Controlling Electron Spin Decoherence in Nd-based Complexes via Symmetry Selection.

Long decoherence time is a key consideration for molecular magnets in the application of the quantum computation. Although previous studies have shown that the local symmetry of spin carriers plays a crucial part in the spin-lattice relaxation process, its role in the spin decoherence is still unclear. Herein, two nine-coordinated capped square antiprism neodymium moieties [Nd(CO3)4H2O]5- with slightly different local symmetries, C1 versus C4 (1 and 2), are reported, which feature in the easy-plane magnetic anisotropy as shown by the high-frequency electron paramagnetic resonance (HF-EPR) studies. Detailed analysis of the relaxation time suggests that the phonon bottleneck effect is essential to the magnetic relaxation in the crystalline samples of 1 and 2. The 240 GHz Pulsed EPR studies show that the higher symmetry results in longer decoherence times, which is supported by the first principle calculations.

Introduction

Single-molecule magnets (SMMs) (Sessoli et al., 1993) are promising candidates as the quantum bits (qubits), the basic building blocks of a quantum computer according to Leuenberger and Loss's proposal (Leuenberger and Loss, 2001), in which they show slow spin relaxation behaviors between the bistable ground states with an energy barrier. However, the large zero-field splittings of SMMs result in low population in high-energy levels at low temperatures, which hinders the application of SMMs as qubits (Takahashi et al., 2008, Takahashi et al., 2009, Takahashi et al., 2011, Wang et al., 2011). In recent years, there has been a drive to achieve smaller energy splittings using the mesoscopic n class="Gene">spin states produced by hyperfine interactions between the electron and nuclear spins as a substitution, which are termed as qudits (Aguilà et al., 2014, Atzori et al., 2016a, Atzori et al., 2016b, Atzori et al., 2017, Fataftah et al., 2016, Graham et al., 2014, Martinez-Perez et al., 2012, Pedersen et al., 2016, Shiddiq et al., 2016, Tesi et al., 2016, Thiele et al., 2014, Yu et al., 2016, Zadrozny et al., 2017). Yet, the strong decoherence must be overcome to implement the envisaged application. Specific design criteria, such as nuclear-spin-free ligands (Yu et al., 2016), clock transitions (Zadrozny et al., 2017, Shiddiq et al., 2016), and low-energy vibrations (Atzori et al., 2017) have been developed to improve the quantum coherence time and temperature. As a matter of fact, the symmetry plays a vital part in spin-lattice relaxation process (namely, T1) (Ding et al., 2016). Consequently, it is very probable that the local symmetry of a spin carrier is equally important to the spin-spin relaxation process (namely, T2). However, the relationship between the decoherence and the local symmetries of spin carriers is still unclear.

Sessoli (Atzori et al., 2016a, Atzori et al., 2016b, Atzori et al., 2017, Tesi et al., 2016) and Freedman (Fataftah et al., 2016, Graham et al., 2014, Yu et al., 2016, Zadrozny et al., 2017) have contributed greatly to the development of molecular qubits with 3d transition metal ions as n class="Gene">spin carriers. Nevertheless, up to now, only a few 4f metal centers showing the quantum coherence properties (Aguilà et al., 2014, Pedersen et al., 2016, Martinez-Perez et al., 2012, Thiele et al., 2014, Shiddiq et al., 2016) have been reported. Herein, we report two capped square antiprism neodymium complexes, [C(NH2)3]5[Nd(CO3)4H2O]∙2H2O (1) and [C(NH2)3]4[H3O][Nd(CO3)4H2O]∙9.5H2O (2), which have different local symmetries, C1 (1) versus C4 (2), in the neodymium moieties [Nd(CO3)4H2O]5–. Both neodymium-based complexes are easy-plane magnetic anisotropic and show field-induced slow magnetic relaxation behaviors, which is rare in lanthanide complexes. The quantum coherence phenomenon was observed by the 240 GHz pulsed EPR spectroscopies at low temperatures on undiluted complexes. Our work indicates that the higher symmetry results in the longer decoherence times, which is explained by the first principle calculations.

Results and Discussions

X-Ray Structural Studies

Complexes 1 and 2 were synthesized according to a modified method reported with different rare-earth salts (Runde et al., 2000, Goff et al., 2010). The single crystal XRD reveals that 1 and 2 crystallize in the orthorhombic Pna21 and tetragonal P4/n space groups, respectively (Table S1). The n class="Chemical">[Nd(CO3)4H2O]5– anion consists of four chelated CO32− anions and a bonded H2O molecule with a real C1 symmetry in 1 and C4 symmetry in 2 (Figure 1). For 1, five guanidinium cations are arranged around the anion, forming a hydrogen-bonding network with two free water molecules in the crystal lattice (Figure S1), which makes the crystal stable in the air. For the lanthanide anion, the Nd(III) ion is in the plane of C1, C2, and C4 atoms (from carbonate) and the C3 atom is out of the plane with a mean deviation of 1.047 Å due to the steric hindrance. The related Nd-O lengths are in the range of 2.46–2.54 Å (Table S2). The coordinated water molecule is located on the quasi-C4 axis with a longest Nd-O length (2.63 Å). For 2, four guanidinium and one H3O+ cations behave as charge-balanced ions, arranged around the anion, forming the hydrogen bonding with other ten water molecules in the crystal lattice, which is similar to the reported complex [C(NH2)3]4[H3O][Dy(CO3)4H2O]∙13H2O (Goff et al., 2010) as supported by thermogravimetric analysis (Figure S2). The crystal of 2 is unstable in the air at the room temperature but is quite stable below the ice point, which can be attributed to the large amount of lattice water molecules in the crystal lattice. In the lanthanide anion, the four C atoms from CO32− are in the same plane and the Nd(III) ion is out of the plane with a mean deviation of 0.335 Å. The lengths of Nd-O bonds are in the range of 2.477–2.508 Å. The coordinated water molecule is located on the C4 axis with a Nd-O length of 2.43 Å. More structural parameters are summarized in Table S2. If the carbonate anion is regarded as one coordination site, structure 2 could be considered to have the quasi-tetragonal pyramid symmetry with a C4 axis.

Figure 1

The Structures of 1 and 2

Crystallographically determined molecular structure of the [Nd(CO3)4(H2O)]5− anion of 1 (left) and 2 (right). Neodymium, yellow; oxygen, red; carbon, gray. See also Figures S1, S2, S16, and S17.

Crystallographically determined molecular structure of the [Nd(CO3)4(n class="Chemical">H2O)]5− anion of 1 (left) and 2 (right). Neodymium, yellow; oxygen, red; carbon, gray. See also Figures S1, S2, S16, and S17.

Magnetic Properties

The direct-current (dc) magnetic susceptibilities of 1 and 2 were measured under 0.1 T in the temperature range of 1.8–300 K for 1 and 1.8–260 K for 2 (2 is unstable above 260 K) (Figure S3). Notably, the χMT value is 1.50 cm3mol−1K at 300 K for 1 (1.47 cm3mol−1K at 260 K for 2), lower than the expected value of one isolated Nd(III) ion (1.64 cm3mol−1K, for J = 9/2, gJ = 8/11) (Wada et al., 2017). Given that the crystal of 1 is stable, regular, and big enough, magnetization measurements at low temperatures were collected along three different orientations (a, b, and c) to determinate the susceptibility tensor (Figure S4). Through the single-crystal XRD analysis, the magnetization is 1.44, 1.54, and 0.59 NμB along the unit axes a, c, b, respectively. The magnetizations of a and c orientations increase quickly at low n class="Chemical">dc fields, slowly reaching to similar maximum values, whereas the magnetization of b orientation increases slowly in the whole field range, indicating that 1 is an “easy-plane” system. Treating the Nd(III) ion as an effective spin-1/2 ion, we could obtain the g-factor gx = 2.97(1), gy = 2.68(2), and gz = 1.23(1) by fitting the magnetizations with the Brillouin function (Darby, 1967). Considering the similarity of local symmetries between 1 and 2, their magnetic anisotropy might be similar. In order to confirm the “easy-plane” anisotropy of 1 and 2, HF-EPR measurements on polycrystalline samples were conducted at 4.2 K and in the frequency range of 60–253 GHz (Figure S5) (Wang et al., 2012; Nojiri and Ouyang, 2012). From the HF-EPR spectra, the relevant g-factors were obtained as gx = 3.00(2) and gy = 2.56(2) for 1 and gx = gy = 2.79(2) for 2, respectively, manifesting the “easy-plane” magnetic anisotropy for 1 and 2. The resonance signals of gz were not observed up to 22 T probably because the signals are too broad and hence weak due to the fast spin relaxations (Figure S6).

To study the origin of their magnetic properties, we performed the ab initio calculations (Karlstrom et al., 2003) for 1 and 2. The results are summarized in Tables S6 and S7. The calculated magnetic axes of the ground state are shown in Figure S17. The calculated g values within CASSCF are gx = 3.10(5), gy = 3.01(7), and gz = 0.86(8) for 1 and gx = 3.12(1), gy = 3.07(1), and gz = 1.31(3) for 2, which are in line with the aforementioned results from magnetization and HF-EPR measurements. The splitting of the two lowest Kramers doublets (KDs) for 1 was 99.5 cm−1 within CASSCF (128.6 cm−1 for 2). Based on the observed g value of the lowest KDs, the ground state of 1 is mixed by several mJ states severely (Table S7), which may induce a large QTM between these states (proved by the high-field magnetization measurements as presented in Figure 3). The alternating-current (ac) magnetic susceptibility measurements were performed on 1 and 2 with polycrystalline samples at low temperatures. Without the external n class="Chemical">dc field, no out-of-phase susceptibility (χM″) signal appeared as predicted by the ab initio calculations (Figures S7 and S10). This could be ascribed to the strong quantum tunneling of magnetization (QTM) at zero dc field, which is common in easy-plane-type systems as observed in the high-field magnetization measurements (Figure 3). When a small external dc field was applied, obvious signals in the frequency dependence of χM″ were clearly observed in 1 and 2 (Figures S7 and S10), which might be due to the magnetic field-suppressed QTM or the strong phonon bottleneck effect. However, the maximum can only be observed in high frequency range under 1.5–2.0 kOe external dc field in 2. This is the second time to discover that the light lanthanide complexes with the easy plane magnetic anisotropy can show slow spin relaxations (Table S8).

Figure 3

The Magnetic Hystersis Loop of 1 and 2

Magnetization versus pulsed magnetic field at 2 and 10 K for a powder sample of 1 (left) and 2 (right). The loop labeled by the asterisk is due to the experimental error when subtracting the background from the sample holder. Inset: Magnetic fields as a function of time. See also Figure S15.

The temperature-dependent ac susceptibilities were measured under 1.5 kOe dc field (Figure 2, Figures S8 and S11). The relaxation times (τ) were obtained by fitting the Cole-Cole curves with the CCFIT program (Guo et al., 2011) (Figures S9 and S12). Complex 1 shows the slow magnetic relaxation in the temperature range of 1.8–4.4 K with relaxation times ranging from 4.39 ms at 1.8 K to 0.112 ms at 4.4 K. Complex 2 shows the slow magnetic relaxation in a lower temperature range (1.8–3.0 K), and the relaxation times (0.29–0.046 ms) are substantially lower than those of 1 (Figure S13). Complex 1 exhibits an exponential relationship for ln(τ) versus temperature, demonstrating that multiple relaxation mechanisms coexist in the relaxation process. By fitting the linear part in high temperature range (4.0–4.4 K for 1 and 2.4–3.0 K for 2) with Arrhenius law τ = τ0 exp(–Ueff/kBT), thermal energy barriers were obtained as Ueff = 30.7 K with τ0 = 1.05 × 10−7 s for 1 and Ueff = 9.25 K with τ0 = 2.09 × 10−6 s for 2. The barriers are much smaller than the calculated energy splitting between ground state and the first excited state (Table S6). As a result, Raman process might dominate in the whole relaxation process, which is common in the easy-plane systems. Fitting of the complete temperature range data to a sum of direct and Raman processes with Equation 1:affords A = 125.89 K−1s−1, C = 0.89K−6.08s−1, and n = 6.08 for 1 (A = 0, C = 413.85 K−3.63s−1, and n = 3.63 for 2). Interestingly, the exponent n of 3.63 for 2 approaches the value of 3 as predicted in the case that both the acoustic and optical vibrations are important in the n class="Gene">spin dynamic process. For 1, the n of 6.08 is close to 9, indicating that the acoustic vibration is dominated in the relaxation process (Abragam and Bleaney, 2012).

Figure 2

The Magnetic Relaxation of 1 and 2

Frequency dependence ac magnetic susceptibilities for 1 (left) and 2 (right) obtained under 1.5 kOe dc field. See also Figures S7–S14.

Phonon-bottleneck effect (PB effect) usually plays a crucial part in the slow magnetic relaxation of “easy plane” system (Zadrozny et al., 2012). For a phonon-supported relaxation process, the energy exchange occurs through two processes (Abragam and Bleaney, 2012): from n class="Gene">spin to phonon (τsp) and from phonon to heat bath (τpb). So the relaxation time (τ) can be described in Equation 2:where Cs and Cp represent the heat capacity of spins and phonons in crystals, respectively. For an SMM, the phonon in crystal is abundant, so the rate-determining step in the relaxation process is τsp. Hence, the relaxation time τ ≈ τsp. The PB describes the situation that the number of spins is much larger than that of available phonons, in which the Cs/Cp can be in the order of 104–106. The theoretical PB relaxation time can be calculated through the following Equation 3:where the τph is the mean lifetime of lattice phonon, v is the averaged sound speed in the crystal, N is the spin carrier density, and ω is the resonant frequency of vibration modes. For complexes 1 and 2, the vibration modes would be nearly the same due to the similar chemical composition. If the magnetization relaxation phenomena of 1 and 2 were supported by PB effect, the relaxation time τ would be sensitive to the spin density. From the crystal parameters, the spin density is N1 = 0.00144 Nd/Å3 in 1 and N2 = 0.00111 Nd/Å3 in 2. Here, N1/N2 > 1, τ1 would be longer than τ2 at the same temperature, which is in agreement with the experimental results. On the other hand, in the PB-dominated relaxation process, the diffusion to the crystal boundary should be taken into consideration. The relaxation time, τ, is proportional consideration to the crystal size, L1 or L2. When the crystal is ground to a smaller size (Figure S14), the relaxation time would be short.

To justify the magnetization dynamic mechanism, ac measurements were tested on a ground powder sample of 1 at 2 K (Figures S7 and S14). After grinding, the size of the crystals became smaller, thus affecting only the single phonon process (Scott and Jeffries, 1962, Pedersen et al., 2015), so that Raman and Orbach processes would not be influenced. However, the maximum of χM″ for the ground powder sample shifts to the higher frequency range, indicating that the magnetic relaxation is sensitive to the low energy region of the phonon spectrum and/or the scattering of phonons on the crystal boundaries (Orendáč et al., 2016). Accordingly, the phonon bottleneck effect dominates the magnetic relaxation in the crystalline sample of 1. Owing to the air instability, the ac susceptibility of the ground powder sample of 2 could not be explored.

The magnetic relaxation properties of 1 and 2 are further studied by the high-field magnetization measurements with a pulsed magnetic field (3000 T s−1 averagely, Figure 3) (Saito and Miyasata, 2001). At 2 K, by sweeping the pulsed field upwardly (A→B), the magnetization gradually increases to 1.5 NμB at 20 T. In the down sweep (B→C), the magnetization decreases with a slower rate compared with upward sweeping, resulting in a pronounced hysteresis loop. A similar hysteresis loop is also observed in the negative field range. The hysteresis loops are due to the slow magnetic relaxations as observed in the ac susceptibility studies. The hysteresis loops were also tested at the same temperature using Squid VSM in a low field sweep rate (100 Oe/s) (Figure S15). No open loop was observed in this situation, which means that both 1 and 2 are not the magnet above 2 K. Thus, it is the pulsed magnetic field that makes it possible to observe the slow relaxation behaviors of 1 and 2 considering the extremely fast scan rate. Under the high sweep rate, the spin relaxation is in an adiabatic process, which limits the energy exchange between the phonons of the crystals and the envn class="Chemical">ironment (Lopez et al., 2009, Schenker et al., 2005). As a result, the butterfly loops were observed, indicating the strong phonon bottleneck effect in these complexes, which is in accordance with the ac susceptibility measurements.

However, it seems that phonon bottleneck effect mechanism has not been valued by chemists, although it might play an important role in many complexes with slow magnetic relaxation behaviors. For some special systems, for example, S = 1/2 systems like Cu(II) (Boča et al., 2017), isotropic systems like Gd(III) (Holmberg et al., 2015), also show slow n class="Gene">spin relaxation under low temperature with external field. To explain the special slow spin relaxation in those systems, chemists often attribute the relaxation behavior to Raman process without further discussions, which deserve further studies. This work opens a way to discuss the nature of the slow spin relaxation behaviors, especially to understand the origin of the slow magnetic relaxation behaviors in S = 1/2 and isotropic systems.

Quantum Coherence

Similar magnetization hysteresis loops were observed on a [Cu3] spin triangle complex whose life time is long enough to be detected by the pulsed EPR, proving [Cu3] complex a good candidate for qubit (Choi et al., 2012). So we studied the quantum decoherence properties of 1 and 2 on a 240 GHz pulsed EPR spectrometer at the National High Magnetic Field Laboratory, in Tallahassee, Florida, USA (van Tol et al., 2005, Morley et al., 2008). Measurements were done on single crystal samples, and the temperature dependence of quantum coherence time (T2) has been collected with the magnetic field along the x axes considering the easy-plane magnetic anisotropy for 1 and 2. The n class="Gene">spin coherence time was measured by a Hahn echo sequence (π/2 – τ – π – τ – echo), with the delay time τ varied during the measurements (Wernsdorfer et al., 2000; Schweiger and Jeschke, 2001). The widths of the pulses were tuned to maximize the echo signals and were typically between 100 and 150 ns. Figures 4A and 4B present the echo area as a function of 2τ at magnetic field 5.70 T for 1 and 5.81 T for 2 at different temperatures. The spin decoherence time T2 was extracted from the decay rate of the echo area, which was well fit by a single exponent function (exp(–2τ/T2)). Above 1.82 K for 1 (1.90 K for 2), T2 becomes too short to give spin echoes with the limited time resolution of the pulsed spectrometer. Taking into account the measurement temperature range (1.67–1.82 K for 1 and 1.67–1.90 K for 2), it is clear that the echo decays are strongly temperature dependent (Figure 4C) and T2 decreases from 130(5) ns at 1.67 K to 91(5) ns at 1.82 K for 1 (150(5) ns at 1.67 K to 100(5) ns at 1.90 K for 2).

Figure 4

The Quantum Coherence of 1 and 2

(A and B) Echo signals as a function of 2τ at different temperatures and 240 GHz for 1 and 2, respectively. Solid lines are the fits using a single exponential.

(C) Temperature dependence of the spin-spin relaxation time, T2, for 1 and 2.

(C) Temperature dependence of the spin-n class="Gene">spin relaxation time, T2, for 1 and 2.

The spin decoherence time of 1 and 2 is close to that of the first single crystal SMM qubit [Fe8] (Takahashi et al., 2009). At the low temperature of 2 K and the strong magnetic fields (H = 5.70 T for 1 and 5.81 T for 2), more than 99% of Nd n class="Gene">spins are polarized to the lowest lying spin state, which suppresses the spin flip-flop process significantly. That is the most important reason why the echo can be observed under concentrated samples. The strong temperature dependence of T2 can be ascribed to spin bath (Takahashi et al., 2008) fluctuation dominated by an energy-conserving spin flip-flop process. To our knowledge, many factors like hyperfine coupling (Wernsdorfer et al., 2000), the distance of spin carriers, and so on, influence the decoherence path. Considering that more H2O molecules are around the [Nd(CO3)4H2O]5– ion in 2 than in 1, one would expect faster spin decoherence in 2. However, the opposite was observed in our study, to put it more clearly, the spin decoherence time of 2 is longer than 1 at the same temperatures. According to the conclusion of Takahashi (Takahashi et al., 2011), the effect of nuclear spin and magnetic exciton on decoherence is much smaller than the phonon effect under high magnetic field and ultralow temperature. Here, the [Nd(CO3)4H2O]5– cluster in 2 is C4 symmetric, whereas 1 is C1 symmetric. Therefore, the phonon spectrum of 1 would be more complicated than that of 2 because the local vibration modes in 1 are more than in 2. The more complicated phonon freedom in 1 would increase the spin decoherence probability. From this point of view, the spin decoherence time of 2 with the high local symmetry should be longer at the same temperature and magnetic field.

The First Principle Calculations

To investigate the origin of the difference in the spin relaxation rate between 1 and 2, we perform the first principle calculations for the eigenstates and eigenvalues of these two samples (Mendeley Data). It is found that, for both 1 and 2, the magnetic moments mainly come from the f states of Nd atoms at the top of the valence band (Figure S18). The corresponding eigenstates, however, are very different for 1 and 2. For 1, the eigenstate at the top of the valence band is extended to both the Nd atom and the nearby n class="Chemical">four CO32− ligands, but for 2, this eigenstate is localized within the Nd atom (Figure 5). This can be well understood from symmetry point of view. The Nd-O-C structure in 2 is C4 symmetric, so that the hopping channels of electrons between the Nd atom and its four nearby CO32− ligands can be suppressed most by the destructive interference, whereas such a suppression of hoppings does not happen in 1 as it has C1 symmetry. The different symmetry can lead to remarkable difference in the spin relaxation rate between 1 and 2, because the spin-orbit interaction (SOI) together with phonons or charge fluctuations exert considerable influence in the spin relaxation processes (Khaetskii and Nazarov, 2001), and the hoppings between the Nd atom and its CO32− ligands contribute largely to the orbital motion of electrons in the spin state both in structures 1 and 2. In this spin relaxation channel, the SOI provides the spin flip mechanism during the electron orbital motion, whereas the phonons or charge fluctuations cause the dissipation. The relaxation rate of the spin-flip process is proportional to the square of the absolute value of the spin-flip matrix element (Khaetskii and Nazarov, 2001), i.e.

Figure 5

The Spatial Distributions of the Spin States of 1 and 2

The spatial distributions of the spin states at the Fermi level in Nd atom and its four ligands for both 1 and 2. Green balls are for Nd atoms, red for O, blue for N, gray for C, and white for H. See also Figure S18.

The spatial distributions of the spin states at the Fermi level in Nd atom and its n class="Chemical">four ligands for both 1 and 2. Green balls are for Nd atoms, red for O, blue for N, gray for C, and white for H. See also Figure S18.

In the present case, the main part of the orbital motion is due to the hopping processes between the Nd atoms and their ligands. Through the SOI, the spin-flip occurs during these hopping processes. Based on the above-mentioned results of the first principle calculations, the hopping processes between the Nd atom and its ligands are strongly suppressed in structure 2 due to the C4 symmetry, whereas in 1 such suppression does not happen. Therefore, in 2 the n class="Gene">spin-flip processes due to the SOI are also suppressed, leading to the much longer spin-flip relaxation time in structure 2 compared with structure 1, T1(2) > T1(1). At the same time, T2 ≤ 2T1 (Golovach et al., 2004) is still valid for both structures 1 and 2. It is also shown that, for localized spin states, such as the case in the quantum dots, one has T2 = 2T1 for all SOI mechanisms in leading order of the electron-phonon interaction (Golovach et al., 2004). The localization of the spin state in structure 2 is very similar to the case in the quantum dots, in which T2(2) = 2T1(2) holds. Hence T2(2) > T2(1) is reasonable as observed in our experiments.

Conclusions

In summary, typical magnetic relaxation behaviors have been observed for two mononuclear Nd(III) complexes 1 and 2 with strong easy-plane magnetic anisotropy due to the strong phonon bottleneck effect. The spin decoherence studies reveal that the higher symmetry results in longer decoherence times, which is explained by the first principle calculations. Furthermore, consistent with the work of [GdW10] reported by Coronado (Martinez-Perez et al., 2012), we believe that the easy-plane magnetic anisotropy and high symmetry are extremely important factors to enhance n class="Gene">spin decoherence time of molecular spin carriers. Further studies of spin decoherence in other lanthanide complexes are in progress in our laboratory.

Limitations of the Study

This work demonstrates that the symmetry is an important factor to develop potential qubits with the improved performance and the higher symmetry results in longer decoherence times. This discovery provides a specific design criteria to develop potential qubits with improved performances. However, the decoherence times of complexes 1 and 2 are too short to application. And we need more couples of samples and more accurate physical model to understand the deep influence of symmetry in quantum coherence.

Methods

All methods can be found in the accompanying Transparent Methods supplemental file.