PHYSICAL REVIEW B 87, 195412 (2013)

Landau-Zener tunneling of a single Tb3+ magnetic moment allowing the electronic read-out ofa nuclear spin

M. Urdampilleta,1 S. Klyatskaya,2 M. Ruben,2,3 and W. Wernsdorfer1

1Institut Neel, CNRS et Universite Joseph Fourier, BP 166, F-38042 Grenoble Cedex 9, France2Institute of Nanotechnology (INT), Karlsruhe Institute of Technology (KIT), 76344 Eggenstein-Leopoldshafen, Germany

3Institut de Physique et Chimie des Materiaux de Strasbourg (IPCMS), CNRS-Universite de Strasbourg, 67034 Strasbourg, France(Received 30 October 2012; revised manuscript received 13 February 2013; published 9 May 2013)

A multiterminal device based on a carbon nanotube quantum dot was used at very low temperature to probea single electronic and nuclear spin embedded in a bis-(phthalocyaninato) terbium (III) complex (TbPc2). Aspin-valve signature with large conductance jumps was found when two molecules were strongly coupled tothe nanotube. The application of a transverse field separated the magnetic signal of both molecules and enabledsingle-shot read-out of the terbium nuclear spin. The Landau-Zener (LZ) quantum tunneling probability wasstudied as a function of field sweep rate, establishing a good agreement with the LZ equation and yielding thetunnel splitting �. It was found that � increased linearly as a function of the transverse field. These studies arean essential prerequisite for the coherent manipulation of a single nuclear spin in TbPc2.

DOI: 10.1103/PhysRevB.87.195412 PACS number(s): 75.50.Xx, 81.07.Nb, 81.07.Ta, 85.75.−d

I. INTRODUCTION

The detection and manipulation of single localized spins is arapidly developing field of nanoscience, enabling the creationof quantum bits and memories. Among the large variety ofspin systems, electron spins confined in quantum dots,1–5

impurities,6 defects in semiconductors,7 and nitrogen-vacancycenters8–11 are promising candidates for spin-based quantumcomputation. However, control of the decoherence inducedby the environment12 and control of the coupling betweendifferent spins are still experimental challenges. Single-molecule magnets (SMMs) offer an interesting alternative forthe embedding of spins in a controllable and reproducibleenvironment.13–18

A SMM consists of an inner magnetic core, containing ingeneral several magnetic atoms, and an outer nonmagneticshell, which is chemically tailored through the addition ofligands.19,20 At low temperatures, the magnetic core of anSMM behaves like a single magnetic moment, which can attainspin ground states up to S = 83/2.21 SMMs are characterizedby a broad range of quantum effects, ranging from quan-tum tunneling of magnetization (QTM)22,23 to Berry-phaseinterference.24 Their spin coherence time is expected to reachthe microsecond range,25,26 and the perspective of entan-glement between SMMs27 makes them good candidates forquantum spintronics14 and quantum computations.28 Amongthe large variety of known SMMs, rare earth based SMMsare of particular interest because the high intrinsic anisotropyof rare earth elements can lead to larger zero-field splittingsthan for transition metal based SMMs.29 Furthermore, thestrong hyperfine interaction of rare-earth elements leads towell separated resonant tunnel transitions for each nuclearspin state.30 Because of the excellent intrinsic isolation ofnuclear spins, they are promising candidates for spin quantumbits.31

In this paper, we demonstrate the probing of singleelectronic and nuclear spins in rare earth based SMMsadsorbed onto the sidewall of a carbon nanotube. The presentstudy focuses on the so-called terbium “double decker” orbis-(phthalocyaninato) terbium (III) complex, referred to as

TbPc2 in the following, which was synthesized using themethod reported in Ref. 32.

II. SPIN HAMILTONIAN AND LANDAU-ZENERTUNNELING IN TbPc2

The TbPc2 complex is composed of two organic phthalo-cyanines (Pc2) coordinating a single terbium ion in a squareantiprismatic geometry [Fig. 1(b)]. The 4f 8 electronic shellof the Tb3+ leads to a total magnetic moment J = L + S =3 + 3. In addition, the Tb ion has a nuclear spin I = 3/2. Theπ -type electronic structure of Pc2 permits coupling betweenthe Tb and the environment.33,34 In order to calculate theZeeman diagram of TbPc2 [Fig. 1(a)], we used the ligand fieldHamiltonian expressed in the Stevens operator formalism,35

Hlf = αA02O

02 + β

(A0

4O04 + A4

4O44

) + γA06O

06 , (1)

and the Zeeman Hamiltonian,

HZeeman = gJ μ0μBJ · H, (2)

where the Okq are the Stevens operators,36 Ak

q the ligand fieldparameters from Ref. 37, α, β, and γ the Stevens constanttabulated in Ref. 38, gJ the gyromagnetic factor of Tb,J the electronic magnetic moment operator of Tb, and Hthe applied magnetic field vector. Numerical diagonalizationof HZeeman + Hlf gives |J = 6,Jz = ±6〉 as a ground statedoublet, isolated from the first excited state |J = 6,Jz = ±5〉,by about 600 K [Fig. 1(a)]. Therefore, the Tb magneticmoment is aligned with the quantization axis (easy axis)within the low temperature and low magnetic field regime(T < 1 K, μ0H < 2 T) and is considered as an Ising-type spinin the following. The presence of a transverse anisotropy inthe ligand field Hamiltonian (A4

4O44 ) couples the two states

|Jz = ±6〉, giving rise to an avoided level crossing with atunnel splitting � ≈ 1 μK [Fig. 1(c)]. As a result, quantumtunneling of magnetization (QTM) can occur in TbPc2 whenthe magnetic field is swept through this avoided level crossing.The tunneling probability is given by the Landau-Zener

195412-11098-0121/2013/87(19)/195412(6) ©2013 American Physical Society

URDAMPILLETA, KLYATSKAYA, RUBEN, AND WERNSDORFER PHYSICAL REVIEW B 87, 195412 (2013)

0

1050-5-10

E/k

B (K

)

(a)

200

400

600

800

(c)

50-5

0

2

-2

JZ=-6

JZ=+6

JZ=-5JZ=+5

E/k

B (µ

K)

easy axis(b)

µ0H (T) µ0H (10-7T)

DTQTM Δ1 PQTM

1-PQTM

-0.4

0

0.4

E/k

B (

K)

-100 -50 0 50 100

µ0H (mT)

Iz = -3/2

Iz = -1/2

Iz = +1/2

Iz = +3/2

(d)

FIG. 1. (Color online) (a) Zeemandiagram of the ground state J = 6. |J =6,Jz = ±6〉 is the low lying ground statedoublet, which is separated by about600 K from the first excited state |J =6,Jz = ±5〉. The two mechanisms formoment reversal are quantum tunnelingof magnetization (QTM) occurring atzero magnetic field and the direct tran-sition (DT) occurring at finite magneticfield. (b) Structure of the TbPc2 complex.The black, blue, white, and purple atomscorrespond to C, N, H, and Tb, respec-tively. The magnetic easy axis and thehard plane are respectively orthogonaland parallel to the phthalocyanines. (c)Zoom on the avoided level crossing labelby QTM in (a). � is the tunnel splitting,and PQTM is the tunnel probability. (d)Zeeman diagram of the ground doublet|Jz = ±6〉 split by the strong hyperfineinteraction of Tb. The color code illus-trates the different nuclear spin states andthe circled intersections are avoided levelcrossings.

equation,39

PQTM = 1 − exp

[−π�2

4hgJ μB | Jz | μ0dHz

dt

], (3)

where dH/dt is the magnetic field sweep rate and Jz = ±6.When the electronic moment does not tunnel at the avoidedlevel crossing, it ends up in the excited state and can relax to theground state via a direct transition (DT). This process involvesa spin-phonon interaction in which the Zeeman energy isreleased via a phonon. It occurs in the presence of a magneticfield typically larger than 0.2 T.30

An important property of the Tb ion is the presence ofa I = 3/2 nuclear spin. The strong hyperfine interactionwith the electronic moment splits each ground state |J =6,Jz = ±6〉 into four substates. We considered the followingspin Hamiltonian containing the ligand field interaction, theZeeman term, the hyperfine interaction, and the quadrupolarterm:

H = HLF + HZeeman + AI · J + P(I 2z − 1

3 (I + 1)I), (4)

where A is the hyperfine constant, P the quadrupolar con-stant, and I the Tb nuclear magnetic moment operator. Thecorresponding Zeeman diagram of the ground multiplet isplotted in Fig. 1(d). Among the sixteen level intersections,only the four encircled intersections correspond to avoidedlevel crossings. Tunneling can take place at one of theseavoided level crossings. It changes the electronic magneticmoment by �Jz = ±12 but conserves the nuclear spin state.As a consequence, the measurement of the field position ofQTM in TbPc2 enables a direct read-out of the Tb nuclear spinstate.

III. SAMPLE FABRICATION AND COULOMB BLOCKADE

QTM was studied in individual TbPc2 complexes, laterallycoupled to a carbon nanotube quantum dot. Catalyst islands ofnanoporous alumina with Fe/Mo were first patterned on a SiO2

surface with a metallic back gate by optical lithography usingLOR3A resist. The nanotubes were then grown in a FirstnanoCVD oven at 750 ◦C, whereby methane was used as a carbonsource. Source, drain, and side gate electrodes were patternedby aligned electron-beam lithography, defining 200 nm longCNT junctions. The residual resist was removed by annealingthe sample at 300 ◦C under Ar flow. The bis-(phthalocyaninato)Tb(III) complexes were functionalized with one pyrene andsix hexyl groups in order to improve their grafting onto thecarbon nanotube32,40 and prevent their crystallization. Theywere suspended in dichloromethane and drop-casted onto thenanotubes. Samples with large source-drain resistance (>100k� at room temperature) were microbonded and measured ina dilution fridge with a base temperature of about 40 mK. Theelectronic temperature of the source and drain electrodes wasestimated to be about 150 mK. The setup was equipped withtwo magnets, generating up to 1 T in two orthogonal directions.Magnetic moment reversal of the TbPc2 was detected by meansof conductance measurements through the carbon nanotube,as proposed previously.14 The differential conductance wasmeasured as a function of side gate and back gate with anAdwin real-time acquisition system, programed in a lock-inmode (100 μV and 67 Hz) at zero bias. Figure 2 presents thedifferential conductance measurements of this four-terminalsdevice, shown in the inset of Fig. 2. The conductance linescorrespond to the Coulomb peaks (charge degeneracy state),and the zero-conductance region corresponds to Coulombblockade. Both the side gate and the back gate are used totune the chemical potential of the quantum dot, and to modifyits coupling with the leads and the TbPc2 molecules.

195412-2

LANDAU-ZENER TUNNELING OF A SINGLE Tb3+ . . . PHYSICAL REVIEW B 87, 195412 (2013)

dI/dV (nS)100500

0

Vbg (V)-3

-3

3

0

V sg

(V)

3-2 -1 1 2

-2

-1

1

2

250 nm

FIG. 2. (Color online) Stability diagram of the device shown inthe inset. The red lines are degeneracy points in the Coulomb blockaderegime. Coupling between the leads and the quantum dot can be tunedby moving along one of these lines, by means of the back gate andside gate voltages (Vbg and Vsg, respectively). Inset: scanning electronmicrograph of the studied device. The green electrodes correspond tothe source and drain, and the blue electrode corresponds to the sidegate.

IV. SPIN-VALVE FEATURES

We recently discovered a surprisingly strong interactionbetween the charge carriers of a carbon nanotube and TbPc2

molecules grafted onto its sidewall.16 We found that thisinteraction gives rise to a spin-valve signature in the magneto-transport measurements: when the magnetic moment of twomolecules are aligned in the parallel configuration the conduc-tance is maximum while in the antiparallel configuration theconductance is minimum. Even though two theoretical studiesproposed an interpretation of the underlying mechanism,41,42

more studies are needed to fully understand this effect.For the present study, the conductance of the device in

Fig. 2 was recorded as a function of the magnetic field for eachnonzero conductance region. The spin-valve feature presentedin Fig. 3(a) was observed at Vbg = 0 V and Vsg = 0 V. Alongthe corresponding degeneracy line, the signal vanished. Onereason for this gate dependence could be that the couplingbetween the molecule and the nanotube, mediated by the π

type electrons of the Pc2, was changed by the electric field.Indeed, it was shown by Lodi Rizzini et al.33 that the exchangecoupling between the Tb and the environment is very sensitiveto the oxidation state of the Pc2. This possible explanationhas to be strengthened by further experiments and theoreticalcalculations.

This spin dependent effect is best visible when the twomolecules have different coercive fields: in this case, eachmagnetic moment reversal induces a sharp change in theconductance of the device at a different field. In the presentsample, one of these molecules (referred to as molecule A inthe following) had a QTM probability close to one, inside theexperimentally accessible sweep rate window, while the other

FIG. 3. (Color online) (a) Magnetoconductance recorded at asweep rate of 50 mT s−1. The blue curve corresponds to themeasurement made from −0.35 T up to +0.35 T, and the red curvecorresponds to the one made from +0.35 T down to −0.35 T. Eachof the sharp changes in conductance corresponds to the signature of asingle TbPc2. The reversal, which takes place under a small magneticfield, is due to QTM, whereas that which takes place under a highermagnetic field is due to DT. (b) Magnetoconductance recorded at asweep rate of 20 mT s−1; the color code is the same as for (a). Thesharp conductance changes occur close to zero magnetic field and arethe consequence of both molecules experiencing QTM.

molecule (molecule B) had a sweep rate dependent probability.It is not surprising that both molecules had a different tunnelsplitting amplitude, since any deformation of the molecule(e.g., a slight twist between both Pc ligands) generates a sig-nificant change in the longitudinal and transverse anisotropy.43

For instance, for rather fast field sweep rates, A experiencedQTM while B experienced a DT. The correspondingmagneto-transport measurement are presented in Fig. 3(a).For slow field sweep rates, the magnetic moment of bothmolecules reversed by tunneling [Fig. 3(b)] and it wastherefore difficult to distinguish between molecule A and B.However, because the molecules had their easy axes aligned intwo different directions [Fig. 4(a)], with a constant transversefield H⊥, orthogonal to the easy axis of molecule B, we coulddiscriminate between the QTM positions of molecules A andB. Indeed, as depicted in Fig. 4(a), the longitudinal field feltby molecule A is H⊥sinθ + H‖cosθ while the field felt bymolecule B is simply H‖, where θ is the angle between the easyaxes of molecules A and B. More details concerning the mag-netic field alignment are presented in the following section.

V. DETERMINATION OF THE EASY AXIS OFMAGNETIZATION

In order to determine the easy axis of magnetization ofmolecule B, the spin-valve effect was measured as a function ofthe applied magnetic field angle. For each angle we measured

195412-3

URDAMPILLETA, KLYATSKAYA, RUBEN, AND WERNSDORFER PHYSICAL REVIEW B 87, 195412 (2013)

0°

45°

90°

135°

180°

Δ(d

I/dV

) (n

S)

-20

20

0

0 0.5 1

225° 315°

µ0H (T)

90°

θ H//

HA B

(b)

(a)

FIG. 4. (Color online) (a) Schematic of the relative orientation ofthe two molecules on the nanotube. We define the longitudinal fieldH‖ and transverse field H⊥ with respect to the molecule B which hasits easy axis perpendicular to the nanotube axis [see Fig. 4(a)], andθ the angle between the easy axis of B and A. (b) Anisotropy of thedirect transition. The difference between the trace (from −1 T to +1T) and the retrace (from +1 T to −1 T) is recorded as a functionof angular orientation. The locus of the direct transition reversal isindicated by dashed lines. It is important to note that the reversal closeto the 90◦ orientation cannot occur because of the field limitationsimposed by the magnetic field coils in the experimental setup.

the magnetoconductance for the trace (from −1 T to +1 T)and the retrace (from +1 T to −1 T). A relative fast sweeprate of 50 mT s−1 was used to have a high probabilityfor a DT of molecule B. Figure 4(b) shows the differencebetween retrace and trace for positive magnetic fields andthe difference between trace and retrace for negative fields.The red areas between −70◦ and +70◦ correspond to curvessimilar to Fig. 3(a), whereas the region between +70◦ and+110◦ correspond to a situation where the magnetic field isnot strong enough to allow a DT [see Fig. 3(b) of Ref. 44]. Fromthis characterization, the easy axis of molecule B was foundto be orthogonal to the nanotube axis. As a consequence, thetransverse field was therefore applied along the nanotube axis.

VI. ELECTRONIC READ-OUT OF ASINGLE NUCLEAR SPIN

Figure 5(a) shows measurements recorded with a constanttransverse field of 0.35 T and presenting only QTM features,the ones presenting DT being rejected. The conductance sud-denly decreases, between the magnetic field values of −50 mTand +50 mT (QTM of B), and then increases in all cases aboveapproximately +50 mT (QTM of A). This measurement wasrepeated 3500 times with a 100 mT s−1 sweep rate. WheneverDT failed to occur, the longitudinal position of the QTM inmolecule B was stored in the histogram plotted in Fig. 5(b):four peaks emerge with a FWHM of approximately 10 mT,and a mean peak-to-peak separation of 25 mT.

-0.4

0

0.4

E/k B

(K)

(a)

(b) 60

40

20

0

cou

nt /

3500

-100 -50 0 50 100

µ0H// (mT)

60

40

00 0.20.1-0.1-0.2

µ0H// (T)

dI/d

V (n

S)

20

B

Aµ0H┴

= 0.35 T

FIG. 5. (Color online) (a) Magnetoconductance curves recordedunder a 0.35 T transverse field, at a 100 mT s−1 sweep rate. Therelative QTM position for molecules A and B are clearly split;the angle between their easy axes is then estimated around 15◦.(b) Histogram of the QTM position of molecule B, for 3500consecutive traces recorded under the same conditions as thoseapplied in (a). The position of these peaks corresponds to theavoided level crossing shown on the Zeeman diagram of the grounddoublet Jz = ±6 split by the hyperfine coupling with the nuclear spinI = 3/2.

In order to explain these results, we compare the positionof these peaks with the Zeeman diagram. Figure 5(b) showsthe very good correspondence between the four peaks and theavoided-level crossing of the Zeeman diagram. This diagramis slightly different from the one presented in Fig. 1(c) sincewe took into account that the easy axis is not lying exactly inthe plane (H‖,H⊥). As evident from the comparison betweenthe histogram and the diagram, each of these peaks can beattributed to a particular nuclear spin state.

Similar results have recently been demonstrated at thesingle molecule level by Vincent et al.,17 in a moleculartransistor configuration. It is important to note that in thepresent case the FWHM is larger than in the case of Vincentet al. leading to a lower fidelity in the single-shot read-out measurement. One reason for this could be that thecurrent tunneling through the carbon nanotube is interactingmore strongly with the TbPc2 molecules. Nevertheless, oneadvantage of the present device is the very large variationof conductance induced by the spin reversal (200% in thepresent case versus 1% in the work of Vincent et al.17), which

195412-4

LANDAU-ZENER TUNNELING OF A SINGLE Tb3+ . . . PHYSICAL REVIEW B 87, 195412 (2013)

1.0

0.8

0.6

0.4

0.2

0.0100806040200

Transverse field: 0 T 0.1 T 0.2 T 0.3 T

dt/d(µ0H) (s/T)

P QTM

3

2

0.20.0

Δ (1

0−6 Κ

)

µ0Htransverse (T)

FIG. 6. (Color online) QTM probability as a function of theinverse of the sweep rate, for different transverse fields. Theexperimental data are fitted by exponential curves, in accordancewith Eq. (3). The inset indicates the variation of tunnel splitting as afunction of the transverse field; the red line is a guide for the eye.

makes the measurement very easy since it does not require anyspecific filtering (physical or numerical).

In order to confirm that only one QTM position exists pernuclear spin state, we measured the tunneling probability as afunction of the sweep rate and compared it with the Landau-Zener theory. 100 magnetoconductance measurements wererecorded for a given sweep rate. The tunneling probabilityPQTM of molecule B can be obtained by PQTM = 1 − PDT,where PDT is the probability of a DT [Fig. 1(c)]. PQTM is plottedin Fig. 6 where the experimental data were fit with Eq. (3),from which a tunnel splitting of � = 1.7 μK was extracted.As a consequence, the monoexponential behavior clearlydemonstrates that only the four circled level crossings inFig. 1(d) are avoided level crossings. This is in agreement withthe work of Vincent et al.17 but not with the measurement doneon a single crystal of TbPc2. The latter case presents QTM at allintersections of the diagram depicted in Fig. 1(d); see the workof Ishikawa et al.30 In an assembly of molecules, coupled withweak dipole interactions, multispin tunnel effects45–47 mightbe responsible for this observation but further investigationsare needed to better understand this issue.

VII. LANDAU-ZENER TUNNELING IN A SINGLE TbPc2

An applied transverse field tunes the tunnel splittings viathe H⊥(J+ + J−) term of the Hamiltonian. In order to studythis effect on the different level crossings, we measured thetunneling probability for several constant transverse fields andfield sweep rates. The symbols in Fig. 6 correspond to theexperimental points and the continuous lines are least-square

fits using Eq. (3). The data agree very well with the Landau-Zener behavior, which suggests that no other measurableavoided-level crossings are induced by the application of atransverse field. The tunnel splitting amplitudes were extractedfrom the fits, and then plotted as a function of the transversefield (inset of Fig. 6). This behavior cannot be explainedby using the parameters of Ishikawa et al.30 First, theligand field might be different from the mean bulk valuebecause the grafted molecules are probably slightly distortedand their anisotropy modified.43 Secondly, the ligand fieldHamiltonian does not predict a linear increase as observed forour measurements, which were confirmed on other moleculardevices. This observation needs further experimental andtheoretical investigations. In particular, we believe that, in thecase of single molecules, the angular moment conservationhas to be taken into account. When the latter is not conserved,the Landau-Zener equation is not valid.48 Nevertheless, ourstudies showed that the hyperfine interaction is a robustfeature allowing us to read the nuclear spin state regardlessof the deposition and measurement techniques. Moreover, thepossibility of tuning the tunnel splitting is very convenient forexperiences of coherent nuclear spin manipulation since theread-out mechanism needs the right value: not too small (noread-out) and not too large (DT possible, reducing the read-outfidelity).

VIII. CONCLUSION

We quantitatively investigated quantum tunneling of mag-netization at the single-molecule level and confirmed thata nanotube-based device is one of the best means for thedetection of single magnetic moment reversal. Furthermore,we demonstrated that the nanotube device used for thesemeasurements was able to read-out in a single shot the stateof an individual nuclear spin. The demonstrated tunability ofthe tunnel splitting, achieved by applying a transverse field,provides the possibility to adjust the quantum dynamics ofsuch quantum spintronic devices.

ACKNOWLEDGMENTS

This work was partially supported by the DFG programsSPP 1459 and TRR 88 3Met, ANR-PNANO project Mol-NanoSpin No. ANR-08-NANO-002, ERC Advanced GrantMolNanoSpin No. 226558, STEP MolSpinQIP, and theNanosciences Foundation of Grenoble. The samples weremanufactured at the NANOFAB facilities of the Neel Institute.We wish to thank B. Fernandez, G. Julie, S. Dufresne,T. Crozes, and T. Fournier for their assistance in manufacturingthe nanotube device. We also wish to thank F. Balestro,J.-P. Cleuziou, D. Feinberg, M. Ganzhorn, N.-V. Nguyen,T. Novotny, and R. Vincent for fruitful discussions, andE. Bonet, C. Thirion, and R. Piquerel for the softwaredevelopment. We also thank E. Eyraud, R. Haettel, C. Hoarau,D. Lepoittevin, and V. Reita for their technical support.

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