F. Branzoli¶, P. Carretta¶, M. Filibian¶ , S. Klytaksaya‡ and M. Ruben‡

¶Department of Physics "A.Volta", University of Pavia-CNISM, Via Bassi 6, I-27100, Pavia (Italy)

‡Institute of Nanotechnology, Forschungszentrum, Hermann-von-Helmholtz-Platz 1, 76344 Eggenstein-Leopoldshafen (Germany)

Low energy Low energy excitations in the excitations in the neutral [LnPcneutral [LnPc22]]00 single molecule single molecule

magnets from magnets from μμSR SR and and NMRNMR

• New class of Single Molecule MagnetsSingle Molecule Magnets (SMM) [1];

• Highest magnetic anisotropymagnetic anisotropy ever achieved in a molecular magnet [2];

• Slow relaxationSlow relaxation and quantum tunnelingquantum tunneling of magnetization;

• Energy separation ∆Energy separation ∆ between the double degenerate ground state and the first excited levels of several hundreds of kelvin;

• Correlation timeCorrelation time ((ττcc)) for the spin fluctuations reaching several µs at liquid nitrogen temperatures ;

• Applications:Applications: - molecular spintronic devices [3]- logic units in quantum computers [4]- contrast agents [5].

Fig. 1: Schematic rapresentation of the TbPc2 molecule. On the left, the energy level succession of Tb3+ ion ground multiplet.

[LnPc[LnPc22 ] ]0 0 SMM:SMM:

• The time decay of the muon polarization Pmuon polarization Pµµ(t)(t) shows a different behaviour for temperatures above and below T* ≈ 90 K for [TbPc2]0 and T* ≈ 60 K for [DyPc2]0 :

− T > T* T > T* →→ high frequency regime high frequency regime::

with β ≈ 0.5 and A ≈ 20 %.

− T < T* T < T* → → onset of very low-frequency fluctuations:onset of very low-frequency fluctuations: only the long tail of a static Kubo-Toyabe function can be detected due to the too fast initial decay of Pµ(t):

with α(H) the field dependent initial asymmetry.

0 3 60.0

0.2

0.4

0.6

0.8

1.0

[TbPc2]0

H = 1 kGauss

2 K

60 K70 K

80 K

52 K

Pt

P

t (s)

300 K

Fig 2: Time evolution of the muon polarization in [TbPc2]0 sample normalized to its value for t 0 at six selected temperatures.

))(exp()( tAtP

Muon polarizations PMuon polarizations Pμμ(t)(t)

))(exp()()( tAHtP )2(

)1(

0 60 120 180 240 300

1

10

100

1000

10000

1H NMR

1/

T1 (

s-1

)

T (K)

[TbPc2]0

[DyPc2]0

[YPc2]- TBA+

• [TbPc[TbPc22]]00 and [DyPc and [DyPc22]]0 0 NMR relaxation rates: NMR relaxation rates:

− The linewidth at half intensity increases from about 37 kHz at 202 K to a few MHz at 19 K evidence of the slowing down of the spin dynamics;

− T > 130 KT > 130 K → 1/T1 progressively increases on cooling;

− T < 40 KT < 40 K → 1/T1 slowly decreases on cooling.

− 140 K ≥ T ≥ 40 K140 K ≥ T ≥ 40 K → the observation of the signal is prevented by the too short relaxation time.

• The small peak in 1/T1 for [YPc2]-TBA+ sample should be ascribed to the TBA+ molecular motions. In fact, the Y3+ ion is non magnetic and no high intensity peak is expected.

Fig. 3: T dipendence of 1H NMR 1/T1 in [TbPc2]0

(black diamond), [DyPc2]0 (green squares) and [YPc2]-

TBA+ (blue circles) for H = 1 T.

Spin-lattice relaxation rates 1/TSpin-lattice relaxation rates 1/T11 from from 11H H NMR NMR [6][6]

0 100 200 300

0.01

0.1

1

10

[TbPc2]0

[DyPc2]0

H = 1000 Gauss

s

-1

T (K)

SR• [TbPc[TbPc22]]00 and [DyPc and [DyPc22]]00 μμSSR relaxation rates:R relaxation rates:

− Peak at Tm ≈ 90 K for [TbPc2]0 and Tm ≈ 60 K for [DyPc2]0 , in agreement with NMR findings.

− The intensity of the peak is observed to scale with the inverse of the field intensity the frequency of the fluctuations is close to Larmor frequency ωL at T = Tm.

Fig. 4: T dependence of the muon longitudinal relaxation rate in [TbPc2]0 and [DyPc2]0 for H = 1000 Gauss (black and green circles) and for H = 6000 Gauss (blue squares).

Spin-lattice relaxation rates Spin-lattice relaxation rates λλ from from μμSRSR [6] [6]

• Owing to spin-phonon scattering processes each CF level is characterized by a finite life-time τm which yields a lorentian broadening

with <∆h2┴> the mean-square amplitude of the hyperfine

field fluctuations and Em the eigenvalues of the CF levels.

)3(

• The life-time τm can be expressed in terms of the transition probabilities between m, m ± 1 levels:

with C the spin-phonon coupling constant.

• If ∆ >> T over all the the explored T range, Eq. (3) can be simplified in the form:

with )./exp(

13

TCc

)4(

)5(

The µSR (1H NMR ) spin-lattice relaxation rate λ (or 1/T1) can be written as [7]:

)6(

Estimating the correlation time of [LnPcEstimating the correlation time of [LnPc2 2 ]]00 SMM SMM

, /1 1,1, mmmmm pp

• [TyPc[TyPc22]]00 correlation time: correlation time:

− T > 50 K → high T activated regimeT > 50 K → high T activated regime:: it describes spin fluctuations among m = ± 6 and m = ± 5 levels.

− T < 50 K → low T regime:T < 50 K → low T regime: the constant correlation time signals tunneling processes among m = +6 and m = -6 levels.

The sum of the fluctuation rates associated with each process gives:

tunnc

T

tunncactcc

eC

1111 /3

• [[DyPcDyPc22]]0 0 correlation time:correlation time:

− T > 40 K T > 40 K high T activated regime: high T activated regime:

∆ ≈ ∆ ≈ 610 K610 K and C ≈ 1780 Hz/KC ≈ 1780 Hz/K33 .

Energy barrier: ∆ ≈ 880 K∆ ≈ 880 K

Spin-phonon coupling constant: C ≈ 3000 Hz/KC ≈ 3000 Hz/K33

Tunneling rate: (1/(1/ττcc))tunntunn≈ 11 ms≈ 11 ms-1-1

)7(

0 50 100 150 200 250 300

1E-5

1E-4

1E-3

0.01

0.1

1

10

100

TbPc2

DyPc2

c (

s)

T (K)

Fig. 5: T dependence of the correlation time for the spin fluctuations in [TbPc2]0 and [DyPc2]0 (for T > T*), derived from λ data reported in Fig. 4 on the basis of Eq. (5). The solid lines are the best fits according to Eq. (7).

− T < 40 K T < 40 K Eq. (3) should be resorted: the two lowest doublets are separated by an energy barrier in the tens of K range. The high energy barrier extimated corresponds to the one between the first and the second excited doublets.

Estimating the correlation time of [LnPcEstimating the correlation time of [LnPc2 2 ]]00 SMM SMM

0 50 100 150 200 250 300

10

12

14

measured calculated

T (

emu*

K/m

ol)

T (K)

[DyPc2]0

Fig.6: T dependence of χT in [DyPc2]0 for H = 1000 gauss (open circles) and calculated curve (line).

• The analysis of the static uniform spin susceptibility allows to determine the low energy level structure of the Dy3+ ion spin multiplet J = 15/2.• The lowest energy splittings which allow to better reproduce the experimental data for χT are:

- ∆- ∆1 1 = 115 K;= 115 K;

- ∆- ∆22 = 547 K; = 547 K;

- ∆- ∆33 = 57 K. = 57 K.

These values are larger than the ones deduced for [DyPc2]- on the basis of crystal field calculations [8].

• The muon relaxation rate behaviour can be correctly modelled with Eq. (3) by considering the first four CF levels and by using three different spin-phonon constants CC1 1 0 Hz/K 0 Hz/K33, CC2 2 ≈ 2400 Hz/K≈ 2400 Hz/K33 and CC3 3 ≈ 3100 Hz/K≈ 3100 Hz/K33,

associated with the four lowest transitions.

Spin susceptibility Spin susceptibility

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