Quantum Spin Dynamics of Molecular Spintronic Devices Based on Single-Molecule Nanomagnets

Quantum Spin Dynamics of Molecular Spintronic

Devices Based on Single-molecule Nanomagnets

Author Supervisor

Kieran HYMAS Assoc. Prof. Alessandro SONCINI ORCID: 0000-0003-1761-4298

Submitted in total fulfilment of the requirements of the degree of Doctor of Philosophy

School of Chemistry The University of Melbourne September 19, 2020 This page is intentionally left blank Abstract

In recent years, molecular analogues to electronic devices have been sought after to remedy the practical limitations imposed on classical circuits by Moore’s law leading to the inception of the multidisciplinary research field of molecular electronics. In addition to the miniaturisation of current electronic technologies, researchers have sought also to exploit the interplay between the spin degree of freedom inherent to magnetic molecules embedded in devices and to the local electronic currents to which they are coupled. Single- molecule magnets (SMMs), metal complexes with large magnetically anisotropic spin moments that exhibit slow relaxation effects, have enjoyed a position in the subfield of molecular spin-electronics (spintronics) as magnetic units that may act as elements in new molecular-scale spintronic technologies. In this thesis, three projects composed of theoretical models of spin transport through single-molecule magnet-based spintronic devices are presented which serve to predict and explain the quantum spin dynamics exhibited by novel device set-ups that are based on current state-of-the-art experimental systems.

In the first project, I have contributed to the development of two models of spin-polarised transport through a general molecular nanomagnet device that is perturbed either by some time dependent, resonant perturbation or by a static perturbation. In the former case, a study of the time evolution of the quantum states of the nanomagnet revealed Rabi oscillations between spin states that are resonantly coupled by the perturbation, suggesting that these states could behave as a molecular qubit for quantum computation that is addressed with a spin-polarised current. In the steady-state limit of the time-dependent model the device functions as a spin current pump, amplifier and inverter which could be potentially useful for logic gates in novel circuitry based on the spin degree of freedom of an electronic current rather than on the charge; these effects are preserved in time-averaged current measurements of the device operating under a pulsed radiation regimen. In the second model, the spin inversion property of the device is preserved even when using a static rather than a time-dependent perturbation owing to a mixing between electric current blockaded and non-blockaded states.

In the second project, I have contributed to the theoretical description of electron transport through a molecular break junction device housing a single terbium bis-phthalocyaninato (TbPc2) nanomagnet. The

i ii

model developed in this project is shown to capture all experimental properties measured for the single-

molecule device, in particular, its magneto-conductance dependence on the applied magnetic field, gate

and bias voltage. Crucially, using the model it was possible to confirm that different states of the molecular

magnet give rise to disparate signals in the magneto-conductance which may be used to perform an electrical

read-out of the molecular states of the device. At variance with previous interpretations that advocated for

a strongly coherent regime of electron transport through the device, the behaviour of the experimentally

observed magneto-conductance is shown here to be fully captured within the incoherent sequential tunnelling

regime.

In the third project, I have contributed to the understanding of an experimentally realised molecular spin

valve by providing the first simulations of the hysteresis of the differential magneto-conductance for a hybrid

molecular-quantum point contact three-terminal device, triggered by the slow relaxation of the SMMs grafted

to the device in a time-dependent sweeping magnetic field. The transport dynamics were modelled here in

a completely incoherent transport regime without necessitating the tenuous assumption of spin dependent

Fano-resonance interference that were invoked in previous theoretical studies of the devices. The signature

of the slow relaxation of two or more TbPc2 single-molecule magnets manifests in magneto-conductance measurements owing to a phonon-mediated direct relaxation process between the Tb electronic states leading to the transient population of non-conducting anti-parallel configurations of the TbPc2 magnetic moments. The model developed here was also able to capture well the temperature dependence of the experimentally measured molecular spin valve magneto-conductance as well as its dependence on bias voltage in the static

field regime. Declaration

In this declaration, I certify that this Ph.D. Thesis is comprised only of my original work except where otherwise stated. Appropriate credit has been given in this thesis whenever work of others has been ref- erenced. I also declare that this thesis is less than 100,000 words in length, exclusive of tables, maps, bibliographies and appendices.

Kieran Hymas

Date iii Preface

This Ph.D. has been funded by the Australian Government through an Australian Government Research

Training Program Scholarship from 2016 to 2019. Participation in international conferences has been fa- cilitated by funding from the University of Melbourne through a Science Abroad Travelling Scholarship

(2017).

The following papers have been published by the candidate and have been included as chapters in this thesis:

ˆ K. Hymas and A. Soncini, “Molecular spintronics using single-molecule magnets under irradiation”,

Phys. Rev. B, 99, 245404, (2019)

DOI: https://doi.org/10.1103/PhysRevB.99.245404

ˆ K. Hymas and A. Soncini, “Mechanisms of spin-charge conversion for the electrical read-out of 4f-

quantum states in a TbPc2 single-molecule magnet spin transistor”, Phys. Rev. B, 102, 045313, (2020)

DOI: https://doi.org/10.1103/PhysRevB.102.045313

ˆ K. Hymas and A. Soncini, “Origin of the hysteresis of magneto-conductance in a supramolecular spin

valve based on a TbPc2 single-molecule magnet”, Phys. Rev. B, in press (2020)

Kieran Hymas

Date iv Acknowledgements

The title doctor of philosophy (PhD) signifies that its bearer has contributed original research to their particular academic field of interest. More than that though, it represents passion, persistence, skill and integrity; it represents a constitution for creativity and a disposition for diligence. It is an object of prestige purchased only with late nights, many failures, few successes and almost certainly with some degree of caffeine addiction. While my postgraduate pursuit has been an incredible experience overall, I would feel disingenuous if not to say that it has been a true test of my character. Now, as I stride towards the end of this momentous journey and look back at how far I have come, I think back to a particular line pertaining to the philosophy of self-overcoming from Friedrich Nietzsche’s magnum opus: Thus Spoke Zarathustra. The quote reads:

Ready must thou be to burn thyself in thine own flame; how couldst thou become new if thou

have not first become ashes!

Friedrich Nietzsche, Thus Spoke Zarathustra

Much like The Creating One who is idealised by Zarathustra in the passage above, after this academic trial by fire, I have appeared on the other side anew as a much better version of myself. It goes without saying that the pursuit of a PhD never occurs in a vacuum and therefore I have many people to thank for their support along the way.

Among those that have shaped me into a better scientist, thinker and person, I must give special mention to my supervisor Associate Professor Alessandro Soncini. Alessandro’s passion for science and his drive remain, to this day, unmatched by anybody that I have met. As well as his tremendous laughter, his bottomless motivation and unabashed passion for science are positively infectious and I would always leave his office cheery, reinvigorated and with a new perspective for the problem at hand. On top of this, Alessandro’s academic history represents an intimidating tour de force in this challenging field and, on more than one occasion, I have found great inspiration for my own work while perusing his publications. I regularly half- joked throughout my PhD that I have learned more about science during my time in Alessandro’s research

v vi group than during my entire undergraduate degree. While this may be somewhat hyperbolic, I believe that the way I have “learned to learn” new skills and information during my PhD will stay with me for a great deal longer than any obscure chemical structure committed to memory during my undergraduate years. So then, I can’t thank Alessandro enough for providing me with a position as a PhD student in his group, introducing me to this phenomenal field and offering his guidance and patience over these last four years.

Furthermore, I count myself superbly lucky to have been a member of a research group filled with such kind, talented and interesting individuals. I thank Matteo Piccardo for the many coffees, beers and (most importantly) stimulating scientific and philosophical conversations, Shashank Rao for entertaining not only my scientific curiosity but also my interest in classical literature and philosophy, Simone Calvello and Haibei

Huang for their many enlightening seminars and Jared Ashtree for his ruthless attention to detail when proof reading my writing or critiquing my practice talks.

Throughout my postgraduate studies I have also received support from people outside of our research group. Asim Najibi has been my desk neighbour and closest friend throughout this degree and, in spite of his love for arcane middle-eastern music and his tendency towards dreadful puns, I must thank him extensively for making even the most frustrating days bearable. Furthermore, I would like to thank both Dr Lars Goerigk and Dr Wallace Wong for appearing with Alessandro on my PhD committee. By unhesitatingly assuming the role of committee chair, Lars also took on an extra degree of responsibility for which I am grateful.

Without a shadow of a doubt, my family have also played a significant role in supporting me during my studies and I feel privileged to always find them within arms reach whenever I need them; I know many for whom this luxury is not available. I would like to thank my parents, Ryan and Lisa, for raising me to be the man that I am today and for looking after me even after I have flown the coop. Finally I would like to thank my sister Kimberley for her support over the years and for consistently putting a smile on my face. List of Publications

The work developed during my Ph.D. is featured in several publications, given below.

ˆ K. Hymas and A. Soncini, “Molecular spintronics using single-molecule magnets under irradiation”,

Phys. Rev. B, 99, 245404, (2019)

DOI: https://doi.org/10.1103/PhysRevB.99.245404

ˆ K. Hymas and A. Soncini, “Mechanisms of spin-charge conversion for the electrical read-out of 4f-

quantum states in a TbPc2 single-molecule magnet spin transistor”, Phys. Rev. B, 102, 045313, (2020)

DOI: https://doi.org/10.1103/PhysRevB.102.045313

ˆ K. Hymas and A. Soncini, “Origin of the hysteresis of magneto-conductance in a supramolecular spin

valve based on a TbPc2 single-molecule magnet”, Phys. Rev. B, in press (2020).

vii Contents

Abstract i

Declaration iii

Preface iv

Acknowledgements v

List of Publications vii

List of Figures xi

Prologue 1 Single-molecule Magnets ...... 2 Memory Effects in Molecular Nanomagnet Spintronic Devices ...... 6 Theoretical Models of Nanomagnet Spin Dynamics in Molecular Devices ...... 10 Research Questions ...... 12 Project 1 ...... 13 Project 2 ...... 13 Project 3 ...... 13 Thesis Outline ...... 14

1 Quantum Rate Equations for Transport through a Nanomagnet Spintronics Device 16 1.1 Equations of motion for the Density Operator ...... 16 1.2 Dissipative Dynamics from the Von Neumann Equation ...... 18 1.3 The Secular Approximation for the Reduced Density Matrix ...... 25 1.4 Recoupling Populations and Coherences with a Resonant Perturbation ...... 27

I Manipulating Spin Currents in Single-molecule Magnet Spintronic De- vices through the Perturbation of Individual Quantum Spin States 29

2 Molecular Spintronics Using Single-molecule Magnets Under Irradiation 30 2.1 Abstract ...... 31

viii ix CONTENTS

2.2 Introduction ...... 31 2.3 Theoretical Model ...... 33 2.3.1 Model Hamiltonian ...... 33 2.3.2 Master Equation in a Time-dependent Resonant Field and Stationary Current . . . . 35 2.4 Results and Discussion ...... 37 2.4.1 Continuous Radiation ...... 38 2.4.2 Pulsed Radiation ...... 40 2.4.3 Candidate Magnets for the Device ...... 42 2.5 Conclusion ...... 42 2.6 Appendix A: Alternate Resonant Perturbation Coupling Schemes ...... 43 2.7 Appendix B: Non-secular Rate Equation ...... 44 2.8 Appendix C: Device Operation Without Ferromagnetic Spin Injection ...... 45

3 Spin Current Switching with a Single-Molecule Magnet Immersed in a Static Transver-

sal Magnetic Field 47 3.1 Introduction ...... 47 3.2 Theoretical model ...... 49 3.2.1 Device Hamiltonian ...... 49 3.2.2 Quantum Master Equation and Stationary Spin Currents ...... 50 3.3 Results and discussion ...... 52 3.4 Conclusion ...... 53

4 Conclusions and Future Work 55

II Addressing the Quantum States of a Single Nanomagnet Break Junction 57

5 Mechanisms of Spin-Charge Conversion for the Electrical Read-out of 4f-Quantum

States in a TbPc2 Single-molecule Magnet Device 58 5.1 Abstract ...... 59 5.2 Introduction ...... 59 5.3 Theoretical Model ...... 61 5.3.1 Coulomb Blockade Transport Model ...... 62 5.3.2 Temperature, magnetic field and bias voltage dependence of the conductance . . . . . 68 5.3.3 Coherent Corrections to Transport ...... 70 5.4 Conclusions ...... 72 5.5 Appendix A: General model - Hyperfine coupling and asymmetric coupling to the leads . . . 73 5.5.1 Hyperfine Levels ...... 73 5.5.2 Asymmetric coupling ...... 75 CONTENTS x

5.6 Appendix B: Conductance formula at zero bias ...... 75

6 Conclusions and Future Work 77

III Hysteresis Loops of Magneto-conductance in a Driven Single-molecule Magnet Molecular Spintronic Device 79

7 Origin of the hysteresis of magneto-conductance in a supramolecular spin valve based

on a TbPc2 single-molecule magnet 80 7.1 Abstract ...... 81 7.2 Introduction ...... 81 7.3 Theoretical Model of the Spin Valve ...... 85 7.3.1 Spin Valve Hamiltonian ...... 85 7.3.2 Time-dependent transport in the adiabatic approximation ...... 88 7.4 Results and Discussion ...... 90 7.4.1 Hysteresis of the Magneto–Conductance and Dynamical Spin–Valve Effect ...... 90 7.4.2 Generalised Dynamical Model: Multiple Nanomagnets, Steady-state Conductance and

Higher Charge States ...... 93 7.5 Conclusions ...... 98

7.6 Appendix A: Model of disordered TbPc2 quantisation axes ...... 100 7.7 Appendix B: Exact Diagonalisation of the M = 2 case ...... 102

7.8 Appendix C: Nature of the Direct Relaxation Process in TbPc2 ...... 103 7.9 Appendix D: Energies and Amplitudes of N + 2 states defined in Eq. (III.7.9) ...... 105 7.10 Appendix E: Electron-Phonon Coupling and Limiting Cases of the Hopping Integral . . . . . 106

8 Conclusions and Future Work 108

IV Appendices and Bibliography 110

Appendix A: The Many Pictures of Quantum Mechanics 111

Appendix B: Perturbation Theory Using the T-Matrix Approach 114

Bibliography 117 List of Figures

1 General mechanism for single-molecule magnet relaxation. In each of the plots, the hori-

zontal lines represent spin angular momentum projections of the giant S manifold split by

2 the uniaxial potential −|D|Sz . Solid red arrows denote direct relaxation processes while the dashed red arrow indicates a quantum tunnelling of the magnetisation (QTM)...... 3

2 Energy splitting diagram of the lowest lying electronic states of a dysprosium complex. The

many-body 4f electronic states are split by a hierarchy of correlation, spin-orbit and crystal

field effects leading to well isolated crystal-field split spin-orbit multiplets that generally span

≤ 103 cm−1. Image used with permission from Associate Prof. A. Soncini...... 5

3 A) Schematic representation of Coulomb blockade in a spintronics device composed of two

electrodes and a neutral and charged state. A gate voltage can be applied to bring the

neutral and charged states to level degeneracy or a bias voltage can be applied to enlarge

the conduction window and allow transport. B) Conductance through a two level quantum

dot as a function of gate voltage Vg and bias voltage Vb. The light lines correspond to regions of high conductance and form diamond lineshapes characteristic of the Coulomb Blockade

transport regime that can each be attributed to the N, N+1 and N+2 electron ground states

of the device. C) Schematic representation of a possible coherent cotunnelling process in

the aforementioned spintronics device. D) Conductance through a two level quantum dot

as a function of gate voltage Vg and bias voltage Vb now with coherent transport processes included. Note the persistence of a zero-bias signal in the N+1 diamond characteristic of

coherent transport...... 8

4 Schematic diagrams of the TbPc2 molecular spintronics junctions that are to be investigated

in this thesis. The left figure shows a single TbPc2 molecule connected to a broken gold

nanowire that is mounted on a HfO2 substrate. The right figure illustrates two TbPc2 molecules grafted to a carbon nanotube which is itself contacted with two Pd electrodes

and mounted on a SiO2 wafer...... 10

xi LIST OF FIGURES xii

4 Schematic depiction of A) closed and B) open quantum systems. The open quantum system

is allowed to exchange energy and/or particles (red arrows) with the thermal bath which is

assumed to exhibit faster relaxation dynamics than the quantum system. As a result, the

open quantum system is influenced by the dissipative effects of the bath whereas the closed

quantum system is not...... 18

I.2.1 A schematic representation of electron transport from a ferromagnetic lead through a quan-

tum dot that is antiferromagnetically coupled to a SMM subject to resonant radiation.

Energy is supplied to the system to tilt the giant spin of the SMM (thick, red) allowing a

spin majority electron to charge the device from the ferromagnetic source. On relaxation,

the SMM aligns against the longitudinal field reversing the spin of the conduction electron

as it is emitted to the non-magnetic drain...... 32

I.2.2 Energy levels of the SMM-dot hybrid described by the Hamiltonian given in Eq. (I.2.3)

calculated using parameters chosen above. The neutral states are represented by black dots

and the plus (minus) charged states by upward-facing, red (downward-facing, blue) triangles. 38

I.2.3 Time evolution of the ρs, ρs−1/2 and ρs−1 density matrix elements obtained by numerical

integration of Eq. (I.2.7) at Vb = 0 with a ferromagnetic source...... 39 I.2.4 The stationary charge current (left) and spin currents at source and drain (right) flowing

through the device as a function of applied bias voltage...... 40

I.2.5 The time-averaged charge current flowing through the device at Vb = 0 as a function of

various pulse times tp and wait times tw...... 41 I.2.6 Zero bias steady state spin currents at the source and drain electrodes for SMM devices

with various spin quantum numbers s and resonant perturbations VN (t)...... 44 I.2.7 The stationary charge current (left) and spin currents at source and drain (right) flowing

through the device as a function of applied bias voltage when both electrodes are non-

magnetic (i.e. PS = PD =0)...... 46

I.3.1 (Left) Exact energies of the eigenstates of HS (labelled by their spin expectation value hSzi) obtained from numerical diagonalisation of the quantum system Hamiltonian; uncharged

(charged) states are shown as blue (red) circles. The states that are the most relevant for

transport for the parameter set chosen above are boxed. (Right) Steady-state spin currents

at the ferromagnetic source (blue) and the non-magnetic drain (red) as a function of the

applied bias voltage Vb using the exact eigenstates and energies obtained from numerical

diagonalisation of HS...... 52 xiii LIST OF FIGURES

II.5.1 (color online) A) Zeeman diagram of the lowest lying levels of the device in the large

exchange coupling regime B) Conductance as a function of magnetic field at 100 mK for the

two orientations of the TbPc2 magnetic moments m = J (spin up, blue arrow) and m = −J (spin down, red arrow) in the large exchange coupling regime. Note the conductance peaks

at the magnetic field values B = ±2δVg/gµB, at which values the gate voltage-detuned

energies of the neutral and reduced states of the TbPc2 are brought back to charge resonance. 65

II.5.2 (color online) A) Zeeman diagram of the lowest lying levels of the device in the weak

exchange coupling regime where a < δVg B) Conductance as a function of magnetic field

at 100 mK for the two orientations of the TbPc2 magnetic moments m = J (spin up, blue arrow) and m = −J (spin down, red arrow) in the weak exchange coupling regime.

Note that in this case, the conductance peaks at the magnetic field values B = ±aJ/gµB, at which values the exchange coupling split energies of the ferromagnetic and antiferromagnetic

reduced states of TbPc2 become degenerate...... 67

II.5.3 Differential conductance averaged over both orientations of the Tb moment as a function

of temperature using a = 0.02 meV. Best agreement with experiments was obtained for

Γ/~ = 6.6 × 108 s−1, and η = 65 µeV...... 69

II.5.4 Contour plot of conductance averaged over both orientations of the Tb moment as a function

of bias voltage (in units of the occupation energy of the dot) and magnetic field, for a = 0.02

meV, Γ/~ = 6.6 × 108 s−1, and η = 65 µeV...... 69

II.5.5 Conductance as a function of magnetic field arising from sequential and cotunnelling pro-

cesses at T = 100 mK for the two orientations of the TbPc2 magnetic moments m = J (spin up, blue arrow) and m = −J (spin down, red arrow) in the weak exchange coupling

regime...... 72

II.5.6 A) Zeeman diagram of the lowest lying hyperfine-split levels of the device |m, mI , σi in the large exchange coupling regime B) Conductance as a function of magnetic field at 100 mK

for the two orientations of the TbPc2 magnetic moments m = J (spin up, blue arrow) and m = −J (spin down, red arrow)...... 74

II.5.7 Magneto-conductance of the device when m = J in the large exchange coupling regime at

various ratios of lead-dot coupling ΓD/ΓS using the same parameter set as for Figure II.5.1

with ΓS =Γ...... 75 LIST OF FIGURES xiv

III.7.1 A schematic for the supramolecular spin valve device, where M TbPc2 nanomagnets are adsorbed on a chain of M coupled quantum dots forming the nanoconstriction junction.

Conduction electrons tunnel onto the device incoherently from nearby electrodes at a rate

Γ and hop coherently along the chain of dot-nanomagnets with amplitude th, experiencing

a local Ising exchange interaction ai with the 4f-quantum states of each Tb(III) centre. . . 83

III.7.2 Zeeman energies of the lowest lying energy levels of the TbPc2 molecular spin valve. The solid black lines represent uncharged states of the device with the magnetic moments of each

TbPc2 nanomagnet labelled with thick, red arrows. The blue (red) dashed lines represent the corresponding ferromagnetic states for the parallel |−J, −J, ↓i (|J, J, ↑i) states. Orange

mechanistic arrows are included to illustrate population transfer between the states when

a magnetic field is traced from negative to positive values across the device thus leading

to the two sharp jumps in magneto-conductance, characteristic of the molecular spin valve

experiments...... 91

III.7.3 Conductance loops of hysteresis on field tracing (blue) and retracing (red) obtained at

different field sweep rates ω. The top (bottom) curves have been shifted by +(−)0.8eΓ for

clarity...... 92

III.7.4 δg for a given field tracing and retracing event calculated at various temperatures with a

sweeping rate of ω = 0.005 s−1...... 93

III.7.5 δg as a function of the dynamically sweeping field amplitude Bz and gate voltage detuning

δVg away from the E0(J, J) = E1(J, J, ↑, −) level degeneracy...... 94

III.7.6 A) Magnetic field dependence of the lowest lying energy levels of the TbPc2 spin valve device with M nanomagnets in series. As in Figure 2, the solid black lines represent uncharged

states of the device and are labelled by the spin-polarisation of the TbPc2 units (red arrows, some configurations have been omitted to ease visualisation). The dashed lines (blue and

red) represent the corresponding ferromagnetic reduced states for the parallel configurations

|±J, . . . , ±J, ↑ (↓)i. Orange arrows along and between energy levels indicate population

transfer between the states on a tracing field; the largest orange arrow is symbolic of a

sequence of phonon-emission events required to relax the system back to the conducting

parallel ground state orientation. B) Loops of hysteresis in the calculated conductance on a

trace (blue) and retrace (red) of a longitudinal magnetic field for the TbPc2 molecular spin

valve consisting of M TbPc2 nanomagnets...... 95 xv LIST OF FIGURES

III.7.7 Calculated steady-state conductance at T = 30 mK as a function of bias voltage at the

N/N + 1 level degeneracy when the device is subject to various strengths of static, longi-

tudinal magnetic field...... 96 III.7.8 Zeeman diagram of the most relevant lowest lying spin valve states for the reduced (N +

1, red) and doubly reduced (N + 2, blue) device obtained by the application of a gate

detuning and diagonalisation of Eq. (III.7.1) on the product basis of the nanomagnet

angular momentum projections and dot spin states...... 97 III.7.9 Hysteresis of the magneto-conductance upon tracing (blue) and retracing (red) the longi-

tudinal magnetic field, obtained at the N + 1/N + 2 level degeneracy at T = 30 mK and

with a field sweep rate ω = 0.005 s−1...... 98 III.7.10Calculated steady-state conductance as a function of bias voltage with and without the

application of a static magnetic field Bz = 0T (red) and Bz = 1T (blue)...... 99

III.7.11Schematic depiction of the M = 2 TbPc2 molecular spin valve wherein the nanomagnets do not share the same quantisation axes. WLOG we take the first nanomagnet in the series

to be quantised along the z axes and the second along the z0 axes which is obtained by a

rotation of the z axis by an angle θ about the in-plane transverse axis of the device. . . . . 100 III.7.12δg calculated for various tilting angles θ with a field sweeping rate ω = 0.005 s−1 and

T = 0.1 K. As θ increases, the spin valve effect is quenched as the reduced states for the

anti-parallel configuration of the TbPc2 magnetic moments become stabilised and therefore begin to participate in transport...... 102 Prologue

As electronic technologies become smaller and smaller, limitations are imposed upon device fabrication related not only to the precision of instrumentation required for microscopic device manufacture but rather more strikingly, the new regime of molecular electronics is approached in which non-negligible quantum mechanical effects become manifest. Nevertheless, a great many molecular electronic devices that utilise quantum mechanical degrees of freedom have been realised experimentally including but not limited to: molecular switches,1–3 molecular break junctions4, 5 and organic spin valves.6, 7 A subset of these devices take advantage of the spin quantum number of electrons for their operation thus giving rise to the field of research known as molecular spin electronics or simply molecular spintronics.8

In a pursuit parallel to the ultimate miniaturisation of electronic devices, the manipulation and read out via electric or spin currents of the coherent superposition of molecular quantum states is also a key step in the development of molecular qubits for quantum computing technologies. The two spin states of an

1 electron ms = ± 2 provide the prototypical physical implementation of a qubit as, under an appropriate perturbation e.g. the combined effect of a longitudinal magnetic field and a resonant transverse microwave pulse, this computational basis can be made to undergo an arbitrary unitary transformation, encoding all the qubit states on the Bloch sphere.9 Importantly, any such implementation of one-qubit quantum gates can only be operative within the characteristic spin-lattice and spin-spin relaxation time scales arising from the interaction of the electron spin with its specific surrounding environment, a time frame also known as the decoherence time. Since the nature of the system-bath interactions are usually very local in character, coupling several electronic spins within a single ion, or across several metal ions in a polynuclear complex, as is typical in molecular nanomagnets, offers a useful strategy to introduce collective spin relaxation bar- riers and increase decoherence times of molecular nanomagnet-based spin qubits. Several theoretical works have discussed in this regard, the plausibility for magnetic molecules to act as molecular qubits in quantum information processing algorithms.10–13 Furthermore, a promising electron paramagnetic resonance exper- iment demonstrated the persistence of coherent Rabi oscillations between the spin states of an irradiated

14 V15 complex that persisted for times on the microsecond time scale at liquid helium temperatures. In

1 Prologue 2

a separate experiment, the quantum superposition of spin states in a magnetically anisotropic vanadium

complex was reported and showed particularly long decoherence times even at room temperature.15, 16 In a recent advancement, Grover’s quantum search algorithm has been performed on a terbium nanomagnet molecular break junction using the four nuclear spin states of the terbium ion as a molecular qudit op- erated on by microwave pulsed gates and read-out via an electric current.17 These recent results suggest that single-molecule magnets may play a significant role in the formulation of future quantum information technologies.

Single-molecule Magnets

The first single-molecule magnet to be reported in the literature was the mixed-valence dodecametallic . manganese-acetate cage [Mn12O12(CH3COO)16(H2O)4]·2CH3COOH·4H2O = Mn12Ac that displayed hys- teresis of the magnetisation reminiscent of bulk ferromagnets. However, by removing intermolecular mag- netic coupling via magnetic dilution, the hysteresis effects were shown to be rooted at the single-molecule level rather than a property of the bulk system.18–20 The superparamagnetic slow relaxation behaviour of this transition metal single-molecule magnet complex originates from the interplay between the large molec- ular spin state, occurring as a consequence of the strong exchange coupling between on-site spin moments of the coupled open-shell metal ions, and on-site magnetic anisotropy, also known as zero-field splitting. In the case of Mn12Ac (and many other transition metal clusters), ligand field effects almost entirely quench the orbital angular momentum of electrons in the valence shell of the individual ions, thus removing the

2L + 1 degeneracy of the ground Hund’s rule Russell-Saunders term 2S+1L, which is thus strongly split into

2Si+1 2Li + 1 crystal field terms Γ (where Si and Li label the local total spin and orbital angular momentum,

th 2Si+1 respectively, of the i metal ion). The resulting single ion ground crystal field term Γ0 thus features a pure spin 2Si + 1 degeneracy. Magnetic anisotropy arises from the fact that the ground crystal field term

2Si+1 Γ0 is then coupled in second order via weak spin-orbit coupling effects and, in low enough symmetry, such coupling leads to a zero-field splitting of the single ion 2Si + 1 spin states, which can be described in terms of effective even-rank (usually second-rank) single-ion spin operators. Moreover, the pure spin degen-

2Si+1 erate ground terms Γ0 of the metal ions of the complex are exchange coupled into a total molecular P spin S (in the ferromagnetic case, S = i Si, but in Mn12Ac the situation is more complex, with S = 10 resulting from the antiferromagnetic coupling of mixed valent Mn(III) and Mn(IV) ions).21 Finally, in the strong exchange coupling regime, the on-site spin-orbit coupling mixing of molecular crystal field terms is in fact a perturbation with respect to ion-ion coupling, so that projection of the local zero-field splitting effective spin Hamiltonians onto the ground total spin (i.e. first order degenerate perturbation theory) often 3 Prologue

M=0 M=0 M=0

Field on Field off Direct Energy M=S QTM M=-S M=S M=-S M=S

M=-S

Figure 1: General mechanism for single-molecule magnet relaxation. In each of the plots, the horizontal lines represent spin angular momentum projections of the giant S manifold split by the uniaxial potential 2 −|D|Sz . Solid red arrows denote direct relaxation processes while the dashed red arrow indicates a quantum tunnelling of the magnetisation (QTM).

suffices to describe the splitting in zero field of the giant spin ground state, which in efficient SMMs results

in a dominant easy-axis type magnetic anisotropy. In complexes with the capacity to act as single-molecule

magnets, the magnetic anisotropy leads to an energy splitting of the giant spin ground state such that the

states with maximal spin projection |S, M = ±Si form the bistable ground states of the molecule now sep-

arated by a energetic barrier to spin reversal consisting of excited giant spin states with intermediary spin

projection quantum numbers M = −(S − 1),...,S − 1 with respect to the easy axis of the molecule. To a

2 first approximation, the height of this intrinsic spin reversal barrier can be written as Ueff = |D|S where D is the uniaxial anisotropy constant obtained from projecting the single-ion zero-field splitting terms onto

22 the ground total spin. EPR experiments conducted on Mn12Ac revealed a uniaxial anisotropy constant

−1 −1 23 D ∼ −0.46 cm giving rise to a modest barrier height of Ueff = 46 cm .

In an ensemble of Mn12Ac single-molecule magnets such as in a crystal, each molecule is most likely to exist in one of its bistable ground states as shown schematically in figure 1 provided that the temperature

is smaller than the anisotropic barrier to spin reversal of the S = 10 giant spin moment. The non-specific population of either ground state leads to a net zero magnetisation for the ensemble. Through the application of a magnetic field, one breaks the degeneracy of the bistable ground states and preferentially orients the giant spin moments of SMMs in the ensemble against the field. If the temperature is lower than the energy barrier for magnetisation, when the field is removed the net magnetisation of the ensemble is retained as each molecule is unable to instantaneously reorient its spin moment due to the energetic cost attributed to traversing the anisotropy barrier. As shown in the final panel of Figure 1, the ensemble will not remain magnetised indefinitely and, given time, will repopulate both sides of the anisotropy barrier through direct transitions or via quantum tunnelling effects.24 Due to the aforementioned slow-relaxation between the bistable ground states of the molecules, hysteresis loops of magnetisation arise when a magnetic field (oriented Prologue 4

along the easy axis of the SMM) is traced back and forth across the ensemble. Sessoli, Gatteschi and

others20, 25 were among the first to investigate these loops of hysteresis using SQUID (superconducting

quantum interference device) magnetometers.

As a consequence of the anisotropy barrier to spin reversal, the Mn12Ac single-molecule magnet was suggested to act as a molecular memory unit in which bits of information would be encoded by preferential

population of one of the SMM’s ground states. The time scale on which this information could be preserved

in the magnet is inextricably linked to the relaxation time of the molecule and thus to the height of the

barrier. One of the largest barrier heights to be reported for a 3d transition metal-based single-molecule . magnet was the hexametallic complex [Mn6O2(sao)6(O2CPh)2(EtOH)4] = Mn6 (saoH2 = salicylaldoxime) with a giant spin ground state of S = 12 and a uniaxial anisotropy constant D ∼ −0.43 cm−1 giving rise

−1 26 −1 to a barrier height of Ueff = 62 cm . Even a barrier height of 62 cm is still too low for the realisation of room temperature molecular memory. The strategy to achieve a large barrier, for a while was to take

2 advantage of the quadratic S dependence of Ueff and generate single-molecule magnets with incredibly large spin ground states. However, it was soon realised that increasing the total molecular spin S was not a viable approach, as while projecting the on-site single ion zero field splitting anisotropy on the large spin manifold produces a barrier proportional to S2 in the easy-axis case, the projected coupling constant D in fact is per se inversely proportional to S2 cancelling out the beneficial effect.22 As a matter of fact, a

Mn25 single-molecule magnet [Mn25O18(OH)2(N3)12(pdm)6(pdmH)6]Cl2 · 12MeCN ·12MeCN and a Mn19

(III) (II) 1 magnetic cluster [Mn12 Mn7 (µ4-O)8(µ3, η -N3)8(HL)12(MeCN)6]Cl2 · 10MeOH·MeCN with spin ground states of S = 51/2 and S = 83/2 respectively were synthesised in this vein however both molecules exhibited

−1 paltry barrier heights with the Mn25 molecule showing a barrier of Ueff = 14 cm and the Mn19 complex

−1 Ueff = 4 cm . An alternative strategy for developing single-molecule magnets with large anisotropy barriers emerged in the early 2000s whereby single lanthanide ions were favoured as the building blocks for new molecular nanomagnets due to their intrinsically large magnetic anisotropy.27–29 Unlike transition metal ion complexes where crystal field effects overcome spin-orbit coupling mixing, in lanthanide ions Russell-Saunders L-S terms can be mixed by spin-orbit coupling, so that the ground state results in a spin-orbit multiplet with the total angular momentum J = L + S good quantum number, which is weakly split by crystal field effects. Thus

2S+1 weak ligand field (crystal field) interactions with the ion partially removes the degeneracies of each LJ multiplet leaving well isolated |J, MJ i manifolds with energy barriers qualitatively quite different from those of transition metal SMMs. In particular, since the symmetry of lanthanide complexes are usually much lower than for transition metal complexes, in principle, a greater deal of mixing between the |J, MJ i angular momentum states is often observed. Figure 2 provides a representative example of the energy splitting 5 Prologue hierarchy of the electronic states in a trivalent dysprosium ion. While crystal field effects prove to be the

2S+1 weakest perturbation on the electronic states of lanthanide ions, the crystal field splitting of a LJ multiplet can be orders of magnitude larger than the splitting of |S, Mi states in transition metal SMMs.30

It is this that makes lanthanide ion single-molecule magnets attractive candidates for high-temperature nanomagnets. In recent years the pursuit towards lanthanide-based single-molecule magnets has seen an explosion in the synthesis of high-temperature molecular nanomagnets.31 The current record is held by a trivalent Dy(III) complex that displays hysteresis loops of magnetisation at temperatures exceeding the boiling point of liquid nitrogen, reaching a blocking temperature as high as 80K when a magnetic field was traced across the sample at a rate of 25 Oe s−1.32 Interestingly, the relaxation time τ for this particular

−1 −1 n −1 complex was obtained with a fitting function τ = τ0 exp(−Ueff/kT ) + CT + τQTM which illustrates that more than just the barrier height Ueff contributes to the slow-relaxation of the complex and that both the rates of Raman processes and quantum tunnelling of magnetisation, respectively, also play a role in

Figure 2: Energy splitting diagram of the lowest lying electronic states of a dysprosium complex. The many- body 4f electronic states are split by a hierarchy of correlation, spin-orbit and crystal field effects leading to well isolated crystal-field split spin-orbit multiplets that generally span ≤ 103 cm−1. Image used with permission from Associate Prof. A. Soncini. Prologue 6

determining the relaxation time of the nanomagnet.

To further the design of better single-molecule magnets, ab initio computer modelling based on Complete

Active Space Self Consistent Field (CASSCF) calculations, followed by a State Interaction with Spin-Orbit

Coupling mixing (RASSI-SO) approach has become an indispensable tool for understanding the electronic

structure of lanthanide SMMs.33–44 In fact, our research group has proposed, in 2016, a novel approach to the ab initio calculation of the electronic structure and magnetic properties of lanthanide complexes based on a Configuration Average Hartree-Fock (CAHF) orbital optimisation followed by the determination of a Complete Active Space Configuration Interaction with Spin-Orbit coupling (CASCI-SO) multielectron wavefunction, which has been implemented in a novel quantum chemistry code named CERES (Computa- tional Emulator of Rare Earth Systems), fully developed in the Soncini Research Group at the University of

Melbourne.45–47

Memory Effects in Molecular Nanomagnet Spintronic Devices

While the design and discovery of better single-molecule magnets is an important step in the development of hyper-dense molecular memory and new information technologies, attaining larger barrier heights becomes a moot pursuit if the information stored in the nanomagnets can not be addressed in an efficient and reliable manner. By integrating single-molecule magnets into molecular electronic junctions, various experiments have demonstrated the fingerprints of anisotropy in electrical conductance measurements suggesting that electric currents could be used to read out the quantum states of a single magnetic molecule (or few molecules) embedded in the junction.48–52

The set-ups are generally composed of tens of single-molecule magnets grafted to an electrically conduct-

ing nanostructure that is weakly contacted with source and drain electron reservoirs (electrodes) that enable

transport through the device. Owing to the confinement of electrons within the nanostructure, Coulomb

blockade diamonds similar to those found in quantum dot transport experiments are often observed in the

stability diagrams of the devices.53–55 The current blockade effect stems from the large energy (on the or- der of meV for quantum dots and eV for molecular devices56) required for an extra conduction electron to charge the nanoarchitecture that connects the source and drain reservoirs. The charging energy can be paid by either i) applying a bias voltage across the device or ii) by the application of a gate voltage as shown schematically in Figure 3A. This dependence on the bias and gate voltages results in a diamond lineshape of conductance as shown in Figure 3B where transport through a single level quantum dot has been simulated as a representative example. Since the non-zero conductance signal in bias-gate voltage space marks the ad- dition or removal of an extra electron to the nanoarchitecture, the interior of each diamond can be identified 7 Prologue

with a parameter set where N electrons occupy the device. In the case of a strong hybridisation between the electrodes and the nanoarchitecture, the Coulomb blockade diamonds can be strongly modified at low tem- peratures by leakage currents within the blocked regions (see Figures 3C and 3D) due to coherent coupling between different charge states of the dot (e.g. cotunnelling effects in low order perturbation theory of the dot-lead hybridisation Hamiltonian,57 or the Kondo effect in the non-perturbative regime58, 59), or they can

disappear altogether at higher temperatures if the hybridisation is stronger than electron correlation effects

on the device. In the latter case, the dot-lead interactions can be treated in the mean field regime, and

the level broadening of the nanoarchitecture induced by the strong hybridisation overcomes the charging

energy leading to a ballistic regime of transport.56 The presence of Coulomb blockade diamonds in the

stability diagrams of single-molecule magnet molecular spintronic devices thus provides evidence for a weak

electrode-substrate coupling regime in which conduction electrons occupy the device on a time scale long

enough to interact with the giant spin moment of the nanomagnets as well as evidence for strong electron

correlation effects on the molecule (again consistent with the presence of a strongly correlated open shell

metal participating in the transport process).

In order to pave the way for a new generation of devices based on spin carriers rather than charge carriers,

the ability to address and manipulate the quantum states of single or few molecules grafted to a conducting

surface using pure spin or spin polarised currents remains a much sought after goal for researchers in the field

of molecular spintronics. A current avenue of investigation is the use of single-molecule magnets as molecular

bits for hyper-dense data storage60 where binary information may be stored in the bistable ground state of

individual molecules. Of course, to realise these SMM memory technologies, one must be able to read and

write information to the quantum spin states in an efficient and controllable manner. To write information,

theoretical spin transport models have demonstrated that a spin transfer torque enforced on the giant spin of

a single-molecule magnet via a spin polarised current can controllably switch the orientation of a SMM’s spin

moment thereby preferentially populating one of the bistable ground states of the molecule.61–66 This spin

torque switching effect has in fact been demonstrated for iron-based nanomagnet clusters in spin polarised

scanning tunnelling microscopy experiments67, 68 where also a magnetoelectric coupling effect was shown to reversibly control the anisotropic barrier height for the spin reversal of the nanomagnet thus increasing the efficiency of writing information to the molecule using the spin polarised current.69

An alternative approach towards developing novel molecular spintronic devices lies in using the unique properties of magnetic molecules embedded in spintronic set-ups to manipulate the polarisation of nearby electric currents. A particularly striking spintronics set-up was reported in 2011 by G¨ohleret al.70 where monolayers of deoxyribose nucleic acid (DNA) were shown to function as extraordinarily efficient spin filters at room temperature; generating a spin polarised current as a consequence of the helicity of the strands. Prologue 8

D) C)

N+1

N N+2

Figure 3: A) Schematic representation of Coulomb blockade in a spintronics device composed of two elec- trodes and a neutral and charged state. A gate voltage can be applied to bring the neutral and charged states to level degeneracy or a bias voltage can be applied to enlarge the conduction window and allow trans- port. B) Conductance through a two level quantum dot as a function of gate voltage Vg and bias voltage Vb. The light lines correspond to regions of high conductance and form diamond lineshapes characteristic of the Coulomb Blockade transport regime that can each be attributed to the N, N+1 and N+2 electron ground states of the device. C) Schematic representation of a possible coherent cotunnelling process in the aforementioned spintronics device. D) Conductance through a two level quantum dot as a function of gate voltage Vg and bias voltage Vb now with coherent transport processes included. Note the persistence of a zero-bias signal in the N+1 diamond characteristic of coherent transport. 9 Prologue

This effect was extended to DNA oligomers in atomic force microscopy junctions by Xie et al.71 and has

sparked a rich branch of theoretical works to explain the origin of this effect in mathematical detail.72–75

In a separate experiment by Butcher et al., spin polarised scanning tunnelling microscopy tips were used to demonstrate the potential for cobalt phthalocyanine molecules to act as spin filters when adsorbed to cobalt nanoislands in when a specific bias voltage was applied across the junction.76 As well as spin filters, spin current inverters that switch the polarisation of an already polarised current could also function in novel logic circuits based on spin carriers. Molecular triangles with non-collinear spin textures have been proposed as efficient spin switching devices using theoretical models of spin current injection into quantum nanoarchitectures.77, 78 The efficient switching effect also provides a means to prepare the molecule in one of its toroidal quantum states hinting at the possibility that this triangular nanomagnet could be implemented as a toroidal qubit for quantum computing.

Recent experiments have sought to utilise the interaction between the magnetic moment arising from the

7 . crystal field-split F6 spin-orbit multiplet of a terbium phthalocyaninato [Tb(C32H18N8)2]= TbPc2 single- molecule magnet and the spin of conduction electrons in spintronics junctions. When the TbPc2 nanomagnets were embedded on an sp2 hybridised carbon substrate (a graphene nanoconstriction or carbon nanotube) and driven by an oscillating magnetic field, researchers were able to identify an anisotropic, dynamic, spin valve-like signature in the magneto-conductance of the system.79–81 The memory effect originating from the slow-relaxation of the TbPc2 single-molecule magnets grafted to the spintronic set-up thus presents the potential for the device to act as a molecular analogue of the macroscopic spin valve that can currently be found in many modern technologies. In a separate experiment, TbPc2 was embedded into a Au-nanowire and a clear read-out of the Tb nuclear spin states with electrical conductance measurements was demonstrated in the presence of a tracing magnetic field.82, 83 It was later demonstrated that, resonant microwave pulses that coupled to the nuclear spin states of the nanomagnet via the hyperfine stark effect could be administered directly to the Tb3+ nucleus such that the coherent dynamics of the nuclear spin states could be controllably manipulated and read-out in conductance measurements.84 In a subsequent study, the coherent dynamics of the Tb nuclear spin states were then used in conjunction with the fast electronic read-out through the spintronics junction in order to perform Grover’s quantum search algorithm and fully demonstrate the

17 potential for the TbPc2 molecular break junction to play a role in novel quantum information technologies. While these single-molecule magnet-based molecular spintronic set-ups are particularly inspiring for prob- ing the spin dynamics of the TbPc2 4f electrons in order to achieve the ultimate miniaturised spin valve technology or, in the case of the break junction, implement the read-out of a nuclear spin qudit, their rela- tively sparse theoretical characterisation leaves their mechanistic underpinnings open to interpretation. For instance, crucial questions that have emerged from these landmark experimental studies, that to date have Prologue 10

A d

A A

Figure 4: Schematic diagrams of the TbPc2 molecular spintronics junctions that are to be investigated in this thesis. The left figure shows a single TbPc2 molecule connected to a broken gold nanowire that is mounted on a HfO2 substrate. The right figure illustrates two TbPc2 molecules grafted to a carbon nanotube which is itself contacted with two Pd electrodes and mounted on a SiO2 wafer.

received scant attention from the theoretical perspective, are: how sensitive are the molecular spin valve

experiments to the number of TbPc2 molecules grafted on the carbon nanostructure device? Would one

79–81 TbPc2 molecule suffice, or would one need to graft at least two or more molecules? Also, what is the

17, 82–85 relevant transport regime in the TbPc2 break junction device ? For instance, it was assumed in a preliminary interpretation of the experimental work that a highly coherent transport regime such as the

Kondo regime, was a necessary condition for the read out operation however this would seem to entail rather

strict requirements on the engineering of the molecule-lead coupling.86 Is it possible then, to reproduce the main features observed in the experiments within a more common non-coherent transport regime? These and other questions triggered our interest in these experiments, to develop a theoretical framework able to capture these results, while elucidating the microscopic mechanisms responsible for the observation of spin dynamics effects on the measured magneto-conductance.

Theoretical Models of Nanomagnet Spin Dynamics

The description of magnetic relaxation in single-molecule magnets has been aided greatly by theoretical models as they can often provide specific mechanistic explanations of phenomena that are not immediately accessible from experiment alone.87 The quantum rate equation approach is one such archetypal framework that is often used in the description of relaxing quantum systems that are coupled to some dissipative bath.88

For instance, a comprehensive study of spin relaxation in the previously discussed Mn12Ac single-molecule magnet using quantum rate equations for the giant spin states of the magnet revealed relaxation pathways in the magnet arising from phonon-induced spin transitions and quantum tunnelling events between degenerate

89–91 levels. Adiabatic rate equations have also been used to model the magnetic slow relaxation of CrDy6 ferrotoroidic clusters when subject to an oscillating magnetic field, which uncovered a microscopic mechanism 11 Prologue for the throttling of population transfer between the ground states of the molecular cluster that culminated in hysteresis loops of magnetisation.92 Further theoretical models of frustrated magnetic systems have also been proposed that serve to augment experimental observations.93, 94

Quantum rate equation models have also been utilised in studies of electron transport through molecular devices. As well as explaining aspects of scanning tunnelling microscopy experiments,65, 95, 96 approaches based on rate equations have predicted novel spin-polarised current-induced magnetic switching,62, 64, 66 ge- ometric current blockade effects63 and spin amplification61 in nanomagnet spintronics devices. Theoretical models of spin-polarised transport have also been constructed in our group to describe spin current ma- nipulation effects using molecular wheels77, 78 and general single-molecule magnet set-ups.97 The quantum rate equation approach is aptly suited for the investigation of magnetic molecules in spintronics junctions as it provides directly, the time evolution, and therefore the spin dynamics, of the magnetic molecules interacting with the dissipative environment. Furthermore, when combined with ab initio methods and/or non-equilibrium Green function density functional theory (as implemented in the SMEAGOL package98), the quantum rate equations become a powerful framework for the realistic modelling of spin transport through molecular systems.99, 100 A full derivation of the quantum rate equations pertaining specifically to electron transport through a single-molecule magnet device are presented in the next chapter.

To understand the fundamental yet specific aspect of field-induced spin reversal via a Landau-Zener transition in the continuously measured TbPc2 break junction device, in a joint experimental and theoretical work, Troiani et al. employed an adiabatic rate equation in Lindblad form to describe the time-evolution of the ground state populations of the terbium magnet.101 Interestingly, with this adiabatic rate equation approach, the deviation observed in experiment from the Landau-Zener dynamics of a closed quantum system was captured and explained tentatively in terms of dephasing processes occurring in the quantum system.

Notably, the exact nature of the dephasing processes postulated in the device were obscured by the non- specific definitions of the time-averaged Lindblad operators and hence the exact mechanism for dephasing in the TbPc2 junction remains an open question. A rate equation equivalent approach has also been utilised in the description of tunnel magnetoresistance

(TMR) effects in a general single-molecule magnet supramolecular spin valve exposed to a static magnetic

field.102 Furthermore, a joint experimental and theoretical work pertaining more directly to the static field operation of the TbPc2 spin valve on carbon nanotubes used a Coulomb blockade transport model combined with a spin-dependent Fano resonance approach to explain the field-sensitive high and low conduction mea- surements through the device in terms of the nanomagnet orientations. While both of these models forward our understanding of TMR in the molecular spin valve set-ups when a static magnetic field is applied, im- portantly simulations of hysteresis loops of magneto-conductance as a response to a driving magnetic field, Prologue 12 have yet to be published.

Research Questions

Beside the more applicative aspects of molecular spintronics described earlier in this introduction, which make our theoretical investigation of interest to further expand our knowledge of the transport mechanisms in SMM-based devices, one key general aspect of molecular spintronics on which we focused our efforts in this thesis, concerns the use of electric currents or spin currents to probe the quantum dynamics of SMM spin magnetisation, at the single molecule level. Usually measurements of SMM spin dynamics are performed on bulk samples and thus provide a less direct access to the pristine single molecule properties whereas SMM molecular electronics presents a mechanism for the interrogation of single-molecule properties. Accordingly, a common thread of the three projects presented in this thesis is: What is the signature of the SMM quantum spin dynamics in the conductance or magneto-conductance probed in a lab for a SMM-based molecular junction or spin valve?

In this thesis, we addressed this central question at first by setting up an ad hoc theoretical spintronics model in which a coherent spin dynamics in the SMM device is triggered by a resonant radiation (project

1) and is not directly related to the slow-relaxation properties of the SMM. Most of the work presented in this thesis, however, has focused on a few published spintronics experiments which probed a lanthanide- based SMM molecular spin junction (project 2) and molecular spin valve (project 3) device, in which the slow-relaxing spin dynamics is instead triggered more traditionally via a sweeping magnetic field.

Addressing this central question, in its various declinations explored in projects 1-3, first of all presented some interesting theoretical challenges in the choice and set up of the model, both in methodological terms, and in the choice of the essential microscopic ingredients necessary to capture the essential physics of each system, as will be explained in greater detail in the following chapter and, to a lesser degree, in the appendices

A and B. The models developed here were then used to simulate the transport properties of the device as a function of bias and gate voltage, of temperature, of external magnetic field and of the sweeping rate of the external oscillating magnetic field. This allowed us to address, to an unprecedented level of detail, open questions such as (i) which is the most relevant transport regime in a device, (ii) how the specific microscopic electronic structure properties of the molecular device influence the observed dynamics of the magneto-conductance, and (iii) how this reflects on the slow relaxation or more generally the spin dynamics of the SMM. More specifically, the three projects presented here are described in the remaining part of this section. 13 Prologue

Project 1

In this project we construct theoretical models of spin current injection into a general single-molecule

magnet spintronic device whereby a coherent dynamics of the giant spin states of the molecule has been

activated by means of a resonant radiative perturbation. In order to present a detection scheme of the

quantum spin dynamics of the nanomagnet in an experimental set-up, we seek to uncover a signature of

the abnormal single-molecule magnetic relaxation (resulting from the coherent, resonant perturbation rather

than the intrinsic slow-relaxation of the molecule) in both the steady-state spin currents as well as in time-

averaged current measurements when pulses of radiation are applied. Furthermore, we seek to investigate if

similar signatures manifest in the steady-state spin currents of the device as a result of a time independent

perturbation. Given the generality of the SMM models proposed here, the conclusions of this project are

expected to provide broad scope to the design of novel nanomagnet-based molecular spintronic experiments

that ultimately give rise to new spin-based technologies.

Project 2

In this project we address the operation of a terbium molecular break junction focusing particularly on

the mechanism for magneto-conductance read-out of the electronic states of the single TbPc2 nanomagnet embedded in the device. As discussed above, the nature of the coupling between the molecule and the

electronic leads is a determinant of the transport regime in which the device operates that, in an experimental

study of the electronic read-out,86 was suggested to occur in the strongly coherent Kondo regime. Without a full theoretical characterisation of transport through the device though, whether an invocation of the strongly coherent, non-perturbative Kondo regime is required in order to observe the magneto-conductance read-out of the Tb electronic states obtained in experiment, remains an open question. To address this point, we set out to provide a counter-example of Kondo regime transport by means of simulating the results of a recent study of the break junction86 using only an incoherent rate equation model that captures

transport in the most commonly encountered regime of molecular electronics, the Coulomb blockade regime.

Consequently, a simple theoretical model of transport through the break junction is shown to uncover a

microscopic mechanism for the read-out of the device that may be tested experimentally, furthering our

understanding of this SMM junction.

Project 3

In this project we turned our attention to modelling the TbPc2 SMM spin dynamics and their effect on the magneto-conductance of recently realised terbium molecular spin valve devices constructed on a Prologue 14

graphene nanoconstriction79 and on carbon nanotubes.80, 81, 103 Our investigation centred around elucidating

the microscopic mechanism, and thus the minimal requirements, by which anisotropic hysteresis loops of the

magneto-conductance could manifest as reported in the experimental set-ups. We set out to address the open

question: what is the minimal number of TbPc2 molecular magnets required for the operation of the molecular spin-valve and how would the addition of superfluous molecular magnets affect magneto-conductance signals in the presence of an oscillating field? As a necessary precursor to answering this question we are also in a position to investigate the most dominant processes of slow-relaxation in the nanomagnets leading to their spin reversal in the magnetic field. Thus we are able to test the current dogma expressed in the literature, wherein it is postulated that spin reversal in a chain of nanomagnets in series proceeds by an initial quantum tunnelling of the magnetisation in the first member of the chain followed by a direct relaxation process of the subsequent magnets.104 With a theoretical microscopic model of the molecular spin valve in tow, the precise behaviour of the device as a function of system parameters, such as gate and bias voltage as well as temperature, and their relation to the anisotropic signals of magneto-conductance may be expounded upon in order to further characterise the device.

Thesis Outline

Before presenting the original research conducted during this PhD, in the next chapter the theoreti- cal framework that underpins the spin transport models discussed thereafter is developed in a clear and comprehensive manner. The chapter begins with a brief review of some fundamental definitions and re- sults from quantum statistical mechanics followed by a discussion of open and closed quantum systems; a molecular electronic device is identified with the former. From there, the secular quantum rate equations for the reduced density matrix of the device are developed and accompanied by discussions of the Born,

Markoff and secular approximations that lead to the final result. By finding the reduced density matrix for the nanomagnet-based spintronic set-up, macroscopic quantities of the system can then be calculated from the model which may be compared and contrasted with experimental observations thus leading to a more comprehensive understanding of each device. Modifications to the quantum rate equations are also examined in order to correctly describe the interplay between the dissipative dynamics of the device and coherent transitions in the nanomagnet driven by exposure to a magnetic perturbation.

As noted above, the thesis consists of three projects that will be presented sequentially in three separate parts of the thesis. Each part will contain a presentation of the main work followed by a recapitulation of the results and a discussion of any remaining open questions that could be addressed in future endeavours.

In part I, two novel theoretical models of nanomagnet-based molecular spintronics devices are presented 15 Prologue

that serve to invert an injected spin polarised current using a particularly general single-molecule magnet

set-up. Chapter 2 consists of a published work97 detailing a nanomagnet spintronics device operating under

the influence of a continuous time-dependent resonant perturbation. In chapter 3, a second single-molecule

magnet-based molecular spintronic set-up is theoretically investigated where, instead of a resonant time-

dependent perturbation, the nanomagnet is immersed in a static transversal field. To end the discussion

of this project, conclusions are drawn in chapter 4 followed by some brief comments on the experimental

viability of the devices hypothesised herein and further avenues that may be pursued in nanomagnet-based

molecular spintronics.

Part II pertains to a discussion of the TbPc2 molecular break junction with a focus towards understanding the low temperature magneto-conductance behaviour reported in recent experiments.82, 86 In chapter 5, a published work is presented in which, for the first time, electron transport through the TbPc2 molecular break junction is modelled using the Coulomb blockade transport regime. Conclusions to this project are drawn then in chapter 6 where the implications and limitations of the spin transport model through the

TbPc2 molecular break junction are re-emphasised and open questions that may provoke future research are discussed.

In part III, an investigation into the molecular spin valve devices based on the nanomagnet TbPc2 grafted to an sp2 hybridised carbon substrate is presented. Chapter 7 consists of a published work reporting, for the first time in the literature, time-dependent simulations of hysteresis loops of conductance in the TbPc2 nanomagnet spin-valve. In chapter 8 conclusions are drawn as in previous parts with a recapitulation of the main results from the original work followed by a discussion of open questions that still dangle elusively in the fray of this fascinating area of research.

Useful, additional information pertaining to this thesis can be found in the appendices. Appendix A contains a discussion of the many pictures of quantum mechanics that are made use of in the theoretical prologue when deriving the quantum rate equations for the reduced density matrix of a molecular spintronic device. A table of the key relations between mathematical objects in each picture can be found here for the reader’s convenience. In appendix B, the T-matrix approach for describing transitions driven by a perturbation V is proffered to motivate the cotunnelling formulae utilised in part II of the thesis. Contained in this section is a microscopic derivation of the Fermi golden rule transition rate followed by corrections to these rates that are higher order in V eventually leading to the generalised Fermi golden rule. Chapter 1

Quantum Rate Equations for

Transport through a Nanomagnet

Spintronics Device

1.1 Equations of motion for the Density Operator

The non-relativistic theory of quantum mechanics was extended to deal with ensembles of particles

105 characterised by a statistical mixture of N state functions |ψn(t)i in 1927 by John Von Neumann. The focus of this new frontier lies in finding or constructing the density operator N X ρ(t) = wn |ψn(t)i hψn(t)| . (1.1) n=1

for the quantum system of interest where each |ψn(t)i is a state vector that evolves in time according to the Schr¨odingerequation d |ψ (t)i i n = H(t) |ψ (t)i (1.2) ~ dt n and appears in the ensemble with the statistical weight wn. The weights wn represent classical probabilities and thus sum to unity. It should be stated explicitly that there are two notions of probability at work here:

i) the quantum probability that is encoded within each state function |ψn(t)i and ii) the classical probability

wn denoting the likelihood of choosing a member of the ensemble described by |ψn(t)i. The probability of finding the system in an eigenstate |ai defined by the eigenvalue equation A |ai = a |ai is then simply the

diagonal element N X 2 ha| ρ(t) |ai = wn|ha |ψn(t)i| (1.3) n=1

16 17 1.1. EQUATIONS OF MOTION FOR THE DENSITY OPERATOR

taken from the matrix representation of ρ(t) (the density matrix) on the basis {|ai}. The diagonal elements of

the density matrix hence accrue a useful physical meaning and are called populations whereas the off diagonal

matrix elements ha| ρ(t) |a0i with a 6= a0 are referred to as coherences and represent a superposition of the eigenstates |ai and |a0i.88 Observables in the quantum statistical theory are described by the expectation values of a set of Hermitian operators {Bi}i∈I as in the quantum theory of pure states however the notion of an expectation value now carries extra mathematical structure. Suppose the Hermitian operator B obeys the eigenvalue equation B |bi = b |bi then its expectation value in the quantum statistical theory is given by N X X X 2 hBi = Tr {ρ(t)B} = hb| ρ(t)B |bi = wn|hb |ψn(t)i| b (1.4) b n=1 b P where Tr {... } = b hb| ... |bi is a trace over any complete set of eigenstates; here the eigenstates |bi have

been chosen for convenience. Note that the individual eigenvalues b are weighted by the classical weight wn as well as the square of the quantum amplitude hb |ψn(t)i for each member of the ensemble. To calculate observables in the theory then, one must first find the density operator ρ(t) that characterises the system.

In order to find the density operator for a quantum system, the equation of motion technique is often employed.88, 106, 107 Given a Hamiltonian H(t), the equation of motion for the density operator is easily obtained by differentiating Eq. (1.1) with respect to time and substituting Eq. (1.2) and its complex conjugate appropriately; this yields

N   X d |ψn(t)i d hψn(t)| −i ρ˙(t) = w hψ (t)| + |ψ (t)i = [H(t), ρ(t)] (1.5) n dt n n dt n=1 ~ where the Schr¨odinger equation d |ψ(t)i /dt = (−i/~)H(t) |ψ(t)i and its complex conjugate have been used. Eq. (1.5) is the Von Neumann equation (sometimes also known as the quantum Liouville equation) for the density operator ρ(t). The main focus of this thesis will pertain to quantum systems described by

Hamiltonians of the form H = H0 + V where V is some weak perturbation in comparison to H0 and therefore it will be convenient to separate out the dynamics induced by each part of the Hamiltonian by applying a unitary transformation and switching to Dirac’s interaction picture of quantum mechanics (see appendix A). In the interaction picture, the Von Neumann equation reads

N d X −i ρ˙ (t) = w |ψ (t)i hψ (t)| = [V (t), ρ (t)] (1.6) I dt n I,n I,n I I n=1 ~ where the subscript I denotes operators or wavefunctions evolving in the interaction picture. When Eq.

(1.6) is integrated between times t0 and t a formal solution for the density operator at time t is obtained Z t i 00 00 00 ρI (t) = ρI (t0) − dt [VI (t ), ρI (t )] . (1.7) ~ t0 1.2. DISSIPATIVE DYNAMICS FROM THE VON NEUMANN EQUATION 18

Figure 4: Schematic depiction of A) closed and B) open quantum systems. The open quantum system is allowed to exchange energy and/or particles (red arrows) with the thermal bath which is assumed to exhibit faster relaxation dynamics than the quantum system. As a result, the open quantum system is influenced by the dissipative effects of the bath whereas the closed quantum system is not.

By inserting Eq. (1.7) back into Eq. (1.6) the integrodifferential equation obtained is Z t −i 1 00 00 00 ρ˙I (t) = [VI (t), ρI (t0)] − 2 dt [VI (t), [VI (t ), ρI (t )]] (1.8) ~ ~ t0

which is a non-local equation of motion for the density operator ρI (t). Before proceeding further with Eq. (1.8), the type of quantum system that will be of concern throughout the thesis shall be discussed in more

detail.

1.2 Dissipative Dynamics from the Von Neumann Equation

The molecular spintronic set-ups that are considered in this thesis are archetypal examples of open

systems, that is, they are quantum mechanical systems that are allowed to exchange energy and/or particles

with a statistical/thermal reservoir. Figure 4 shows schematically the difference between a closed and an

open quantum system by illustrating the potential for the open quantum system to exchange energy and/or

particles (red arrows) with the thermal bath whereas the closed quantum system and the thermal bath evolve

in isolation from one another. As a consequence of coupling to a thermal reservoir, the dynamical evolution

of open and closed quantum systems can show a marked disparity. While the closed quantum system should

evolve only according to the unitary dynamics set by the Schr¨odingerequation, open quantum systems often

display non-unitary time evolution as a result of dissipative effects from the thermal bath.88, 108

The molecular spintronics experiments that are modelled in this thesis are composed of a molecular quantum system which is allowed to exchange electrons with two electronic leads; a source electrode and a drain electrode. In the density matrix formulation of the nanomagnet spintronics problem, the leads are treated as semi-infinite non-interacting electron reservoirs labelled α = S, D with an energy spectrum defined P † by HL = αkσ αkσaαkσaαkσ that are allowed to exchange electrons with wavevector k and spin σ with the 19 1.2. DISSIPATIVE DYNAMICS FROM THE VON NEUMANN EQUATION

molecular system via the Hamiltonian61, 62, 109–111

X ∗ † † HT = Tαkσaαkσcσ + Tαkσcσaαkσ. (1.9) αkσ

The quantum system Hamiltonian HS is quite mutable and is constructed to capture the pertinent physics of the nanomagnet coupled to a radical participating in electron transport through the device. For example this could occur via a direct reduction of the nanomagnet species by the conducting electron as suggested

48 for Mn12 spintronics devices or, if the conducting electron is stabilised in the ligand molecular orbitals,

86, 112 through an exchange interaction as expected in TbPc2. In this thesis, we generally consider system Hamiltonians that commute with the z projection of the total spin operator for the nanomagnet and radical and thus are often able to enumerate the eigenstates of HS with the good total spin quantum number of the device. The Hamiltonians for the isolated leads and the isolated quantum system are both considered to be exactly solvable and hence are collected together into H0 = HL + HS respectively. While the tunnelling

Hamiltonian HT facilitates electron transport through the system, it complicates the model by mixing the wavefunction of the quantum system with the wavefunction of the leads. A commonly encountered regime of electron transport however, is one in which the coupling between the reservoirs and the mesoscopic quantum system is weak (i.e. transport is dominated by the on-site Coulomb repulsion of electrons in the confined

113 quantum system) and hence progress at this theoretical impasse can be made by identifying HT as a perturbative part of the full Hamiltonian H = H0 + HT . With this, the integrodifferential equation for the interaction picture density operator described in Eq. (1.8) can be specialised to the case of a molecular spintronic device by the identification V 7→ HT Z t −i 1 00 00 00 ρ˙I (t) = [HT,I (t), ρI (t0)] − 2 dt [HT,I (t), [HT,I (t ), ρI (t )]] . (1.10) ~ ~ t0

In framing the problem this way, HT is considered as a perturbation which is turned on at time t0 before

which the leads and the quantum system are completely uncorrelated. Before t0 the occupation of states

in the leads are expected to follow a Boltzmann distribution around the chemical potential µα and thus

i −β(HL−µiN ) the density operator simply reads ρL(t0) = e /Zp where the Einstein summation convention is

i implied for the product of chemical potential and number operator for the electrodes such that µiN =

n i o −β(HL−µiN ) µSNS + µDND, Zp is the leads partition function Zp = Tr e and β = 1/kBT is the inverse temperature. Furthermore, owing to the many degrees of freedom associated to the leads, any effects manifested by switching on the tunnelling Hamiltonian HT will quickly dissipate away and thus have no bearing on the quantum subsystem. With this in mind, it is assumed that the leads will relax back to a

Boltzmann distribution faster than the time scale of individual electron tunnelling events and hence the density operator for the leads may be approximated at any time after t0 also as the constant time operator 1.2. DISSIPATIVE DYNAMICS FROM THE VON NEUMANN EQUATION 20

i −β(HL−µiN ) 88, 114 ρL = e /Zp; this is the Born approximation. With this assumption the full density operator

is ρI (t) = ρL ⊗ σI (t) where σI (t) = TrL {ρI (t)} is the interaction picture reduced density operator for the

quantum subsystem and TrL {... } denotes a tracing operation over the states in the leads only.

Eq. (1.10) may be further simplified by ignoring memory effects from the time evolution of ρI (t) by

00 making the substitution ρI (t ) → ρI (t) under the integral sign; this second approximation is known as the Markoff approximation. The validity of this replacement follows from the rapid decay of the leads correlation

functions. After making this replacement what remains is a differential equation for ρ(t) that is local in time

Z t −i 1 00 00 ρ˙I (t) = [HT,I (t), ρL ⊗ σI (t0)] − 2 dt [HT,I (t), [HT,I (t ), ρI (t)]] . (1.11) ~ ~ t0

A curious artefact of this derivation arises when higher order corrections to the density operator are considered via the iteration procedure set out in the last section. Returning for a moment to Eq. (1.6), if

Eq. (1.7) is continuously imported back into this expression an integrodifferential equation is obtained of the form Z t −i 1 00 00 ρ˙I (t) = [VI (t), ρI (t0)] − 2 dt [VI (t), [VI (t ), ρI (t0)]] + ~ ~ t0 m (1.12) −i Z t Z tm−1 ··· + dt1··· dtm [VI (t), [..., [VI (tm), ρI (tm)] ... ]] ~ t0 t0 where only the integral that is leading order in VI (t) contains the density operator at a general time t; all lower order terms are only influenced by the density operator evaluated at some initial time t0. One may proceed alternatively by considering the formal solution to the Von Neumann equation integrated over the time domain τ ∈ [t, t0] Z t0 0 i ρI (t ) = ρI (t) − dτ [VI (τ), ρI (τ)] . (1.13) ~ t and import this (with the appropriate renaming of variables) into Eq. (1.8) so that Z t −i 1 00 00 ρ˙I (t) = [VI (t), ρI (t0)] − 2 dt [VI (t), [VI (t ), ρI (t)]] ~ ~ t 0 (1.14) Z t Z t0 i 00 000 00 000 000 + 3 dt dt [VI (t), [VI (t ), [VI (t ), ρI (t )]]] . ~ t0 t0

The aforementioned procedure can be reapplied to include higher order perturbative effects in VI (t) in the Von Neumann equation in a manner consistent with experiment.115 Including these higher order effects

however leads to cumbersome and overtly non-intuitive equations of motion for the reduced density matrix.

An alternative approach that is consistent with a fully microscopic derivation of the rate equations is to

calculate cotunnelling processes between non-interacting product states of the leads and the quantum system

directly using the T-matrix approach and then incorporate these processes consistently into the secular

rate equations for the reduced density matrix discussed later.111, 115–117 The full derivation of generalised

transition rates calculated in the T-matrix approach is given in appendix B for the convenience of the reader

and are used in part II of this thesis. 21 1.2. DISSIPATIVE DYNAMICS FROM THE VON NEUMANN EQUATION

By including terms which are third order in VI (t) the non-local density operator contributions have been shifted to the highest order term and hence, after dropping this term on the grounds of weak coupling

(V 3  V 2), the integrodifferential equation (1.11) has been recovered without explicitly employing the

Markoff approximation.115

Returning from this Markovian digression, after choosing t0 → −∞, relabelling the integration variable t00 → t − s and tracing over the states in the leads, Eq. (1.11) gives −i 1 Z ∞ σ˙ I (t) = TrL {[HT,I (t), ρL ⊗ σI (−∞)]} − 2 ds TrL {[HT,I (t), [HT,I (t − s), ρL ⊗ σI (t)]]} . (1.15) ~ ~ 0 By virtue of the simplistic density operator chosen for the leads, the first term in Eq. (1.15) is exactly zero due

† to trivial expectation values of the lead electron creation and annihilation operators: haαkσiL = haαkσiL = 0 appearing in the sum. Thus sixteen terms that originate from expanding the commutator are left under the integral sign

−1 X X Z ∞ σ˙ I (t) = 2 ds TrL { ~ αkσ βlλ 0  T ∗ T a† (t)c (t)c† (t − s)a (t − s)ρ ⊗ σ (t) αkσ βlλ I,αkσ I,σ I,λ I,βlλ L I    +T T ∗ c† (t)a (t)a† (t − s)c (t − s)ρ ⊗ σ (t) αkσ βlλ I,σ I,αkσ I,βlλ I,λ L I   ∗ † †  −T Tβlλa (t)cI,σ(t)ρL ⊗ σI (t)c (t − s)aI,βlλ(t − s) αkσ I,αkσ I,λ   ∗ † †  −TαkσTβlλcI,σ(t)aI,αkσ(t)ρL ⊗ σI (t)aI,βlλ(t − s)cI,λ(t − s) Type I −T ∗ T c† (t − s)a (t − s)ρ ⊗ σ (t)a† (t)c (t) αkσ βlλ I,λ I,βlλ L I I,αkσ I,σ   ∗ † †  −TαkσT a (t − s)cI,λ(t − s)ρL ⊗ σI (t)c (t)aI,αkσ(t) βlλ I,βlλ I,σ   ∗ † †  +TαkσTβlλρL ⊗ σI (t)cI,λ(t − s)aI,βlλ(t − s)aI,αkσ(t)cI,σ(t)   ∗ † †  +TαkσTβlλρL ⊗ σI (t)aI,βlλ(t − s)cI,λ(t − s)cI,σ(t)aI,αkσ(t)  +T ∗ T ∗ a† (t)c (t)a† (t − s)c (t − s)ρ ⊗ σ (t) αkσ βlλ I,αkσ I,σ I,βlλ I,λ L I    +T T c† (t)a (t)c† (t − s)a (t − s)ρ ⊗ σ (t) αkσ βlλ I,σ I,αkσ I,λ I,βlλ L I   ∗ ∗ † †  −T T a (t)cI,σ(t)ρL ⊗ σI (t)a (t − s)cI,λ(t − s) αkσ βlλ I,αkσ I,βlλ   † †  −TαkσTβlλcI,σ(t)aI,αkσ(t)ρL ⊗ σI (t)cI,λ(t − s)aI,βlλ(t − s) Type II −T ∗ T ∗ a† (t − s)c (t − s)ρ ⊗ σ (t)a† (t)c (t) αkσ βlλ I,βlλ I,λ L I I,αkσ I,σ   † †  −TαkσTβlλc (t − s)aI,βlλ(t − s)ρL ⊗ σI (t)c (t)aI,αkσ(t) I,λ I,σ   ∗ ∗ † †  +TαkσTβlλρL ⊗ σI (t)aI,βlλ(t − s)cI,λ(t − s)aI,αkσ(t)cI,σ(t)   † †  +TαkσTβlλρL ⊗ σI (t)cI,λ(t − s)aI,βlλ(t − s)cI,σ(t)aI,αkσ(t) } (1.16) 1.2. DISSIPATIVE DYNAMICS FROM THE VON NEUMANN EQUATION 22

that have been conveniently grouped into “type I” and “type II” terms. Before proceeding further, define

the projection operators PN for the quantum subsystem that, when sandwiched between an operator C →

PN CPN that acts on the quantum subsystem (e.g. σI (t)) project C onto the manifold of quantum subsystem states containing N electrons. Clearly, these projection operators can not change the number of electrons in the quantum subsystem or in the leads and therefore will commute with both parts of the exactly solvable

Hamiltonian [HL,PN ] = [HS,PN ] = 0 as well as obey the two relations

† † PN cσ = cσPN−1 PN cσ = cσPN+1. (1.17)

By applying the projection operators PN to both sides of Eq. (1.16) and making use of the relations in Eq. (1.17) the reason for the separation of type I and type II terms becomes transparent; type I terms contain

(N) contributions from diagonal blocks of the density matrix e.g. σI (t) = PN σI (t)PN while the type II terms contain contributions from the coherences between different charge states of the device e.g. PN σI (t)PN+2.

0 In a weakly coupled system, the off-diagonal blocks of the density matrix PN σI (t)PN 0 with N 6= N are expected to decohere rapidly owing to the continuous electrical measurement of the quantum system118 and

(N) thus these terms may safely be neglected from the time evolution of σI (t).

(N) (N) Now consider the time evolution of a general matrix element σI,mn = hN, m| σI (t) |N, ni where |N, ni

and |N, mi are N electron eigenstates of the quantum subsystem Hamiltonian such that HS |N, mi = (N) Em |N, mi. It will often be the case that the subsystem Hamiltonian HS commutes with the giant spin

projection operators of the nanomagnet Sz so that the good quantum numbers that characterise the spin

projection of the nanomagnet can be used to enumerate the energy eigenvalues of HS. So as not to crowd each line with much of the same algebra, the development of a representative term on the right hand side of

Eq. (1.16) is presented and all other type I terms will be reintroduced back into the final result. The time

(N) evolution of σI,mn(t) is governed by

Z ∞ (N) −1 X X ∗ † (N) † σ˙ I,mn(t) = 2 TαkσTβlλ ds hN, m| cI,σ(t)cI,λ(t − s)σI (t) |N, ni haI,αkσ(t)aI,βlλ(t − s)iL + ... ~ αkσ βlλ 0 −1 N+1 N X X X X ∗ i∆mq t/~ † (N) = 2 TαkσTβlλe hN, m| cσ |N + 1, pi hN + 1, p| cλ |N, qi σI,qn(t) ~ p q αkσ βlλ Z ∞ −i∆pq s/~ † × ds e haI,αkσ(t)aI,βlλ(t − s)iL + ... 0 (1.18)

where a resolution of identity was inserted between the subsystem creation, annihilation and reduced

(N) (N) density operators, ∆mq = Em − Eq is the energy gap between the |N, mi and |N, qi states, ∆pq = (N+1) (N) † Ep − Eq is the energy gap between the |N + 1, pi and |N, qi states and haI,αkσ(t)aI,βlλ(t − s)iL = 23 1.2. DISSIPATIVE DYNAMICS FROM THE VON NEUMANN EQUATION

n † o < TrL aI,αkσ(t)aI,βlλ(t − s)ρL is the lesser Green function iGαβ(t, t − s) for the leads written in the inter- action picture.

The Green function for the leads can be calculated exactly owing to the simple form of the leads density

operator chosen above. Using the cyclic permutation property of the trace88

n i o < −1 † −β(HL−µiN ) iGαβ(t, t − s) = Zp TrL aI,αkσ(t)aI,βlλ(t − s)e

n i o −1 † −β(HL−µiN ) (1.19) = Zp TrL aI,αkσ(s)aβlλe

< = iGαβ(s, 0)

119 † and using the equation of motion technique for the operator aI,αkσ(s) leads to

i † † † † iαkσ s/~ a˙ I,αkσ(s) = αkσaI,αkσ(s) =⇒ aI,αkσ(s) = aαkσe . (1.20) ~

The greater Green function for the leads appears in other type I expressions from Eq. (1.16) and can be

obtained by a similar means to the lesser function. Both lesser and greater functions are

< iαkσ s/~ † iαkσ s/~ iGαβ(s, 0) = e haαkσaβlλiL = e f(αkσ − µα)δαβδklδσλ (1.21) > −iαkσ s/~ † −iαkσ s/~ iGαβ(s, 0) = e haαkσaβlλiL = e [1 − f(αkσ − µα)] δαβδklδσλ

−1  β( −µα) where f(αkσ − µα) = 1 + e αkσ is the Fermi-Dirac distribution. Now the time evolution of the reduced density matrix element in Eq. (1.20) is governed by

−1 N+1 N (N) X X X 2 i∆mq t/~ † (N) σ˙ I,mn(t) = |Tαkσ| e hN, m| cσ |N + 1, pi hN + 1, p| cσ |N, qi σI,qn(t) ~2 p q αkσ (1.22) Z ∞ i(αkσ −∆pq )s/ × f(αkσ − µα) ds e ~ + ... 0

(†) 0 i∆ t/ (†) 0 where hN, a| cσ (t) |N , bi = e ab ~ hN, a| cσ |N , bi has been employed. Unfortunately, the integral that appears in Eq. (1.22) does not converge as s → ∞ but instead oscillates indefinitely. To remedy this problem a damping factor lim e−ηs is included to ensure convergence, so that η→0+

Z ∞ Z ∞ ds ei(αkσ −∆pq )s/~ = lim ds ei(αkσ −∆pq +i~η)s/~ 0 η→0+ 0 i~ = lim (1.23) η→0+ αkσ − ∆pq + i~η  1  = π~δ(αkσ − ∆pq) + i~P αkσ − ∆pq 1.2. DISSIPATIVE DYNAMICS FROM THE VON NEUMANN EQUATION 24

where the Sokhotski-Plemelj theorem120 has been used in the final step and P denotes the Cauchy principal

value. This integral has a real dissipative part and an imaginary energy shift part; the latter shall be

discussed later. To proceed with Eq. (1.22) the sum over wavevectors is replaced with an integral over the P R energies  weighted by the density of states in the leads k 7→ dDασ() and αkσ has been replaced by  to yield

−1 N+1 N (N) X X X i∆mq t/~ † (N) σ˙ I,mn(t) = e hN, m| cσ |N + 1, pi hN + 1, p| cσ |N, qi σI,qn(t) ~ p q ασ Z    2 1 × d|Tασ()| Dασ()f( − µα) πδ( − ∆pq) + iP + .... (1.24)  − ∆pq N+1 N   X X i∆ t/ 1 N,N+1 N,N+1 (N) = − e mq ~ Γ −→ + iδ −→ σ (t) + ... 2 mppq mppq I,qn p q

where terms contributing to the dissipative part and the energy shifts have been temporarily collected into

N,N+1 N,N+1 the tensors Γmppq−→ and δmppq−→ respectively, with

N,N+1 2π X 2 † Γmppq−→ = |Tασ| Dασf(∆pq − µα) hN, m| cσ |N + 1, pi hN + 1, p| cσ |N, qi ~ ασ Z (1.25) N,N+1 1 X 2 † f( − µα) δ −→ = |T | D hN, m| c |N + 1, pi hN + 1, p| c |N, qi P d mppq ασ ασ σ σ  − ∆ ~ ασ pq

61, 111, 121 where Tασ and Dασ are assumed to remain constant over the region of integration and the arrow −→ over subscripts pq indicates that one should consider the gap ∆pq.

Following the same treatment with the other seven type I terms in Eq. (1.16) provides the fully coupled

(N) quantum rate equation for the density matrix element σI,mn(t)

(N) σ˙ I,mn(t) = N+1 N+1   X X i(∆ +∆ )t/ 1  N,N+1 N,N+1  N,N+1 N,N+1 (N+1) e mp rn ~ Γ −→ + Γ←− + i δ −→ − δ←− σ (t) 2 mprn mprn mprn mprn I,pr p r N−1 N−1   X X i(∆ +∆ )t/ 1  N−1,N N−1,N   N−1,N N−1,N  (N−1) + e mp rn ~ Γ˜←− + Γ˜ −→ − i δ˜ ←− − δ˜ −→ σ (t) 2 rnmp rnmp rnmp rnmp I,pr p r N "N+1   N−1  # X i∆ t/ X 1 N,N+1 N,N+1 X 1 N−1,N N−1,N (N) − e qn ~ Γ←− − iδ←− + Γ˜ −→ + iδ˜ −→ σ (t) 2 qppn qppn 2 rnqr rnqr I,mq q p r "N+1   N−1  # i∆ t/ X 1 N,N+1 N,N+1 X 1 N−1,N N−1,N (N) +e mq ~ Γ −→ + iδ −→ + Γ˜←− − iδ˜ ←− σ (t) 2 mppq mppq 2 rqmr rqmr I,qn p r (1.26) 0 N,N 0 N,N N,N 0 N,N 0 where Γ˜ −→ and δ˜ −→ are obtained from Γ −→ and δ −→ respectively with the substitution f() 7→ abcd abcd abcd abcd 25 1.3. THE SECULAR APPROXIMATION FOR THE REDUCED DENSITY MATRIX

[1 − f()]. In principle, one may convert back to the Schr¨odingerpicture and solve the system of coupled

equations given in Eq. (1.26) for each matrix element of the density matrix. In practice however, there is

(N) a further simplification that is often implemented to decouple the time evolution of the coherences σI,m6=n (N) from the populations σI,mm; the secular approximation. This approximation will be developed and its applicability discussed in the next section.

1.3 The Secular Approximation for the Reduced Density Matrix

(N) . (N) To motivate the secular approximation first consider the time evolution of the population σI,mm = σI,m set by Eq. (1.26)

(N) σ˙ I,m(t) = N+1 N+1   X X i∆ t/ 1  N,N+1 N,N+1  N,N+1 N,N+1 (N+1) e rp ~ Γ −→ + Γ←− + i δ −→ − δ←− σ (t) 2 mprm mprm mprm mprm I,pr p r N−1 N−1   X X i∆ t/ 1  N−1,N N−1,N   N−1,N N−1,N  (N−1) + e rp ~ Γ˜←− + Γ˜ −→ − i δ˜ ←− − δ˜ −→ σ (t) 2 rmmp rmmp rmmp rmmp I,pr p r N "N+1   N−1  # X i∆ t/ X 1 N,N+1 N,N+1 X 1 N−1,N N−1,N (N) − e qm ~ Γ←− − iδ←− + Γ˜ −→ + iδ˜ −→ σ (t) 2 qppm qppm 2 rmqr rmqr I,mq q p r "N+1   N−1  # i∆ t/ X 1 N,N+1 N,N+1 X 1 N−1,N N−1,N (N) +e mq ~ Γ −→ + iδ −→ + Γ˜←− − iδ˜ ←− σ (t). 2 mppq mppq 2 rqmr rqmr I,qm p r (1.27)

Note that each term in Eq. (1.27) is weighted by a fast oscillating phase factor except when the energy gap ∆ in the exponent is zero. The secular approximation consists of discarding all terms with an oscillatory phase factor from the right hand side of Eq. (1.27) under the assumption that these contributions undergo many cycles during the course-grained evolution of the quantum system and thus their effects cancel out upon integration.88 The validity of the secular approximation is discussed for a theoretical model proposed in part

I of this thesis whereby we found that the non-secular oscillatory terms contribute nothing to the dynamical evolution of the populations thus allowing them to be neglected safely. Performing this approximation reduces Eq. (1.26) to

N+1 N−1 "N+1 N−1 # (N) X N,N+1 (N+1) X ˜N−1,N (N−1) X N,N+1 X ˜N−1,N (N) σ˙ I,m(t) = Γmppm−→ σI,p (t) + Γrmmr−→ σI,r (t) − Γmppm−→ + Γrmmr−→ σI,m(t) (1.28) p r p r 1.3. THE SECULAR APPROXIMATION FOR THE REDUCED DENSITY MATRIX 26

for the populations and

"N+1 (N) (N) X 1  N,N+1 N,N+1  N,N+1 N,N+1 σ˙ (t) = −σ (t) Γ −→ + Γ −→ − i δ −→ − δ −→ I,mn I,mn 2 nppn mppm nppn mppm p (1.29) N−1 # X 1  N−1,N N−1,N   N−1,N N−1,N  + Γ˜ −→ + Γ˜ −→ + i δ˜ −→ − δ˜ −→ 2 rnnr rmmr rnnr rmmr r for the coherences. Note that the small energy shifts δ now only appear in the time evolution of the coherences. The evaluation of a representative shift is calculated following Engels and Loss:111

Z Z µα Z ∆pn−Λ Z µα N,N+1 f( − µα) d d d δ −→ ∝ P d ≈ P = lim + nppn  − ∆  − ∆ Λ→0  − ∆  − ∆ pn 0 pn 0 pn ∆pn+Λ pn (1.30)   µα + ∆pn ∆pn = log ≈ log = 0 ∆pn ∆pn and thus they shall be dropped from here on.

Finally, the rate equations for the reduced density matrix elements of the quantum system coupled

(N) to the dissipative leads are obtained by transforming back into the Schr¨odingerpicture with σI = eiH0t/~σ(N)e−iH0t/~ so that

N+1 N−1 "N+1 N−1 # (N) X p→m (N+1) X r→m (N−1) X m→p X m→r (N) σ˙ m (t) = W σp (t) + W σr (t) − W + W σm (t) p r p r (1.31) "N+1 N−1 # (N) i h (N) i 1 (N) X m→p n→p X m→r n→r σ˙ mn (t) = − HS, σ (t) − σmn (t) (W + W ) + (W + W ) mn 2 ~ p r

where a more streamlined notation has been adopted

  0 † 2 0  hN ; b| cσ |N; ai fα(∆ba) N > N a→b 2π X 2  W = |Tασ| Dασ (1.32) ~ ασ  0 2 0 |hN ; b| cσ |N; ai| [1 − fα(∆ab)] N < N which is more in-keeping with the remainder of this thesis and the current molecular spintronics litera- ture.61, 92, 97 The upper case of Eq. (1.32) is the transition rate between two states |N; ai and |N 0; bi from

different redox manifolds of the quantum system induced by an electron from the leads charging the quantum

system; the lower case to electron discharging. The two expressions in Eq. (1.31) can be expressed succinctly

in the single equation

N+1 N−1 ! (N) i h (N) i X p→m (N+1) X r→m (N−1) (N) σ˙ mn (t) = − HS, σ (t) + δmn W σp + W σr − γmnσmn (t) (1.33) mn ~ p r 271.4. RECOUPLING POPULATIONS AND COHERENCES WITH A RESONANT PERTURBATION

where "N+1 N−1 # 1 X X γ = (W m→p + W n→p) + (W m→r + W n→r) (1.34) mn 2 p r is often referred to as the decoherence rate.

The secular rate equations presented in Eq. (1.33) form the bedrock of the theoretical models investigated in this thesis. After solving these equations analytically or numerically, the reduced density matrix elements can be used to compute macroscopic observables of the quantum system such as its magnetisation, its electrical conductivity and the spin currents flowing through the system using Eq. (1.4).

1.4 Recoupling Populations and Coherences with a Resonant Per-

turbation

To effectively model transport through a nanomagnet spintronic junction coupled to a resonant pertur- bation, one must modify the quantum rate equations that were derived in the previous section, consistently.

The Hamiltonian that properly describes the nanomagnet-based molecular spintronics device coupled to a resonant radiation is

0 H(t) = HL + HS + V (t) + HT = H0 + V (t) + HT = H0(t) + HT (1.35)

where, as in the previous sections, HS, HL and HT are Hamiltonians for the quantum system, the leads and the tunnelling. Unlike in the previous section another part to the Hamiltonian V (t) that describes the interaction between a semi-classical radiation and the nanomagnet spin must be included. The explicit time dependence of V (t) complicates the unitary transformation required to turn the density operator from the Schr¨odingerpicture to the interaction picture. The correct unitary transformation (see appendix A) to

† h −i R t 0 i actualise this picture change is ρI (t) = S (t)ρ(t)S(t) with S(t) = T exp dτH (τ) and where T is the ~ 0 time ordering operator. The developments leading to the second line of Eq. (1.18) are no longer straight forward to apply as the basis for the reduced density matrix chosen for the quantum rate equations are the energy eigenstates of HS and not of HS + V (t). If, however, the amplitude of the radiation described by

V (t) is smaller than the energy scale set by HS then it is possible to approximate the action of S(t) on an energy eigenstate |N, pi by

−i R t 0 (N) dτH0+V (t ) −iE / S(t) |N, pi = T e ~ |N, pi ≈ e p ~ |N, pi (1.36) 1.4. RECOUPLING POPULATIONS AND COHERENCES WITH A RESONANT PERTURBATION 28

and so the derivation proceeds approximately as presented in the previous section. In transforming the

density matrix elements defined by Eq. (1.28) and Eq. (1.29) back into the Schr¨odingerpicture, one

acquires an extra term in the differential equations for both the populations and the coherences which may

be written in the succinct form

N+1 N−1 ! (N) i h (N) i X p→m (N+1) X r→m (N−1) (N) σ˙ mn (t) = − HS + V (t), σ (t) + δmn W σp + W σr − γmnσmn (t). mn ~ p r (1.37)

The usual quantum rate equations for the density matrix in the basis of HS are hence obtained (with a retention of the good quantum numbers that index each energy eigenstate) but now with an extra term proportional to the time-dependent perturbation V (t) that recouples the evolution of the populations and the coherences even after the secular approximation has been performed; this approach will be implemented in chapter 2 of part I of the thesis. Part I

Manipulating Spin Currents in

Single-molecule Magnet Spintronic

Devices through the Perturbation of

Individual Quantum Spin States

29 Chapter 2

Molecular Spintronics Using

Single-molecule Magnets Under

Irradiation

Authors: Kieran Hymas, Alessandro Soncini Affiliation: School of Chemistry, University of Melbourne, Parkville, 3010 Published: 10/6/2019 Journal: Physical Review B

30 31 2.1. ABSTRACT

2.1 Abstract

We theoretically investigate a single-molecule magnet (SMM) grafted to a quantum dot in contact with metallic leads and interacting with a resonant electromagnetic radiation. We explore both the explicit time- dependent behaviour and the steady state current-voltage characteristics of the device when the source lead is ferromagnetic. At zero bias voltage a net current is pumped through the device with the source spin current being reversed and amplified in the drain lead; this effect also persists for non-zero bias. We explain this effect in terms of spin transitions in the nanomagnet induced by the resonant radiation followed by their subsequent relaxation via spin-asymmetric charge transfer processes. We demonstrate that the same effects are recovered in the time-averaged current when the device interacts with pulsed resonant radiation.

Moreover, within the pulsed irradiation regime, appropriate choices of pulse length and wait times are shown here to allow the detection of coherent Rabi oscillations of the SMM spin states, via time-averaged spin current measurements.

2.2 Introduction

Single molecule magnets (SMMs) are magnetically anisotropic inorganic complexes with large spin mo- ments that display a slow relaxation of the magnetisation below a given blocking temperature.122 When grafted to graphene quantum point contacts or carbon nanotubes, single molecule magnets have been shown to impart highly anisotropic magneto-conductance hysteresis fingerprints on local electric currents, providing compelling evidence for the existence of an exchange interaction between the giant spin of the SMM and the spin of conduction electrons of the carbon nanostructure79, 80 or phthalocyaninato quantum dots in the case

82, 84 113 TbPc2 break junction devices. SMMs have been studied in the context of molecular spintronics and show potential as molecular memory units60 and spin valves80, 103 that may eventually form the foundations of complex spintronic technologies or even more ambitiously, quantum computers.

Recent spin-polarised STM studies of quantum magnets on surfaces have demonstrated that polarised spin currents can influence and even flip the nanomagnet’s spin moment via a spin transfer torque effect.67, 68

This effect could be used to read or write bits of information to single nanomagnets in spintronics devices.

A crucial challenge in the development of molecular quantum spintronics consists of injecting a spin current into a SMM-based device. To date, a spintronics experiment with this format has not yet been realised.

A feasible strategy to achieve coupling between a spin current and the quantum spin states of a single- molecule magnet is to graft SMMs onto the surface of a graphene quantum point contact, since (i) efficient spin injection in graphene has already been achieved123, 124 and (ii) coupling between SMMs and a graphene 2.2. INTRODUCTION 32

Figure I.2.1: A schematic representation of electron transport from a ferromagnetic lead through a quantum dot that is antiferromagnetically coupled to a SMM subject to resonant radiation. Energy is supplied to the system to tilt the giant spin of the SMM (thick, red) allowing a spin majority electron to charge the device from the ferromagnetic source. On relaxation, the SMM aligns against the longitudinal field reversing the spin of the conduction electron as it is emitted to the non-magnetic drain.

quantum dot device has been demonstrated.79

In this paper we propose and theoretically study a molecular spintronics set-up based on a SMM device under resonant irradiation. The aim is to perturb the populations of SMM spin states by inducing simple coherent spin dynamics behaviour in the SMM and assess its influence on the spin current flowing through a device via the aforementioned exchange interaction, so that the spin current effectively measures the dynamics of the SMM spin states under irradiation. In SMM-based transport experiments a sweeping magnetic field is often used in this spirit to probe the incoherent dynamics related to the slow-relaxation of the nanomagnet79, 80, 101 but here, by using resonant electromagnetic radiation we are able to study also the coherent oscillatory dynamics of the magnetic subsystem and its interplay with the dissipative dynamics of the leads.

While the spectroscopy of nanomagnets in the bulk phase is relatively commonplace, addressing single

(or few) molecules in a spintronic device with radiation is not at all trivial. Recently, scanning tunnelling microscopy (STM) tips have been employed in this vein to induce atomically localised time-dependent modulations to the crystal field of magnetic atoms adsorbed to a MgO/Ag(001) substrate.125 Another approach to achieve coherent transitions within a SMM device was demonstrated by Thiele et al.84 whereby the nuclear spin states of a single TbPc2 molecule in a molecular break junction were coupled to resonant microwave signals via the hyperfine Stark effect. While the experimental details of inducing resonant coherent transitions in a nanomagnet spintronics device are intricate and system specific, a radiation-magnetic dipole coupling is archetypal of more general coupling schemes (discussed in appendix A) that may be utilised in an experimental nanomagnet spintronics set-up. In this manuscript we focus on this simple regime of 33 2.3. THEORETICAL MODEL radiation-dipole coupling in order to illustrate the interesting phenomena that can arise from a nanomagnet spintronic device subject to a resonant, time-dependent perturbation.

We contribute to the already extensive nanomagnet-based spintronics literature58, 61, 62, 65, 66, 126–128 by considering a SMM configuration with the potential to work as a spin pump and spin switch. Although non-collinear magnetic molecules have been previously presented as efficient spin-switching devices77, 78 the possibility of inducing spin current switching is presented here via a more general SMM system (i.e. without invoking specific non-collinear spin configurations). Finally, we discuss the possibility of reading out Rabi oscillations between spin states via time-averaged spin current measurements, a result already observed in experiment between the nuclear spin states of a TbPc2 molecule, in which case however the device also required a sweeping magnetic field.84

In section 2.3 we present a model describing the operation of our SMM-based spintronic device under irradiation utilising the density matrix formalism. In section 2.4 we show results from our model when both continuous and pulsed radiation are applied and discuss the underlying mechanism that leads to pumping, switching and amplification of the spin current. Finally, in Sec 2.5 we recapitulate and make concluding remarks.

2.3 Theoretical Model

2.3.1 Model Hamiltonian

We consider a device (Figure I.2.1) consisting of a SMM grafted to a quantum dot that is weakly coupled to two metallic leads. We include an interaction with a static longitudinal magnetic field and a gate electrode.

At sufficiently low temperatures, we assume that the device operates in the Coulomb blockade regime such that charging and discharging to and from the dot occurs sequentially. We suppose that the on-site Coulomb repulsion between electrons on the dot is large enough to exclude doubly charged states from participating in transport through the device. We also include a coupling between the total spin of the device and the magnetic component of applied radiation.

The total Hamiltonian for the device reads

H(t) = HL + HS + V (t) + HT (I.2.1) where:

X † HL = (αkσ − µα) aαkσaαkσ (I.2.2) αkσ 2.3. THEORETICAL MODEL 34

(†) is the isolated source and drain Hamiltonian, in which aαkσ destroys (creates) an electron in lead α with wavevector k, spin σ and energy αkσ. Here µα corresponds to the chemical potential of electrons in the Fermi level of lead α which is often modulated in experiment by the application of an antisymmetric bias

voltage Vb such that µL = Vb/2 and µR = −Vb/2. The system Hamiltonian is

2 X † HS = −DSz + ( − eVg)cσcσ + µBBz (g1Sz + g2sz) − JS · s (I.2.3) σ

(†) where S = (Sx,Sy,Sz) is the SMM spin operator, cσ annihilates (creates) an electron on the dot with

spin σ and s = (sx, sy, sz) is the spin operator for the aforementioned radical. D is the uniaxial anisotropy

characterising the zero-field splitting of the SMM spin states, g1 and g2 are the g-factors for the SMM and

the dot respectively, µB is the Bohr magneton, Bz is the amplitude of a static longitudinal magnetic field, 

is the one–electron dot–orbital energy, Vg is the magnitude of an applied gate voltage and J is the exchange coupling between the SMM and an electron on the dot. The tunnelling Hamiltonian is simply

X ∗ † † HT = Tαaαkσcσ + Tαcσaαkσ (I.2.4) αkσ

where Tα are the tunnelling amplitudes for charging and discharging events between lead α and the dot; we neglect the possibility of direct tunnelling between source and drain leads.

We discuss here the simplest radiation-dipole coupling regime that can induce magnetic dipole-allowed

resonant transitions in the ground spin multiplet of the nanomagnet. We approximate the magnetic com-

ponent of radiation propagating along the easy axis of the nanomagnet as a rotating transverse magnetic

field that couples to the giant spin of the SMM by a Zeeman interaction. We take the field to be rotating

clockwise with a frequency ω in the plane perpendicular to the easy axis of the SMM so that

V (t) = g1µBB⊥[Sx cos(ωt) − Sy sin(ωt)] (I.2.5)

where B⊥ is the amplitude of the magnetic component of the radiation.

After noting that the z-component of the total spin operator (defined by St = S + s) commutes with

t HS, it is convenient to enumerate the energy eigenstates of HS with the eigenvalues of Sz. We use a notation where |n, mi denotes an electronic state of the SMM-dot hybrid with a total spin m and with n

electrons occupying the LUMO of the dot. The energy eigenstates of the neutral and charged system are

± ± ± |0, mi ≡ |mi ⊗ |vaci and |1, mi ≡ Am |m + 1/2i ⊗ |↓i + Bm |m − 1/2i ⊗ |↑i respectively; the fully polarised ± ± states are simply |1, s + 1/2i ≡ |si ⊗ |↑i and |1, −s − 1/2i ≡ |−si ⊗ |↓i. The coefficients Am and Bm are of 35 2.3. THEORETICAL MODEL the form p ± |J| 2∆(m) ∓ [(2D − J)m − µBBzδg] Am = ± J 2p∆(m) (I.2.6) p 2 ± |J| s(s + 1) − m + 1/4 Bm = p p 2 ∆(m) 2∆(m) ∓ [(2D − J)m − µBBzδg]

2 2 2 2 1/2 with ∆(m) = [(µBBzδg/2) +µBBzδg(2D−J)m/2+D(D−J)m +(J/4) (2s+1) ] and δg = g1 −g2. The

2 ± energies of the electronic states of the SMM-dot hybrid are E(0, m) = −Dm − g1µBmBz and E(1, m) =

2  − Vg + J/4 − D(m + 1/4) − g1µBmBz ± ∆(m). The energies of the fully polarised charged states are given by E(1, ±s ± 1/2)+ when 2D0 − J ≥ 0 and E(1, ±s ± 1/2)− otherwise.

From here we shall be concerned with the 2D − J > 0 regime in which the charged ground states are the antiferromagnetic |1, ±s ∓ 1/2i− states. Note that the exchange part of the Hamiltonian in Eq. (I.2.3) mixes states of the SMM-dot hybrid that conserve the axial projection of the total spin of the device. Thus, in the antiferromagnetic coupling regime the charged ground states are linear combinations of SMM spin states; this is a crucial condition for the operation of the device as discussed later. We choose Bz < 0 to lift the degeneracy of both neutral and charged spectra but are careful not to choose |Bz| so large that the ferromagnetic |1, s + 1/2i state becomes the new ground state of the charged system. Finally, we impose a

− level degeneracy condition between the |0, si and |1, s − 1/2i states by choosing a suitable gate voltage Vg so that |E(0, s) − E(1, s − 1/2)−| = 0.

2.3.2 Master Equation in a Time-dependent Resonant Field and Stationary Current

The reduced density matrix describing the electronic spin states of the SMM-dot hybrid is defined by

tot tot ρ(t) = TrL{ρ (t)} where ρ (t) is the density matrix for the entire device and TrL{... } denotes a trace over states in the leads. A system of differential equations for ρ(t) is obtained within the Born-Markoff approximation by making standard manipulations88 to the Von Neumann equation however neglecting the effect of V (t) in the unperturbed propagators used to transform the equations of motion of the density matrix into the interaction picture. It is self consistent to neglect the effect of the radiation in the definition of the interaction picture provided that the transitions caused by V (t) are much slower than the decay of

111 correlations in the leads induced by HT . After retaining only the secular terms in the resultant master equation (the validity of which is investigated in Appendix B), the evolution of a reduced density matrix element is governed by

−i X l→m ρ˙mm0 = [HS + V (t), ρ]mm0 + δmm0 W ρl − γmm0 ρmm0 (I.2.7) ~ l 2.3. THEORETICAL MODEL 36

0 where ρmm0 = hn, m| ρ(t) |n, m i is a matrix element between eigenstates of HS (we do not consider co- herences between states from different charge spaces and so unambiguously drop the index n in ρmm0 ),

1 P m→l m0→l l→m P l→m γmm0 = 2 l W + W is the total decoherence rate and W = ασ Wασ are rates of charg- ing/discharging (summed over leads and spin) from a state |n, li to a state |n0, mi given by62  2  n→n+1  cσ,ml fα(∆ml) l→m Γα(1 + 2σPα)  Wασ = (I.2.8) 2~  2  n→n−1  cσ,ml [1 − fα(∆lm)]

where the upper case applies for charging transitions (n0 = n + 1) and the lower case applies for discharging

0 −1 transitions (n = n − 1). In the expression above, fα(∆) = [1 + exp(β(∆ ∓ Vb/2))] is the Fermi-Dirac distribution for electrons in lead α, the argument ∆ is the energy difference between the relevant charged

and neutral states, −(+)Vb corresponds to the applied bias voltage at the source (drain) lead, β = 1/kBT

where T is temperature and kB is Boltzmann’s constant, Γα is the coupling strength between lead α and

n→n+1 0 † the SMM-dot hybrid and Pα is the spin polarisation inherent to lead α. Finally, cσ,ml = hn , m| cσ |n, li n→n−1 0 and cσ,ml = hn , m| cσ |n, li are the charging and discharging transition amplitudes respectively. Note that W l→m is only non-zero when the number of conduction electrons is changed by one and the total spin of the

SMM-dot hybrid is changed by one half i.e. |n0 − n| = 1 and |l − m| = 1/2.

In the Coulomb blockade regime, at low temperatures and bias voltages, only the |0, si, |0, s − 1i and

|1, s − 1/2i− states make significant contributions to the current flowing through the device and so we focus

only on the evolution of these states. Since the |1, s − 1/2i+ state will not participate in transport, we will

from now on unambiguously refer to |1, s − 1/2i− as |1, s − 1/2i in order to ease notation. Due to the presence

of V (t) inside the commutator in Eq. (I.2.7) we obtain rate equations with an explicit time dependence in the

coefficients of the density matrix elements. This explicit time dependence can be eliminated111 by changing

imωt iω(m−m0)t R to the rotating reference frame |n, miR = e |n, mi so that ρmm0 = e ρmm0 . In the rotating frame the relevant rate equations take the form

√ R 2sg1µBB⊥ R s−1/2→s R s→s−1/2 R ρ˙s = Im{ρs−1,s} + W ρs−1/2 − W ρs ~ √ R 2sg1µBB⊥ R s−1/2→s−1 R s−1→s−1/2 R ρ˙s−1 = Im{ρs,s−1} + W ρs−1/2 − W ρs−1 ~

R s→s−1/2 R s−1→s−1/2 R  s−1/2→s s−1/2→s−1 R ρ˙s−1/2 = W ρs + W ρs−1 − W + W ρs−1/2 (I.2.9) √ R i 2sg1µBB⊥ R R
R R ρ˙s−1,s = ρs−1 − ρs − i (∆s−1,s − ω) ρs−1,s − γs−1,sρs−1,s 2~ √ R i 2sg1µBB⊥ R R
R R ρ˙s,s−1 = ρs − ρs−1 − i (∆s,s−1 + ω) ρs,s−1 − γs,s−1ρs,s−1 2~ 37 2.4. RESULTS AND DISCUSSION

where ∆s−1,s = [E(0, s − 1) − E(0, s)] /~. We approximate the coherences in the rotating frame by setting R R R ρ˙s−1,s =ρ ˙s,s−1 = 0 so that by inverting the last two equations in Eq. (I.2.9) we obtain expressions for ρs−1,s R and ρs,s−1. The imaginary parts of the coherences are Lorentzian lineshapes multiplied by the difference in the non-equilibrium populations of the two states involved in the coherent superposition, and are given by

√ R R
R 2sg1µBB⊥ γs−1,s ρs−1 − ρs Im{ρs−1,s} = 2 2 (I.2.10) 2~ (∆s−1,s − ω) + γs−1,s

R R with Im{ρs,s−1} = − Im{ρs−1,s}. Note that the Lorentzian lineshapes appearing in Eq. (I.2.10) are broadened

by the total decoherence rate γs−1,s, and peaked at ω = ∆s−1,s, thus defining the resonance condition for the dissipative system. After inserting the imaginary part of the coherences into the top two expressions in

Eq. (I.2.9) we obtain a 3 × 3 system of differential equations containing only the diagonal components of the

reduced density matrix in the rotating reference frame. To explore the stationary current limit we invoke

a further steady state approximation and solve for the long time behaviour of the diagonal components of

the density matrix. The solutions may be transformed back into the rest frame trivially as the diagonal

components of the density matrix do not pick up an explicit time dependence when shifting between frames.

We calculate the total current and the spin current at lead α with

(α) (α) It = ±e(Iα↑ + Iα↓) Is = ±e(Iα↑ − Iα↓) (I.2.11)

respectively, where the plus (minus) sign is used for the source (drain), e is the elementary charge,

s→s−1/2 s−1→s−1/2  s−1/2→s s−1/2→s−1 Iασ = Wασ ρs + Wασ ρs−1 − Wασ + Wασ ρs−1/2 (I.2.12)

and ρs, ρs−1, ρs−1/2 are the rest frame reduced density matrix elements obtained above.

2.4 Results and Discussion

For the purpose of our calculations we have chosen some reasonable parameters describing an easy-axis spin system containing all the necessary properties to behave as a SMM with s = 6, D = 0.02 meV and

−3 −3 J = −0.06 meV. We further choose Bz = −0.2 T, B⊥ = 2 × 10 T, ΓS = ΓD = 10 meV, T = 10 mK and

11 −1 ω = ∆s−1,s = 3.5 × 10 s . Vg is always chosen to impose a level degeneracy condition between the ground states of the neutral and charged manifolds rendering  an arbitrary parameter. We consider a system with g1 = g2 = 2 but note that the implications of our model are not restricted by this choice. Variation of g1 will change the position of the level degeneracy; this can be compensated for by adjusting Vg. With this 2.4. RESULTS AND DISCUSSION 38

Figure I.2.2: Energy levels of the SMM-dot hybrid described by the Hamiltonian given in Eq. (I.2.3) calcu- lated using parameters chosen above. The neutral states are represented by black dots and the plus (minus) charged states by upward-facing, red (downward-facing, blue) triangles. choice of parameters, the resulting energy levels of the neutral and singly charged states of the device have the structure presented in Figure I.2.2. It is particularly important to note that due to the antiferromagnetic coupling assumed here the lowest lying exchange coupled state is |1, s − 1/2i while the ferromagnetic state is thermally inaccessible for charge transport. We consider the case of an idealised spintronics experiment in which the source lead is ferromagnetic and spin injection is 100% effective (PS = 1) while the drain remains non-magnetic (PD = 0).

2.4.1 Continuous Radiation

In order to investigate the time dependent coherent dynamics of the magnetic system induced by the resonant radiation we first performed brute force numerical integration of Eq. (I.2.7). In addition, numerical integration of Eq. (I.2.7) provides a means to assess the robustness of the approximations leading to the analytic steady state solutions obtained for Eq. (I.2.9). Figure I.2.3 shows the time evolution of the relevant diagonal elements of ρ(t) obtained at Vb = 0 when the SMM-hybrid is initially prepared in the |0, si state. The radiation induces damped Rabi oscillations between |0, si and |0, s − 1i that quickly decay to a steady state due to decoherence introduced by the incoherent charge transfer process between the the leads and the open quantum system. We find that the long-time behaviour of these solutions agree with the analytical solutions obtained from our treatment of the master equation above therefore corroborating the steady-state approximations leading to Eq. (I.2.10). The rate of population transfer between |0, si and |0, s − 1i at steady state is related to the imaginary part of the off-diagonal matrix element given in Eq. (I.2.10) and is thus maximal when ω = ∆s−1,s. The energy supplied to the device via continuous irradiation drives a population imbalance in the neutral manifold leading to the manifestation of several interesting steady state transport 39 2.4. RESULTS AND DISCUSSION

Figure I.2.3: Time evolution of the ρs, ρs−1/2 and ρs−1 density matrix elements obtained by numerical integration of Eq. (I.2.7) at Vb = 0 with a ferromagnetic source.

effects.

Figure I.2.4 shows the stationary charge and spin currents as a function of applied bias voltage flowing through the device. A net current is pumped through the device at zero bias voltage with the majority spin current injected from the ferromagnetic source being completely reversed and amplified at the drain. When the SMM is prepared in the |0, si ground state via an external magnetic field along the easy axis but is not irradiated then charging from the source can not occur as the ferromagnetic reduced state |1, s + 1/2i of the device is thermally inaccessible for transport (see Figure I.2.2). One may view this configuration as the high resistance state of a molecular spin valve where the single molecule magnet acts as a spin analyser. When energy is supplied to the system by resonant electromagnetic radiation (see Figure I.2.1 for a schematic) then the giant spin of the SMM is tilted via transfer of population to the excited |0, s − 1i state. A spin majority

† electron may now charge the device owing to the non-zero amplitude h1, s − 1/2| c↑ |0, s − 1i between the − − |0, s − 1i and |1, s − 1/2i = As−1/2 |si ⊗ |↓i + Bs−1/2 ⊗ |s − 1i |↑i states. The only non-zero discharging process that can take place from |1, s − 1/2i is one in which the SMM is returned to its maximal spin ground state |si, and therefore only discharging of spin minority electrons is possible, due to the coherent superposition structure of |1, s − 1/2i; crucially, this can occur only at the drain owing to the fully spin- polarised character of the ferromagnetic source lead. Thus even at zero bias voltage a spin-switched current is pumped through the device due to energy supplied via the resonant radiation and the spin-asymmetric charge transfer processes at the ferromagnetic source and non-magnetic drain. We note that at low temperatures the

|0, s − 1i state lies outside of the conduction window provided that Vb < 2∆s−1,s−1/2 = D(2s − 1) − g1µBBz. As a consequence, when the |0, s − 1i is populated as a result of the resonant electromagnetic radiation, the device may also be charged by electrons from the drain that also undergo a spin reversal before being emitted back to the drain. Though this process does not contribute to the net charge current flowing through 2.4. RESULTS AND DISCUSSION 40

Figure I.2.4: The stationary charge current (left) and spin currents at source and drain (right) flowing through the device as a function of applied bias voltage.

the device, it does provide an additional contribution to the negative spin current at the drain resulting in

an amplification of the drain spin current. These effects persist for non-zero bias voltage provided that

the bias is not so large as to activate the ferromagnetic |1, s + 1/2i charged state or to include |0, s − 1i

in the conduction window. While the charge pumping described here is reminiscent of the photon assisted

tunnelling already observed in quantum dots,129, 130 we stress that in this set-up it is the SMM that absorbs the radiation in order to overcome the current blockade rather than the conduction electron.

2.4.2 Pulsed Radiation

The continuous irradiation model described in the previous section may present practical challenges in attaining constant temperature of the system due to heat dissipation involved by the absorption process.

Thus we also explore a perhaps more easily realisable experimental set-up, investigating the spintronics problem under pulsed radiation. Accordingly, we define a time scale tp+w = tp + tw corresponding to a single pulse-wait sequence. During the interval t ∈ [0, tp] the radiation is switched on and V (t) is given by

Eq. (I.2.5) whereas in the interval t ∈ [tp, tp+w] the radiation is switched off and V (t) = 0; this sequence is

repeated for multiples of tp+w. We calculate the average current through the device by numerical integration of the master equation followed by averaging of the time dependent current over an arbitrary number of

pulse-wait sequences occurring after the initial pulse. For clarity, we define the time-average of a function

f(t) over the time domain T = {t ∈ R | ta ≤ t ≤ tb} by

1 Z tb hfiT = f(t) dt. (I.2.13) tb − ta ta 41 2.4. RESULTS AND DISCUSSION

Figure I.2.5: The time-averaged charge current flowing through the device at Vb = 0 as a function of various pulse times tp and wait times tw.

We focus on the case when Vb = 0 and investigate the dependence of the time-averaged current on the pulse and wait times tp and tw respectively.

Figure I.2.5 shows the time-averaged current flowing through the device at zero bias for values of tp and tw. Even here we obtain a finite time-averaged charge current for all values of tp 6= 0 which tends towards saturation as tp → ∞ and manifests an oscillatory behaviour as tp → 0. By increasing the wait time in between pulses we see that the average charge current per tp+w cycle diminishes and tends to zero for tw → ∞.

As noted previously, the resonant radiation causes damped Rabi oscillations between elements of the density matrix (see figure I.2.3) which is consequently reflected in the time dependent current. When tp

tot is shorter than the decay of the damped oscillations, hI iT provides piece-wise measurements of the time evolution of the Rabi oscillations between |0, si and |0, s − 1i. Conversely, when tp is longer than the decay of the damped Rabi oscillations, the system is able to reach a quasi-steady state limit (as in the continuous irradiation model) within the pulse phase of each tp+w cycle and therefore the oscillations are

tot averaged out in hI iT . During wait sequences (where V (t) = 0) the coherences in Eq. (I.2.7) become completely decoupled from the diagonal elements of the density matrix and the master equation becomes completely soluble up until the next pulse. Specialising to the 3 × 3 system discussed above we solve
T ρ˙ = Mρ over tp ≤ t ≤ tp+w where ρ = ρs, ρs−1, ρs−1/2 and M is the time independent rate matrix describing charging and discharging processes between the dot and the leads. A great deal of simplification

s−1/2→s−1 s→s−1/2 s−1/2→s can be made when Vb = 0 as W ≈ 0 and W = W , leading one to discover the eigenvalues of M as {0, −2W s→s−1/2, −W s−1→s−1/2}. Recalling that the long-time limit of the system in the absence of resonant radiation leads to a blockage of current we see that, regardless of tp, for tw > max(−2W s→s−1/2, −W s−1→s−1/2) no current flows through the device resulting in a diminishing value of

tot hI iT as tw → ∞.

Although in this section we have focused only on the time-averaged charge current in the pulsed radiation 2.5. CONCLUSION 42

regime, we note that the time-averaged spin currents (not shown) are also switched and amplified at the

drain as in the continuous radiation model. The behaviour of the time-averaged spin currents as functions

of tp and tw is mirrored in the discussion above and so we omit it here. For the device to function optimally, tp and tw should be chosen such that the |0, s − 1i state is sufficiently populated on each tp+w cycle and also such that heat acquired from the resonant radiation diffuses away from the SMM before the next pulse. We

do not investigate the added complexity of heat diffusion in this manuscript.

2.4.3 Candidate Magnets for the Device

In the model presented above we have made no mention of the specific SMM that should be used in the

junction as we predict the pumping, switching and amplification effects described above to be achievable with

any nanomagnet that is well described by the Hamiltonian given in Eq. (I.2.3). In a practical setting however,

the choice of SMM is far from arbitrary as the frequency of radiation required for the m = s → s−1 transition

may also couple to vibrational modes in the molecule or contribute to other undesirable interactions. Fe4 based nanomagnets could be prime candidates for the device proposed above as their magnetic properties are

retained following surface deposition131 and have been shown to be robust under successive oxidation and

50, 52 reduction in three terminal devices. A first-principles theoretical study of an Fe4 nanomagnet attached

to metallic leads has furthermore indicated that the magnetic properties of Fe4 are likely to be preserved in such a junction and may enjoy a modest increase in uniaxial anisotropy on reduction.132 The aforementioned

52 theoretical work by Nossa et al. partially corroborates the assumption made by Burzuri et al. in that Fe4 acquires a S = 9/2 ground state on reduction, implying an antiferromagnetic coupling between the giant

spin of the magnet and the radical. In addition, the gap between the ground and excited state on graphene

has been reported ∼ 1cm−1 which could be probed with microwave radiation.24, 133 Octanuclear Fe(III)

−1 nanomagnets are also good candidates for the device since the gap ∆s−1,s ∼ 4 cm is also amenable to

microwave radiation. In fact, the m = s → s−1 transition in Fe8 SMM crystals has already been probed with

134–138 pulsed microwave radiation in previous studies. The required radiation-induced transition in Mn12

−1 could also be achieved with microwave radiation as it has been reported to possess a ∆s−1,s of ∼ 9cm

23 . Cr7M (M = Cd, Mn, Ni) molecular wheels may also be excellent candidates for our device given their stability on surface deposition139, 140 and microwave radiation.141, 142

2.5 Conclusion

We have proposed a model for electron transport through a SMM nanostructure under irradiation in the Coulomb blockade regime. We demonstrated that a spin current is pumped through the device at zero 43 2.6. APPENDIX A: ALTERNATE RESONANT PERTURBATION COUPLING SCHEMES bias voltage when coupled to a ferromagnetic source as a result of radiation induced transitions in the SMM followed by spin-asymmetric discharging at the source and drain leads. In addition to this spin pumping effect, we find that the spin polarised current pumped from the source is reversed and amplified at the drain even when Vb 6= 0. We also investigated the behaviour of the device under pulsed irradiation and discussed the time-averaged current as a function of pulse length and wait time. We find that for long enough pulse lengths and short wait times the stationary current results are recovered. Interestingly however, for short pulse lengths and long wait times we also find that the proposed device can be used to measure coherent

Rabi oscillations between the SMM spin states, which could offer an as yet unexplored strategy to integrate

SMM-based spin qubits into spintronics circuits.

2.6 Appendix A: Alternate Resonant Perturbation Coupling Schemes

Throughout this manuscript we have discussed the spin transport dynamics imparted to a SMM-dot hybrid device with a specific radiation-dipole coupling scheme (see Eq. I.2.5). A more general coupling between the magnetic states of the SMM and a resonant coherent perturbation can be achieved with the

Hamiltonian

 N iωt N −iωt VN (t) = ν S+ e + S− e (I.2.14) where N ∈ N and ν is some constant specific to the applied resonant perturbation in a given experimental set-up. Note that Eq. I.2.5 is recovered from Eq. I.2.16 when N = 1 and ν = g1µBB⊥. To investigate the consequences on the spin transport dynamics of the device when the ground spin state |0, si is coherently and resonantly coupled to an excited spin state |0, s − Ni with N > 1, we proceeded as in the main text and developed a master equation for the reduced density matrix elements akin to Eq. I.2.7 but with the substitution V (t) 7→ VN (t). Using the parameters from section 2.4 with ω = ∆s−N,s, we computed the zero bias steady state spin currents at the ferromagnetic source and non-magnetic drain electrodes for several values of N and nanomagnet spin quantum numbers s.

If N > s, the resonant perturbation will induce population transfer over the nanomagnet anisotropy barrier thus quenching the spin pumping, switching and amplification effects described above. Multiple excitations caused by VN (t), while occurring on the slowest time scales can, in the steady state limit, also result in population transfer over the barrier even when N < s; these processes are suppressed however with increasing s or D. Provided that N is not too large with respect to the barrier height, Figure I.2.6 demonstrates that coupling the ground state |0, si to excited states with axial spin projections less than s − 1 with a resonant perturbation can still give rise to the current pumping, switching and amplification 2.7. APPENDIX B: NON-SECULAR RATE EQUATION 44

Source Drain

Figure I.2.6: Zero bias steady state spin currents at the source and drain electrodes for SMM devices with various spin quantum numbers s and resonant perturbations VN (t).

effects described in the main text. As N is increased, these effects are augmented owing to the multi-step charging/discharging cascade that is required to relax the nanomagnet from the excited state |0, s − Ni to the ground state |0, si. For example, consider a resonant perturbation that couples |0, si and |0, s − 2i: after each excitation from the ground state to |0, s − 2i relaxation proceeds via the charging/discharging cascade |0, s − 2i → |1, s − 3/2i− → |0, s − 1i → |1, s − 1/2i− → |0, si. In this example two electrons much sequentially charge and discharge from the device both with their individual spin moments switched hence leading to a larger spin current than observed in the N = 1 case. Finally, we note that for the nanomagnet spintronics set-up outlined here with a resonant perturbation VN>1(t), calculation of the steady state spin currents can be reduced to that of the effective three state model described in the main text with only the lead-dot coupling Γα renormalized to account for the multi-step charging/discharging cascade.

2.7 Appendix B: Non-secular Rate Equation

In order to confirm the validity of the secular approximation leading to Eq. I.2.7 we performed a numerical integration of the full non-secular quantum master equation. The evolution of a reduced density matrix element is −i ρ˙mm0 = [HS + V (t), ρ]mm0 + (Rρ)mm0 (I.2.15) ~ 45 2.8. APPENDIX C: DEVICE OPERATION WITHOUT FERROMAGNETIC SPIN INJECTION where R controls the full non-secular dynamics of the device owing to the dissipative effects from coupling to the leads and is given explicitly by

S,D ↑↓ (n+1 X X Γα (1 + 2σPα) X n+1→n n→n+1 (Rρ) = × c ρ c [2 − f (∆ ) − f (∆ 0 )] mm0 4 σ,ml lk σ,km0 α lm α km α σ ~ kl n−1 n n+1 X n−1→n n→n−1 X X  n+1→n n→n+1 n+1→n n→n+1 + cσ,ml ρlkcσ,km0 [fα(∆ml) + fα(∆m0k)] − fα(∆lk) cσ,ml cσ,lk ρkm0 +ρmkcσ,kl cσ,lm0 kl k l n n−1 X X  n−1→n n→n−1 n−1→n n→n−1o − [1 − fα(∆kl)] cσ,ml cσ,lk ρkm0 + ρmkcσ,kl cσ,lm0 . k l (I.2.16)

We found the numerical solutions to Eq. I.2.15 to be indistinguishable from those that result from a quantum rate equation containing only secular terms (Figure I.2.3). The secular approximation is thus justified for this system as the non-secular dynamics of the coherences induced by the dissipative effect of the leads occurs on a much faster time scale than the time evolution of the populations ρm and as such are cancelled out upon integration.

2.8 Appendix C: Device Operation Without Ferromagnetic Spin

Injection

The motivation for proposing the novel spintronics device above was to design a molecular junction that could manipulate an injected ferromagnetic spin current however, in order to complete the description of our radiation-coupled nanomagnet device, we also investigated its operation in the presence of a non-magnetic source electrode. Implementing both non-magnetic source and drain electrodes corresponds simply to the parameter choice PS = PD = 0 in Eq. (I.2.8) where −1 ≤ Pα ≤ 1 is the spin polarisation of lead α. We note en passant that this limit is representative of essentially all instances of our model for PS 6= 1. This choice of spin polarisation is more typical of the precious metal and graphene electrodes used in previously constructed molecular break junctions84, 85 and nanoconstriction set-ups.79

The total electrical current flowing through the device and the spin currents at the non-magnetic source and drain electrodes obtained using the same parameters specified in section 2.4 are shown in Figure I.2.7. By replacing the ferromagnetic source electrode with a non-magnetic alternative the device no longer pumps an electric current nor amplifies spin current at zero bias voltage. Furthermore, while the spin current signals at the source and drain electrodes remain distinct from one another, they exhibit a markedly different behaviour as a function of bias voltage than in the ferromagnetic source electrode case.

Unlike in the ferromagnetic source scenario, electron transport from the source is not blocked in the 2.8. APPENDIX C: DEVICE OPERATION WITHOUT FERROMAGNETIC SPIN INJECTION 46

0.15 0.00

) - 0.05 Source )

0.10 / ℏ / Γ (e

e £ ( -I tot - 0.10

I ¢ 0.05 I

- 0.15 Drain 0.00 0 2 4 6 8 10 0 2 4 6 8 10 Vb (μV) Vb ( V)

Figure I.2.7: The stationary charge current (left) and spin currents at source and drain (right) flowing through the device as a function of applied bias voltage when both electrodes are non-magnetic (i.e. PS = PD = 0).

absence of the resonant radiation as spin down electrons are now present at the Fermi level of the source

to facilitate |0, si ↔ |1, s − 1/2i− charging and discharging transitions. As a result, these charging and

discharging processes are now in competition with the resonant radiation mediated population transfer from

|0, si → |0, s − 1i and the subsequent relaxation of the |0, s − 1i through a similar charge transfer cascade to that described above. Owing to the manifest symmetric character of the source and drain electrodes at zero bias voltage, both of these processes occur with equal probability at each electrode thus resulting in a net zero electric current through the device at Vb = 0 despite supplying energy to the system via the resonant radiation. The splitting between source and drain spin currents at zero bias voltage can be explained solely in terms of the charging/discharging mediated relaxation of the |0, s − 1i state since direct charge transfer between |0, si and |1, s − 1/2i− does not contribute to the zero bias spin currents at either electrode. As in the ferromagnetic case, consider an excitation from |0, si to |0, s − 1i resulting from the absorption of a resonant photon. Relaxation from the excited |0, s − 1i state occurs via charging the SMM-dot hybrid

† from either electrode with a spin α electron owing to the non-zero amplitude h1, s − 1/2| c↑ |0, s − 1i thus − − populating the |1, s − 1/2i = As−1/2 |si ⊗ |↓i + Bs−1/2 |s − 1i ⊗ |↑i state in which the conduction electron’s spin becomes entangled with the nanomagnets spin moment. In contrast to the ferromagnet source case, a

β electron may now discharge to either source or drain electrode in order to return the device to its |0, si neutral ground state. The conduction electron spin flip process described here enters the expression for the source and drain spin currents (Eq. (I.2.11))with equal weight but opposite sign thus resulting in the splitting between the two measurements observed in Figure I.2.7. At zero applied bias then, the device operates as a sink (emitter) for spin α (β) electrons at both source and drain electrodes however this sink

(emitter) behaviour becomes quenched by the application of a finite bias voltage owing to competition with direct charging/ discharging with spin β between the |0, si and |1, s − 1/2i− states. Chapter 3

Spin Current Switching with a

Single-Molecule Magnet Immersed in

a Static Transversal Magnetic Field

3.1 Introduction

In the previous chapter, spin current injection into a single-molecule magnet based spintronic device

was discussed wherein the nanomagnet was subjected to a resonant time dependent perturbation. By

manipulating the spin states of the nanomagnet with resonant radiation, spin current pumping, inversion

and amplification effects were observed whereas in the absence of the radiation the device experienced a

current blockade. A natural question follows from this model: is it possible to enforce the same spin state

manipulations within a general nanomagnet set-up using a static perturbation and again provoke pumping,

inversion and amplification effects in a spin current that has been injected into the device from a ferromagnetic

electrode?

Instead of provoking transitions in the manifold of nanomagnet spin states in order to manipulate spin

currents in the device with a time-dependent resonant radiation, an alternative approach is to perturb the

spin system with a time independent perturbation V such as the transversal component of a magnetic field.

As the perturbation is not time-dependent, the quantum rate equations for the reduced density matrix of the quantum system retain the same form as derived in the previous section only with S(t) = eiHS t/~ replaced

with S(t) = ei(HS +V )t/~ in the derivation. In this case, however, the good quantum number characterising

47 3.1. INTRODUCTION 48

the spin projection of the nanomagnet is lost and a diagonalisation of the entire nanomagnet spin manifold

is, in general, mandatory. If the amplitude of the static transversal field is small compared to the energy scale

set by HS however, then one may approximate the energies and the eigenstates of the Hamiltonian HS + V using the Rayleigh-Schr¨odingerperturbation theory.143 To first order in V , the approximate eigenvalues and

eigenstates of HS + V are

(N) (N) Ep = Ep,0 + 0 hN, p| V |N, pi0 + ...

(I.3.1) N X 0 hN, k| V |N, pi |N, pi = |N, pi + 0 |N, ki + ... 0 EN − EN 0 k6=p p,0 k,0

respectively, where |N, pi0 is the N electron many-body state of the nanomagnet exchange coupled to the spintronics device in the absence of a transversal field and is characterised by the eigenvalue equation

(N) HS |N, pi0 = Ep,0 |N, pi0. The relations above can then serve to preserve some degree of analyticity in theoretical models of electron transport through the nanomagnet device.

Intuition gained from the previous model suggests that the device should again be prepared in a current blockade using a longitudinal magnetic field and a perturbation V should be chosen so as to mix the ground

|N, si magnetic state of the neutral manifold with the first excited spin state |N, s − 1i thus creating a non- zero amplitude between the ground states of the neutral and reduced manifolds when charging the device with a spin majority electron from the ferromagnetic source. A natural candidate for such a perturbation is the transversal component of a magnetic field that couples spin states with |∆ms| = 1 in first order perturbation theory via the spin ladder operators S±. It is worth noting here that for a weak static transversal field, one would not expect a significant mixing of spin states with |∆ms| > 1 as this would require high-order terms in the perturbation theory weighted with increasingly large powers of the weak transversal field. It is possible that even in high-order perturbation theory, degenerate spin states may be strongly coupled with only a weak perturbation however this possibility is discounted here owing to the removal of all spin state degeneracies in the nanomagnet by the longitudinal field.

In section 3.2 a model for spin transport through the nanomagnet subsystem immersed in a static transversal magnetic field is introduced wherein a relevant low temperature regime elucidating the underlying spin switching mechanism at play is explored analytically. In section 3.3 the steady-state spin currents for a full numerical treatment of the model are presented and are shown to be in good agreement with the analytical model. Finally, the origin of spin switching and the applicability of this model is discussed before

finally concluding in section 3.4. 49 3.2. THEORETICAL MODEL

3.2 Theoretical model

3.2.1 Device Hamiltonian

To describe the dynamics of the nanomagnet spintronics device immersed in a transversal magnetic field,

we begin similarly to section 2.3 by employing a tripartite Hamiltonian H = HL + HS + HT that describes the energetic structure of the leads, the nanomagnet-quantum dot hybrid and the hybridisation between the

subsystems of the device. As in the previous section, we take the leads to be non-interacting electron gases

held at some chemical potential µα given by Eq. (I.2.2) and the hybridisation between the leads and the SMM-dot hybrid by the Anderson Hamiltonian given in Eq. (I.2.4). We depart from the theoretical model in the previous section by introducing a new Hamiltonian HS

2 X † HS = −DSz + ( − eVg)cσcσ + µBB · (g1S + g2s) − JS · s (I.3.2) σ where we take B to be a magnetic field that couples to the longitudinal and transverse components of the spin moment of the SMM-quantum dot hybrid. This Zeeman term can be separated into its transverse and longitudinal components explicitly as

µBB⊥  −iφ iφ
µBB · (g1S + g2s) = µBBz (g1Sz + g2sz) + e (g1S+ + g2s+) + e (g1S− + g2s−] | {z } 2 (I.3.3) Longitudinal part | {z } Transverse part

q 2 2 where B⊥ = Bx + By is the amplitude of the transverse component of the field and 0 ≤ φ < 2π defined by tan(φ) = By/Bx is the in-plane angle of its orientation. Without the transverse component of the field, the Hamiltonian in Eq. (I.3.2) can be diagonalised exactly (as in section 2.3) and so, as we expect the transverse

field to be small in comparison to the uniaxial splitting of the nanomagnet D and the amplitude of the longitudinal field Bz, we will treat the transverse component of the field as a perturbation to the system (0) Hamiltonian so that HS = HS + V where V is the “Transverse part” from Eq. (I.3.3) and approximate 143 the spectra of HS to first order in B⊥ using the Rayleigh-Schr¨odingerperturbation theory. For example, (0) consider the effect of V on the eigenstate of HS corresponding to the maximal spin projection of the nanomagnet |N, si when a longitudinal field is applied

  √ µBg1B⊥ 2s 2 |N, si = |N, si0 + |N, s − 1i0 + O(B⊥). (I.3.4) 2 D (2s − 1) − µBg1Bz

As the perturbed state |N, si retains a large proportion of the unperturbed state |N, si0, the quantum

number characterising the spin projection of the nanomagnet ms = s has been approximately retained and 3.2. THEORETICAL MODEL 50

will be used to index other eigenstates of the Hamiltonian HS. To first order in the transversal field, the energies of all redox states in the device remain unchanged under the influence of the transversal field as the

(0) diagonal matrix elements of V on the basis of eigenstates of HS are exactly zero. As in the previous chapter, we assume that the nanomagnet is antiferromagnetically coupled to the conducting substrate (i.e. 2D − J > 0) such that even in the presence of the longitudinal field, the ground state of the reduced device is given by

− − − − |N + 1, s − 1/2i ≈ |N + 1, s − 1/2i0 = As−1/2 |si ⊗ |↓i + Bs−1/2 |s − 1i ⊗ |↑i (I.3.5)

± ± where the Am and Bm coefficients are defined as in Eq. (I.2.6). 3.2.2 Quantum Master Equation and Stationary Spin Currents

Unlike in the previous chapter, the derivation of the quantum rate equations presented in the theoretical prologue to this thesis requires no alterations to account for the static transversal field and is used here with only some small changes of notation. The time evolution of the population for state |N, mi is governed by

X l→m X m→l ρ˙m = W ρl − W ρm (I.3.6) l l where ρm is the diagonal element of the reduced density matrix for the quantum system hN, m| ρ(t) |N, mi and W m→l are charging/discharging transition rates defined in Eq. (I.2.8). For the low temperatures and bias voltages typical of molecular spintronics experiments, only the ground states of the neutral and reduced manifolds of the SMM-hybrid device are likely to contribute significantly to transport therefore we consider only the rate equations for these two states. Furthermore, the static perturbation V does not induce any novel phenomena in the time evolution of the populations and hence we shall immediately specialise to the steady-state limit of Eq. (I.3.6) in order to compute the steady-state spin currents flowing through the device. The steady-state rate equations for the ground states of the neutral and reduced device are

s→s−1/2− s−1/2−→s −W ρs + W ρs−1/2− = 0 (I.3.7) s→s−1/2− s−1/2−→s W ρs − W ρs−1/2− = 0

which may be solved trivially alongside the normalisation condition ρs + ρs−1/2− = 1 to yield

− W s−1/2 →s 2q (1 − f ) − (q + q )(1 − f ) ρ = = ↑ S ↑ ↓ D s s→s−1/2− s−1/2−→s W + W 3q↑ + q↓ (I.3.8) s→s−1/2− W 2q↑fS − (q↑ + q↓)fD ρ − = = s−1/2 s→s−1/2− s−1/2−→s W + W 3q↑ + q↓ 51 3.2. THEORETICAL MODEL

− † 2 where qσ = hN + 1, s − 1/2| cσ |N, si is the modulus squared amplitude for charging the device with an −1 electron with spin σ and fα = (1 + exp(±Vb/2kBT )) is the Fermi-Dirac distribution at the source (drain) with a positive (negative) bias voltage applied; note that the neutral and reduced ground states are assumed to be degenerate through the action of the gate voltage in Eq. (I.3.2). Using these populations, the spin currents flowing through the device and measured at electrode α take on the particularly simple form

↑ ↓ h s→s−1/2− s→s−1/2−   s−1/2−→s s−1/2−→s i Iα − Iα = ±e Wα↑ − Wα↓ ρs − Wα↑ − Wα↓ ρs−1/2− (I.3.9)

where the ± corresponds to the spin current at the source/drain. By setting the lead polarisations PS = 1 and PD = 0 such that the source lead is ferromagnetic and the drain non-magnetic, these equations can be simplified to yield   ↑ ↓ eΓ q↑ (q↑ + q↓) Vb IS − IS = tanh ~ 3q↑ + q↓ 4kBT (I.3.10)   ↑ ↓ eΓ q↑ (q↑ − q↓) Vb ID − ID = tanh ~ 3q↑ + q↓ 4kBT where e is the elementary charge 1.602 × 10−19 C. From these equations, two important observations can

be made: i) at zero bias voltage no spin polarised current flows through the device owing to the hyperbolic

tangent dependence of the voltage and, ii) at finite bias voltage a spin polarised current injected from the

ferromagnetic source will be inverted at the drain provided that q↑ − q↓ < 0 and q↑ 6= 0. The square modulus overlaps can be approximated to first order in the transversal field using the Rayleigh-Schr¨odinger perturbation theory on the eigenstates of HS

2 − 2 2s 2 µBg1B⊥ Bs−1/2 q ≈ and q ≈ − . (I.3.11) ↑ 2 ↓ As−1/2 2 [D(2s − 1) − µBg1Bz]

So then, all other approximations holding, a spin polarised current injected from the ferromagnetic source

will be inverted at the drain provided that

2 [D(2s − 1) − µ g B ]2 B− < B 1 z (I.3.12) s−1/2  1 2 2 2s 2 µBg1B⊥ + [D(2s − 1) − µBg1Bz]

2 2 where use has been made of the normalisation property − = 1 − − . As−1/2 Bs−1/2 3.3. RESULTS AND DISCUSSION 52

2 ) 2 1 Source ×10

/ 0

(e £ -I

¢ - 1

I Drain

- 2 0 2 4 6 8 10

Vb ( V)

Figure I.3.1: (Left) Exact energies of the eigenstates of HS (labelled by their spin expectation value hSzi) obtained from numerical diagonalisation of the quantum system Hamiltonian; uncharged (charged) states are shown as blue (red) circles. The states that are the most relevant for transport for the parameter set chosen above are boxed. (Right) Steady-state spin currents at the ferromagnetic source (blue) and the non-magnetic drain (red) as a function of the applied bias voltage Vb using the exact eigenstates and energies obtained from numerical diagonalisation of HS.

3.3 Results and discussion

In order to validate the approximations leading to Eq. (I.3.10) and to vindicate the use of the Rayleigh-

Schr¨odingerperturbation theory to describe the effect of the transversal field on the energy eigenstates of

HS, HS was also diagonalised numerically and the secular quantum rate equations for both redox manifolds solved to compute the spin currents flowing through the device. For the following calculations reasonable parameters are chosen that again describe an easy-axis spin system containing all of the properties to behave as a SMM: S = 4, D = 0.02 meV and J = −0.1 meV. We further choose Bz = −1 T, B⊥ = 0.5 T,

−3 ΓS = ΓD = 10 meV, T = 10 mK and φ = 0. As discussed previously, the gate voltage is selected to bring the ground states of the neutral and reduced manifolds of the device to level degeneracy (see Fig. I.3.1) and therefore renders the value of  an arbitrary parameter. As discussed above, the case in which the source electrode is ferromagnetic and the drain electrode is non-magnetic is considered here which corresponds to the choices PS = 1 and PD = 0 for the lead polarisations.

The left sub figure of Fig. I.3.1 denotes the energies that result from an exact numerical diagonalisation of Eq. (I.3.2) with the parameter set chosen above plotted against the total spin expectation value of the eigenstates. Unlike in Fig. I.2.2, the characteristic double well symmetry of the SMM is lost owing to the application of the transversal field. The states with hSzi < 0 exhibit the largest degree of mixing as their relative energy gaps have been reduced through the application of the longitudinal field. In contrast, the states with hSzi > 0 remain relatively pure as energy gaps between these states have increased with the application of the longitudinal field thus reducing the mixing induced by the transversal field. This validates 53 3.4. CONCLUSION the perturbative expansion in powers of the transversal field performed above for states with spin projection ms > 1 however a perturbative expansion to only first order in B⊥ of, for example, the |N, −si state (with the current parameter set) is likely to be inaccurate.

The spin current that is injected from the ferromagnetic source electrode and measured at the drain electrode is presented in the right sub figure of Fig. I.3.1. These spin currents were simulated using the exact eigenstates obtained through numerical diagonalisation of HS, by solving numerically the full set of secular quantum rate equations for both redox manifolds of the device and summing all spin-dependent charging and discharging processes into and out of the SMM-quantum dot hybrid weighted by the corresponding density matrix elements. The simulated spin currents in Fig. I.3.1 are in good agreement with the analytical expressions obtained in Eq. (I.3.10) and hence validate the assumptions incumbent in their derivation.

From Eq. (I.3.10) and Eq. (I.3.11) one immediately sees that in the absence of a transversal field a current blockade exists in the device as q↑, the probability of charging the device from the source with a spin majority electron, is exactly zero. However, by switching on the transversal field, the ground state of the neutral manifold manifests a small |N, s − 1i0 character thus allowing a spin majority electron to charge the device from the ferromagnetic source and thus facilitate a transition to the |N + 1, s − 1/2i− charged state whereby the spin of the conduction electron is entangled with the spin of the nanomagnet. The electron that charged the device now, with the assistance of a finite bias voltage, is emitted to the drain with spin up or spin down occurring respectively with the probabilities q↑ and q↓. If parameters are chosen such that Eq. (I.3.12) is satisfied then the emission of spin down electrons at the drain will be more likely to occur than the emission of spin up electrons and thus the spin polarised current injected at the source will be successfully inverted after passing through the device.

The experimental realisation of the nanomagnet-based molecular spintronics set-up presented in this chapter is likely to be feasible as its fabrication is dependent only on the ability to couple a single-molecule magnet antiferromagnetically to a molecular spintronics device and to inject a spin polarised current through the device. The application of a transversal magnetic field, crucial for the operation of our device, has already been demonstrated in nanomagnet-based spintronics junctions using three dimensional vector magnets.79, 82

3.4 Conclusion

By coherently coupling a single-molecule magnet to the transversal component of a static magnetic field we have demonstrated the potential for a novel molecular spintronics device that is able to reverse the polarisation of an injected spin current when a finite bias voltage is applied. The operation of the device was explained in terms of a virtual spin transition induced by the transversal component of the magnetic field 3.4. CONCLUSION 54 followed by asymmetric charging and discharged of spin majority and minority electrons at the source and drain; this explanation was elucidated with assistance from an analytical model. Unlike the device operating under irradiation that was discussed in the previous chapter, this device does not exhibit spin pumping or amplification effects owing to the time independent nature of the perturbation. Chapter 4

Conclusions and Future Work

Using spin polarised currents to manipulate the quantum states of single or few molecules grafted to a

conducting surface is a sought-after goal in the field of molecular spintronics. The use of a spin polarised

current to reverse the giant spin of both a single-molecule magnet and a magnetic adatom has been de-

scribed with theoretical models62, 65 and has been performed experimentally using spin-polarised scanning

tunnelling microscopy tips.67, 68 Alternatively, one may desire to use the molecules themselves as units that controllably modify the polarisation of currents so as to develop novel circuitry dependent upon the spin degree of freedom rather than charge. Large biomolecule monolayers have been used in this vein to generate spin polarised currents at room temperature when embedded in atomic force microscopy set-ups.71 Also,

theoretical spin transport models of molecular wheels exhibiting toroidal moments have been shown to in-

vert spin polarised currents owing to the non-collinear spin texture of the nanomagnet77, 78 however their

deposition into molecular spintronic devices has yet to be realised experimentally. Since an efficient coupling

between simpler single-molecule magnets and molecular electronic circuits has been actualised in previous

experiments48, 52, 79, 144 and the possibility of spin injection into graphene substrates at room temperature

has been achieved,124 the development of single-molecule magnet spintronic models could guide the design of

soon-to-be realised transport experiments as well as the development of more general SMM-based spintronic

devices.

In chapter 2, a published work was presented in which a general single-molecule magnet spintronic device

that interacted with a continuous, resonant radiation functioned as a spin current pump, amplifier and

inverter. The aforementioned spin current manipulations were attributed to radiative excitations in the

nanomagnet that relaxed via spin asymmetric charging and discharging processes at the source and drain

electrodes. In light of experimental considerations, a pulsed radiation scheme was also investigated in which

55 Conclusions and Future Work 56 the pumping, amplification and inversion effects exhibited by the device were recovered in time-averaged measurements of the current. Furthermore, sequential time-averaged measurements of the electrical current using different radiation pulse lengths were shown to reconstruct manifestations of Rabi oscillations between the spin states of the nanomagnet that were coherently coupled by the resonant radiation thus suggesting the possibility that these quasi-degenerate spin states could act as a qubit for quantum computation. Following this, a discussion of suitable nanomagnet candidates for the device was presented based on the limited parametric constraints required from the model.

Another such model was presented in chapter 3 whereby a single-molecule magnet exchange coupled to a spintronics device was immersed in a static magnetic field that coupled both to the transversal and longitudinal components of its spin moment. While the device was initially prepared in a current blockade state, the transversal component of the field induced virtual transitions between the spin levels of the nanomagnet that facilitated electron transport through the device in proportion to the square amplitude of the applied field. Owing to the antiferromagnetic coupling assumed between the conducting electron and the nanomagnet spin, the spin polarised current that was injected into the device from the ferromagnetic source emerged at the drain inverted when a finite bias voltage was applied to the system.

The future of this project lies in establishing collaborations with experimental molecular spintronics specialists that are interested in constructing and testing the devices proposed herein. Either vindication or rejection of these set-ups by experiment will incite refinements to the theoretical models proposed and hence will inevitably lead to a deeper understanding of spin current injection and manipulation in nanomagnet- based molecular spintronics junctions. Part II

Addressing the Quantum States of a

Single Nanomagnet Break Junction

57 Chapter 5

Mechanisms of Spin-Charge

Conversion for the Electrical

Read-out of 4f-Quantum States in a

TbPc2 Single-molecule Magnet Device

Authors: Kieran Hymas, Alessandro Soncini Affiliation: School of Chemistry, University of Melbourne, Parkville, 3010 Published: 31/7/20 Journal: Physical Review B

58 59 5.1. ABSTRACT

5.1 Abstract

We present a theoretical study exposing the dominant microscopic electronic transport mechanisms un- derlying a recent molecular spin-transistor experiment [C. Godfrin et al., ACS Nano 11, 3984 (2017)], where purely electrical read-out of the spin of a Tb(III)-based single-molecule magnet was achieved. To identify the relevant spin-to-charge conversion mechanisms enabling opposite spin-polarisations of the Tb(III) ion

4f-electrons to generate different magneto-conductance responses, we investigate both incoherent sequential tunnelling charge transport, and coherent cotunnelling corrections. Contrary to previous interpretations invoking the highly coherent Kondo transport regime, we find that all reported experimental observations, including the temperature and magnetic field dependence of the differential conductance, can be reproduced reasonably well within a sequential tunnelling transport regime explicitly accounting for broadening of the device energy levels due to molecule-lead coupling.

5.2 Introduction

Single-molecule magnets (SMMs) have been proposed as candidates for molecular memory,60 molecular qubits13, 145 and for novel molecular spintronics applications97, 146 owing to their large magnetic anisotropy, stability upon surface and thin film deposition and their unique, rich, quantum properties. The bis-

(phthalocyaninato) terbium nanomagnet (TbPc2) in particular, has recently enjoyed a great deal of popu- larity in molecular spintronics set-ups, such as in molecular spin valve experiments on graphene surfaces79 and carbon nanotubes,80, 103 when probed in thin-films via scanning tunnelling microscopy tips147, 148 and also in molecular break junctions.82, 84, 85

The break junction device has become a system of keen interest due to the potential of the weakly de- cohering 159Tb nuclear states to act as a qudit computational basis for molecular quantum computation technologies.17, 83 The electrical read-out of the 159Tb nuclear spin computational basis is fundamentally rooted in a two-step coupling mechanism: (i) the 159Tb nuclear spin is hyperfine-coupled to the doubly degenerate mJ = ±6 4f-electron states of the Tb(III) ion, (ii) the mJ = ±6 states are, in turn, ferromag- netically exchange coupled to a radical s = 1/2 spin hosted by the Pc2 organic ligands of the nanomagnet, which are coupled to the Au-nanowire break-junction, thus part of a sequential tunnelling current in and out of the leads. The coupling between the sequential tunnelling conduction electron hosted as a Pc2 s = 1/2 radical, and the 4f-hyperfine states, enables the transfer of the nuclear spin states quantum information to the device current, resulting in a read-out process.82–84

Several experimental works reporting highly anisotropic hysteresis loops of conductance measurements 5.2. INTRODUCTION 60

82 104, 149 recorded from the break junction and from similar TbPc2 set-ups have demonstrated that the bistable electronic ground state of the TbPc2 may be read out electronically by virtue of an exchange coupling between the Tb(III) electronic states and conduction electrons that transiently occupy the Pc ligands of the molecule.

Despite the many experimental results concerning this terbium molecular break junction device, to our knowledge, fewer theoretical investigations have been undertaken to model electron transport through the system, and to understand the microscopic mechanism for the resultant read-out of the Tb(III) electronic states. In a recent joint theoretical and experimental study of the device by Troiani et al.101 the Landau-

Zener-like tunnelling dynamics of the Tb 4f-electron states was investigated under continuous measurement from a local electric current, and simulated via a Lindblad-type master equation. The work focused in particular on identifying the signature of decoherence in the 4f-electron tunnelling dynamics, as measured by the transport experiments, using a phenomenological simulation of the coupling to the environment.

However, the specific microscopic mechanisms of the transport measurement process, as those of the coupling to the environment, were not the object of that work.

In all previous experimental studies,82, 84–86, 101 the flipping of the terbium moment between the the mJ = 6 and mJ = −6 ground states was detected by measuring the differential conductance of the device as a function of an applied magnetic field, which was shown to give rise to disparate conductance signals for different initialisations of the Tb 4f-state. The transport measurements were always interpreted within a highly coherent and strongly correlated transport regime (Kondo transport), which entails the assumption of a strong coupling between the Au-nanowire junction and the Pc ligands of the TbPc2 molecule. The main evidence for a highly coherent transport regime, presented especially by Godfrin et al.,86 consisted of the following points (i) the conductance was probed at zero bias voltage in gate-voltage detuned conditions i.e. away from charge resonance points, which should suppress sequential tunnelling via Coulomb blockade, (ii) the observed temperature dependence of the differential conductance displayed features that are reminiscent of Kondo transport through a simple s = 1/2 quantum dot device, e.g. the conductance is

maximal at the lowest temperatures, only to decay at higher temperatures.

We note that in reference,86 in order to model the observed magnetic field dependence of the differential

conductance, it was necessary to assume that the anisotropic exchange coupling between the conduction

electron and the Tb 4f-electrons features a strong component perpendicular to the TbPc2 easy-axis (roughly 60% of the parallel component). However, such an assumption appears to be at variance with previous

150 151–153 0 experimental and ab initio results on 4f-Pc radical exchange coupling in [TbPc2] , which instead all corroborate a more likely Ising purely axial anisotropic exchange mechanism, having negligible perpendicular component. Furthermore, the stability diagram reported by Godfrin et al.86 appears to only weakly depart from a clear cut Coulomb blockade diamond diagram, as the gate-detuned conductance appears to die off 61 5.3. THEORETICAL MODEL

quite quickly as a function of gate voltage instead of clearly being established in the Coulomb blockaded

dark regions. Finally, we also note that the assumption of strong molecule-lead coupling is not commonly

observed in nanomagnet-based spintronics set-ups.48, 52, 79

Prompted by these observations, in this paper we take a rather different interpretation of the transport experiments reported for the TbPc2 spin transistor device, and we present a theoretical model that primarily describes transport within the sequential tunnelling regime, in the presence of broadening arising from the coupling to the leads. Using our model, we show that sequential tunnelling indeed suffices to explain most features of the magneto-conductance reported in previous works,82, 86 thus suggesting that electrical read-out

of a single spin can be achieved even without assuming a strong molecule-lead coupling, arguably an easier

to attain hence more common experimental outcome in device fabrication. Finally, we also explore coherent

cotunnelling corrections to the transport problem and discuss the significance and limitations of our model.

5.3 Theoretical Model

In the simplest approximation, the TbPc2 nanomagnet molecular break junction consists of two electronic leads assumed here to be weakly hybridised with a read-out quantum dot (the phthalocyaninato ligands of

the nanomagnet), which in turn coordinate the central Tb(III) ion. Also, recent experimental works82, 84 have

159 demonstrated a sizable hyperfine coupling between the TbPc2 nucleus and its |mJ = ±6i 4f-electronic states, however we find that our results remain invariant to the inclusion of this coupling for the magnetic field

strengths explored herein (see Appendix A) and so, for simplicity, we proceed by neglecting this coupling from

our model. To model low energy electron transport through the TbPc2 device we partition the Hamiltonian as

H = HL + HS + HT , which describes the two non-interacting electronic leads HL, the nanomagnet exchange

coupled to the read-out dot HS and the electron tunnelling between each subsystem HT , respectively. P † More specifically, the leads Hamiltonian reads HL = αkσ αkσaαkσaαkσ and describes the non-interacting (†) electrons in lead α ∈ {S, D} with wavevector k, spin σ and energy αkσ. The aαkσ hence form a set of annihilation (creation) operators that act on the single particle states |αkσi of each electrode. The

P ∗ † † hybridisation between the read-out dot and the leads is given by HT = αkσ Tαkσaαkσdσ + Tαkσdσaαkσ (†) where dσ annihilates (creates) an electron with spin σ on the read-out dot and Tαkσ represents an amplitude that quantifies the strength of the coupling between the read-out dot and the leads. The low energy spectrum of the terbium nanomagnet exchange coupled to a read-out dot is modelled effectively by

X † HS = (D − eVg) dσdσ + µBB (gJ Jz + gsz) − aJzsz (II.5.1) σ 5.3. THEORETICAL MODEL 62

where D is the energy of the lowest unoccupied molecular orbital (LUMO) of the phthalocyaninato ligands which constitute the read-out dot and is modulated by some local gate voltage Vg, Jz and sz are angular momentum operators that retrieve the projection of the total angular momentum in the ground state spin- orbit multiplet of Tb(III) along the TbPc2 magnetic anisotropy axis (z-axis), and the spin projection along the same axis of the unpaired conduction electron hosted on the Pc2 ligand read-out dot, respectively, µBB is the amplitude of a longitudinally applied magnetic field pre-multiplied by the Bohr magneton, gJ is the

7 Land´eg factor for the ground F6 spin-orbit multiplet of the Tb(III) ion, g is the g factor for a free electron, and finally, a is the ferromagnetic coupling constant (a > 0) describing the Ising exchange coupling150, 153 between 4f-electrons on the Tb(III) ion and unpaired electrons on the Pc2 ligands (i.e. the read-out dot).

For Eq. (II.5.1) to give a faithful representation of the low-energy spectrum of the TbPc2 device, a few assumptions have been made that are well justified for this molecular device. Firstly, we assume that

−1 the TbPc2 magnet retains the large splitting (>400 cm ) between the ground and first excited crystal

7 30, 151–154 field states within the lowest F6 spin-orbit multiplet when embedded in the break junction device. This assumption is consistent with the fact that magnetic hysteresis measurements indicate the preserva- tion of the nanomagnet’s magnetic anisotropy axis.82 Owing to the sub-Kelvin temperatures explored in these experiments, this assumption allows us to safely discard all but the two maximal total angular mo- mentum projections |mJ = ±Ji from our model and consider the Tb(III) moment as a semi-classical Ising spin. Secondly, on the basis of experimental evidence150 and high-level scalar relativistic multireference ab

153 initio calculations, we describe the Tb(III) 4f-electron-Pc2 radical exchange coupling in terms of a purely axial Ising exchange coupling Hamiltonian, which would result from projection of, e.g. an isotropic Heisen- berg exchange Hamiltonian, onto the doubly-degenerate thermally isolated ground state |mJ = ±6i of this molecule. This SMM-radical exchange coupling scheme then offers the simplest paradigm in which to capture theoretically the physics of the TbPc2 molecular break junction. As a consequence of these assumptions,

HS is diagonal on the product basis of the nanomagnet’s bistable ground states and the spin states of the read-out dot |m; σi ≡ |m = ±JiSMM ⊗ |σi which we will utilise for the rest of the manuscript.

5.3.1 Coulomb Blockade Transport Model

We now discuss the theoretical framework in which we model the most relevant experimental quantity of the device: the zero bias differential conductance. In order to study the relationship between the orientation of the Tb magnetic moment and conductance measurements at finite field i.e. the very origin of the electrical read-out mechanism observed in experiments,82, 84, 86, 101 we compute contributions to the conductance from each orientation separately, and average the two signals when appropriate. The linear response differential 63 5.3. THEORETICAL MODEL

conductance for one of the two possible orientations (m = ±6) of the semiclassical TbPc2 Ising magnetic moment m is defined by the derivative of the steady-state current with respect to bias voltage

  dIm d dWm dPm gm = = e (Wm · Pm) = e · Pm + Wm · (II.5.2) dV dV dV dV b Vb=0 b Vb=0 b b Vb=0

T where Pm = (pm, pm;↑, pm;↓) contains the non-equilibrium populations of the electronic states of the device

for a given orientation of the Tb moment m and Wm is a vector of transition rates between the redox states of the device obtained from the Fermi Golden rule. As a result of weak but non-negligible coupling

of the read-out Pc2 ligand-dot orbital state to the continuum of states in the leads, the molecular energy levels obtain a finite linewidth proportional to the imaginary part of the self energy, arising as a correction

to the effective molecular Hamiltonian to account for the coupling to the leads states, after eliminating

the leads manifold from the full molecule-lead partitioned Hamiltonian.119 The ensuing broadening of the

molecular effective Hamiltonian eigenvalues (energy levels) is encoded in the spectral density function of the

read-out dot, which can be approximated as a Lorentzian lineshape centred at the LUMO energy of the non-

interacting Pc2 ligand. We include this important effect into our model phenomenologically by expressing charge transfer processes as convolutions of the leads thermal functions with a Lorentzian lineshape centred

at the charging energy155 Z m→m;σ0 Γδσσ0 f( − µα)ηd Wασ = 2 2 . (II.5.3) ~π ( − ∆m;σ0,m) + η

2 Here, Γ = 2πραkσ|Tαkσ| is the coupling strength between the leads and the dot that, to a good approx-

156, 157 imation, can be taken as constant over the energy range explored herein, δσσ0 is a Kronecker delta

function accounting for the overlap between the incoming electron spin and the reduced state, ∆m;σ0,m is the energy gap between the relevant reduced and uncharged states of the device, η is the hybridisation-induced

−1 broadening of the molecular energy levels and f( − µα) = [1 + exp(( − µα)/kBT )] is the Fermi-Dirac

distribution of electrons in lead α at some temperature T and chemical potential µα = ±Vb. The discharging

mσ0→m rate of a spin σ electron to lead α is given by Wασ and is readily obtained from Eq. (II.5.3) with the

substitution f( − µα) → [1 − f( − µα)].

In order to evaluate the conductance formula in Eq. (II.5.2) we compute the populations of the electronic

states of the device from a quantum rate equation that describes the non-equilibrium dynamics imparted

on the molecular system as a result of coupling to the leads.61 The time evolution of each population is

determined by X m;σ→m m→m;σ p˙m = W pm;σ − pmW σ (II.5.4) m→m;σ m;σ→m p˙m;σ = W pm − pm;σW 5.3. THEORETICAL MODEL 64

m→m;σ P m→m;σ where the charging rates summed over leads and spin are W = ασ0 Wασ0 and likewise for the discharging rates. As we are interested only in the steady-state limit of Eq. (II.5.4) we solve the linear

system that originates when P˙ m = 0 with the additional normalisation condition pm + pm;↑ + pm;↓ = 1.

A final simplification to our model can be made by noting that if our device is invariant under a parity transformation, which in our simple model is tantamount of assuming that the molecular device is sym- metrically coupled to left and right leads, then the non-equilibrium populations of the single-level quantum dot are invariant under reversal of the bias voltage. On the other hand, the bias voltage drop across the device is by definition odd under a parity transformation. Hence the Taylor series expansion about Vb = 0 of the populations in powers of the external bias voltage must necessarily be an even polynomial in the bias voltage, so that all odd derivatives of the populations with respect to the bias evaluated at zero bias must be identically zero. In particular, we have dPm/dVb = 0 at zero bias, which simplifies Eq. (II.5.2), leading to the compact formula   dWm gm| = e · Pm (II.5.5) Vb=0 dV b Vb=0 for the zero bias differential conductance. An alternative proof of Eq. (II.5.5) is presented in Appendix B.

We note that the fabrication of a molecular device with a perfectly symmetrical source/drain coupling (i.e.

ΓS = ΓD = Γ) is somewhat unlikely, hence limiting the scope of Eq. (II.5.5) for realistic devices. However, introducing such a lead-dot coupling asymmetry does not change the essential physics exposed by our model

(see Appendix A) and so we proceed with this none-too-restrictive assumption for simplicity.

We now discuss in some detail sequential tunnelling transport occurring via two different exchange cou-

pling regimes in which, through different mechanisms, it is possible to explain the observed electric read-out

of the quantum states of the TbPc2 nanomagnet embedded in the molecular break junction device.

Large exchange coupling regime: gate detuning driven read-out mechanism

We begin with a study of the coupling between the Tb 4f-electrons and the unpaired conduction elec-

tron hosted on the Pc2 dot in the large ferromagnetic exchange regime (6a >> kBT , where 6a is the exchange energy gap according to Eq. (II.5.1)), so that the antiferromagnetically coupled reduced states

|m = ±J, σ = ∓1/2i are thermally isolated and do not participate in electron transport. We choose a gate

(0) (0) voltage Vg = Vg + δVg, where Vg is the gate that brings to degeneracy at zero field the ferromagnetic reduced state |m = ±J, σ = ±1/2i and the uncharged state |m = ±Ji. As shown in the Zeeman energies

plot in Figure II.5.1A , the effect of δVg is to shift the system away from charge resonance (gate detuning), preparing the system in a the ferromagnetic reduced ground state. Note that, in our model, the role of

detuning is to shift the charge resonance degeneracies between the uncharged and reduced ground states, 65 5.3. THEORETICAL MODEL

hence the peaks of the sequential tunnelling current, to non-zero values of the magnetic field, having opposite

signs for opposite orientations of the Tb magnetic moment (circled in black in Figure II.5.1A).

For a given orientation of the terbium magnetic moment m the conductance (obtained from Eq. (II.5.5))

takes on the particularly simple form

Z   eΓ ηd ∂f( − Vb) ∂f( + Vb) gm|V =0 = − . (II.5.6) b 2π 2 2 dV dV ~ ( + δVg − gµBBσ) + η b b Vb=0

It is instructive to take the zero temperature limit of Eq. (II.5.6), so that the Fermi-Dirac functions become

step functions whose derivatives are Dirac delta functions centred at Vb = 0. In this limit the integral in Eq. (II.5.6) can be evaluated exactly, and it can be readily seen that the zero-temperature limit of the

conductance for each orientation of the TbPc2 moment as a function of the magnetic field is proportional

to a Lorentzian lineshape peaked at B = ±2δVg/gµB, where the ± corresponds to the m = ±J orientation of the Tb magnetic moment. From this analysis we can ascribe the splitting of the conductance signals

0.15 A) 0.10

0.05

0.00 E (meV) - 0.05 | | - 0.10 > > | - 0.15 | > 0.55 > B)- 200 - 100 0 100 200 B (mT) 0.54

S) 0.53 g (

0.52

0.51 - 200 - 100 0 100 200 B (mT)

Figure II.5.1: (color online) A) Zeeman diagram of the lowest lying levels of the device in the large exchange coupling regime B) Conductance as a function of magnetic field at 100 mK for the two orientations of the TbPc2 magnetic moments m = J (spin up, blue arrow) and m = −J (spin down, red arrow) in the large exchange coupling regime. Note the conductance peaks at the magnetic field values B = ±2δVg/gµB, at which values the gate voltage-detuned energies of the neutral and reduced states of the TbPc2 are brought back to charge resonance. 5.3. THEORETICAL MODEL 66 in an applied magnetic field for each transport channel m = ±J at zero bias and low temperature to an off-resonance phenomenon that originates from detuning the electronic levels of the Tb-dot hybrid away from level degeneracy with a gate voltage, then restored by a magnetic field together with the peaks of maximal differential conductance.

With the above mechanism in mind, we calculate the conductance as a function of magnetic field for

86 both orientations of the TbPc2 moment at the finite temperature T = 100 mK used in the experiment by numerical integration of Eq. (II.5.6). We obtain best agreement with the experiments for molecule-leads tunnel-coupling Γ/~ = 6.5 × 108 s−1, and using a broadening factor η = 55 µeV, which is of the order of magnitude of the broadening used to model electron transport through quantum dots.155, 158 Furthermore, in order to reproduce the conductance peaks at the experimentally observed fields of B = ±100 mT, within the strong coupling regime the detuning gate voltage must be fixed at δVg ≈ 0.005 meV. Figure II.5.1B shows the calculated conductance as a function of magnetic field for each orientation of the Tb magnetic moment. The peaks in the conductance associated to each orientation of the Tb moment clearly originate from a recovery of the level degeneracy condition restored via the magnetic field. In this regime, applying a static magnetic field to the device leads to two disparate conductance signals that provide an electronic read-out of the spin state of the terbium nanomagnet, explaining the observed read-out experiments86, 101 within the Coulomb blockade transport regime.

We note that recent multifrequency EPR experiments on single crystals of this spin transistor molecular

0 unit [TbPc2] measured an intramolecular 4f-electron-Pc2 radical Ising exchange energy gap of 6a = 0.11 meV (a ≈ 0.02 meV).150 Given that the thermal energy available at T = 100 mK (≈ 0.01 meV) is ten times smaller than the experimental exchange gap, we would expect the strong exchange limit discussed in this section to be the relevant exchange coupling regime to describe the read-out mechanism observed for this molecular spin transistor. This could be easily verified within the spin transistor experimental set-up by monitoring as function of applied gate voltage the magnetic field values for which the conductance peaks are observed. We have not so far been able to find these data in the literature.

Weak exchange coupling regime: exchange-driven read-out mechanism

We now proceed with a discussion of the weak exchange coupling regime (6a . kBT ), where both the ferromagnetic and antiferromagnetic reduced states participate in electronic transport through the device.

We set the exchange coupling constant to a = 2 × 10−3 meV, which is one order of magnitude smaller than that measured in single crystal experiments,150 hence comparable to the thermal energy available at

T = 100 mK. The ensuing Zeeman spectrum of the device, assuming a detuning gate voltage δVg = 0.02 meV, is reported in Figure II.5.2A. 67 5.3. THEORETICAL MODEL

A) | | 0.10 > > 0.05 | > | > 0.00 | | E (meV) > - 0.05 >

- 0.10

- 0.15 0.56 B) 0.55

0.54 S)

g ( 0.53

0.52

0.51 - 200 - 100 0 100 200 B (mT)

Figure II.5.2: (color online) A) Zeeman diagram of the lowest lying levels of the device in the weak exchange coupling regime where a < δVg B) Conductance as a function of magnetic field at 100 mK for the two orientations of the TbPc2 magnetic moments m = J (spin up, blue arrow) and m = −J (spin down, red arrow) in the weak exchange coupling regime. Note that in this case, the conductance peaks at the magnetic field values B = ±aJ/gµB, at which values the exchange coupling split energies of the ferromagnetic and antiferromagnetic reduced states of TbPc2 become degenerate.

In order to simulate the finite temperature sequential tunnelling transport, we use Eq. (II.5.5) as function of the external magnetic field, and plot the differential conductance as function of field in Figure II.5.2B.

The best agreement with experiments was obtained for Γ/~ = 6.7 × 108 s−1 and η = 75 µeV.

As shown in Figure II.5.2B, also in the weak exchange coupling regime the two opposite spin polarisations of the Tb(III) nanomagnet give rise to well defined disparate conductance signals at finite applied field, peaked at the same but opposite non-zero values of the external magnetic field, reproducing the experimental observations (peaks are centred at B = 100 mT). However, inspection of the Zeeman energy diagram reported in Figure II.5.2A, immediately shows that the peaks of conductance in this case coincide with field-induced level crossings within the same redox manifold i.e. at B = 100 mT we recover degeneracy between the ferromagnetic and antiferromagnetic reduced states only, as opposed to the large exchange coupling regime reported in Figure II.5.1, where the peaks correspond to charge resonance points at level degeneracies between different charged and uncharged redox states. The reason why the conductance peaks at the level crossing 5.3. THEORETICAL MODEL 68

between same-redox states, occurring at field values B = ±aJ/gµB (circled in black in Figure II.5.2B), is that at these crossings the steady-state populations of the reduced states become equal, and hence contribute

simultaneously and maximally to transport through the device.

As the amplitude of the magnetic field required to bring the reduced states to degeneracy is unaffected

by gate detuning, the positions of the differential conductance signals in this regime are sensitive only to

the value of the exchange gap 6a, hence implementing an exchange-driven read-out mechanism, which is expected to be less sensitive to the detailed value of the detuning gate voltage used in the experiment. In this limit, the transport experiments would then provide a direct measure of the exchange coupling strength between 4f-electrons and the sequential tunnelling electrons.

5.3.2 Temperature, magnetic field and bias voltage dependence of the conduc- tance

To study the temperature, bias voltage and magnetic field dependence of the device differential conduc- tance we simulate sequential tunnelling transport using the exchange coupling a observed in single crystal

EPR experiment 150 and corroborated by multiconfigurational151–153 and multireference ab initio calcula-

tions,153 corresponding to a ≈ 0.02 meV, and leading to an exchange energy gap ≈ 0.1 meV, ten times smaller than the thermal energy available at the operating temperature. While for this choice of coupling both of the aforementioned mechanisms could in principle play a role, we expect the gate detuning-driven mechanism discussed for the limit 6a >> kBT to dominate, with only a negligible amount of population transfer to the antiferromagnetic reduced states of the device when the field is applied. We find that the effect of such population transfer is to shift the centre of the peaks of differential conductance shown in

Figure II.5.1B by a few tens of mT to higher fields.

In Figure II.5.3 the differential conductance in the absence of a magnetic field is plotted as a function of temperature. A plateau is observed in the conductance at low temperatures until kBT ∼ δVg wherein the electronic states of the device thermally equilibrate and the conductance signal begins to fall off to zero. We note that the behaviour of the differential conductance as a function of temperature in our model captures the temperature dependence of the molecular break junction device as reported in Ref.,86 except that in our microscopic model of sequential tunnelling conductance we do not need to invoke Kondo coherent transport in order to reproduce this behaviour. This specific lineshape then, appears to be a necessary but evidently not sufficient condition to infer the strongly correlated Kondo transport regime in this molecular device.

In Figure II.5.4 we report the differential conductance as a function of magnetic field and bias voltage.

The strong, broad resonance about B = 0 and Vb = 0 can be attributed to the averaged conductance signals 69 5.3. THEORETICAL MODEL

0.55

0.50

0.45 ) S μ ( 0.40 g

0.35

0.30

0.25 0.01 0.05 0.10 0.50 1 T (K)

Figure II.5.3: Differential conductance averaged over both orientations of the Tb moment as a function of temperature using a = 0.02 meV. Best agreement with experiments was obtained for Γ/~ = 6.6 × 108 s−1, and η = 65 µeV.

that appear for each orientation of the Tb moment (see Figure II.5.1B). At zero bias voltage and larger values of the magnetic field the level degeneracies between the uncharged states and the reduced states of the device are lost and the conductance signal falls to zero. For magnetic fields greater than the exchange coupling, the ferromagnetic reduced state becomes the ground state of the device which may transfer excess charge to electrodes and thus reinstate a sequential tunnelling electric current through the device only when the bias conduction window is wide enough so as to include the uncharged ground state of the TbPc2 molecule.

g (¢S) 2 0.6 1 0.5

D ¡ / b 0 0.4 V

- 1 0.3

- 2 0.2

- 4 - 2 0 2 4 0.1 B (T)

Figure II.5.4: Contour plot of conductance averaged over both orientations of the Tb moment as a function of bias voltage (in units of the occupation energy of the dot) and magnetic field, for a = 0.02 meV, Γ/~ = 6.6 × 108 s−1, and η = 65 µeV. 5.3. THEORETICAL MODEL 70

5.3.3 Coherent Corrections to Transport

To investigate the extent of the coherent character of the conductance in the TbPc2 break junction, we consider corrections to the Coulomb blockade models presented above to second non-vanishing order

(cotunnelling) in the hybridisation Hamiltonian HT . To calculate the cotunnelling rates we employ a T- matrix approach which is known to be consistent with a full microscopic derivation of the transport prob-

lem.111, 116, 117 In this approach, there are three types of cotunnelling processes that may contribute to the

conductance within our model and for the experimental choice of gate-voltage detuning (i) elastic transitions

in the uncharged manifold, (ii) elastic transitions in the reduced manifold and (iii) inelastic transitions in

the reduced manifold. The most general expressions for the cotunnelling rates in the neutral and reduced

manifolds respectively are159

2 2 Z † N,i→f Γ X hf| dσ |νi hν| dσ0 |ii W 0 0 = df () [1 − f 0 ( − ∆ )] aσ;a σ 2π α α fi  − ∆ + iγ ~ ν νi (II.5.7) 2 2 Z † N+1,i→f Γ X hf| dσ |νi hν| dσ0 |ii W 0 0 = df () [1 − f 0 ( − ∆ )] . aσ;a σ 2π α α fi − + ∆ + iγ ~ ν fν

where the sums run over all virtual states |νi of the device that differ in electron number from the final

and initial states |fi and |ii by 1. The finite lifetime γ for the virtually populated state has been included

to regularise the denominators and, to a first approximation, the broadening of the molecular energy levels

induced by the coupling to the leads is absorbed into the finite lifetime of the virtual transition so that

γ = η. The quantum rate equations given in Eq. (II.5.4) for the reduced states must now be amended to

account for population transfer as a result of inelastic cotunnelling transitions (elastic cotunnelling transition

by definition do not change the populations of the states). The new rate equations for the reduced states

are given by

m→m;σ N+1,m;¯σ→m;σ  m;σ→m N+1,m;σ→m;¯σ p˙m;σ = W pm + Wcot pm;¯σ − pm;σ W + Wcot (II.5.8)

N+1,m;σ→m;¯σ P N+1,mσ→m;¯σ where |m;σ ¯i denotes the reduced state other than |m; σi and Wcot = α0α00σ0σ00 Wα0σ0;α00σ00 are all possible inelastic cotunnelling processes that transfer population from the state |m; σi to |m;σ ¯i. Like-

wise, the expression for the conductance is now recast to include all cotunnelling contributions to electronic

transport through the device. Again, using the parity invariance arguments outlined above, one obtains the

compact formula  seq cot  dWm dWm gm| = e + · Pm (II.5.9) Vb=0 dV dV b b Vb=0 71 5.3. THEORETICAL MODEL for the zero bias steady-state conductance of the device where the appropriate transition rates for sequential

seq cot tunnelling and cotunnelling have been collected into the vectors Wm and Wm respectively.

Large exchange coupling regime

We consider again the large exchange coupling regime in which the antiferromagnetic reduced states of the device are thermally inaccessible for transport. As a consequence of this large coupling, all inelastic cotunnelling transitions between the ferromagnetic and antiferromagnetic reduced states are suppressed and the steady-state quantum rate equations for all states in the device remain identical to the purely incoherent regime.

The sequential tunnelling contribution to the zero-bias steady-state conductance is

seq Z β dWm eΓ e d η · Pm = (II.5.10) dV π k T (1 + eβ)2 2 2 b Vb=0 ~ B ( − ∆m;σ,m) + η while the elastic cotunnelling contribution is

cot 2 Z β dWm eΓ e d η · Pm = (II.5.11) dV 2π2η k T (1 + eβ)2 2 2 b Vb=0 ~ B ( − ∆m;σ,m) + η

where β = 1/kBT and ∆m;σ,m is the energy gap between the ferromagnetic reduced state and the neutral state of the device with Tb orientation m. Combining Eq. (II.5.10) and Eq. (II.5.11) results in a formula for the conductance that is identical to the pure Coulomb blockade transport model described above however now with a renormalised coupling constant Γ 7→ Γ(1 + Γ/2πη) between the dot and the leads. As before, even after including cotunnelling processes into the model, the disparate signals of conductance that can be attributed to the individual quantum states of the nanomagnet result from an off-resonance effect induced by detuning the system from level degeneracy with a gate voltage.

Weak exchange coupling regime

We briefly return to the weak exchange coupling regime but now with the cotunnelling corrections that were discussed above, included into the model. Using the same parameter set as discussed in the previous weak coupling section we simulated the differential conductance as a function of magnetic field for each orientation of the terbium magnetic moment as shown in Figure II.5.5. With coherent corrections included to second non-vanishing order of the perturbation theory, we observed no change to either the conductance in a magnetic field or to the temperature dependent conductance as the cotunnelling transition rates appeared two orders of magnitude smaller than the sequential charging and discharging rates. 5.4. CONCLUSIONS 72

0.54

S) 0.53 g ( 0.52

0.51

- 200 - 100 0 100 200 B (mT)

Figure II.5.5: Conductance as a function of magnetic field arising from sequential and cotunnelling processes at T = 100 mK for the two orientations of the TbPc2 magnetic moments m = J (spin up, blue arrow) and m = −J (spin down, red arrow) in the weak exchange coupling regime.

5.4 Conclusions

In this work we presented two sequential tunnelling theoretical models, which were shown to capture recent low temperature experimental observations of off-charge resonance differential conductance as function of magnetic field, temperature and bias voltage. The two separate mechanisms we have identified for the electric read-out of the magnetic quantum state of the TbPc2 in differential conductance measurements at finite magnetic field depend on the system’s parameters in a fundamentally different way, which provides an

experimental handle to check the prevalence of each regime in a given device.

Specifically, the peaks of the differential conductance read-outs as function of magnetic field, for the two

opposite orientations of the magnet in the large exchange coupling regime, are found to be linearly dependent

on the magnitude of the gate voltage shift from the N/N+1 charge resonance point. Conversely, the position

of the same peaks of magneto-conductance in the weak exchange coupling limit, are mainly an expression

of the Ising exchange coupling strength between the Tb(III) nanomagnet 4f-electrons and the sequential tunnelling electron spin hosted by the molecule’s Pc2-ligand, and are not expected to change significantly on scanning a gate voltage across the charge resonance.

On the basis of the experimental value of the exchange coupling energy gap (6a ≈ 0.1 meV) for the

0 [TbPc2] molecule in the crystal phase, assuming it is not significantly affected by coupling to the break junction device or by the application of gate voltages, we argue that the gate detuning driven read-out mechanism identified here in the large exchange coupling regime is expected to be the most prominent for explaining the origin of the disparate signals of differential conductance measured for each of the 4f-quantum ground states of the TbPc2 nanomagnet. On the basis of the good performance of our sequential tunnelling model, including its simulation of the 5.5. APPENDIX A: GENERAL MODEL - HYPERFINE COUPLING AND ASYMMETRIC 73 COUPLING TO THE LEADS temperature dependence of the differential conductance lineshape which was argued to be associated to a

Kondo transport regime, we have provided evidence that the disparate conductance signals measured for each Tb magnetic moment orientation in a longitudinal magnetic field can be explained solely on the basis of sequential tunnelling processes in the Coulomb blockade transport regime, with no essential need to invoke coherent transport regimes, as also shown by the negligible influence of the coherent cotunnelling corrections explored here.

As such, while we cannot exclude that higher order coherent conduction mechanisms might improve quantitative agreement between theory and experiment, we posit here that incoherent charge tunnelling processes and the associated Coulomb blockade physics contributes dominantly to the low temperature conductance of this device, which may contribute to explaining the microscopic mechanisms of dephasing in

101 the Landau-Zener-like 4f spin-tunnelling dynamics identified in recent studies of the TbPc2 break junction. This has significant implications for future studies of molecular spin transistors based on single-molecule magnets, as weak molecule-lead coupling represents a more common scenario in the fabrication of these devices, where selective control of the interactions between molecule and leads cannot as yet be easily achieved.

5.5 Appendix A: General model - Hyperfine coupling and asym-

metric coupling to the leads

In this appendix we show that the relaxation of some of the constraints in our model (namely the neglect of 159Tb hyperfine coupling, and the symmetric coupling to the source and drain leads) does not change any of the conclusions achieved with the simpler and more symmetric model.

5.5.1 Hyperfine Levels

To include the hyperfine structure of the 159Tb nanomagnet into our model we append to Eq. (II.5.1) the

2
Hamiltonian Hnuc = µBBgI Iz +AIzJz +P Iz + I(I + 1)/3 which accounts in the first term for the Zeeman interaction of the 159Tb nucleus, in the second term for the hyperfine interaction between the nucleus and the electronic angular momentum of the nanomagnet and in the third term for the non-spherical, quadrupolar structure of the nucleus. Here, gI = 1.34 is the nuclear g-factor for Terbium, A = 2.14 µeV is the hyperfine

160 0 coupling constant and P = 1.24 µeV as reported by Ishikawa et al. The new Hamiltonian HS = HS +Hnuc is diagonal on the product basis of the nanomagnet’s bistable ground states, nuclear spin states and the spin

−3 −1 1 states of the read-out dot |m, mI , σi = |m = ±Ji ⊗ |mI i ⊗ |σi where mI can take on the values: 2 , 2 , 2 5.5. APPENDIX A: GENERAL MODEL - HYPERFINE COUPLING AND ASYMMETRIC COUPLING TO THE LEADS 74

3 and 2 . The energies of these states are plotted as functions of magnetic field in the upper panel of Figure

II.5.6A after detuning the system from level degeneracy with a gate voltage δVg = 0.005 meV. Note that as a result of this detuning, the level degeneracies between the neutral and charged states with matching electronic and nuclear angular momentum quantum numbers are shifted (black circles in Figure II.5.6A) to

B = ±2δVg/gµB irrespective of the nuclear spin quantum number of the states; note that the ± sign relates to the orientation of the TbPc2 magnetic moment m = ±J.

We proceed from here as in the main text by developing master equations for the populations of each

orientation of the TbPc2 magnetic moment and solve for the steady-state populations which are then used to compute the conductance from Eq. (II.5.5). In Figure II.5.6B we plot the resultant curves of magneto-

conductance that arise for each orientation of the TbPc2 moment utilising the parameter set used to produce Figure II.5.1 in the main text. The invariance of our model upon the introduction of hyperfine coupling

0 results from the inability of neutral and charged states with mI 6= mI to participate in charging/discharging events thus the only level degeneracies left that may contribute to the device conductance are unaffected

by interactions with the 159Tb nucleus. A similar argument can be made for the weak exchange coupling

| , -3/2, >| , -1/2, >| , 1/2, >| , 3/2, > | , -3/2> | , -1/2> | , 1/2> | , 3/2> 0.20 A) 0.15 | , 3/2, >

0.10 | , 3/2> | , 1/2, > 0.05 | , 1/2> 0.00

E (meV) | , -1/2, > - 0.05 | , -1/2>

- 0.10 | , -3/2, > | , -3/2> - 0.15 - 200 - 100 0 100 200 0.55 B) B (mT) 0.54

S) 0.53 g ( 0.52

0.51

- 200 - 100 0 100 200 B (mT)

Figure II.5.6: A) Zeeman diagram of the lowest lying hyperfine-split levels of the device |m, mI , σi in the large exchange coupling regime B) Conductance as a function of magnetic field at 100 mK for the two orientations of the TbPc2 magnetic moments m = J (spin up, blue arrow) and m = −J (spin down, red arrow). 75 5.6. APPENDIX B: CONDUCTANCE FORMULA AT ZERO BIAS

regime wherein the magneto-conductance curves for each orientation of the TbPc2 magnetic moment again remain invariant upon the inclusion of the hyperfine states of the device.

5.5.2 Asymmetric coupling

If the lead-dot coupling to source and drain is allowed to be different, we note that our simple expression

for the conductance (given in Eq. (II.5.5)) is no longer valid. To ascertain the consequences of introducing

such an asymmetry we implemented the full calculation of the conductance accounting for the derivative of

the populations with respect to bias voltage (Eq. (II.5.2)) and report the results of the magneto-conductance

for one orientation of the TbPc2 magnetic moment as a function of magnetic field in Figure II.5.7. Notably, by varying the ratio between the lead-dot coupling constants for the source and drain, the magneto-conductance

becomes diminished in magnitude but still retains its broad, Lorentzian lineshape centred at B = 2δVg/gµB.

Therefore, the appearance of a small asymmetry in the coupling constants ΓS and ΓD does not affect the read-out mechanisms discussed in the main text and can be accounted for in our more symmetric model by

adjusting the value of Γ.

0.55 ¢ ¢ D/ S=1

0.50 ¢

¢ D/ S=0.9 ¢ ¢ D/ S=0.8

S) 0.45 ¢ ¢ D/ S=0.7 g (

0.40 ¢

¢ D/ S=0.6 ¢ 0.35 ¢ D/ S=0.5

- 200 - 100 0 100 200 B (mT)

Figure II.5.7: Magneto-conductance of the device when m = J in the large exchange coupling regime at various ratios of lead-dot coupling ΓD/ΓS using the same parameter set as for Figure II.5.1 with ΓS = Γ.

5.6 Appendix B: Conductance formula at zero bias

We will now prove the validity of Eq. (II.5.5) from Eq. (II.5.2) which amounts to showing that the term proportional to dPm/dVb on the RHS of Eq. (II.5.2) vanishes when Vb = 0. The population vector Pm is obtained by solving the rate equations presented in Eq. (II.5.4). For ease of notation we cast these equations 5.6. APPENDIX B: CONDUCTANCE FORMULA AT ZERO BIAS 76

into a matrix form and consider a derivative with respect to bias voltage Vb

dPm ∂ dPm ∂ = MPm =⇒ = (MPm) dt ∂Vb dt ∂Vb (II.5.12)   d ∂Pm ∂Pm ∂M =⇒ = M + Pm. dt ∂Vb ∂Vb ∂Vb

At zero bias voltage, the last term on the RHS of Eq. (II.5.12) is exactly zero. To see why this is the case, consider the derivative with respect to bias voltage of a representative coefficient of the matrix M

m→m;σ0 m→m;σ0 Z   ∂W X ∂W ∂f( + Vb) ∂f( − Vb) = ασ = dµ() + (II.5.13) ∂V ∂V ∂V ∂V b ασ b b b

 2 2 where dµ() = d (Γ/~π) η/( − ∆mσ0;m) + η . Notably, in this step we have assumed a symmetric cou-

pling strength between the dot and both leads i.e. ΓS = ΓD = Γ. Clearly, when Vb = 0 the integrand vanishes due to the derivatives of the Fermi functions. The same calculation can be made for entries of M

containing discharging rates thus reducing Eq. (II.5.12) to

d ∂P  ∂P m = M m (II.5.14) dt ∂Vb ∂Vb

when Vb = 0. Eq. (II.5.14) has the exact form of the master equation for the populations Pm and thus

P i possesses a steady-state limit i.e. the matrix M has one zero eigenvalue. Secondly, note that i ∂Pm/∂Vb = P i ∂/∂Vb i Pm = ∂/∂Vb(1) = 0, where we have used that the populations sum to unity. It can be readily

verified then that the steady state limit of ∂Pm/∂Vb that is obtained from solving the linear system

∂P M m = 0 (II.5.15) ∂Vb

P i with the additional condition i ∂Pm/∂Vb = 0 is ∂Pm/∂Vb = 0 and thus, at zero-bias voltage, Eq. (II.5.5) is retrieved. Chapter 6

Conclusions and Future Work

The terbium molecular break junction first reported by Vincent et al. in 201282 represents an important step forwards in molecular spintronics, molecular quantum computing and fundamental physics. While the device has been thoroughly investigated in transport experiments, theoretical modelling of the device appears absent from the literature if not for a joint theoretical and experimental paper that investigated a very particular aspect of the device: a Landau-Zener-like tunnelling of the quantum system under continuous measurement.101 Models of electron transport through the magnetic system that capture other aspects of the many experiments concerning the device can be useful to fully theoretically characterise the terbium molecular break junction and lead to refinements that could propel the nanomagnet-based device towards useful commercial applications.

In chapter 5, a published work was presented in which transport through the terbium molecular break junction was studied in the Coulomb blockade regime. With a simple rate equation model, two exchange coupling regimes were identified that each resulted in disparate electrical conductance signals in the presence of a longitudinal magnetic field that also appeared in a recent experimental investigation of the device by

86 Godfrin et al. In both cases each signal could be attributed to one of the bistable TbPc2 ground states such that electrical conductance measurements provide a means to address the individual quantum states of the nanomagnet directly. In the large exchange coupling regime, the two conductance signals at opposing positive and negative values of the magnetic field stemmed from an off-resonance gate voltage effect whereby the magnetic field restored the level-degeneracy between the ground states of the neutral and reduced device when the terbium moment was oriented against the field. In the small exchange coupling regime, the peaks occurred at level crossings between the ferromagnetic and antiferromagnetic reduced states of the device and proved sensitive to the value of the exchange parameter suggesting that the position of the peaks may provide

77 Conclusions and Future Work 78

a measure of the Tb-radical exchange coupling in a laboratory setting. Importantly, based on recent high field

EPR150 and ab initio calculations,153 we suggested that a ≈ 0.02 meV was a likely value for the exchange

1 coupling between the Tb 4f electrons and the spin 2 radical delocalised in the Pc ligands of the device. While this corresponds in our model to the large exchange coupling mechanism, it still remains smaller than earlier predictions of the exchange coupling made by Trojan et al.161, 162 Cotunnelling contributions to the

conductance were investigated to second order in the tunnelling Hamiltonian using a T-matrix approach and

were shown to be negligible if only to contribute fine details to the differential conductance lineshapes. For

cotunnelling to determine the device physics, a much weaker (or absent) broadening of the molecular energy

levels of the device would be required however, this would come at the cost of a temperature dependence on

the device conductance far-removed from the observations of experiment.

The work conducted in part II of this thesis has demonstrated that incoherent transport through the break

junction marks a dominant contribution to the disparate conductance signals measured from the device,

used subsequently to read-out the 4f-quantum state of the Tb(III) ion. This new evidence goes against current dogmas concerning the electronic transport regime of the device (in previous works transport was thought to occur within the highly coherent Kondo regime). Incoherent transport processes, then, become strong candidates for a dephasing mechanism of the TbPc2 spin states postulated by Troiani et al. and, consequently, are likely to impact the interpretations of the Landau-Zener-like tunnelling of the device

observed in the experiment.101 To understand the fundamental physics of non-adiabatic transitions in open

quantum systems, a study of the spin dynamics of the TbPc2 break junction in a time-dependent tracing magnetic field could be a pivotal future research project now that incoherent processes have been shown to

dominate charge transfer in the device for the particular parametric regime explored in experiment.86, 101

Furthermore, the hyperfine levels of the 159Tb nuclei in the device were excluded from models presented in

the previous chapter for simplicity however their inclusion in future microscopic models of transport could

serve to explain the mechanistic underpinnings of the nuclear spin read-out experiments conducted on the

device.84, 85 Part III

Hysteresis Loops of

Magneto-conductance in a Driven

Single-molecule Magnet Molecular

Spintronic Device

79 Chapter 7

Origin of the hysteresis of magneto-conductance in a supramolecular spin valve based on a

TbPc2 single-molecule magnet

Authors: Kieran Hymas, Alessandro Soncini Affiliation: School of Chemistry, University of Melbourne, Parkville, 3010 Journal: Physical Review B

80 81 7.1. ABSTRACT

7.1 Abstract

We present a time-dependent microscopic model for Coulomb blockade transport through an experi-

mentally realised supramolecular spin valve device driven by an oscillating magnetic field, in which the

4f-electron magnetic states of an array of TbPc2 single-molecule magnets (SMMs) were observed to modu- late a sequential tunnelling current through an underlying substrate nanoconstriction. Our model elucidates

the dynamical mechanism at the origin of the observed hysteresis loops of the magneto-conductance, sig-

nature of the SMM-modulated spin valve effect, in terms of a phonon-assisted multi-spin-reversal cascade

relaxation process, which mediates the switching of the device between the two conductive all-parallel spin

configurations of the SMM array. Moreover, our proposed model can explain the zero-bias giant magnetore-

sistive transport gap measured in this device, solely within the incoherent transport regime, consistently with

the experimental observations, as opposed to previous interpretations invoking Fano-resonance conductance

suppression within a coherent ballistic transport regime. Finally, according to the proposed Coulomb block-

ade scenario, the SMM-mediated giant magnetoresistance effect is predicted to increase with the number

of SMMs aligned on the nanoconstriction surface, on account of the increased number of intermediate non-

conducting spin-flip states intervening in the phonon-assisted multi-spin-reversal cascade relaxation process

necessary to switch between the two conducting all-parallel SMM spin configurations.

7.2 Introduction

In the wake of Fert and Gr¨unberg’s late-1980s discovery of the giant magnetoresistance effect,163, 164 thin-

film spin valves have undergone a rapid transition from the laboratory to their incorporation into commercial devices such as magnetic random access memory devices and read-heads for high density hard drives. To further the pursuit for the ultimate miniaturisation of electronic devices, molecular analogues of the spin valve have been studied where one or both of the bulk ferromagnetic domains of the device are replaced by a single (or few) magnetic molecules.165–169

Molecular spin valve devices based on the lanthanide single-molecule magnet (SMM) bis-(phthalocyaninato)

27, 30, 170 79 terbium(III) (TbPc2) have been experimentally realised in graphene nanoconstriction and car- bon nanotube (CNT)80, 81, 103 spintronics set-ups wherein highly anisotropic hysteresis loops of magneto- conductance emerge when a magnetic field aligned along the easy axis of the molecules is traced and retraced over the device. Given the anisotropic nature of the magneto-conductance signals observed in the terbium

84, 85 spin valve experiments as well as in other TbPc2 molecular devices, the existence of an exchange inter- action between the terbium ion and conduction electrons delocalised across the phthalocyanine ligands of 7.2. INTRODUCTION 82

the molecule has been proposed.

The sign and magnitude of this exchange interaction has been probed in spintronics set-ups,82 in high

field EPR experiments,150 and corroborated by multiconfigurational152 and multireference153 ab initio cal-

1 culations, all revealing a ferromagnetic coupling between Tb(III) and a spin 2 radical delocalised across the phthalocyanine ligands of the order of 0.1cm−1. The presence of a radical spin delocalised over the

0 Pc2 organic ligands in the neutral oxidation state [TbPc2] of this SMM, and its ferromagnetic exchange coupling to the 4f-electrons of the Tb(III) ion, has been argued to mediate spin interactions between the localised Ln(III) magnetic moment and underlying substrates, such as magnetic thin films,151, 154, 171 or the conduction electrons carrying the electric current in the carbon nanostructures underpinning some molecular spintronics devices.79–81, 84, 85, 103, 172 Notably, this exchange coupling has been argued to be at the origin of

17 a weakly decohering read-out of a single TbPc2 nuclear qudit. In the spin valve set-up of interest in this work, the ferromagnetic exchange coupling has been implicated in a microscopic spin transport mechanism that gives rise to a strongly anisotropic giant magnetoresistance effect, accounting for sharp jumps of the conductance of the carbon nanoconstriction as a function of a time-dependent sweeping longitudinal magnetic field, which orients the magnetic moments of the grafted

79, 80, 149 TbPc2 molecules parallel to the field, and to each other. Crucially, the observed effect is intrinsically kinetic, displaying also a memory dependence in that, for a given sign of the magnetic field, two sharp jumps of the magneto-conductance are only observed upon tracing the field (i.e. while the field grows from zero to its maximal value along a given direction), and not upon retracing (when the field is decreased from its maximal value back to zero). A symmetric situation is then observed upon dynamical reversal of the field direction. This gives rise to the typical hysteresis loop of the magneto-conductance which, together with its strongly anisotropic character, is the hallmark of the SMM signature on the transport properties of the carbon nanoconstriction.

Early interpretations reported in the original experimental works79–81 pointed at the need to have at least two SMMs grafted on the carbon nanoconstriction device in order to observe the spin valve effect.

The reasoning was based on similar arguments explaining the workings of thin film spin valves, thus on the expectation that the anti-parallel orientation of the molecular magnetic moments expected in low magnetic

fields, where the two conductance jumps are observed, leads to reduced transmission of charge carriers with either spin projection, resulting in a blocking of the conductance and the onset of giant magnetoresistance.

Conversely, a strong magnetic field will orient the molecular spins parallel to the field (and hence to each other), thus quenching the destructive interference between opposite spin carriers and greatly enhancing the differential conductance of the device.

These early intuitions were at first rationalised by Hong and Kim in 2013 using broken-symmetry DFT 83 7.2. INTRODUCTION

Figure III.7.1: A schematic for the supramolecular spin valve device, where M TbPc2 nanomagnets are adsorbed on a chain of M coupled quantum dots forming the nanoconstriction junction. Conduction electrons tunnel onto the device incoherently from nearby electrodes at a rate Γ and hop coherently along the chain of dot-nanomagnets with amplitude th, experiencing a local Ising exchange interaction ai with the 4f-quantum states of each Tb(III) centre.

calculations to simulate ballistic transport linear response of a model (6,6)-nanotube device, consisting of 24

unit cells in the scattering region, with two vanadocene complexes with spin polarisation forced to be either

parallel or anti-parallel to each other.173 The authors found that while oppositely spin polarised molecules

would induce transmission dips and corresponding halving of conductance for scattering electrons of both spin

polarisations provided they have an energy matching the molecule-projected density of states resonances, two

vanadocene molecules with parallel spin polarisation (simulating high magnetic field conductance) could only

reduce the transmission of like-spin conduction electrons, and leave unaffected the transmission of electrons

with opposite spin polarisation. The effect was interpreted as a spin-dependent Fano-resonance suppression

of conductance.

Questions remain as to whether a purely coherent transport model, which is a necessary condition in

order to invoke a destructive interference effect such as spin-dependent Fano-resonance, and which implies

a strong coupling between leads and device, can be fully consistent with the experimental observation of

a Coulomb blockade transport regime in molecular spin valve set-ups79–81 that epitomises a quintessential

non-coherent sequential tunnelling transport regime.

The question of integrating non-coherent transport into the proposed Fano-resonance model was later

addressed in a 2017 work by Krainov et al.,149 where additional experimental features such as the opening up

of a transport gap in the Coulomb blockade stability diagram in zero field were highlighted and rationalised

by further elaborating on the spin-dependent Fano-resonance model proposed in 2013.173 In particular,

a spin-dependent Fano-resonance with a state localised within a ballistic one-dimensional double barrier

scattering problem was invoked both as a transport mechanism explaining the transport gap in zero-field, but

also as the mechanism responsible for the partitioning of the CNT device into sequences of spin-dependent 7.2. INTRODUCTION 84

quantum dot islands. Such quantum dot islands were then mapped into a system of classical capacitors

in a phenomenological model, establishing the energetics of sequential charging and discharging processes,

then utilised to set up a somewhat featureless spin-dependent sequential tunnelling transport model in the

presence or in the absence of a magnetic field.

While previous theoretical models102, 149, 173 captured specific features of the reported supramolecular

spin valve transport experiments,79, 80 these models did not integrate a comprehensive description of all

available experimental information. In particular, previous works did not address some key aspects related

to the TbPc2 SMM electronic structure and ensuing dissipative spin dynamics, which are arguably significant in this particular transport experiment, performed at cryogenic temperatures and in the presence of a time-

dependent oscillating magnetic field: (i) the first aspect is the role played by the 4f-Pc2 radical exchange energy scale in the microscopic transport mechanism, (ii) the second aspect consists of the time-dependent

nature of the dissipative SMM spin dynamics affected by spin-phonon coupling, which results in the observed

hysteresis of the magneto-conductance (iii) the third and final aspect consists of the dependence of the spin

valve transport dynamics on the number of aligned molecular nanomagnets in the SMM array, and the

discussion of the underlying microscopic mechanism.

In the attempt to formulate a full comprehensive picture of quantum transport in this molecular spin

valve device that is consistent with all experimental observations reported thus far, we propose here a time-

dependent dissipative microscopic model based on quantum master equations, which fully accounts for the

observed non-coherent Coulomb blockade transport regime and for the observed hysteresis of the magneto-

conductance signature of a kinetic molecular spin valve effect. The proposed model explicitly includes

all those ingredients that were disregarded in previous theoretical works, namely (i) the energy scale of

the Tb(III) 4f-Pc2 radical/conduction electron ferromagnetic exchange coupling, explicitly included in the microscopic Hamiltonian of the device quantum states, (ii) the effect on the non-coherent transport processes

of the interplay between the time-dependent driving force of the oscillating longitudinal magnetic field, and

the dissipative SMM spin-phonon relaxation dynamics (iii) the study of the transport dynamics as function

of the size of the SMM array.

As in the model proposed by Krainov et al.,149 also in our model we assume the device is partitioned into a series of quantum dots, each dot defined by the presence of one TbPc2 SMM. However, here we assume that the source of incoherent hopping in the transport process originates solely at the lead-device interface (thus

assuming weak lead-device coupling), while the sequence of quantum dots in which the whole supramolecular

nanojunction structure is partitioned by the interaction with the grafted TbPc2 nanomagnets, is assumed to be described in terms of coherent multielectronic states, whose occupation fluctuates during the sequential

tunnelling transport process. In other words, the sequential array of hybridised TbPc2-dot states in our 85 7.3. THEORETICAL MODEL OF THE SPIN VALVE

model is not represented in terms of a number of incoherently coupled quantum dot islands,149 but rather

they are described as a chain formed by an arbitrary number of localised orbital basis functions, generalising

recently proposed electronic structure setups limited to two molecules only102 to an arbitrary number of localised electronic states, which can be linearly combined into states delocalised over the full extent of the nanoconstriction.

We show that our dissipative dynamical model is capable of reproducing the observed hysteresis of the magneto-conductance, offering a novel explanation of the kinetic molecular spin valve effect as manifested through the observed jumps of the magneto-conductance, one that is based on the interplay between multiple

Tb(III) spin relaxation processes in the driving sweeping field. Furthermore, we will also show that our model is able to reproduce the transport gap observed at all gate voltages in zero magnetic field without invoking ballistic Fano-resonance transport events, and to provide some predictions as to the scaling of the magneto- resistance effect with the number of molecules in the spin valve device.

7.3 Theoretical Model of the Spin Valve

7.3.1 Spin Valve Hamiltonian

2 The device is comprised of a chain of M TbPc2 molecules adsorbed to an sp hybridised surface (modelled here as several quantum dots in series) and subject to an oscillating magnetic field B(t) as well as a static electric potential Vg from a non-local gate electrode. A bias voltage Vb is applied to two leads (electrodes) that sandwich the scattering region, facilitating electron transport through the device. A schematic of the device for M molecules grafted to the nanoconstriction is shown above in Figure III.7.1.

In particular, we approximate the relevant electronic states of the nanojunction with M TbPc2 molecules sequentially grafted onto its surface in terms of a set of M states sequentially localised between the two leads, each subband hybridised with one singly-occupied molecular orbital (SOMO) state of the Pc ligand of the TbPc2 molecule grafted on that section of the substrate. These one-electron dot-molecule states are assumed to be orbitally non-degenerate, hence they can be empty (N electron charge state), occupied by one electron of either spin (N + 1 electron charge state), or occupied at most by two spin paired electrons (N + 2 electron charge state). The spin moment of the electrons occupying the dot-molecule hybridised states, in turn, are exchange coupled to the local Tb(III) 4f-electrons, described by the doubly-degenerate ground

Ising doublet |J = 6, mJ = ±6i, which is well known to be thermally well isolated from other excited crystal

7 27, 30, 152, 153, 160 field states belonging to the same strongly split F6 spin-orbit multiplet. The energy scale of the exchange coupling between the spin of the fluctuating charge on the dot-Pc subband and the Tb(III) 7.3. THEORETICAL MODEL OF THE SPIN VALVE 86

th 4f-electrons for the i site, is described by the coupling constant ai, and assumed to be ferromagnetic as in

0 150, 152, 153 the case of the isolated [TbPc2] molecule thus favouring parallel orientation between the spin on the Pc ligand (here delocalised over the local dot-Pc state), and the 4f-electron magnetic moment.

With this assumption the low energy spectrum of a chain of M TbPc2 molecules adsorbed onto the series of underlying quantum dots and subject to an oscillating magnetic field is captured with the Hamiltonian

M X h † i i i i
 X ∗ † Hs(t) = (i − eVg) ciσciσ + Uni↑ni↓ − aiJzsz + µBBz(t) gJ Jz + gsz + thcjσciσ + H.c. (III.7.1) i=1 hiji

where the first sum runs over all M dot-molecule sites and the second sum runs over nearest neighbour

dot-molecule sites, i.e. it describes coherent hopping with amplitude th along the chain of dot-molecule site orbitals. The first term represents the single electron occupancy energy of each quantum dot modified by

(†) th the gate voltage Vg. Here, ciσ annihilates (creates) an electron with spin σ on the i quantum dot. The second term is weighted by an on-site Coulomb integral U accounting for electronic repulsion between two

† spin-paired electrons at a given site; niσ = ciσciσ are the number operators for spin σ at site i. The third term describes the exchange interaction between the Tb(III) nanomagnet at site i and the spin of the ith

i i quantum dot. Jz and sz are the total and spin-only angular momentum projection operators along the TbPc2 magnetic anisotropy axis for the Tb(III) 4f-electrons and the Pc-dot electron, respectively (a Heisenberg

exchange Hamiltonian would be projected into this simpler Ising coupling Hamiltonian due to the highly

axial nature of the Tb(III) ground state), while ai denotes the magnitude and sign of the 4f-Pc radical exchange coupling at the ith site. Notably, we have assumed here for simplicity that all of the nanomagnets

in the chain share the same quantisation axis (the device z axis). This is a reasonable assumption owing

to the favourable interaction between the sp2 hybridised substrate and the bulky pyrene anchors that are

covalently attached to the Pc ligands of each nanomagnet.174 Nonetheless, in Appendix A we relax this

constraint and show that the inclusion of disordered nanomagnet quantisation axes can lead to a quenching

of the kinetic spin valve for large tilting angles. The penultimate term is the Zeeman interaction between

the total magnetic moment of the device and the longitudinal component of an external oscillating magnetic

field Bz(t).

To explicitly account for the interplay between the spin dynamics driven by the oscillating magnetic

field, the sequential tunnelling dynamics, and the slow magnetic relaxation of the TbPc2 nanomagnets, we also introduce a model Hamiltonian accounting for the coupling between the nanomagnets spin and the

acoustic phonons of the underlying 2D substrate. Thus the quantised lattice vibrations of the substrate

couple weakly to the angular momentum states of TbPc2 via small perturbations to each molecule’s ligand 87 7.3. THEORETICAL MODEL OF THE SPIN VALVE

field,175 as embodied to first order in the spin-phonon coupling Hamiltonian42

M i X X ∂Hcf i Hsp = Q (III.7.2) ∂Qi q i=1 q q 0

i th where Qq corresponds to the contribution of the active molecular normal mode of vibration for the i i molecule to the device phonon eigenmode of wavevector q and Hcf is the equilibrium crystal field Hamiltonian for the ith nanomagnet. The device is coupled also to two electrodes modelled here as semi-infinite, non- P † (†) interacting electron reservoirs with Hl = αkσ εαkσaαkσaαkσ where aαkσ destroys (creates) an electron in lead α with wavevector k, spin σ and energy αkσ. We describe tunnelling of electrons into and out of the scattering region with the Hamiltonian111

M X X ∗ † HT = Tα,i aαkσciσ + H.c. (III.7.3) i=1 αkσ

In principle, electrons may tunnel from an electrode to any site i on the substrate (and vice versa) with amplitude Tα,i however we restrict our model to the most likely case for the given experimental set-up, namely where Tα,i = 0 ∀i 6= 1,M.

In this work we investigate the Coulomb blockade transport regime for a device described by the Hamil- tonians Eq. (III.7.1 - III.7.3), exploring in particular two N/N + 1 and N + 1/N + 2 charge resonances giving rise to a sequential tunnelling current. Furthermore, we shall investigate the effect on our model of increasing the chain length M. To understand the simplest regime, we place great emphasis on the

N/N + 1 resonance obtained by assuming that only M = 2 nanomagnets are grafted on the nanojunc- tion, which will allow us later to draw generalised conclusions for a larger number of nanomagnets. In the case M = 2, Eq. (III.7.1) can be diagonalised exactly, and we are able to obtain the most pertinent physics of the molecular spin valve. When both dots in the scattering region are vacant the instanta- neous eigenstates of Hs(t) are uncorrelated products of the terbium spin states |mJ,1, mJ,2i where each mJ,i = ±J with time-dependent Zeeman energy E0(mJ,1, mJ,2) for a non-zero oscillating magnetic field in the adiabatic approximation. When the device is charged by a conduction electron with spin σ, due to the Ising character of the exchange coupling and the assumption that the external field is aligned along the magnetic anisotropy axes of the nanomagnets, all spin and total angular momenta quantum numbers

σ, mJ,1 and mJ,2 remain good quantum numbers (i.e. the Hamiltonian in Eq. (III.7.1) commutes with

J1, J2 and σ angular momentum and spin operators) so that the instantaneous eigenstates of Hs(t) become

± ± |mJ,1, mJ,2, σ, ±i = |mJ,1, mJ,2i×[u (mJ,1, mJ,2, σ) |σi1 |0i2 + v (mJ,1, mJ,2, σ) |0i1 |σi2] where |σii denotes the occupancy of site i with an electron of spin σ; the energy of these states are denoted E1(mJ,1, mJ,2, σ, ±) 7.3. THEORETICAL MODEL OF THE SPIN VALVE 88

and can be found in Appendix B. Consequently, a conduction electron that charges the scattering region

(regardless of spin) becomes delocalised across both sites 1 and 2 interacting with both Tb moments via the

exchange coupling ai. If we consider a1 = a2 = a > 0 then the fully polarised (parallel) configurations of the Tb moments |J, J, ↑, −i and |−J, −J, ↓, −i become the degenerate ground states of the singly charged mani-

fold at zero applied longitudinal field. The exact, generalised expressions for the energies and amplitudes of

these redox states are given in Appendix B.

7.3.2 Time-dependent transport in the adiabatic approximation

To model the transport dynamics of the spin valve in the presence of a time-dependent longitudinal mag-

netic field we develop adiabatic rate equations92 to first (non-vanishing) order in the spin-phonon Hamiltonian from Eq. (III.7.2), and the transport Hamiltonian from Eq. (III.7.3), for the diagonal elements of the reduced density matrix ρ expressed on the basis of the eigenstates of Hs(t). The rate equation for the population of an Hs(t) eigenstate |qi reads

X r→q r→q X q→r q→r ρ˙q = (W + Ω ) ρr − (W + Ω ) ρq. (III.7.4) r r

The time evolution of the diagonal elements of the reduced density matrix in Eq. (III.7.4) is governed by two processes: (i) the rates of charging and discharging of electrons from the leads onto the device W r→q and (ii) the phonon-mediated Tb magnetic moment flipping rates Ωr→q, which are both time-dependent quantities owing to the oscillating field. More explicitly, the charging transition rate between a state |qi and a state |ri (differing by one electron on the device) induced by HT is given by

2 Z ∞  −ωrq 2 X Γα,i d − W q→r = c† √ e ξ f( − µ ) (III.7.5) 2 iσ,rq 2πξ α αiσ −∞

with the rates summed over the two leads α = L, R, sites i and conduction electron spin σ. Here, Γα,i =

2 th 2πDα|Tα,i| is the coupling strength between the i dot and lead α with the constant density of states † † Dα and ciσ,rq = hr| ciσ |qi is the transition amplitude. The transition rates depend on the number of

available electrons with energy  in lead α which is given by the Fermi-Dirac distribution f( − µα) = −1 [1 + exp(( − µα)/kBT )] evaluated at the bias voltage µα = ±Vb/2; the upper (lower) sign refers to the source (drain) electrode. The electronic levels of the molecular system are broadened by a Gaussian lineshape

centred at the time-dependent energy gap ωrq in order to account for the finite linewidth acquired by coupling the system to the continuum of states in the leads.112 The discharging rate from |ri to |qi is obtained from

† Eq. (III.7.5) with the substitutions ciσ,rq 7→ ciσ,qr and f( − µα) 7→ [1 − f( − µα)]. Incoherent transitions 89 7.3. THEORETICAL MODEL OF THE SPIN VALVE induced by the absorption or emission of lattice phonons are described with a two dimensional Debye model and are calculated using the formula

2 q→q0 ωq0q Ω = ±γsp (III.7.6) 1 − exp(ωq0q/kBT )

where the ± takes into account phonon absorption or emission and γsp is the spin-phonon coupling constant. The microscopic nature of this direct transition is discussed in Appendix C.

We assume here that only single phonons can be absorbed (emitted) from (to) the lattice such that only a single Tb moment may be flipped at a time92 so, for example, Ω|J,Ji→|−J,−Ji = 0 whereas Ω|J,Ji→|J,−Ji 6= 0.

The matrix elements for each transition are absorbed into the coupling constant γsp and are not evaluated explicitly. Finally, note that the phonon transition rates, like the charging/discharging rates, depend on the gaps ωqq0 hence they are also time-dependent quantities.

In Eq. (III.7.4) we have neglected the contribution of non-secular terms (that is, terms that couple the populations and coherences of the density matrix) to the evolution of ρq on the grounds that away from degeneracies these contributions oscillate for several cycles on the timescale of the course-grained evolution assumed for the Markovian master equation in Eq. (III.7.4) and thus integrate to zero. At degeneracy however, where non-secular terms can not in general be ignored, we discard these contributions to the spin dynamics on alternative grounds. The non-secular terms originating from the spin-phonon coupling in Eq. (III.7.2) are weighted by thermal factors that, at the low temperatures explored in molecular spintronics experiments, greatly suppress their contributions around the degeneracy. For the N electron manifold, non-secular terms arising from the hybridisation to the leads due to Eq. (III.7.3) are proportional

P † to σ hr| cσ |νi hν| cσ |qi where |ri is a state from the same charge manifold as |qi. Since charging and discharging an electron via the reduced state |νi can not modify the configuration of the terbium moments these terms are only non-zero when the Hs(t) eigenstates |ri and |qi are one in the same; such a scenario corresponds to a secular term which is defacto included in Eq. (III.7.4). A similar argument can be made for non-secular corrections to the dynamics of states in the N+1 manifold.

Due to the complexity of the ensuing adiabatic rate equation we resort to numerical integration of Eq.

(III.7.4) to find the time evolution of the diagonal elements ρq. For sufficiently long times the solutions to Eq. (III.7.4) tend to a limit cycle, recurring with the period of the oscillating field; in other words, they display a hysteresis dynamics which is independent of the initial conditions. With the solutions to the adiabatic rate equations in tow, the time-dependent magneto-conductance g of the molecular spin valve is evaluated 7.4. RESULTS AND DISCUSSION 90

by taking the derivative with respect to bias voltage of the time-dependent current given by61, 62

X q→r IL = e (Nr − Nq) ρqWL , (III.7.7) qr

q→r where Nq and Nr are the number of electrons on the device for the states |qi and |ri, respectively, and WL are charging/discharging rates from/to the left electrode only.

7.4 Results and Discussion

7.4.1 Hysteresis of the Magneto–Conductance and Dynamical Spin–Valve Ef- fect

For the purposes of our calculations we choose a = 0.02 meV as obtained from transport experiments

82 on a terbium molecular spin transistor, we set 1 = 2 = , th = 5 meV, ξ = 0.05 meV and explore the

operation of the device in the presence of a sinusoidally oscillating magnetic field Bz(t) = Bz sin(ωt) aligned

along the easy axes of M = 2 Tb nanomagnets with amplitude Bz = 500 mT. With this choice of parameters, the most relevant states for low temperature and low bias electron transport through the spin valve have

been plotted in Figure III.7.2. The coupling strength between the carbon nanoconstriction region and the

leads can be expressed in matrix form as

   

 ΓS,1 ΓS,2   1 0      Γ =   = Γ   (III.7.8)         ΓD,1 ΓD,2 0 1

assumed here to be weighted by a lead-junction tunnelling rate Γ = 104 s−1. We note that our calculations

remain largely invariant, if only for a renormalisation of the magneto-conductance magnitude, to the choice

of Γ provided that single electron charging and discharging rates are much faster than the sweeping rate of

−2 the magnetic field (i.e. Γ >> ω). Calculations were performed with a spin-phonon coupling γsp = 1 meV s−1 and at T = 0.5 K. As will become apparent in the following discussion, the particular value of the

spin-phonon coupling γsp is not crucial for the observation of hysteresis loops of the magneto-conductance

in our calculations, but rather it is the ratio between γsp and the magnetic field sweeping rate ω that gives rise to the loops of conductance.

To begin our discussion of the molecular spin valve we specialise to the N/N +1 resonance and choose Vg so as to bring the N +1 electron ferromagnetic ground states for the reduced system to level degeneracy with 91 7.4. RESULTS AND DISCUSSION

Figure III.7.2: Zeeman energies of the lowest lying energy levels of the TbPc2 molecular spin valve. The solid black lines represent uncharged states of the device with the magnetic moments of each TbPc2 nanomagnet labelled with thick, red arrows. The blue (red) dashed lines represent the corresponding ferromagnetic states for the parallel |−J, −J, ↓i (|J, J, ↑i) states. Orange mechanistic arrows are included to illustrate population transfer between the states when a magnetic field is traced from negative to positive values across the device thus leading to the two sharp jumps in magneto-conductance, characteristic of the molecular spin valve experiments.

the ground states of the neutral system at zero field. In Figure III.7.3 we report the calculated magneto- conductance as a function of the oscillating magnetic field, highlighting the tracing (blue) and the re-tracing

(red) sweep phases, for several field sweep rates. When the magnetic field is traced across the device, the conductance remains constant until an abrupt decrease and subsequent recovery is observed after crossing zero field. The same behaviour is observed on retracing the longitudinal field, hence giving rise to loops of hysteresis in the conductance. We thus immediately note that our model is capable of capturing the spin valve-like dynamic hysteretic behaviour observed in experiments.79, 80 As the rate of the sweeping field is increased, the loops of hysteresis become broader and their minima occur at larger magnitudes of the applied

field. When the rate of the sweeping field is decreased the converse is true and, in fact, the conductance begins to fall before zero field.

To understand the origin of the loops of hysteresis in the calculated conductance shown in Figure III.7.3, consider a field tracing event in the long-time limit. At the beginning of the trace we have Bz sin(ωt) < 0, so that the ground state configuration of the spin valve consists of parallel Tb moments aligned against the

field: |J, Ji. As can be observed from Figure III.7.2, since E1(J, J, ↑, −) ≈ E0(J, J), the level degeneracy condition is satisfied and a finite conductance is observed through the device. As the field is traced through

Bz sin(ωt) = 0, |J, Ji becomes an excited state of the system and phonon-mediated relaxation to |J, −Ji and |−J, Ji begins to play a role. Owing to the ferromagnetic exchange coupling a, the charged state

|±J, ∓J, σ, −i is thermally inaccessible for transport leading to a suppression of the conductance. At positive 7.4. RESULTS AND DISCUSSION 92

2.5

2.0 -1

✁ = 0.008 s

1.5

) g (e -1

1.0 ✁ = 0.005 s

0.5

-1

✁ = 0.002 s

- 400 - 200 0 200 400

Bz (mT) Figure III.7.3: Conductance loops of hysteresis on field tracing (blue) and retracing (red) obtained at different field sweep rates ω. The top (bottom) curves have been shifted by +(−)0.8eΓ for clarity.

fields |±J, ∓Ji is still an excited state of the device, and a phonon-induced relaxation to the |−J, −Ji ground state is observed, but only for positive magnetic field values of ∼ 100 mT since, as per Eq. (III.7.6), the phonon emission relaxation process grows quadratically with the energy of the released phonon. As in the time-reversed configuration, the |−J, −J, ↓, −i satisfies the level degeneracy condition with |−J, −Ji and the conductance is resumed to its normal value before completion of the trace.

Note that this configures a two-step magneto-conductance drop and then revival, where the drop occurs in the proximity of zero magnetic field as observed in experiment (depending also on the field sweep rate), while the revival of the current occurs for Bz ∼ 200 mT, which compares well with the value observed for the nanomagnet-based spin valve experiments on a graphene nanoconstriction.79 This procedure replays on retracing the field however now with the Tb configurations reversed and the drop in conductance appearing towards the other side of zero field. The timescales of the sweeping field and the phonon-mediated slow relaxation of the spin valve through the intermediate non-conducting |±J, ∓Ji states are hence inextricably linked, as expected for a hysteresis of the sequential tunnelling conductance driven by the slow magnetic 93 7.4. RESULTS AND DISCUSSION

Figure III.7.4: δg for a given field tracing and retracing event calculated at various temperatures with a sweeping rate of ω = 0.005 s−1.

relaxation of the TbPc2 magnetisation. The faster the sweeping field, the further apart the conductance- suppressing and the conductance-reviving phonon transitions appear, leading to broader loops of hysteresis as shown Figure III.7.3.

To further the investigation of the hysteresis loops of conductance it becomes useful to define a quantity

δg = gtrace −gretrace that characterises the sign of the loops for some trace-retrace cycle in the long-time limit. In Figure III.7.4 we show that the opening of the hysteresis loops of conductance is temperature sensitive,

with the loops closing completely around T = 0.7 K as observed in experiment.80 The spin valve effect is quenched when kBT ∼ a as the |±J, ∓J, σ, −i states become thermally accessible for charge transport, hence the anti-parallel Tb configurations become electrically conductive. As all configurations of the nanomagnet moments are electrically conductive above this temperature, the magneto-conductance becomes insensitive to phonon-mediated population transfer between different spin-flip configurations of the nanomagnet array, so that the kinetic spin valve effect is lost. Thus the energy scale of the exchange coupling between 4f electrons and the sequential tunnelling electrons fixes the temperature range for the observation of the supramolecular spin valve effect. We also investigated the effect on δg of detuning the gate voltage Vg by δVg away from the

E0(J, J) = E1(J, J, ↑, −) level degeneracy, and found that the sign of δg is preserved when crossing the level degeneracy, as shown in Figure III.7.5, in agreement with experimental findings.79, 80

7.4.2 Generalised Dynamical Model: Multiple Nanomagnets, Steady-state Con- ductance and Higher Charge States

In addition to the simplest case when M = 2, we also investigated the effect of increasing the SMM array length M. Figure III.7.6B reports the hysteresis of the differential conductance we calculated for the spin valve device encompassing arrays of M TbPc2 dot-molecule units, with M = 2, M = 3 and M = 4. We note 7.4. RESULTS AND DISCUSSION 94

400

δg (eΓ) 200 0.4

) 0.2 mT

( 0 z

B 0

-200 -0.2

-0.4

-400 -2 -1 0 1 2 δVg/ϵ

Figure III.7.5: δg as a function of the dynamically sweeping field amplitude Bz and gate voltage detuning δVg away from the E0(J, J) = E1(J, J, ↑, −) level degeneracy.

that while the field value at which the conductance reaches its minimum upon tracing and retracing remains invariant with increased chain length M, the % drop in conductance increases as more dot-molecule units are included in the model. Thus our model predicts that the giant magnetoresistance effect will increase with the number of nanomagnets aligned along the spin valve device, as predicted in a previous theoretical treatment of the nanomagnet device, which however was based on the ballistic transport regime.173 Furthermore, for

M > 2 we observe a gradual fall in the conductance before zero field upon tracing and retracing following the usual precipitous drops in conductance already described above.

The microscopic mechanism leading to the increased dynamical magneto-resistance effect with increased

M can be readily rationalised as follows. Including M sequential TbPc2 nanomagnets into the molecular spin valve has the effect of introducing 2M − 2 non-conducting configurations of the Tb moments that

must be parsed via phonon-mediated population transfer when switching between the conducting all-parallel

configurations |J, ..., Ji and |−J, ..., −Ji, with the sweeping magnetic field (see Figure III.7.6A). The larger drop in conductance during the phonon-mediated slow relaxation of the spin valve with increased M can then be attributed to greater population transfer out of the conducting parallel configuration and into the set of 2M − 2 spin-flip non-conducting microstates of the device, in a multi-step spin-lattice relaxation cascade.

A similar argument accounts for the pre-zero field decline in conductance whereby phonon absorption occurs to a small degree between the conducting ground state of the device and the M single-Tb-flipped levels e.g. 95 7.4. RESULTS AND DISCUSSION

1.5 A) 1.0

0.5 ... 0.0 ...

E (meV) - 0.5

- 1.0

- 1.5 B)

Figure III.7.6: A) Magnetic field dependence of the lowest lying energy levels of the TbPc2 spin valve device with M nanomagnets in series. As in Figure 2, the solid black lines represent uncharged states of the device and are labelled by the spin-polarisation of the TbPc2 units (red arrows, some configurations have been omitted to ease visualisation). The dashed lines (blue and red) represent the corresponding ferromagnetic reduced states for the parallel configurations |±J, . . . , ±J, ↑ (↓)i. Orange arrows along and between energy levels indicate population transfer between the states on a tracing field; the largest orange arrow is symbolic of a sequence of phonon-emission events required to relax the system back to the conducting parallel ground state orientation. B) Loops of hysteresis in the calculated conductance on a trace (blue) and retrace (red) of a longitudinal magnetic field for the TbPc2 molecular spin valve consisting of M TbPc2 nanomagnets.

|J, ..., −J, ..., Ji.

To further characterise our microscopic Coulomb blockade transport model, we now investigate the bias voltage dependence of the steady-state differential conductance for a range of applied, static magnetic field strengths. In Figure III.7.7, we report the steady-state conductance for the M = 2 molecular spin valve at

T = 30 mK, in line with a recent CNT molecular spin valve experiment.149 We note that at this resonance, the conductance has a pronounced peak about zero bias voltage, which broadens with the application of a magnetic field. The peak at zero bias in Figure III.7.7 can be understood with reference to the magnetic

field dependence of the energies reported in Figure 2 wherein, at Bz = 0, the level degeneracy between the conducting neutral and reduced states is exact, only to be lifted gradually with the application of the 7.4. RESULTS AND DISCUSSION 96

2.5

2.0 0 T 0.1 T S)

¢ 1.5 0.3 T

g ( 0.5 T 1.0 0.7 T 0.9 T 0.5 1 T - 20 - 10 0 10 20

Vb/¡ Figure III.7.7: Calculated steady-state conductance at T = 30 mK as a function of bias voltage at the N/N + 1 level degeneracy when the device is subject to various strengths of static, longitudinal magnetic field.

magnetic field, resulting in a partial suppression of the conductance. We note that this behaviour at the

N/N + 1 charge resonance point of our model is at odds with the experimental observation149 of a drop of magneto-conductance or finite transport gap at zero bias voltage in the absence of a magnetic field, which is then resumed in the presence of a magnetic field.

In an attempt at identifying other transport states of our model device displaying the experimentally observed dynamical spin valve effect, but also reproducing the static transport gap observations,149 we turn our attention to the N + 1/N + 2 charge resonance point, achieved with a gate voltage Vg that brings the |±J, ±J, ↑ (↓), −i reduced states in resonance with the doubly charged states. The N + 2 doubly charged states are obtained by numerical diagonalisation of Eq. (III.7.1) in the strongly interacting regime U >> th. The resulting energies are plotted in Figure III.7.8 as function of the longitudinal magnetic field. In contrast to the N electron case in which the neutral manifold is completely degenerate at zero applied field, with a choice of U = 500 meV consistent with first-principles estimations of the charging energy of the Pc ligands,147, 149 we find that the non-magnetic configurations of the terbium moments become the ground states of the doubly charged device (see Figure III.7.8). The N + 2 spin singlet states take the general form

(i) E h (i) (i) (i) (i) i Φ = |mJ,1, mJ,2i ⊗ a |↑↓i |0i +b |↑i |↓i + c |↓i |↑i + d |0i |↑↓i (III.7.9) mJ,1,mJ,2 1 2 1 2 1 2 1 2 wherein the amplitudes and energies of each state are reported in Appendix D. Strong on-site Coulomb repulsion and ferromagnetic coupling to the Tb magnets imply that the N +2 spin singlet states participating E E (1) (1) in low temperature transport have the approximate form ΦJ,−J ≈ |J, −Ji ⊗ |↑i1 |↓i2 and Φ−J,J ≈ 97 7.4. RESULTS AND DISCUSSION

|−J, Ji ⊗ |↓i1 |↑i2, thus remaining non-conducting at low temperatures and bias voltages, since the singly reduced states |±J, ∓J, ↑ (↓), −i are destabilised by the ferromagnetic coupling, hence thermally inaccessible for transport.

- 9.2 - 9.4 > >

- 9.6 (1) (1)
-J,-J
> (1) J,J - 9.8
> -J,J>

E (meV) - 10.0
(1) - 10.2 J,-J> - 10.4 > > - 400 - 200 0 200 400 B (mT) Figure III.7.8: Zeeman diagram of the most relevant lowest lying spin valve states for the reduced (N +1, red) and doubly reduced (N + 2, blue) device obtained by the application of a gate detuning and diagonalisation of Eq. (III.7.1) on the product basis of the nanomagnet angular momentum projections and dot spin states.

Next, we calculate the time-dependent magneto-conductance at T = 30 mK in a sweeping longitudinal

field traced and retraced across the device at a field sweep speed of ω = 0.005 s−1. The results are reported

in Figure III.7.9, where it is evident that also at this new N +1/N +2 charge resonance point, the differential

conductance displays sharp jumps as a function of the time-dependent tracing and retracing sweeping field,

giving rise to a hysteresis of the magneto-conductance that is akin to what we obtained at the resonance

N/N + 1, and is the hallmark of the dynamical spin valve behaviour of this supramolecular device.

The manifestation of hysteretic magnetoresistance at the N +1/N +2 charge resonance can be understood

in similar terms to that at the N/N + 1 charge resonance. From Figure III.7.8, the ground state of the

device at the beginning of a trace cycle (i.e. at negative magnetic field) is the fully polarised ferromagnetic E (1) reduced state which stays approximately in resonance with the doubly charged ΦJ,J state, thus leading to a non-zero conductance measurement at negative fields. After zero field is crossed, a two-step phonon E (1) emission cascade relaxes the device to the conducting Φ−J,−J state but does so via the non-conducting E (1) Φ±J,∓J states, leading to the dip and subsequent revival of the magneto-conductance on the completion of the trace. A symmetrical mechanism is then observed upon dynamical reversal of the field direction during

the retracing sweep.

Finally, we investigate the steady-state conductance at the N +1/N +2 resonance, when a static magnetic

field is applied to the device. In Figure III.7.10 we report the electrical conductance of the molecular spin 7.5. CONCLUSIONS 98

0.9

0.8 ) Γ

g (e 0.7

0.6

- 400 - 200 0 200 400

Bz (mT) Figure III.7.9: Hysteresis of the magneto-conductance upon tracing (blue) and retracing (red) the longitu- dinal magnetic field, obtained at the N + 1/N + 2 level degeneracy at T = 30 mK and with a field sweep rate ω = 0.005 s−1.

valve device as a function of bias voltage at T = 30 mK and zero magnetic field. Without a magnetic field, E (1) only the non-conducting ground states Φ±J,∓J of the spin valve are populated (see Figure III.7.8), leading to a conductance suppression at zero bias voltage. As bias voltage is increased the conductance grows due to the population of the conducting all-parallel spin configurations of the spin valve. When a magnetic field of

1 T is applied to the device, the fully spin-polarised parallel configuration of the terbium magnetic moments becomes the ground state of the device. At this value of magnetic field, the level degeneracy between the fully spin-polarised N + 2 doubly charged state and the N + 1 singly charged all-parallel spin configuration

of the terbium moments (as per Figure III.7.8) is preserved within the charge transport window allowed by

level broadening and temperature, leading to an enhanced value of the conductance at zero bias with respect

to the zero magnetic field case. Thus our model can capture the transport gap observed in experiments,149 albeit only at the N + 1/N + 2 charge resonance point, crucially, within a fully incoherent transport regime where it is arguably not straightforward to account for a ballistic transport Fano-resonance conductance suppression effect.149, 173

7.5 Conclusions

We have developed and used a time-dependent rate equation model within the adiabatic approximation to calculate the electrical conductance within the Coulomb blockade regime through a sequence of quantum dots coherently hybridised with TbPc2 molecular nanomagnets, and coherently coupled in a chain configuration, in the presence of an oscillating magnetic field. Our Coulomb blockade microscopic time-dependent model of the charge and spin dynamics of the spin valve device, also accounting for dissipative spin-phonon relaxation 99 7.5. CONCLUSIONS

0 T 2.0 1 T

) 1.5 g (e

1.0

0.5 - 15 - 10 - 5 0 5 10 15

Vb/ Figure III.7.10: Calculated steady-state conductance as a function of bias voltage with and without the application of a static magnetic field Bz = 0T (red) and Bz = 1T (blue).

mechanism of the nanomagnets grafted to the nanojunction, was shown here to be capable of capturing the

experimentally observed hysteresis dynamics of the magneto-conductance, including its dependence on the

sweep rate of the magnetic field ω, and on temperature.

To rationalise the occurrence of the hysteresis of the magneto-conductance and its behaviour as function

of external parameters, we identified a microscopic mechanism based solely on phonon-assisted spin flips of

the M TbPc2 Ising magnetic moments, resulting in a sequential cascade of population transfer between the two oppositely polarised conducting configurations (all spins parallel), mediated by the remaining 2M − 2

non-conducting configurations of the supramolecular spin array. The proposed mechanism also allowed us

to explicitly correlate the temperature dependence of the hysteretic spin valve effect with the exchange

coupling strength between Tb(III) 4f-electrons and the sequential tunnelling electron. We also investigated the conductance loops as a function of Tb chain length M and noted that as M increased, the deeper

the conductance troughs became as a function of the tracing field, consistent with the proposed phonon-

assisted multiple spin-reversal relaxation mechanism. Finally, we investigated the steady-state limit of our

model, which could reproduce the presence of a finite transport gap in the magneto-conductance observed

in experiments, without a reference to the coherent ballistic transport Fano-resonance argument utilised in

previous studies.

In conclusion, we believe that our proposed microscopic model of time-dependent dissipative quantum

transport offers a consistent and comprehensive picture of the physics involved in the experimentally observed

supramolecular spin valve effect, which explicitly accounts for the modulation of the transport dynamics of 7.6. APPENDIX A: MODEL OF DISORDERED TBPC2 QUANTISATION AXES 100

the device caused by the slow magnetic relaxation of single-molecule magnets grafted on its surface, which

explains both the microscopic origin of the observed kinetic spin valve effect (hysteresis of the magneto-

conducance), and of the giant magnetoresistance effect observed in static magnetic fields, within a Coulomb

blockade transport picture fully consistent with experimental observations. Our model also predicts a corre-

lation between the observed drop of magneto-conductance in zero field, and the number of grafted molecular

nanomagnets, which could be tested in future experiments.

These findings are expected to be useful for the design and interpretation of future experiments in

molecular electronics advancing the quest for the ultimate miniaturisation of spintronics devices using single

molecule magnets.

7.6 Appendix A: Model of disordered TbPc2 quantisation axes

In the main text we have assumed for simplicity that the nanomagnets grafted to the sp2 hybridised surface share the same quantisation axis however, under experimental conditions this pristine configuration may be difficult to actualise. To investigate the effect of introducing non-collinear quantisation axes for the TbPc2 nanomagnets we return to the prototypical M = 2 nanomagnet model and focus again on the N/N+1 charge resonance. We take the second in the series of nanomagnets to be quantised along the z0 axis that is obtained from the quantisation axis of the first nanomagnet by a rotation of θ about the in-plane transverse axis of the device (see Figure III.7.11).

The Hamiltonian given in Eq. (III.7.1) is modified to account for the non-collinear magnetic anisotropy

θ z z z'

Substrate

Figure III.7.11: Schematic depiction of the M = 2 TbPc2 molecular spin valve wherein the nanomagnets do not share the same quantisation axes. WLOG we take the first nanomagnet in the series to be quantised along the z axes and the second along the z0 axes which is obtained by a rotation of the z axis by an angle θ about the in-plane transverse axis of the device. 101 7.6. APPENDIX A: MODEL OF DISORDERED TBPC2 QUANTISATION AXES

axes of the TbPc2 nanomagnets accordingly

† † 1 1 20 20 Hs(t) = (1 − eVg) c1σc1σ + (2 − eVg) c20σc20σ − a1Jz sz − a1Jz sz

h  1 20   1 20 i + µBBz(t) gJ Jz + Jz cos θ + g sz + sz cos θ (III.7.10)

∗ h  † †   † † i + th cos θ c1↑c20↑ + c1↓c20↓ − sin θ c1↑c20↓ − c1↓c20↑ + H.c. where the primes indicate operators quantised along the z0 axis. Owing to a stabilising interaction between the bulky pyrene substituents (chemically grafted to the Pc ligands of the nanomagnets) and the sp2 hy- bridised surface, only small deviations of the TbPc2 magnetic anisotropy axis from the substrate surface

174 normal are expected. Since θ is small, the inter-ligand hopping integrals th can to a good approximation be considered constant as a function of the canting angle. With this in mind, we proceed as in the main text and compute the magneto-conductance of the canted system noting that now, by virtue of the spin-flip hopping term proportional to th sin θ in Eq. (III.7.10), the eigenstates of the reduced system are linear combinations of both radical spins delocalised across the Pc ligands of each nanomagnet

h (i) (i) (i) (i) i |mJ,1, mJ,2, 1 ≤ i ≤ 4i = |mJ,1, mJ,2i ⊗ e |↑i1 |0i2 + f |↓i1 |0i2 +g |0i1 |↑i2 + h |0i1 |↓i2 (III.7.11) where the time-dependent coefficients e(i), f (i), g(i) and h(i) are obtained from numerical diagonalisation of each 4 × 4 mJ,1, mJ,2 block of the Hamiltonian in Eq. (III.7.10).

In Figure III.7.12 we plot δg = gtrace − gretrace for a range of canting angles θ. Notably, for larger values of the canting angle, the kinetic spin valve effect is quenched. To understand why this is, recall that in the collinear quantisation axes case that is explored in the main text, the conductance troughs characteristic of the kinetic spin valve effect result from the transient population of the states with an anti-parallel TbPc2 magnetic moment configuration (see Figure (III.7.2)). Since the reduced states with anti-parallel TbPc2 magnetic moments are thermally inaccessible for transport (owing to the energetically unfavourable exchange interaction between the radical and one of the nanomagnets), the population of the neutral, non-conducting, anti-parallel configuration of the TbPc2 spin states leads to a drop in the conductance. In the tilted case, the presence of the spin-flip hopping term proportional to th sin θ in Eq. (III.7.10) stabilises the reduced states with anti-parallel TbPc2 orientations and thus the population of these non-magnetic states no longer leads to a drop in the conductance. 7.7. APPENDIX B: EXACT DIAGONALISATION OF THE M = 2 CASE 102

2 15° 0° 1 20° 5°

) 25° 10° ¢ 0

g(e ¡ - 1

- 2 - 400 - 200 0 200 400

Bz (mT) Figure III.7.12: δg calculated for various tilting angles θ with a field sweeping rate ω = 0.005 s−1 and T = 0.1 K. As θ increases, the spin valve effect is quenched as the reduced states for the anti-parallel configuration of the TbPc2 magnetic moments become stabilised and therefore begin to participate in transport.

7.7 Appendix B: Exact Diagonalisation of the M = 2 case

When M = 2 and doubly charged states of the device are neglected, the Hamilton in Eq (III.7.1) can be diagonalised exactly on the basis |mJ,1, mJ,2i ⊗ |σi (mJ,i = ±J). The adiabatic energies EN for each charge state N are:

E0(mJ,1, mJ,2) = gJ (mJ,1 + mJ,2) µBBz(t) (III.7.12) (1) (2) E1(mJ,1, mJ,2, σ, ±) = (E + E )/2 ± ∆mJ,1,mJ,2,σ + (gJ (mJ,1 + mJ,2) + gσ) µBBz(t)

q 1 (1) (2)
2 2 (i) where the splitting is given by ∆mJ,1,mJ,2,σ = 4 E − E + |th| and E = i − eVg − aimJ,iσ. The eigenstates of HS(t) in the neutral case are product states between the Ising-like spin states of the ter-

± bium nanomagnets and in the charged case are |mJ,1, mJ,2, σ, ±i = |mJ,1, mJ,2i ⊗ [u (mJ,1, mJ,2, σ) |σi1 |0i2 ± ± ± +v (mJ,1, mJ,2, σ) |0i1 |σi2]. The coefficients u and v for a given orientation of the Tb moments mJ,1 and mJ,2 and conduction electron spin σ are

± |th| 1 u (mJ,1, mJ,2, σ) = √ 2∆ p∆ ∓ (E(1) − E(2))/2 (III.7.13) (1) (2) ± ∆ ± (E − E )/2 v (mJ,1, mJ,2, σ) = ∓√ . 2∆p∆ ∓ (E(1) − E(2))/2

(1) (2)
where ∆ = ∆mJ,1,mJ,2,σ and Π = sgn (E − E )/2 ∓ ∆ have been introduced for notational convenience. 103 7.8. APPENDIX C: NATURE OF THE DIRECT RELAXATION PROCESS IN TBPC2

7.8 Appendix C: Nature of the Direct Relaxation Process in TbPc2

The simplest effective crystal field Hamiltonian that captures the energy splitting of the lowest J = 6 spin-orbit multiplet of TbPc2 is

0 2 0  0 4 0 Hcf = αA2hr iO2+β A4hr iO4 (III.7.14) 4 4 4 0 6 0 +A4hr iO4 + γA6hr iO6

k q k where α, β and γ are Stevens parameters, Aq hr i are crystal field parameters and Oq s are Stevens oper- 176 177 ators. In a previous experimental work, TbPc2 nanomagnets deposited on a carbon nanotube in a nanoelectromechanical set-up were shown to hybridise strongly to a longitudinal stretching mode in the

CNT (denoted by the displacement function δu(r)) that induced a rotation δφ = ∇ × δu(r) about the z axis of the TbPc2 nanomagnets resulting in vibronic corrections to the crystal field Hamiltonian given in Eq. (III.7.14). Implementing this strong coupling into our model amounts to calculating the vibronically

0 corrected Hamiltonian H = exp(iδφJz/~)Hcf exp(−iδφJz/~) after which (to first order in δφ) we retrieve 4 the equilibrium Hamiltonian Hcf and a spin-phonon coupling term proportional to gs-phO4 where gs-ph is a coupling constant specific to the underlying substrate.

The resultant spin-phonon coupling term causes transitions between the crystal field states of the TbPc2 nanomagnets that we treat within a Fermi golden rule approach and after standard manipulations21 arrive at the formula ±J→∓J 2 2 4 2 Ω = ± 2 4 |gs-ph| h∓J| O4 |±Ji σm~ cs (III.7.15) ω2 × ∓J,±J 1 − exp(ω∓J,±J /kBT ) where σm = m/A is the mass per unit area of the substrate and cs is the speed of sound in the substrate.

Naturally, if |±Ji were a pure mJ = ±J total angular momentum eigenstate, then the matrix element

4 4 4 appearing in Eq. (III.7.15) would be exactly zero. However, by virtue of the non-axial harmonic βA4hr iO4 term appearing in the equilibrium crystal field Hamiltonian from Eq. (III.7.14), the TbPc2 ground doublet contains some small component of the |mJ = ±2i states, even for non-zero magnetic field, hence the matrix element in Eq. (III.7.15) can be non-zero i.e. the |mJ = ±2i components of the ground doublet can be coupled by the spin-phonon Hamiltonian.

The matrix element in Eq. (III.7.15) can be obtained by first calculating the correction to the |±Ji states

4 4 4 at an applied magnetic field to first order in the crystal field operator βA4hr iO4. The corrections to each 7.8. APPENDIX C: NATURE OF THE DIRECT RELAXATION PROCESS IN TBPC2 104 wavefunction are X hm| βA4hr4iO4 |±6i δ |J = ±6i = 4 4 |mi E±6 − Em m6=±6 ±2 (III.7.16) βA4hr4i Y ≈ 4 pJ(J + 1) − m(m ∓ 1) |±2i 2∆E m=±6 where the energy gap between |±6i and |±2i in the denominator remains roughly unchanged from the case of zero applied field and so we have collected it into the constant term ∆E ≈ 60 meV. By identifying the

2 4
2 4 2 prefactor 2/σm~ cs |gs-ph| h∓J| O4 |±Ji with γsp we arrive at the effective expression for the phonon- mediated magnetic moment reversal rate in TbPc2 reported in Eq. (III.7.6). 1057.9. APPENDIX D: ENERGIES AND AMPLITUDES OF N + 2 STATES DEFINED IN EQ. (III.7.9)

7.9 Appendix D: Energies and Amplitudes of N + 2 states defined

in Eq. (III.7.9)

In this appendix we tabulate the numerical eigenvalues (energies) and the amplitudes of the N + 2 quantum dot singlet states of the M = 2 spin valve device defined in Eq. (III.7.9) which were obtained by diagonalising the Hamiltonian with Bz = 0 given in Eq. (III.7.1).

(i) (i) (i) (i) State E2 (meV) a b c d

(1) ΦJ,J −9.92 0.01 −0.71 −0.71 0.01

(2) ΦJ,J −9.72 0 0.71 −0.71 0

(3) ΦJ,J 490.28 0.71 0 0 −0.71

(4) ΦJ,J 490.48 0.71 0.01 0.01 0.71

(1) Φ−J,J −9.98 0.01 −0.34 −0.94 0.01

(2) Φ−J,J −9.66 0.01 −0.94 0.34 0.01

(3) Φ−J,J 490.28 −0.71 0 0 0.71

(4) Φ−J,J 490.48 0.71 0.01 0.01 0.71

(1) ΦJ,−J −9.98 0.013 −0.94 −0.34 0.01

(2) ΦJ,−J −9.66 0.01 0.34 −0.94 0.01

(3) ΦJ,−J 490.28 0.71 0 0 −0.71

(4) ΦJ,−J 490.48 0.71 0.01 0.01 0.71

(1) Φ−J,−J −9.92 0.01 −0.71 −0.71 0.01

(2) Φ−J,−J −9.72 0 0.71 −0.71 0

(3) Φ−J,−J 490.28 0.71 0 0 −0.71

(4) Φ−J,−J 490.48 0.71 0.01 0.01 0.71 7.10. APPENDIX E: ELECTRON-PHONON COUPLING AND LIMITING CASES OF THE HOPPING INTEGRAL 106

7.10 Appendix E: Electron-Phonon Coupling and Limiting Cases

of the Hopping Integral

The hopping integral th is an effective parameter in our model that represents the transfer of a radical between the Pc ligands of the nanomagnets via the conducting orbitals of the sp2 hybridised substrate.

Notably, we have neglected electron-phonon coupling between this conducting radical and the vibrational modes of the substrate from our discussion. To understand why this is reasonable, we first note that the microscopic electron-phonon coupling Hamiltonian preserves the spin of the electron during scattering events119 and thus to observe a spin-flip of the radical via electron-phonon coupling, a rather large on-site spin-orbit coupling would be required to mix the orbital and spin degrees of freedom of the radical when

1 scattering against a substrate phonon. By modelling the radical as a purely axial spin 2 delocalised in the phthalocyanine ligands of the TbPc2, we have implicitly assumed that any spin-orbit coupling of the radical is negligible and thus spin flips mediated by electron-phonon coupling can not occur. On the other hand, a large spin-orbit coupling does exist in the Tb ion to which the radical spin is exchange coupled. Even here though, spin-phonon mediated flips of the Tb magnetic moment are unable to affect the spin of the radical owing to the orthonormality of the radical states when considering matrix elements of the vibronically corrected Hamiltonian (as in Eq. (III.7.15))

0 0 4 0 4 hmJ , σ | O4 |mJ , σi = hmJ | O4 |mJ i δσ0σ (III.7.17) where σ is the radical spin.

In the final part of this appendix we briefly discuss the limits of this parameter and its effect on transport through the molecular spin valve device. In the main text of this chapter we invoked the strong hopping regime wherein th  ai. The consequences of this choice can most easily be understood by investigating the analytical expressions for the energies and eigenstates of the M = 2 device set-up presented in Eq. (III.7.12) and Eq. (III.7.13). In this case, the splitting of the reduced states is dominated by the hopping since √ √ ± ± ∆mJ,1,mJ,2,σ ≈ |th|. Furthermore, the coefficients u (mJ,1, mJ,2, σ) → 1/ 2 and u (mJ,1, mJ,2, σ) → ∓1/ 2 as th → ∞ such that the radical spin becomes delocalised across both Pc ligands indiscriminately.

The inverse of this case is the th  ai limit in which the exchange coupling between the conduction electron and the 4f electrons of the Tb nanomagnet dominate. It is again instructive to consider the M = 2 device set-up and thus the analytical expressions for the reduced state energies and eigenstates presented in

Eq. (III.7.12) and Eq. (III.7.13). Note that, for 1 = 2 =  and a1 = a2 = a as discussed in the main text,

(1) (2) E − E = −aσ (mJ,1 − mJ,2) such that for the parallel configuration of the nanomagnet moments this 7.10. APPENDIX E: ELECTRON-PHONON COUPLING AND LIMITING CASES OF THE HOPPING 107 INTEGRAL term tends to zero, and for the anti-parallel, it tends to ∓2aσJ. The reduced states with parallel nanomagnet moment configurations therefore maintain the form discussed in the strong hopping limit however the reduced states with anti-parallel moments obtain splittings

2 ! 1 |th| ∆±J,∓J,σ = |aσJ| 1 + + ... . (III.7.18) 2 (aσJ)2

In the limit th → 0, the reduced states with anti-parallel nanomagnet moments are composed of nanomagnet product states with the radical strongly localised to only one of the Pc ligands. In this case, the kinetic spin valve effect that is discussed in the main text is recovered, however, now the origin of the conductance troughs lies in the strongly localised character of the radical spin when the anti-parallel configuration of the

TbPc2 magnetic moments is activated by the spin-phonon mechanism discussed in Appendix C. Chapter 8

Conclusions and Future Work

The terbium nanomagnet molecular spin valve behaves as a nanoscopic analogue of the traditional spin

valve actualised after the discovery of the GMR effect by Fert and Gr¨unberg. While the molecular device has

enjoyed a great deal of experimental characterisation only a handful of theoretical studies in the literature

have sought to offer a microscopic mechanism for the effect. So far, there have been no theoretical studies

of the molecular spin valve that include the interaction of the Tb nanomagnet spin moments with a driving,

time dependent magnetic field which appears to be a crucial component for the operation of the device.

Furthermore, previous theoretical studies of the molecular spin valve device have made strong use of the

coherent Fano-resonance transport regime to explain experimentally observed aspects of the steady-state

differential conductance, namely a finite transport gap in the stability diagram of the device, a claim that

appears at odds with the weak lead-substrate coupling implied by the experimental observation of Coulomb

blockade diamonds in conductance measurements as a function of bias and gate voltage.

In chapter 7, a published manuscript was presented in which the first dynamical simulations of the TbPc2 molecular spin valve in the presence of a driving, longitudinal magnetic field were reported. The hysteresis loops of conductance that are characteristic of the device were shown to originate from the phonon-mediated slow relaxation of the Ising-like magnetic moments of the terbium nanomagnets as they adopted conduct- ing (parallel) and non-conducting (antiparallel) configurations in response to the tracing field. Moreover, experimentally observed trends such as the closing of the loops with increased temperature and the finite transport gap in the steady-state currents were captured by the model. By studying the transport properties of the valve for various TbPc2 chain lengths, the number of TbPc2 units participating in the spin valve was shown to influence the depth of the magneto-conductance troughs thus providing a potential measure of participating nanomagnets in experimental results. Crucially, our simulations of the molecular spin valve

108 109 Conclusions and Future Work device were performed with no reference to the ballistic transport regime implied by the invocation of spin- dependent Fano-resonance transport effects that have been reported in previous theoretical characterisations of the device.

While the molecular spin valve effect originates from the sequential slow-relaxation of the terbium spin moments due to the driving magnetic field, the substrate on which the nanomagnets are grafted plays some role in determining the shape of the characteristic magneto-conductance troughs observed in experiments.

When the device is constructed with graphene nanoconstrictions only a short dip in the magneto-conductance is observed79 however when terbium spin valve experiments are conducted using carbon nanotubes the dips extend for larger values of the field and appear as square lineshapes.80 A possible reason for this discrepancy could be due to the varying acoustic phonon densities of states in each of the substrates thus leading to different slow-relaxation timescales in the nanomagnets. To investigate this hypothesis in a future project, the phonon density of states for each of the substrates may be simulated using ab initio software and incorporated into the Fermi golden rule transition rates that appear in the adiabatic quantum master equations for the Tb spin states.

Furthermore, we found that the sign of the static magneto-resistance is reversed upon scanning a gate voltage across the two different charge resonances N/N+1 and N+1/N+2. This property of the spin valve device has not yet been reported in any experimental works however it may be tested by examining the static magneto-resistance response (as a function of bias voltage) of two adjacent charge resonances in the molecular spin valve stability diagram that also both display the kinetic spin valve effect when probed with a magnetic field. Part IV

Appendices and Bibliography

110 Appendix A: The Many Pictures of

Quantum Mechanics

Quantum mechanics may be presented from several equivalent but different perspectives which are often

referred to as pictures.56, 119, 178 The picture that one utilises to solve a problem in quantum mechanics

relates to the specific time evolution of certain mathematical objects in the theory and any particular choice

that is made often results from taste or from an aversion to carrying lengthy expressions through one’s

derivations. As we have dipped in and out of each picture quite capriciously throughout this thesis, this

appendix serves to consolidate the transformations of common quantum mechanical objects from one picture

to another for ease of reference to the reader.

We begin with the most commonly encountered picture of quantum mechanics whereby the state vector

|ψS (t)i evolves in time according to the Schr¨odingerequation with a (possibly time dependent) Hamiltonian H(t) d |ψ (t)i i S = H(t) |ψ (t)i (A1) ~ dt S

where the “S” subscript has been added to indicate that |ψS (t)i corresponds to the Schr¨odingerpicture. Using the time-ordering operator T , the formal solution to Eq. (A1) is

−i R t dτH(τ) |ψS (t)i = T e ~ t0 |ψS (t0)i = US (t; t0) |ψS (t0)i (A2)

thus defining the time evolution operator US (t; t0) in the Schr¨odingerrepresentation. Observables in this

theory are represented by expectation values of (possibly time dependent) operators AS (t) given by

hAS (t)i = hψS (t)| AS (t) |ψS (t)i . (A3)

111 112

To describe statistical mixtures in the Schr¨odingerrepresentation one defines the density operator

X X † † ρS (t) = wn |ψS,n(t)i hψS,n(t)| = US (t; t0) wn |ψS,n(t0)i hψS,n(t0)| US (t; t0) = US (t; t0)ρS (t0)US (t; t0) n n (A4)

where the n subscript enumerates states appearing with weights wn in the statistical mixture.

The next representation we shall discuss is the Heisenberg picture of quantum mechanics. As physical

observables should be independent of our theoretical formulation, we begin our definition of the Heisenberg

picture with Eq. (A3)

† hψS (t)| AS (t) |ψS (t)i = hψS (t0)| US (t; t0)AS (t)US (t; t0) |ψS (t)i = hψH| AH(t) |ψHi = hAH(t)i (A5)

thereby relating the Heisenberg state wavefunction |ψHi to the Schr¨odinger picture wavefunction |ψS (t0)i and † the Heisenberg operators AH(t) to the same operators in the Schr¨odingerpicture by US (t; t0)AS (t)US (t; t0).

Note that now, the state wavefunction |ψHi is independent of time and the operators AH(t) evolve according to the Heisenberg equation of motion

dAH(t) d  †  −i ∂AS (t) = US (t; t0)AS (t)US (t; t0) = [H(t),AH(t)] + (A6) dt dt ~ ∂t

˙ where use has been made of the Schr¨odingerequation US (t; t0) = (= i/~)H(t)US (t; t0) and its Hermitian conjugate in the second step. Finally, the density operator in the Heisenberg picture can be obtained directly

from Eq. (A4) after noting that ρH = ρS (t0).

The final representation that will be discussed here is the interaction picture (also known as the Dirac

picture). In this picture we assume that the Hamiltonian given in Eq. (A1) can be partitioned in such a way

that H(t) = H0(t) + V (t) where H0 and V may or may not have some time-dependence. We then define the

interaction picture state function |ψI (t)i by the unitary transformation |ψS (t)i = S(t) |ψI (t)i. Substituting this transformation into Eq. (A1) gives

d (S(t) |ψ (t)i) dS(t) d |ψ (t)i i I = i |ψ (t)i + i S(t) I = [H (t) + V (t)] S(t) |ψ (t)i (A7) ~ dt ~ dt I ~ dt 0 I

˙ −iH0t we now choose S(t) such that i~S(t) = H0(t)S(t) (if H0 is time independent then S(t) = e ~ ) thus reducing Eq. (A7) to d |ψ (t)i i I = S†(t)V (t)S(t) |ψ (t)i = V (t) |ψ (t)i . (A8) ~ dt I I I

Again, with the help of the time-ordering operator T , we are able to construct the formal solution to Eq. 113

Quantum Mechanical Pictures Object Schr¨odinger Heisenberg Interaction

Wavefunction |ψS (t)i |ψS (t0)i S(t) |ψS (t)i † † Operator AS (t) US (t, t0)AS (t)US (t, t0) S (t)AS (t)S(t) † Density operator ρS (t) ρS (t0) S (t)ρS (t)S(t)

Table 1: A tabular summary of the relationships of common mathematical objects between the three different pictures of quantum mechanics. We assume the Hamiltonian to be of the form H(t) = H0(t) + V (t) and impose that all three pictures coincide at t = t0.

(A8)

−i R t dτVI (τ) |ψI (t)i = T e ~ t0 |ψI (t0)i = UI (t; t0) |ψI (t0)i (A9)

defining the time evolution operator UI (t; t0) in the interaction picture. Finally, we turn our attention to finding an interaction picture density operator that is consistent with the Schr¨odingerpicture version.

Starting from Eq. (A4) we have

X X † † ρS (t) = wn |ψS,n(t)i hψS,n(t)| = wnS(t) |ψI,n(t)i hψI,n(t)| S (t) = S(t)ρI (t)S (t) (A10) n n

therefore defining the density operator in the interaction picture.

For the convenience of the reader, the key results concerning the representations of these foundational

objects in the different quantum mechanical pictures have been collected in Table 1. Appendix B: Perturbation Theory

Using the T-Matrix Approach

For many interesting many-body systems, the exact spectrum of the Hamiltonian H is too difficult to find

analytically and thus Schr¨odinger’sequation becomes quite insoluble. To circumvent the bottomless despair

induced by this fact one may attempt to partition the Hamiltonian into an exactly solvable part H0 for which the full spectrum is known and a part V that is assumed to only weakly perturb the system. Typically, V

includes interactions between the particles that compose the many-body system such as electron-electron

repulsion terms in weakly interacting electron gases,119 fermionic interactions in superconductors179 or it may even describe the hybridisation between a metal and an impurity.180, 181 In this thesis, we regularly take the perturbation V as a part of the Hamiltonian that hybridises the wavefunctions of the leads and the quantum system

X X ∗ † HT = Tαkσ,naαkσcnσ + H.C. (B1) αkσ n

† (†) where aαkσ are annihilation (creation) operators acting on electrons in the electrodes and cnσ are annihilation (creation) operators acting on the mesostructure. We utilise this Hamiltonian in order to described electron

transport between two electronic leads via a weakly coupled quantum mesostructure. Since we are interested

in transitions between the eigenstates of the non-interacting mesostructure induced by the Hamiltonian HT we regularly invoke the T-matrix perturbation theory to investigate the evolution of the mesostructure to

low order in the perturbation HT . This appendix serves to provide the reader with a formal derivation of the T-matrix if they are initially unfamiliar with the technique.

In order to develop a perturbation theory on V we write H = H0 + V where the spectrum defined by

H0 |φν i = Eν |φν i is considered exactly solved and the state vector |ψ(t)i is evolved in the Schr¨odinger

114 115

picture according to the Schr¨odingerequation

d |ψ(t)i i = H |ψ(t)i = (H + V ) |ψ(t)i . (B2) ~ dt 0

−iH0t Switching to the Interaction picture with the transformation |ψ(t)i = e ~ |ψI (t)i eliminates H0 from Eq. (B2) and thus leads to Eq. (A8). Eq. (A8) can be equivalently written for the evolution operator in the interaction picture by substituting the RHS of |ψI (t)i = UI (t; t0) |ψI (t0)i and cancelling |ψI (t0)i from both sides hence leading to the operator differential equation

dU (t; t ) i I 0 = V (t)U (t; t ). (B3) ~ dt I I 0

Integrating both sides of Eq. (B3) with respect to time from t0 to t leads to the recursive equation

i Z t UI (t; t0) = I − dt1VI (t1)UI (t1, t0) (B4) ~ t0

where UI (t0; t0) = I is the identity operator. We suppose an iterative solution to the recursive relation in

th (B4) and accordingly write the (N + 1) correction to the evolution operator UI (t; t0) as

Z t (N+1) i (N) UI (t; t0) = I − dt1VI (t1)UI (t1, t0) (B5) ~ t0

(0) which, along with the ansatz UI (t; t0) = I, gives rise to the sequence of approximations to UI (t; t0)

(0) UI (t; t0) = I Z t (1) i UI (t; t0) = I − dt1VI (t1) ~ t0 Z t Z t Z t1 (2) i 1 UI (t; t0) = I − dt1VI (t1) − 2 dt1dt2VI (t1)VI (t2) (B6) ~ t0 ~ t0 t0 . .

N  n Z t Z tn−1 (N) X −i UI (t; t0) = dt1 ... dtnVI (t1) ...VI (tn). n=0 ~ t0 t0

By homogenising the integrals in the last line of Eq. (B6) with the time ordering operator T , we see that

the N → ∞ limit corresponds to the full time evolution operator UI (t; t0) defined in Eq. (A9). In practice however, we iterate Eq. (B5) to finite order and use this as an approximation to the full time evolution

operator in the interaction picture.

We are now in good shape to calculate transition rates between the eigenstates of H0 induced by the 116

perturbation V . Consider the probability Pi→f of measuring the system in an eigenstate |φf i of H0 at some time t after initially preparing it in the state |φii. The transition rate between these states is then obtained from the time derivative of this probability

2 dPi→f d 2 d −iH0t d 2 Γ = = |hφ |ψ(t)i| = hφ | e ~ |ψ (t)i = |hφ | U (t; t ) |φ i| . (B7) i→f dt dt f dt f I dt f I 0 i

We now consider the effect of V on the transition rate Γi→f power by power in the time evolution operator

UI (t; t0). Before proceeding we will assume that V is turned on adiabatically from some time in the distant past which can established mathematically with the mapping V 7→ lim V eηt.56 To first order in V then, η→0+ we have

2 2 1 d Z t 1 d Z t (1) i(ωfi−i~η)t1 Γi→f = 2 dt1 hφf | VI (t1) |φii = lim 2 dt1e hφf | V |φii ~ dt −∞ η→0+ ~ dt −∞ ! 2π 2 1 ~η (B8) = |hφf | V |φii| lim + 2 2 ~ η→0 π ωfi + (~η)

2π 2 = |hφf | V |φii| δ(ωfi) ~ where ωfi = (Ef − Ei) /~ and δ(ωfi) is the Dirac delta function. The result given in Eq. (B8) is the famed

Fermi golden rule which provides a first approximation to the rate of transitions between |φf i and |φii induced by V.182 The next order correction follows along the same lines but now we insert a resolution of P identity |φν i hφν | = I in the string of VI (ti) operators thusly ν

2 Z t Z t1 (2) 1 d Γi→f = 2 dt1 dt2 hφf | VI (t1)VI (t2) |φii ~ dt −∞ t0 2 1 d Z t Z t1 X i(ωfν −i η)t1 i(ωνi−i η)t2 = lim hφf | V |φν i hφν | V |φii dt1e ~ dt2e ~ η→0+ 2 dt ~ ν −∞ −∞ 2 ! (B9) 2π hφ | V |φ i hφ | V |φ i 1 2 η X f ν ν i ~ = lim 2 η→0+ ω + iη π ω + (2 η)2 ~ ν iν fi ~ 2 2π X hφf | V |φν i hφν | V |φii = δ(ω ). ω + i0+ fi ~ ν iν

One may continue on in this fashion indefinitely to generate the expression56

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Minerva Access is the Institutional Repository of The University of Melbourne

Author/s: Hymas, Kieran

Title: Quantum Spin Dynamics of Molecular Spintronic Devices Based on Single-molecule Nanomagnets

Date: 2020

Persistent Link: http://hdl.handle.net/11343/243008

File Description: Final thesis file

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