting roughly to be ∆ − , ≈ . · − K, which is in good agreement with previouswork . The factor f is indeed on the order of unity.As for the n ≥ n magneti-zation jumps is outside of the scope of the LZS theory.We speculate that when the n ≥ molecular magnet tunnels from a metastable groundstate to an excited state and from there to the stableground state, the temperature does not increase above2 . n ≥ Acknowledgment
We thank Lev Melnikovsky for helpful discussions.This study was partially supported by the Russell BerrieNanotechnology Institute, Technion, Israel Institute ofTechnology. C. Sangregorio, T. Ohm, C. Paulsen, R. Sessoli, and D.Gatteschi,
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