Breaking the ring: ^{53}Cr-NMR on the Cr_{8}Cd molecular nanomagnet
E. Garlatti, G. Allodi, S. Bordignon, L. Bordonali, G.A. Timco, R.E.P. Winpenny, A. Lascialfari, R. De Renzi, S. Carretta
BBreaking the ring: Cr-NMR on the Cr Cdmolecular nanomagnet
E. Garlatti , G. Allodi , S. Bordignon , L. Bordonali , G.A.Timco , R.E.P. Winpenny , A. Lascialfari , R. De Renzi and S. Carretta , Dipartimento di Science Matematiche, Fisiche e Informatiche, Universit`a diParma, Parco Area delle Scienze 7/A, 43124 Parma, Italy. Karlsruhe Institute of Technology, Institute of Microstructure Technology,Hermann-von-Helmholtz-Platz 1 76344 Eggenstein-Leopoldshafen, Germany. School of Chemistry and Photon Institute, The University of Manchester, M139PL Manchester, United Kingdom. Dipartimento di Fisica, Universit`a degli Studi di Pavia, Via Bassi 6, 27100Pavia, Italy and INFN, Milano Unit, Milano, Italy. INSTM, UdR Parma, 43124 Parma, Italy.E-mail: [email protected]
February 2020
Abstract.
An accurate experimental characterization of finite antiferromag-netic (AF) spin chains is crucial for controlling and manipulating their magneticproperties and quantum states for potential applications in spintronics or quan-tum computation. In particular, finite AF chains are expected to show a differentmagnetic behaviour depending on their length and topology. Molecular AF ringsare able to combine the quantum-magnetic behaviour of AF chains with a veryremarkable tunability of their topological and geometrical properties. In this workwe measure the Cr-NMR spectra of the Cr Cd ring to study the local spin den-sities on the Cr sites. Cr Cd can in fact be considered a model system of a finiteAF open chain with an even number of spins. The NMR resonant frequenciesare in good agreement with the theoretical local spin densities, by assuming acore polarization field A C = -12.7 T/ µ B . Moreover, these NMR results confirmthe theoretically predicted non-collinear spin arrangement along the Cr Cd ring,which is typical of an even-open AF spin chain. a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n reaking the ring: Cr-NMR on the Cr Cd molecular nanomagnet
1. Introduction
Quantum properties of matter have been in thespotlight of fundamental research in physics fordecades, but nowadays they also represent essentialand solid resources for the future of nanoscience andnanotechnology. In particular, magnetic properties ofquantum systems are of central interests for the currentresearch in condensed matter physics. Topologicaleffects have a strong influence on low-dimensionalmagnetic systems and on the excitation spectrum andeigenfunctions of antiferromagnetic (AF) Heisenbergspin chains [1, 2]. The experimental observation of theso-called “edge states” in broken S = 1 AF Heisenbergspin chains [3] also triggered the interest in finite open(or “broken”) chains, making them the subject ofseveral theoretical investigations in the past decades.These works showed that the magnetic behaviourof finite open AF chains depends on their length,topology (closed vs open) and parity of the number ofatoms (even vs odd) [4–8]. For instance, finite even-open chains are predicted to show a Non-Collinear(NC) configuration of the local-spin s i expectationvalues for s i > / N ≤ s = 3 / and Cr [13–26] allow the study of closed chains,thanks to their periodic boundary conditions, andto investigate how frustration influence the magneticbehaviour of the ring/chain. The introduction of non-magnetic impurities breaking the cyclic symmetry asin Cr Cd [27, 28] and Cr Cd or Cr Zn [29–32] makesthese rings effective model systems for open chains.Heterometallic rings [33–36] with a magnetic groundstate can be obtained by chemical substitution of oneor two magnetic centers. Among them, Cr Ni has been proposed as a qubit for quantum computation [37–42].Indeed, the investigation of the local spin structureof AF rings not only provides a quantitative insighton the behaviour of AF spin chains, but it is alsovery important in the design of supramolecular chainsfor quantum information processing. For instance,these molecular rings can be linked together, eitherdirectly or through magnetic ions, into supramolecularinteracting dimers [39, 41, 42]. The site dependenceof the local spin density plays a key role in thescheme proposed for obtaining time dependent qubit-qubit couplings in the presence of permanent exchangeinteractions [40]. Thus, it is crucial to understandhow the local spin density depends on the length andtopology of the investigated ring.The local spin density along the odd-memberedopen ring Cr Cd has been previously determinedthrough Cr-NMR [27]. That work showed thatthe local spin density in the ground state is ratheruniformly distributed over the ring with an alternatedstaggered orientation due to the AF coupling. Then,we investigated how the single-ion spin moment isdistributed in an heterometallic ring, where one Cr ion is replaced by a different magnetic ion ratherthan a diamagnetic one [43]. The Cr-NMR spectrameasured at low temperature in a single crystalof the Cr Ni in its S = 1 / ions next tothe Ni ion displaying the greatest component of thelocal spin s i .The spin arrangement for even-open rings like Cr Cdis expected to be different both from the spin-flopconfiguration of even-closed rings, like Cr , and fromthe AF staggered arrangement of odd-open ones, likeCr Cd. Theoretical models in fact predict a NCconfiguration [5, 44]. In this configuration, the spinsat the extremities of the chain have the highest localmagnetic moment along the direction of the appliedmagnetic field, since they have to compete only withone exchange interaction. Moving towards the centreof the chain, nearest-neighbouring spins are alignedin the opposite direction due to AF interactions andthe magnetic moment along the applied field decreases.For the two spins at the exact centre of the even chain,the AF interactions between them and their nearest-neighbours cannot be simultaneously satisfied, leadingto competing interactions.In this work we measure the Cr-NMR spectra ofCr Cd to study the local spin densities on the Crsites, as a model system of a finite Heisenberg AFeven-open chain. Cr Cd has already been studied bypolarized neutron diffraction (PND) [44]. PND allowedto extract the thermal averages of the projected localspin moments along the applied magnetic field, which reaking the ring: Cr-NMR on the Cr Cd molecular nanomagnet [44]. Herewe measure the even-open ring Cr Cd with Cr-NMRto compare it with previous NMR results on otherAF rings and to confirm polarised neutron results.Moreover, with Cr-NMR we are able to obtain theexpectation values of local spin density along thering, together with the core polarisation field of Crnuclei. Our experimental results are in agreementwith theoretical predictions, confirming the NC spinconfiguration for Cr Cd. It is important to notethat NMR is a less resource- and time-consumingtechnique with respect to PND and allows one todetermine local spin moments after a straightforwarddata analysis. NMR experiments similar to the onepresented here on Cr Cd can in fact be devised for morecomplex AF rings or long AF chains, allowing one toobtain local spin moments with the level of accuracyrequired for quantum computation applications. Onthe contrary, the complexity of the interpretationof PND data increases with the complexity of theinvestigated system. For instance, PND data analysisrequires a very accurate determination of the molecularstructure of sample from neutron or X-rays diffractionexperiments as a starting point, on which the outcomeof the analysis depends critically.
2. Experimental results [H N t Bu is Pr][Cr CdF (O CCMe ) ], Cr Cd in short,is a heterometallic AF ring constituted of ninetransition metal ions (see Fig.1-a), obtained from itsparent compound Cr by adding a s = 0 Cd ion tothe eight Cr . We performed our NMR experiment ona single crystal sample, prepared as in Ref. [29]. Theunit cell contains two identical but differently orientedmolecules (see Fig.1-c), each making an angle of 27 ◦ between the perpendicular to the plane of the ring(i.e., z axis of the molecule) and the b axis. Thestatic magnetic field has therefore been applied alongthe b axis of the crystal, in order to ensures equalcomponents of the field along the z -axis for all themolecules in the unit cell.The main difficulty of Cr-NMR measurements is dueto the low natural abundance and low sensitivity of theprobe (9.54 % abundance, γ/ π = 2.406 MHzT − ),together with the millimetric size of the measuredcrystal. Despite this drawback, we managed to observesignals at low temperature in the desired magnetic fieldrange. The experiments have been performed in aMaglab EXA (cid:114) (Oxford Instruments) cold-bore field-sweeping superconducting cryomagnet, with a variabletemperature insert (VTI) as a sample environment,and a “HyReSpect” home-built NMR spectrometer a c b a c b Figure 1.
Panel a: [H N t Bu is Pr][Cr CdF (O CCMe ) ]molecular structure. Cr ions are reported in green, Cd in blue,O in red, F in yellow and C in grey. Hydrogen ions are omittedfor clarity. Panel b: schematic representation of ions along thering, the color-code highlights the magnetically equivalent Cr + ions and the non-magnetic Cd . Panel c: two inequivalentmolecules are clearly distinguishable within the unit cell. [45], equipped with an external rf power amplifier.The magnet allows to cover a of 0-9 T static magneticfield range, while the VTI allows us to reach a basetemperature of 1.4 K, at which all measurements havebeen performed.A ( π/ π ) pulse sequence was employed tocollect Cr Cd NMR spectra with two complementarymethods: by varying the frequency at a fixed externalmagnetic field (frequency- sweep , i.e., the frequency ischanged in fine steps), or by varying the field in smallsteps at fixed frequency (field- sweep ). Signal intensities reaking the ring: Cr-NMR on the Cr Cd molecular nanomagnet Magnetic Field (Tesla)Frequency (MHz)32302725 ab Figure 2. Cr-NMR spectra of Cr Cd at T =1.4 K obtained byvarying the magnetic field at a constant frequency (panel a) andby changing the frequency at a constant magnetic field (panelb). Data have been fit according to a Gaussian profile. have been collected point by point as a function of bothfield and frequency, by integrating the whole spin-echoover time. Our experimental configuration allowed usto detect resonance frequencies in the 17.5-35 MHzrange. The detection of Cr resonances below 17MHz was hindered by the drop of sensitivity ( ∝ ν )and by the dramatic increase of the receiver dead-timeat decreasing frequency, mostly arising from spuriousmagneto-acoustic couplings ( ringing ) of the pick-upcoil. Both field- and frequency- sweep Cr-NMRspectra of Cr Cd are reported in Fig.2. Resonancepeaks have then been fitted with Gaussians line-shapes, in order to extract the resonance frequenciesreported as black scatters in Fig.3, where they are alsocompared with theoretical calculations (more details inthe following section).
Figure 3.
Resonance frequencies calculated with Eq.4 asa function of the external magnetic field in the exploredfrequency range. To each frequency we also associated aGaussian line-shape with σ =1 MHz and amplitude given bythe level population, yielding the above intensity plot. Blackscatters: experimental resonance frequencies obtained by fittingfield- and frequency- sweep Cr-NMR spectra in Fig.2. Thedimension of the scatters reflect the relative intensities of thedetected transitions (in arb. units). Only resonances with asignificant intensity are reported, shaded areas have not beenexperimentally explored.
3. Analysis of the data and discussion
The magnetic properties of each Cr Cd molecule canbe described by the Hamiltonian: H = J (cid:88) i =1 s i · s i +1 + d (cid:88) i =1 [ s z,i − s i ( s i + 1) /
3] (1)+ µ B (cid:88) i =1 g s i · B , where s i is the spin operator of the i th ion in thering. The first term is the dominant nearest-neighbourisotropic exchange interaction, the second one is theuniaxial single-ion zero-field splitting term (where the z axis is perpendicular to the plane of the ring) and thelast term is the Zeeman interaction with an externalmagnetic field B . Spin Hamiltonian parameters havebeen previously determined by INS measurements onthe parent Cr Zn compound [31] and magnetizationmeasurements on Cr Cd [30], yielding J =1.32 meV, d =0.036 meV and g =1.98. Hereinafter, we label Cr Cdeigenstates by their dominant total-spin S . Indeed,even if it is not a good quantum number for Eq.1,Cr Cd low-energy eigenstates are characterised by avery small S-mixing, with dominant component on thetotal-spin basis of >
99% (except for the very smallfield ranges around anti-crossings). This result is a reaking the ring: Cr-NMR on the Cr Cd molecular nanomagnet S eff , where (cid:104) S (cid:105) = S eff ( S eff +1) (calculations are reported as a colour-map in Fig.6).Energy levels as a function of the applied magneticfield with θ = 27 ◦ are reported in Fig.4. At zero-fieldCr Cd has a non-magnetic S = 0 ground state, whichbecomes and S = 1 after a level crossing at 2.6 T andan S = 2 after another crossing at 6.9 T. An anti-crossing between the S = 0 and the S = 2 multipletsis also present at 4.8 T.If we take into account the presence of the magnetic Figure 4.
Low-lying energy levels of Cr Cd as a function of theexternal magnetic field calculated for θ = 27 ◦ . Cr nuclei and their interaction with electronic spins,the full Hamiltonian for a Cr Cd molecule can bewritten as: H = H el + H n , (2)where H el is the electronic spin Hamiltonian in Eq.1and H n for the i th Cr ion in the ring is given by: H n,i = A s i · I i + g I µ N I i · B . (3)The first term in Eq.3 is the hyperfine coupling(assuming an isotropic coupling constant A) while thesecond one is the nuclear Zeeman term. ‡ Eigenvectors of the full Hamiltonian in Eq.2 can bewritten as product states between the electronic andnuclear eigenfunctions, with well-defined electronic andnuclear quantum numbers. Given its low naturalabundance, we can simply assume the presence ofone magnetic Cr nucleus per ring. We label theeigenstates as | α (cid:105) = | SM S , M I (cid:105) , where S labels the ‡ Since we measured only the central line transition ( M I =1 / ↔ M I = − /
2) of the Cr NMR spectrum, we neglectsecond-order effects due to quadrupolar interactions [27, 43]. electronic total-spin multiplet and the quantizationaxis for M S and M I coincides with the directionof the static magnetic field. We have also checkedthat electronic spins are practically aligned with thedirection of the applied field already at the lowestexplored magnetic fields ( > Cr Cr Cr Cr Cr Cr Cr Cr B Figure 5.
Schematic representation of the NC spin arrangementof the Cr Cd ring. Red arrows represent the calculatedexpectation values of the local spins along the direction of thestatic magnetic field z’ at 5 T (when the ground state is an S = 1). Resonance frequencies can be expressed in terms of theLarmor frequency of Cr, shifted by the isotropic corepolarization hyperfine contact term [46]: ν iα,α (cid:48) = γ π ( B z (cid:48) − gA C (cid:104) s z (cid:48) ,i (cid:105) S,M S ) , (4)where z (cid:48) is the direction of the static magnetic field, γ = g I µ N is the Cr gyromagnetic ratio, A C is thecore polarization field, the only free parameter of ourmodel for Cr Cd, and (cid:104) s z (cid:48) ,i (cid:105) S,M S = (cid:104) SM S | s z (cid:48) ,i | SM S (cid:105) is the expectation value of the local spin operator ofthe i th ion in the ring [27, 43].The absorptive part of the dynamic susceptibilityfor the nuclear transitions between two levels | α (cid:105) = | SM S , M I (cid:105) and | α (cid:48) (cid:105) = | SM S , M I ± (cid:105) can be writtenas [47]: χ (cid:48)(cid:48) ( ω ) ∝ (cid:88) α,α (cid:48) n α β ( E α (cid:48) − E α )Γ α,α (cid:48) √ π × (5) × exp( − ( (cid:126) ω − ( E α (cid:48) − E α ) α,α (cid:48) ) ×× {(cid:104) α | g I µ N I x (cid:48) | α (cid:48) (cid:105)(cid:104) α (cid:48) | g I µ N I x (cid:48) | α (cid:105) ++ (cid:104) α | g I µ N I y (cid:48) | α (cid:48) (cid:105)(cid:104) α (cid:48) | g I µ N I y (cid:48) | α (cid:105)} .n α is the thermal population of the level α , β = 1 /K B T and ( E α (cid:48) − E α ) = ± ( Am S + g I µ N B z (cid:48) ) = (cid:126) ω α,α (cid:48) ,where m S is the local spin operator expectation value (cid:104) s z (cid:48) ,i (cid:105) S,M S . At last, Γ α,α (cid:48) is the width of the Gaus-sian line-shape. Thus, for nuclear transitions between reaking the ring: Cr-NMR on the Cr Cd molecular nanomagnet Cd we do expect resonance-frequency shiftsto depend on the expectation value of the local spin op-erator (cid:104) s z (cid:48) ,i (cid:105) and intensities to depend on the thermalpopulation n α , which is mostly set by the electroniccomponent § .In order to determine the core polarization field A C for Cr Cd, we calculated the resonance frequencieswith Eq.4 as a function of the applied static magneticfield. Expectation values of the local spin operators (cid:104) s z (cid:48) ,i (cid:105) S,M S were calculated on each considered eigen-state by diagonalizing the spin Hamiltonian in Eq.1.Fig.5 reports the calculated expectation values for astatic magnetic field of 5 T on the magnetic groundstate, i.e. when it is a magnetic S = 1. The spinsare arranged in a NC configuration. We also simulatedthe intensities of each transition, assuming a Gaussianline-shape with σ =1 MHz and amplitude given by thepopulation n α . Theoretical results are reported in theintensity plot in Fig.3, compared with experimental fre-quencies (black scatters). The latter can be reproducedby assuming a core polarization field A C = -12.7 T/ µ B ,in agreement with the results on the parent compoundsCr Cd and Cr Ni (see Table 1 and Ref. [27, 43]).AF ring A C (T/ µ B )Cr Cd -12.7Cr Cd -12.38Cr Ni -11
Table 1.
Comparison of core polarization constant A C obtainedwith Cr-NMR experiments on Cr Cd (present work), Cr Cd[27] and Cr Ni [43].
The next step was to attribute the detected signals tothe corresponding Cr nuclei along the ring. Fig.6-a shows the calculated frequencies for the Cr -Cr and Cr -Cr magnetically-equivalent sites (see Fig.1-b), compared with the experimental data. It is evi-dent that the signals observed can be totally ascribedto these Cr nuclei. Given the theoretically predictedNC configuration (see Fig.5), Cr -Cr at the extremi-ties of the chain have a significant electronic magneticmoment along the direction of the applied field, induc-ing a high resonance-frequency shift. Their nearest-neighbours Cr -Cr have a smaller electronic moment,but anti-parallel to the applied field. Thus, given A C <
0, the two contributions to Eq.4 add up, yield-ing an increased resonance frequency. Cr -Cr ions,localized in the middle of the ring, are expected tohave a negligible magnetic moment along the directionof the applied field, thus inducing a small resonance-frequency shift. The magnetic moment of Cr -Cr ions § The difference ( E α (cid:48) − E α ) in Eq.5 is weakly field dependent,because the main contribution comes from hyperfine interaction,and it is small with respect to β = 1 /K B T . is instead comparable with the Cr -Cr one. However,electronic spins on these sites are parallel to the appliedfield and thus the two contributions in Eq.4 are oppo-site ( A C < Cr nuclei on 4-5 and 3-6 sites havenot been observed in the spectra. A core polarizationfield twice as high as -12.7 T/ µ B would be necessary toshift them in the experimental frequency window (seeFig.6-b), a value not compatible with previous resultson similar compounds (see Table 1). Therefore, Crnuclei on 1-8 and 2-7 sites are the only ones that can bedetected in the explored frequency range. Thus, Cr-NMR experimental data confirm the NC configurationof the electronic local spin moments along the Cr Cdring. At last, Fig.6-a show that the measured signalsare mainly due to nuclei whose electronic spins are inhas an effective total-spin S ≥ ab Figure 6.
Calculated resonance frequencies for eachmagnetically-equivalent pair of ions: Cr -Cr and Cr -Cr (panel a) and Cr -Cr and Cr -Cr (panel b) (see also Fig.1-b). The colour-map indicates the effective total-spin S eff (where (cid:104) S (cid:105) = S eff ( S eff + 1)) of the populated multiplets consideredin the calculation (only levels with a non-negligible populationat 1.4 K have been considered). Black scatters represent bothfield- and frequency- sweep experimental data. These figuresdemonstrate that only Cr nuclei on 1-8 and 2-7 sites have beendetected in the explored frequency range. Dashed lines indicatethe lower limit of the experimental frequency range. reaking the ring: Cr-NMR on the Cr Cd molecular nanomagnet
4. Conclusions
In conclusion, we have studied the local spin densitiesalong the Cr Cd AF ring with Cr-NMR. Thismagnetically-open ring is a model system to studyopen AF chains with an even number of spins. Fromthe NMR spectra we have extracted low-temperatureresonant frequencies of Cr-NMR at different appliedmagnetic fields. By comparing these results withour calculations, we have obtained a core polarizationfield for Cr Cd of A C = -12.7 T/ µ B , in agreementwith the results on the parent compounds Cr Cd andCr Ni. Moreover, we have investigated the expectationvalues of the local spins (cid:104) s z (cid:48) ,i (cid:105) along the directionof the applied magnetic field, confirming a NC spinconfiguration, when Cr Cd is in a magnetic state.We have also demonstrated that with Cr-NMRit is possible to obtain the same information onthe spin arrangement of AF rings, which can beextracted from more resource- and time consumingtechniques like polarised neutron diffraction. Evenif both experiments require suitably large singlecrystal samples, the flexibility of Cr-NMR and thestraightforward data analysis allows one to performthese kind of experiments on rings/chains with anhigher number of spins or on more complex systems.In order to characterise new compounds with Cr-NMR more data should be collected with respect tothose presented here for the well-known Cr Cd ring.For instance, more information could be obtained bysimply measuring NMR spectra for different directionsof the applied magnetic field. Finally, it is worthunderlining that NMR is a very effective techniqueto characterise molecular nanomagnets in view oftheir application in quantum information processing.Indeed, other experiments can be combined toinvestigate local spin moments on the magnetic ionsexploiting the same NMR apparatus. For instance,measurements of the H spin-lattice relaxation timeT can be exploited to probe electronic relaxationas a function of field and temperature [26, 48, 49].Most importantly, Rabi oscillations of the nuclearmagnetization of a Yb qudit has been recentlymeasured with
Yb-NMR [50], demonstrating thecapability to coherently manipulate qubits and quditswith NMR radio frequencies.
Acknowledgments
Authors gratefully acknowledge financial support fromFIRB Project No. RBFR12RPD1 and PRIN Project2015 No. HYFSRT of the Italian MIUR and from theEuropean project QuantERA 2017 - SUMO.
References [1] Haldane F D M 1983
Phys. Rev. Lett. Phys.Rev. Lett. Phys. Rev. Lett. Phys. Rev. B Phys. Rev. B Phys. Rev. Lett.
Phys. Rev. B Phys. Rev. B Nat. Mater. Nat. Phys. Science
Nature
Phys. Rev. B Phys. Rev. B Phys. Rev. Lett. Nat. Phys. PNAS ¨ udelI S and Mutka H 2009 Phys. Rev. Lett. ¨ assle P L W T P, Mutka H, GChristou,Waldmann O and Schnack J 2012 Phys. Rev. B J. Am. Chem. Soc.
Angew. Chem. Int. Ed. Phys. Rev. B J. Am. Chem. Soc.
Phys. Rev. B Phys. Rev.B Phys.Rev. B Phys.Rev. Lett. reaking the ring: Cr-NMR on the Cr Cd molecular nanomagnet Weihe H, Winpenny R E P and Neese F 2007
J. Am.Chem. Soc.
Chem.Commun.
Phys.Rev. B Phys.Rev. B J. Chem. Phys.
Angew. Chem.
Phys. Rev. B J. Am. Chem. Soc.
Chem. Sci. Phys.Rev. Lett. Phys.Rev. Lett. Nat. Nanotech. Phys. Rev. Lett.
Nat. Commun. Nat.Commun. J. Phys.: Cond. Matter Nat.Commun. Rev.Sci. Instrum. Z. Phys. Rare-Earths Magnetism (Oxford: Clarendon Press)[48] Bianchi A, Carretta S, Santini P, Amoretti G, Lago J,Corti M, Lascialfari A, Arosio P, Timco G and WinpennyR E P 2010
Phys. Rev. B Phys. Rev. B J. Am. Chem. Soc.140