It is of fundamental importance to understand the dynamics of spin states of the magnetic molecules on electrodes for their potential applications in miniaturized molecular spintronic devices [1-4]. Due to the requirement of a long enough spin relaxation time to exceed the duration of spin manipulations , a wide research interest has been stimulated in searching the molecular magnetic systems with long spin relaxation time [6-10]. However, it has been observed that the spin relaxation time of magnetization excitations in magnetic molecules are very short in the presence of electrodes, typically in the order of 10-14 s [11, 12]. This time scale is even shorter by over one order of magnitude than those of the magnetic atoms on metal surfaces [13-17] or in bulks of metals [18, 19] and semiconductors [20-22].
It is known that when individual quantum spins are placed in close proximity to conducting substrates, the dynamic properties of the localized spins can be significantly affected because of the exchange interaction between itinerant conduction electrons and localized d-shells, so-called s-d exchange interaction. In this regime, the relatively short relaxation time of spin states in magnetic atoms can be described in reasonable time scale by considering the s-d exchange interaction [18, 23, 24], although the observed relaxation times still vary in different magnetic atoms or substrates [13-17]. In particular, the substrate-dependences of spin relaxation times of individual magnetic atoms have been interpreted according to some of other additional processes, like Stoner excitations of the itinerant conduction electrons of the substrate  and the spin polarization of nonmagnetic atoms of the substrate . Much longer spin lifetimes have also been observed in the systems with large magnetic anisotropy barrier [15-17, 25-27] and/or by introducing f electrons for the weak coupling of f electrons with both tunneling electrons and itinerant electrons of the conducting substrates [28, 29]. Even in the systems with large magnetic anisotropy barriers, the sequential spin transitions induced by the s-d interaction may still play an important role in the spin lifetime [15, 26]. In comparison, the existences of ligands in molecular systems may introduce additional spin relaxation channels , in which the π electrons in ligands have shown important roles in Kondo physics [31-33]. Understanding the effect of π electrons in ligands on the spin relaxation of magnetic molecules, especially at a quantitative level, is crucial toward their applications in the area of molecular spintronics.
Here, we investigate the relaxation time of spin excited states in trilayer cobalt-phthalocyanine (CoPc) films on the Au (111) surface using scanning tunneling microscopy and spectroscopy (STM/STS). Korrigna-like formula [18, 23, 24, 34] is employed to analyze the spin relaxation time by considering s-d interaction between the localized spin of CoPc molecule and the bulk and surface density of states (DOS) of the Au (111) substrate. This calculation regime gives the consistent spin relaxation time with the experimental result in the case of Fe atom adsorbed on Pt (111) surface. However, there exists huge discrepancy between the experimental and the calculated values for the case of CoPc/Au (111). According to our analyses of various interactions in this adsorption system of magnetic molecule, this discrepancy could be attributed to the π-electron assisted process.Ⅱ. EXPERIMENTS
Our STM experiments were carried out using an ultrahigh vacuum low temperature scanning tunneling microscope (Unisoku) equipped with a sample preparation chamber for film growth with a base pressure of 5 × 10-11 Torr. The Au (111) surface was cleaned by cycles of 1000 eV Ar+ ion sputtering and 650 K annealing. CoPc thin films were deposited on the Au (111) substrates using a Knudsen cell, where the fully covered first layer of CoPc was prepared by keeping the substrates at room temperature, and the second and third island layers were prepared by keeping the substrates at ~150 K. The STM measurements were performed at 4.2 K. An electrochemically etched tungsten tip was used. The dI/dV and d2I/dV2 spectra were measured using a lock-in preamplifier, with a typical sinusoidal modulation of 0.5 mV by root mean square (rms) at 797 Hz.Ⅲ. RESULTS AND DISCUSSION
Figure 1 (a) and (b) show the representative topography images of the CoPc films with different layers on Au (111). The CoPc molecules in the first layer show nearly uniform four-lobe patterns, and those in the second and third layers are tilted with a certain degree with respect to the surface plane. The Co2+ ion centers are much protruded in the second and third layers. In the images, the colored dots indicate the Co2+ ion centers of CoPc in different layers. From the molecules at the island edge and the indicated centers, the Co2+ ion centers of CoPc in the second layer are shifted with respect to the ones in the first layer by about 3.9 Å, accompanied a rotation of 45° with respect to the latter. The Co2+ ion centers of CoPc in the third layer are further shifted with respect to the ones in the second layer by about 2.2 Å, while the molecules in the second and third layers almost have the same orientation. The derived stacking configuration is shown in Fig. 1(c).
The dI/dV spectra measured on top of Co2+ ions of CoPc molecules in different layers on Au (111) substrates are shown in Fig. 2(a). In the first layer, the dI/dV spectra almost show no feature in the concerned energy range, which can be attributed to quenching of magnetic-moment of Co2+ ions in the first CoPc layer . Hence, the first CoPc layer may act as a decoupling layer. There is no feature observed from the second CoPc layer on Au (111), different from the case of the second CoPc layer on Pb (111) in which a Kondo resonance peak appeared . It is noted that the second-layer Co2+ ion centers are shifted with respect to the first-layer Co2+ ion centers for CoPc on Au (111), while in the case of CoPc films on Pb (111) the second-layer Co2+ ions are just located on top of the first-layer ones . The off-top configuration of the second CoPc layer on Au (111) may release the molecular spin from Kondo screening of the substrate and thus the Kondo resonance cannot be observed, as suggested in Ref..
We focus on the relaxation times of the excited spin states measured at the Co2+ ion centers in the third layer. The step-like features in the dI/dV spectra of the Co2+ ion centers in the third layer originate from the magnetization excitation from the antiferromagnetic (AFM) coupling to the ferromagnetic (FM) coupling between the spins of Co2+ ions in the second and third layers . The Co2+ ion of a single CoPc molecule in the second and the third layer has a spin of 1/2. The total spin Sc is thus 0 and 1 for the AFM and the FM coupling, respectively. The spin excitation energy is 20.9 meV on Au (111), which can be much clearly seen in the second derivative spectrum (d2I/dV2 spectrum), as shown in Fig. 2(b). The variation of the peak positions of different Co2+ ion centers in the third layer is observed within 2 meV. The full width at half maximum (FWHM) of the peak reads W=11.1±0.5 meV for CoPc on the Au (111) substrate. Using the following equation ,
where Vmod(=0.5 mV, rms) is the modulation of the bias voltage, e is the electron charge, kB is the Boltzmann constant, Winst (0.15 meV) is the instrumental broadening of our STM , we get the intrinsic line width Win=10.9 meV for CoPc on Au (111) substrates. Considering the uncertainty relation τWin≥ħ/2, where ħ is the Plank constant, we then have the lifetime τ=3.1 × 10-14 s for the magnetization excitations of CoPc on Au (111).
where ρs is the DOS of the conduction electrons at the Fermi level of substrate, Js-d is the s-d exchange integral with a value of about 0.25 eV , △ is the energy difference between the total states of |1〉 and |0〉 and |0〉 by using the observed excitation energies, Sa is the spin matrix. The contributions of the effective DOSs, that is, the conduction electrons around the localized spins, are from both of the surface and bulk states in the case of Au (111) substrate. Here, we use ρs(Au)=4/3πr3nAu-b+πr2nAu-s to count the effective DOSs of the substrates which scatter the localized spin moment of Co2+ ion, where nAu-b eV-1Å-3 and nAu-s eV-1Å-2 are the bulk DOS  and the surface DOS  of Au respectively, and r=1.26 Å is the ionic radius of Co2+ ions in CoPc , giving ρsJs-d=0.049 for Au (111) substrate. We get the calculated lifetimes τ1→0=8.4 × 10−12 for CoPc on Au (111). However, there is a huge discrepancy by about two orders of magnitude between the experimental and calculated values. It is noticed that in the calculations we do not include the contribution of the DOS from the tungsten tip and the decay of the bulk and surface DOSs at the vacuum side . If taking into consideration of the DOS decay, the calculated lifetimes will be longer. On the other hand, since the localized spin of Co2+ ion is more weaker coupled to the tip than to the substrate, the contribution of the tip DOS to the relaxation rate is still too small to compensate the huge discrepancies even if the tip is considered. Therefore, only considering the scattering of the itinerant electrons by the localized spin of Co2+ ion is obviously insufficient.
As a comparison, let us consider the spin relaxation of Fe atom on Pt (111) and Cu (111), where Pt has the bulk DOS  of nPt-b=0.11 eV-1Å-3 and the surface DOS  of nPt-s=0.012 eV-1Å-2, and Cu has the bulk DOS  of nCu-b=0.025 eV-1Å-3 and the surface DOS  of nCu-s=0.017 eV-1Å-2. Then we have ρsJs-d=0.25 for Pt substrate and ρsJs-d=0.074 for Cu substrate respectively. Adopting the measured energies of the first order excitations and the magnetic moments of Sc(Fe/Pt)=5/2 and Sc(Fe/Cu)≈2 for Fe atom on Pt (111) and Cu (111) [13, 14], respectively, we then get the calculated relaxation times as 1.7 × 10-12 and 1.5 × 10-11 s. The calculated relaxation time of Fe atom on Pt (111) is well consistent with the experimental value . Our calculations indicate that the strong s-d interaction due to the relatively large DOSs of Pt can well describe the spin relaxation process of Fe atom on Pt (111). While, the calculated relaxation time of Fe atom on Cu (111) is longer by 2 orders of magnitude than the experimental value , which has been attributed to the contribution of the strong Stoner excitations . Although the decay into the Stoner excitations is a possible channel to shorten the spin relaxation time in the case of CoPc films on Au (111), it is not clear that in what degree of this substrate may participate the process of Stoner excitations.
Alternatively, we here consider the possible effects of Pc macrocycle. It was reported that the spin relaxation time was 5.9 × 10-2 s in the mixture of CuPc/H2Pc films on insulating substrate at 5 K . Except the absence of conduction electrons, this system is very similar to the CoPc films in our experiment. Analogous to analysis of the s-d exchange interaction above, by considering
Phenomenologically, instead of considering the radius of Co2+ ion only, we take into consideration of the Pc macrocycle, by modeling it as a plate with a thickness of 3.0 Å for the CoPc  and a radius of rπ=3.9 Å for the extension of the polarized π electrons, which is adopted by considering the observed extended Kondo resonance over the pyrrole ring in Pc macrocycle . Using Eq.(1), we then obtain the calculated relaxation time of 3.7 × 10-14 s for CoPc on Au (111), with phenomenological dimensionless parameters ρJ ≈ 0.7, which is consistent with the value by considering the spatially extended Kondo state . The calculated relaxation time is now in better agreement with the experimental result. To rationalize this analysis, we may consider a system consisting of a localized spin and delocalized spin (s), where the coupling between the localized spin and the delocalized spins may be either FM or AFM, as shown in Fig. 3. In the presence of itinerant conduction electrons, both of the localized spin and the delocalized spins scatter the conduction electrons, leading to a faster spin relaxation of the whole system. This scenario should depend on the spin polarization of π electrons in the Pc macrocycle and the participation of the polarized π electrons as scattering centers. The observed spatially extended Kondo state at Pc macrocycle in the similar systems [30-32] well supports our scenario, even though the intramolecular π-d coupling can be either FM or AFM in different molecular systems . It is noted that the large value of ρJ ≈ 0.7 may be already beyond the limit of the perturbation approach . For a better understanding of the spin relaxation in such molecular systems, one may need to consider the many-body effect . Nevertheless, our analysis here provides useful information to address the importance of the Pc macrocycle due to the spin polarization of the π electrons.Ⅳ. CONCLUSION
In summary, we measured the relaxation time of the excited spin states in CoPc films on Au (111). The spin relaxation time estimated using the Korringa-like formula by just considering the s-d exchange interaction is much longer than the experimental result by about two orders of magnitude. However, since the intramolecular π-d coupling and other weaker couplings are smaller than the s-d coupling by over 1 orders of magnitude, the huge discrepancy can not be explained by considering these interactions in a single-particle picture. Our estimation by including the spin polarized π electrons in Pc marcocycle as scattering centers can phenomenologically accord with the experimental value. Our analyses srongly suggest the involvement of many-body effect to efficiently relax the spin states through π electrons in such a kind of molecular systems.Ⅴ. Acknowledgments
This work was supported by the National Natural Science Foundation of China (No.91321309, No.91421313, No.21421063, and No.21273210), the "Strategic Priority Research Program" of the Chinese Academy of Sciences (No.XDB01020100), and the Fundamental Research Funds for the Central Universities (No.2340000050 and No.2340000074).
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