E
LECTRON TRANSPORT THROUGH SINGLE
MAGNETIC MOLECULES
E
LECTRON TRANSPORT THROUGH SINGLE
MAGNETIC MOLECULES
Proefschrift
ter verkrijging van de graad van doctor aan de Technische Universiteit Delft,
op gezag van de Rector Magnificus prof. ir. K. C. A. M. Luyben, voorzitter van het College voor Promoties,
in het openbaar te verdedigen op woensdag 28 maart 2012 om 10:00 uur
door
Alexander Sergeyevich Z
YAZIN
Physicist, M.V. Lomonosov Moscow State University, Moskou, Rusland geboren te Obninsk, Sovjet-Unie.
Prof. dr. ir. H. S. J. van der Zant
Samenstelling van de promotiecommissie:
Rector Magnificus voorzitter
Prof. dr. ir. H. S. J. van der Zant Technische Universiteit Delft, promotor Prof. dr. K. Park Virginia Tech, Verenigde Staten
Prof. dr. A. Cornia University of Modena and Reggio Emilia, Italië
Prof. dr. ir. M. R. Wegewijs RWTH Aachen, Duitsland Prof. dr. ir. T. M. Klapwijk Technische Universiteit Delft Prof. dr. ir. L. M. K. Vandersypen Technische Universiteit Delft Dr. J. M. Thijssen Technische Universiteit Delft
Keywords: single-molecule electronics, three-terminal transport, molecular spin-tronics, magnetic anisotropy, spin excitations.
Printed by: Gildeprint Drukkerijen - Enschede
Cover: Photograph of the experimental setup emphasizing rotating sample stage used in experiments described in chapter 6.
Cover design and photo: A. S. Zyazin
Copyright © 2012 by A. S. Zyazin
Casimir PhD Series, Delft-Leiden, 2012-8 ISBN 978-90-8593-119-5
An electronic version of this dissertation is available at .
C
ONTENTS
1 Introduction 1 1.1 Motivation . . . 2 1.2 Molecular electronics . . . 2 1.3 Molecular spintronics . . . 7 1.4 Thesis outline . . . 8 References . . . 9 2 Theoretical background 17 2.1 Three-terminal transport through molecular junctions . . . 182.1.1 Single molecule as a quantum dot . . . 18
2.1.2 Weak coupling: Coulomb blockade and sequential tunneling . 20 2.1.3 Intermediate coupling: elastic and inelastic cotunneling and Kondo effect . . . 24
2.1.4 The role of spin . . . 29
2.2 Single-molecule magnets . . . 32
2.2.1 Superparamagnetism . . . 33
2.2.2 Spin Hamiltonian: zero-field splitting and quantum tunneling of magnetization . . . 34
2.2.3 Beyond the spin Hamiltonian: single-ion levels and interaction 38 2.2.4 An example of SMM: Fe4. . . 40
References . . . 42
3 Device fabrication and measurement setup 49 3.1 Experimental setup . . . 50 3.1.1 Low-temperature insert . . . 50 3.1.2 Sample rotator . . . 51 3.1.3 Measurement electronics . . . 53 3.2 Device fabrication . . . 54 3.2.1 Electron-beam lithography . . . 54
3.2.2 Molecule deposition and electromigration . . . 55
3.3 Measurements technique . . . 56
References . . . 57
4 Magnetic anisotropy in multiple charge states of an Fe4molecule 59 4.1 Introduction . . . 60 4.2 Experimental details . . . 61 4.3 Fitting to a model . . . 65 4.4 Discussion . . . 66 4.5 Supplementary information . . . 67 References . . . 79
5 High-spin Kondo effect and spin transitions 83 5.1 Higher-energy excitations . . . 84
5.2 Spin transition calculations . . . 87
5.3 High-spin Kondo effect . . . 93
References . . . 97
6 Gate-voltage spectroscopy 99 6.1 Measurement and analysis technique . . . 100
6.2 Results and discussion . . . 101
6.3 Numerical calculations . . . 105
6.4 Conclusion . . . 108
References . . . 108
7 Spin blockade in a single-molecule junction 111 7.1 Introduction . . . 112 7.2 Measurements . . . 112 7.3 Model calculations . . . 117 7.4 Discussion . . . 121 References . . . 122 Summary 125 Samenvatting 129 Curriculum Vitae 133 List of Publications 135 Acknowledgements 137
1
I
NTRODUCTION
1.1
M
OTIVATION
T
he idea of using single organic molecules as building blocks for functional elec-tronic devices was first proposed in 1974 by Aviram and Ratner [1]. They sug-gested a molecular system consisting of aπ-conjugated acceptor (tetracyanoquin-odimethane, known in literature as TCNQ) and aπ-conjugated donor (tetrathio-fulvalene or TTF) separated by aσ-bonded (triple methylene) tunneling bridge. In this molecule the TCNQ subunit is electron poor, TTF is electron rich, and the pur-pose of methylene is to prevent theπ-levels of TTF and TCNQ to interact strongly. This structure thus roughly resembles the structure of a conventional semicon-ductor p-n diode [2] with TTF corresponding to the n-doped region, TCNQ to the p-doped region and methylene to the depletion layer. According to Aviram and Ratner’s calculations the response of such a molecule to an applied voltage shows rectifier properties.It took, however, a long time before the technology development allowed this proposal to be verified experimentally. The obvious challenge is to establish robust electrical contacts to a single molecule, which is a nanometer-sized object. It must be said that although significant progress in this area has been made during the last two or three decades, this is still an open problem. A variety of techniques de-scribed in the next section have been suggested and tested in the research labora-tories. Some of them are indeed viable for studying properties of single molecules on chip, but industry standards of yield and reproducibility are still out of reach; commercial single-molecule devices will therefore not appear in the market in the near future. Nevertheless, several fundamental questions of molecular electronics can already be posited. How do the properties of materials change when single molecules are isolated on a surface and attached to the electrodes? How can me-chanical, optical or magnetic properties of single molecules affect electron trans-port and how can electron transtrans-port affect these properties? What functionality can these effects offer? Can we, for instance, store and process information in the single molecules? The work presented in this thesis aims to address some of these questions, in particular relating to the magnetic molecular properties.
1.2
M
OLECUL AR ELECTRONICS
To study electron transport through a single molecule, one first has to fabricate electrical contacts to it. Given the sizes of the molecules which are typically several nanometers, it is necessary to produce electrodes with nanometer-scale separa-tion. This is far beyond the capabilities of conventional mass-production technolo-gies. As of late 2011 the state of the art deep ultraviolet lithography, used in semi-conductor industry, results in a 22 nm node [3]. Further reducing critical dimen-sions requires a transition to extreme ultraviolet lithography [4], which involves
1.2. MOLECULAR ELECTRONICS 3
major technological challenges, requires extremely expensive equipment and still cannot achieve a resolution below 10 nm. Conventional electron-beam lithogra-phy, widely used in the research labs for nanostructure fabrication, is also limited by 10-15 nm due to electron scattering and proximity effects [5]. Hence, it is not surprising that the history of molecular electronics is to a large extent the history of contact fabrication techniques.
The idea of contacting single molecules became feasible after the invention of the scanning tunneling microscope (STM) in the beginning of 1980’s [6]. The STM is based on a tunneling contact between an atomically sharp probe and a conduct-ing sample. It can work in two regimes. In order to image the sample surface a constant tunneling current between the tip and the sample is maintained by a feedback loop during the scanning of the surface. In the other regime the feedback loop is switched off and a current-voltage characteristics is recorded at a given spot of the sample surface. If the metallic substrate is covered by a submonolayer of organic molecules, the tip can be placed on top of a molecule and electrical mea-surements of the current from the tip through the molecule to the surface can be performed [7]. By scanning the surface with a feedback control on, an image of the same molecule can be acquired; this combined inspection is the main advantage of this technique. Apart from conductance measurements the STM can be used as a tool for inelastic tunneling spectroscopy (IETS) [8], where features of d I /dV -curves can be assigned to molecular excitations of vibrational [9] or electronic [10] origin.
The junctions obtained using this approach are inherently asymmetric: the coupling of the molecule to a metallic surface is generally much higher than to the tip. Furthermore, the strong coupling to the surface leads to a broadening and dis-tortion of the molecular orbitals which modifies both imaging and electrical mea-surements. To reduce this effect a thin insulating layer, such as sodium chloride can be introduced between the molecules and the conductive substrate [11]. This tech-nique allows imaging of individual orbitals and improves the resolution of IETS. Spin excitations can then be also resolved in IETS as the insulating layer protects the d -electron states from coupling to the metallic substrate [12].
Another application of a scanning probe technique, the STM break junction, first used by Xu and Tao [13], involves the repeated breaking and formation of a metallic contact between tip and substrate in the presence of molecules in solution. The molecules, with a certain probability, can form a conducting bridge between the metallic surface and the tip when the tip is retracted. The current is recorded at a fixed bias voltage during the retraction and stable conductance values can be attributed to the molecule. This approach is particularly interesting because the molecule is attached to both electrodes (the tip and the surface). Using this tech-nique the conductance as a function of the length of the polymer molecular wire
has been investigated [14]. The current can also be measured as a function of the bias voltage at a fixed length of the junction.
The combination of imaging capabilities and electrical measurements together with the possibility of in situ deposition of the molecules in ultra high vacuum (UHV) makes the STM a powerful tool with a very high degree of control on the measured structures. Remarkably, it can be used even for experiments with single atoms [15–17]. However, the junctions formed by traditional STM approach are in-evitably asymmetric, the stability of the STM break junctions is quite limited and overall, STM-based devices are not scalable: it is difficult to form more than one operational molecular junction on a sample using STM. From this point of view lithography-based approaches, where junctions form a planar structure on a chip are more favourable.
Several approaches have been followed: mechanically controllable break junc-tions (MCBJ) [18–21], electromigration [22–27], electrochemical approaches [28– 30], shadow evaporation [31–34], ’molecular lithography’, using self-assembled monolayers of organic molecules as a mask [35, 36] and ’self-aligned lithography’, using a thick chromium-oxide shadow mask [37–39], to name a few. All of them have their advantages and drawbacks. Here, we will only discuss MCBJ, shadow evaporation and electromigration approaches, which proved to be most suitable for fabrication of single-molecule junctions.
The MCBJ technique is built on a similar principle as the STM break junction discussed earlier. It stems from experiments with notched metallic wires glued on a flexible substrate, bending of which is translated to the stretching of the wire. In their 1992 experiment Muller et al. [40] showed that breaking of the wire caused by such a stretching can lead to a reversible transition from a metallic conductance to a tunneling junction. In 1997 Reed et al. [18] performed a similar experiment, but in the presence of a 1 mM solution of benzene-1,4-dithiol molecules. After evaporation of the solution in an inert atmosphere, they found a gap of 0.7 V in the I (V ) curves, which was interpreted as either a Coulomb blockade signature or a mismatch between the contact Fermi level and the lowest unoccupied molecular orbital (LUMO) of the molecule. The key factor determining the stability of the con-tact or of the tunneling gap is the large reduction factor between the elongation of the piezo element pushing the substrate and electrode separation. In Reed’s exper-iment the spacing between two parts of the wire, set by the piezo voltage, was 8 Å, while the length of the molecule is 8.46 Å. In lithographically defined MCBJ devices, where the two metal electrodes are suspended over a micron-sized trench, the ra-tio of the electrode separara-tion to the piezo elongara-tion can be as low as 10−5[41], leading to a subAngstrom control of the gap size.
If the junctions are broken in a cryogenic vacuum, stable atomically clean metal point contacts result. This makes MCBJ devices an ideal testbed for experiments,
1.2. MOLECULAR ELECTRONICS 5
where simple organic molecules (typically benzene derivatives) are functionalized with different anchoring groups. The results of these tests are very important as the molecules of interest must form strong and electronically transparent chem-ical bonds to the electrodes in order to obtain stable electronic devices. Thiol (SH) [21, 42], amine (NH2) [21] and fullerene (C60) [43] groups have been employed
among others. Although similar experiments can also be done with STM break junctions [44, 45], the results can differ for two techniques. For instance, it was found that while the thiol groups resulted in a spread of molecular conductance in the STM setup [45], this effect was negligible in MCBJ experiments [21, 42]. The spread results from the deformation of electrodes caused by the fact that the sulfur-gold bond is stronger than the sulfur-gold-sulfur-gold bond [46]. In contrast to STM setup such a deformation is negligible in the MCBJ setup because the speed of electrodes re-traction is typically two orders of magnitude smaller in MCBJ.
STM and MCBJ meausurements are typically two-terminal, in which a molecule bridges a gap between two electrodes (source and drain). Introduction of a third electrode, a gate, allows studies in which the molecule can be oxidized and reduced: molecular levels can be brought into and out of resonance with Fermi lev-els of the electrodes and transport through excited and different charge states can be probed [47]. Although a side gate was introduced into an STM setup and sig-natures of Coulomb blockade were measured in transport through single carboran clusters as early as in 1996 [48], these attempts did not receive further develop-ment. A gate electrode has also been introduced in the MCBJ setup [20, 49]. How-ever, these devices suffer from a low gate coupling rendering gating inefficient in a reasonable voltage range. Moreover, it was found that the gate voltage can induce mechanical movement of the source and drain, complicating interpretation of the data [50].
One of the techniques allowing efficient gating of molecular transport is shadow evaporation [31–34]. Gold source and drain electrtodes are evaporated in UHV conditions onto a tilted substrate (which can be a surface of oxidized gate) through a suspended mask. If the tilt angle is high, there is no overlap between source and drain. Reducing the angle decreases the size of the gap. The separation is controlled through an in situ measurement of the tunneling conductance. After the desired gap is obtained, molecules of interest can be deposited into the gap by quench condensation without breaking vacuum. This approach results in clean devices with an efficient gate coupling. However, quench condensation limits the choice of molecules to those with high vapor pressures, like oligophenylenevyni-lene (OPV) derivatives [33, 34].
To date most three-terminal measurements are performed using the electro-migration approach [22–27]. In this approach a thin metallic, typically gold, wire, lithographically defined on top of an oxidized gate, is broken by a current sent
through it [22]. The mechanism, responsible for the breaking is the momentum transfer between conducting electrons and diffusing metal atoms. Electromigra-tion can be done in the soluElectromigra-tion containing the molecules, so that the self-assembly takes place during the breaking. To reduce the risk of violent, abrupt breaking, feedback schemes, monitoring the resistance of the wire in situ, have been intro-duced [24]. Electromigration has several advantages: an efficient gate can be easily fabricated, it is fairly simple compared to the shadow evaporation method, it does not limit the choice of molecules, and the number of junctions that can be fabri-cated on one sample is limited mainly by its surface area. Due to these advantages, the electromigration method has been widely used, however it was found that it can result in the formation of gold nanoparticles in the vicintity of the source-drain gap. These nanoparticles can exhibit Coulomb blockade and the Kondo effect [51], mimicking molecular signatures. Data analysis in principle allows discrimination between samples containing molecules and gold nanoparticles: the latter typically reside on the oxide surface, thus having much stronger gate coupling compared to the ones with a molecule, because the molecule is attached to the source and drain, which partially screen the gate electric field [52]. It was also discovered that electromigration, stopped before the transition to tunneling transport (i.e. a wire with a resistance lower than R0= 1/G0= h/2e2= 12.9 kOhm), induces further
self-breaking of the wire [26]. Such a self-self-breaking prevents nanoparticle formation. Despite the extensive studies to improve its performance, electromigration is still plagued by several drawbacks. First, it is impossible to control the geometry and separation of the electrodes on the atomic scale, while these are key factors de-termining electronic coupling and thus the conductance of the junction [53]. Sec-ond, the yield of electromigration is low: only a few percent of the junctions show spectroscopic molecular signatures. Third, the stability of gold electromigrated junctions is limited: due to the high mobility of gold atoms (which eases elec-tromigration), junctions go through irreversible changes at temperatures higher than about 200 K. The high mobility at high temperatures makes it impossible to perform room temperature measurements or repeated measurements of the same sample after the sample is warmed up. Recently, however, it was found that the use of platinum instead of gold increases stability significantly both at low and room temperatures [27].
Nevertheless, electromigration remains the technique of choice for fabrication of three-terminal single-molecule devices, and it is used, for example, for the fab-rication of all samples discussed in this thesis. Very recently, a similar principle (breaking with a large current) has been applied to fabrication of graphene elec-trodes [54], which is a promising direction in molecular electronics. The differ-ence between the two techniques is that in case of graphene, the mechanism of the breaking is Joule heating in combination with burning, as the presence of oxygen
1.3. MOLECULAR SPINTRONICS 7
is crucial for formation of the gap.
1.3
M
OLECUL AR SPINTRONICS
The field of spintronics, studying active control and manipulation of spin degrees of freedom in solid-state systems, is an established fundamental discipline that found its way to commercial applications in an amazingly short time and holds great expectations for the future [55, 56]. The field of molecular spintronics [57, 58] emerged recently when the development of molecular electronics made possible creation of devices containing single molecules with magnetic properties [59, 60]. Possible applications of these devices include switching [61, 62], high-density data storage [63] and quantum information processing [64, 65]. They can also be used for exploration of a reach fundamental physics of the magnetic molecules: effects like quantum tunneling of magnetization (QTM) [66] are expected to be observed in the electron transport [67, 68]. The variety of molecules that can be employed in these devices include single transition metal atom complexes, single-molecule magnets (SMMs), spin crossover compounds and polyoxometalates(POMs). In this section we briefly review major experimental results in this field.
The first experiments on electron transport through magnetic molecules were reported in 2006 by groups from Delft [59] and from Harvard and Cornell [60]. In these experiments three-terminal devices, containing an individual Mn12 SMM,
were fabricated using electromigration. The results however looked different. In Harvard/Cornell experiment a characteristic zero-field splitting (ZFS) of the ground spin multiplet, caused by the magnetic anisotropy, was observed in four samples as excitations in single-electron tunneling and inelastic cotunneling. The origin of these excitations was confirmed by the measurements in the magnetic field. The value of ZFS varied from 0.25 to 1.34 meV for different samples. Each sample showed two charge states, but ZFS was present only in one of them. In Delft experiment no ZFS was found, however the data exhibited striking features as negative differential conductance and complete current suppression. These effects were explained by the model, taking into account mixing of the two lowest-lying spin multiplets (S = 10 and S = 9 for neutral Mn12molecule). Two years after the
publication of these data it was found that Mn12degrades upon deposition onto a
gold surface [69]. The results of Heersche et al. and Jo et al., however, should not be disproved by this finding as they showed magnetic nature of the observed effects, and although the molecule could undergo some structural changes, the signatures of non-zero spin and magnetic anisotropy were clear.
The next milestone result was published by Cornell group in 2008 [70]. They fabricated three-terminal junctions with endofullerene molecule (N@C60) using
pected to possess magnetic anisotropy. Two charge states with several SET ex-citations were observed. Magnetic field data showed that there was a change of one of the ground states, caused by level crossing at a field of about 6 T. This ob-servation was used to determine the charge and the spin S values for each of the states, starting with an assumption that neutral endofullerene molecule preserves its spin S = 3/2 when deposited on a surface. They found that at zero gate voltage the molecule was twicely reduced (carried two extra electrons) and one electron could be removed by applying a negative voltage to the gate. In the twicely reduced state the spin was S2−= 3/2 and in another charge state at zero field the ground
state spin was S1−= 1. At the field of ∼ 6 T the level crossing occured between
S1−= 1 and S1−= 2 charge states. Thus the data showed for the first time signatures of spin-state transition in a single electron transport. Recently these observations were confirmed by Roch et al. [71], who observed the same transitions in inelastic cotunneling regime.
In 2010 a purely electrically-controlled transition from high-spin (S = 5/2) to low-spin (S = 1/2) in a single Mn2+coordination complex was reported [72]. In this molecule a Mn2+ion is coordinated by two terpydine ligands. The gate voltage induced a reduction of a terpydine moiety which enhanced the ligand field on the Mn ion. An SET transport between these two charge states was suppressed by a spin blockade [73]. Interestingly, that sample also showed strongly gate-dependent singlet-triplet splitting at a low-spin side. The gate-dependence was explained by a difference in the gate coupling for two terpydine moieties, arising from asymmetric nature of an electromigrated junction.
A different kind of device architecture was reported very recently by Urdampil-leta et al. [74]. They presented a spin-valve device in which a gated single-walled carbon nanotube contacted with palladium electrodes, is laterally coupled through supramolecular interactions to TbPc2SMMs, deposited on a surface. The localized
magnetic moments of the SMMs lead to a magnetic field dependence of the elec-tron transport through the nanotube with the measured magnetoresistance values reaching 300% at low bias voltage. The conductance is strongly anisotropic with respect to magnetic field, reflecting anisotropy of TbPc2molecules. This result is
particularly interesting because the spin valve operation is achieved without use of magnetic electrodes, the fabrication of which proved to be extremely challeng-ing [75–77].
1.4
T
HESIS OUTLINE
The content of this thesis is outlined below:
Chapter 2 lays the theoretical foundation for the thesis. It starts with discussion of the three-terminal transport through single molecules in two different regimes:
1.4. THESIS OUTLINE 9
weak and intermediate coupling to the leads. Next, we discuss the interplay be-tween spin properties and electron transport. The following section introduces the physics of single-molecule magnets in the spin Hamiltonian approximation and beyond this model when exchange interaction between constituent ions are taken into account. The chapter finishes with the discussion of properties of an Fe4molecule in the bulk phase, so that these properties can be compared to the
properties of individual molecules studied in the following chapters.
Chapter 3 describes the experimental setup and device fabrication procedures. Electromigration as the technique of establishing single-molecule contacts is dis-cussed in detail. Furthermore, the chapter introduces the measurement methods used in the experiments.
In chapter 4 the effect of electrical gating on the magnetic anisotropy of single Fe4molecules is studied. Characteristic zero-field splitting is observed in
multi-ple charge states and anisotropy parameters are derived. It is found that reduction and oxidation by the gate voltage lead to an enhancement of the anisotropy. More-over, nonlinear Zeeman effect was observed originating from mismatch between anisotropy axis and magnetic field direction. Orientation of the molecule with re-spect to the magnetic field is derived from the fitting to theoretical models.
Chapter 5 discusses higher energy excitations of the samples from the previous chapter. Although the energy scale of these excitations corresponds to the energy of spin transition in the bulk phase, their magnetic field behaviour differs from the trivial Zeeman effect. Calculations of a transport through a model exchange-coupled anisotropic system shows that spin excitations appear in the transport characteristics as a band consisting of several transitions in contrast to the zero-field splitting which appears as a single line. The magnetic zero-field evolution of such a band is found consistent with experimental observations. The chapter finishes with a discussion of two different samples, in which a high-spin Kondo effect was observed.
Chapter 6 presents experiments in which a sample is rotated in the magnetic field. Gate-voltage spectroscopy is introduced as analysis method for determina-tion of anisotropy parameters. This technique based on ground-state-to-ground-state transition observation is found useful when the zero-field splitting cannot be observed. Using this analysis a change of the anisotropy axes orientation was found after the sample rotation.
Chapter 7 concludes the thesis with a discussion of a spin blockade of a cur-rent through an Fe4junction. The blockade in these measurements is slightly
vio-lated and the negative differential conductance (NDC) is observed. This situation is modelled with a five-levels system. A microscopic mechanism of such a blockade is suggested.
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2
T
HEORETICAL BACKGROUND
2.1
T
HREE
-
TERMINAL TRANSPORT THROUGH MOLECU
-L AR JUNCTIONS
2.1.1
S
INGLE MOLECULE AS A QUANTUM DOTS
ingle molecules, objects with nanometer scale dimensions, are essentially quantum-mechanical systems. To get a feel of the energy scales we view them for the time being as small spheres, containing a finite number of electrons (some molecules, such as C60have indeed a spherical shape). In classical electrostaticscharging an isolated metallic sphere with one electron requires an energy
EC= e2/2C = e2/8π²0r, (2.1)
where e is an electron charge, C is a capacitance of the sphere,²0is a dielectric
constant and r is a radius of the sphere. For a diameter 2r = 1 nm, this expression gives a value EC ≈ 1.4 eV. Already at room temperature (T ∼ 300 K) this energy is
much larger than the energy of thermal fluctuations kBT ≈ 25 meV, so the number
of electrons, residing at a molecule is well defined, and its charge is quantized. A spatial confinement of an electron is a well-known in quantum mechanics problem of a particle-in-a-box [1]. The solution yields a discrete set of energy levels that are available for electrons:
Enxnynz= π2 ħ2 2m ( n2x L2x + n2 y L2y + n2 y L2y ), (2.2)
where Lx,y,zare lateral dimensions of a system, nx,y,zare quantum numbers. For
Lx = Ly = Lz = 1 nm this gives a level separation ∆ ∼ π
2ħ2
2mL2 ≈ 0.4 eV. This
en-ergy quantization is a generic phenomenon: it has been also observed in metallic nanoparticles and semiconductor heterostructures. In single molecules this spec-trum is modified by interactions of electrons with nuclei that lead to a formation of molecular orbitals with a level spacing specific for a particular kind of molecules. Furthermore, the spectrum of the molecule trapped in between two metallic elec-trodes is modified compared to a free isolated molecule.
Source and drain electrodes are usually described as bulk metallic electron reservoirs. The number of electrons in these reservoirs is variable as they are con-nected to a bias voltage source. The density of states in the electrodes follows the Fermi-Dirac distribution:
f (E ,µ) = 1
exp[(E − µ)/kBT ] + 1
. (2.3)
Here,µ is the chemical potential of the electrode. At zero temperature and in the absence of a bias voltage the chemical potential by definition equals the Fermi en-ergy: µ = EF. The Fermi-Dirac distribution is then just a step function changing
2.1. THREE-TERMINAL TRANSPORT THROUGH MOLECULAR JUNCTIONS 19
from 1 at energies E < µ to 0 at energies E > µ; all electronic states below Fermi energy are filled and all states above this energy are empty.
Chemical potentials of the electrodes can be shifted by applying a bias voltage Vb, according to
µS− µD= eVb, (2.4)
whereµS(D) is the chemical potential of the source (drain) electrode. In all the
following discussion we assume that the source is grounded as is usually the case in an experiment, i.e.,
VS= 0. (2.5)
In this caseµD= EF+eVbandµS= EF. The inequality between chemical potentials
leads to a difference in the population of the two electrodes and thus to a flow of the electrons (electric current) from source to drain. This flow occurs via electron tunneling. If a small island like a molecule or a nanoparticle is present in a gap be-tweeen two contacts, two tunneling events are needed: first, the electron tunnels from the source to the molecule and second, it is transferred from the molecule to the drain. The tunneling processes are fast and in most cases they can be consid-ered elastic: the energy of electron is conserved during each single tunneling event. The complete transfer of the electron from the source to the drain, however, can be inelastic, with the energy partially absorbed by the molecule, as will be discussed in the following sections.
The current depends both quantitavely and qualitatively on the electronic cou-plings of the molecule to the leadsΓS,D. This parameter characterizes the overlap
of the molecular wavefunction with that of the density of states in the contacts. It can also be introduced as a tunneling rate:
ΓS,D= ħ
τS,D
. (2.6)
HereτS,Dis an average time within which a single electron tunnels from the source
to the molecule/from the molecule to the drain. The total coupling is the sum of couplings to the source and the drain:Γ = ΓS+ ΓD. The coupling leads to a
broad-ening of the molecular energy levels. The levels can also shift compared to an iso-lated molecule due to image charge formation in the contacts and electronic po-larization [2–4]. These two effects have a strong influence on transport properties. Three different regimes are possible: weak (Γ << EC,∆), intermediate (Γ ∼ EC,∆)
and strong (Γ >> EC,∆) coupling. These regimes are discussed in detail in the
2.1.2
W
EAK COUPLING: C
OULOMB BLOCKADE AND SEQUENTIAL TUNNELINGWhen the levels broadening due to interaction with the leads is small (Γ << EC,∆)
one is in the weak coupling limit. In this regime the transport properties are similar to those of a single-electron transistor [5–7]. In these devices a tiny metallic or superconducting island is separated from two electrodes by tunneling barriers of a low transparency. The island is capacitevely coupled to a third electrode, gate, that controls a potential of the island. In molecular devices the island is reduced to a single molecule. The difference in sizes is reflected in the difference in spectra: whereas level separation in the metallic single-electron transistors is typically very small, in the molecules it is not and the separate levels can be distinguished in the transport measurements at relatively high temperatures.
The total charge Q of the molecule is given by
Q = CS(V − VS) +CD(V − VD) +CG(V − VG), (2.7)
where V is the potential of the molecule and VS,Dare the potentials of the source
and the drain, CSis a capacitance to the source, CDto the drain and CGto the gate.
A total capacitance C of the molecule in the three-terminal junction is
C = CS+CD+CG. (2.8)
After substitution (2.8) to Eq. (2.7) we find a potential of the molecule
V =Q C + VS Cs C + VD CD C + VG CG C . (2.9)
The total charge is restricted to integer multiples of the electron charge: Q = Ne. The energy of the system can then be written as a sum of the electrostatic work of adding N electrons to the molecule and the single-particle energies En of all
occupied molecular levels:
U (N ) = −Ne Z 0 V d q + N X n=1 En= (Ne)2 2C − Ne VSCS+ VDCD+ VGCG C + N X n=1 En. (2.10)
The chemical potential is then by definition
µ(N) ≡ U(N) −U(N − 1) = (N − 1/2)e2 C − e
VSCS+ VDCD+ VGCG
C + EN. (2.11)
This derivation is performed within so-called constant interaction model [8]: ca-pacitance values and single-particle levels En are assumed independent of the
2.1. THREE-TERMINAL TRANSPORT THROUGH MOLECULAR JUNCTIONS 21
V
GV
b E / eaddβa
b
N-1
N
N+1
S
D
S
D
μ(N-1) μ(N-1) μ(N) μ(N) Eadd Eadd μ(N+1) μFIGURE2.1: Single-electron transport in the weak coupling regime. (a) Typical d I /dVbversus Vband
VGmap (stability diagram or conductance map) in the absence of excitations. White lines represent peaks in d I /dV due to levels brought into the bias window, in the grey areas d I /dVb= 0. (b) A sketch of SET transport. Source and drain electrodes are represented with their densities of states. Tunneling under the barriers between the electrodes and the molecule is only possible when there is a molecular level present between the Fermi levels of the electrodes.
charge, which is not always a valid assumption for the molecular devices. How-ever, the basic features of transport in the weak coupling regime are described by this model very well and it can always be used as a starting point when analyzing the measurements results.
The mechanism of transport is sketched in the Figure 2.1b. At low bias voltage there is not enough energy to charge the molecule and the current is suppressed: the molecule is in Coulomb blockade. The Fermi levels of the contacts (whose po-sition is controlled by the bias voltage) determine the bias window. Transport is only possible when there is a resonant level of the molecule present in this win-dow. Indeed, the density of states in the source must be nonzero for the energy of the level, otherwise there are no electrons in the source with that energy. The density of states in the drain, however, must be zero to allow tunneling of an elec-tron in, according to Pauli exclusion principle. By application of a gate voltage the molecular levels can be shifted with respect to the contact Fermi levels and thus be brought into the bias window. When the chemical potential of an empty charge state is within the bias window the Coulomb blockade is lifted and current can be measured. At zero bias this condition is
µN +1(VG) = µS= µD. (2.12)
The point at the gate voltage where (2.12) is satisfied is called the degeneracy-point. Figure 2.1a shows a typical transport measurements result represented as a map that plots differential conductance (d I /dVb) versus bias (Vb) and gate (VG)
S
D
S
D
μ(N-1) μ(N-1) μ(N) μ(N) Eexc Eexc Eadd Eadd μ(N+1) μ(N+1)V
GV
b E /eexc E / eaddβa
b
N-1
N
FIGURE2.2: (a)Typical stability diagram with SET excitations present as slanted lines inside SET areas parallel to the diamond edges. (b)A sketch of SET transport through an excited level. Source and drain electrodes are represented with their densities of states. Tunneling through an excited state is only possible when both excited and ground state levels are between the Fermi levels of the electrodes.
voltages (so-called stability diagram). This pattern represents a classical signature of Coulomb blockade.Slanted lines enclose diamond-shaped regions with a zero conductance (Coulomb diamonds). The number of electrons inside the Coulomb diamonds is fixed and every diamond corresponds to a well-defined redox state of the molecule. Going from a negative to a positive gate voltage, one lowers the molecular levels, leading to a consecutive addition of electrons (reduction of the molecule). Going from a positive to a negative gate voltage, the levels are shifted up, electrons are subtracted and the molecule is oxidized. It must be noted, how-ever, that it is not an easy task to tell the exact number of electrons residing on the molecule for each single diamond. The zero gate voltage does not necessar-ily correspond to a neutral state: the molecule can be charged by the electrostatic environment (for instance, defects in the substrate or gate oxide or from another molecule residing in the vicinity of the junction) or due to the interaction with the leads.
The Coulomb diamonds are separated by the regions of single-electron, or se-quential, tunneling (SET), where the current is non-zero. The differential conduc-tance is, however, zero (the current does not vary with the bias voltage) except at the Coulomb diamond edges. The diamond edges mark the onset of SET when the chemical potential of the next charge state matches the the potential of the source (for positively inclined lines) or the drain (for negatively inclined lines): µ(N)(VG) = µS(D)(Vb). At this point the electron can tunnel into the molecule or off
the molecule (see Figure 2.1b), switching the current on. Using (2.11) and (2.5) one can find a general condition for the onset of SET. Forµ(N) = µS(diamond edges
2.1. THREE-TERMINAL TRANSPORT THROUGH MOLECULAR JUNCTIONS 23
with positive slope):
VD= 1 CG+CD (CGVG−(2N − 1)e 2 − ENC e ). (2.13)
Forµ(N) = µD(diamond edges with negative slope):
VD= − 1 CS (CGVG+(2N − 1)e 2 + ENC e ). (2.14)
The slopes of the edges on the conductance map can be used to extract quan-titative information about capacitances. As can be seen from the Eq. (2.13) and (2.14) the positive slope of the edge isα+= CG/(CG+ CD) and the negative slope
isα−= −CG/CS. The gate coupling parameterβ ≡ CG/C = (1/α++ |1/α−|) gives
the ratio between the applied gate voltage and the shift of the molecular levels ∆E: ∆E = βeVG. This expression can be used to determine the addition energy
Ead d≡ µN + 1 − µN , required to add an electron to the molecule: Ead d= βe∆VG,
where∆VG is the distance between two consequtive degeneracy-points.
Alterna-tively Ead d can be found as the distance from the zero-bias axis to the crossing
point of two consequtive diamond, multiplied by an electron charge (see also Fig-ure 2.1a).
At the stability diagram shown in Figure 2.2a there are also lines of non-zero differential conductance, parallel to the diamond edges and terminating on them. These lines mark the onset at which molecular excited states start to contribute to the transport. Here the chemical potential of the excitation matches the poten-tial of the source (for positively inclined lines) or the drain (for negatively inclined lines):µexc(VG) = µS(D)(Vb), and at this point the state becomes accesible for
tun-neling. This situation is shown in Figure 2.2b. The opening of a new tunneling channel leads to an abrupt rise of the current and a peak in d I /dVb. Note that for
excitation to occur it is necessary to have also the ground state in the bias window. That is why the excitation lines are terminating at the diamond edges.
For the SET region separating charge states with N and N + 1 electrons lines terminating at the left edge of the region correspond to the excitations of the charge state N and lines terminating at the right edge to the excitations of the state N + 1. The energy of the excitation Eexccan be read off as the distance from the zero-bias
axis to the crossing point of the ecitation line with a diamond edge (see Figure 2.2a). When the energy spectrum of the molecule in the junction is given, a theoretical description of the transport can be obtained by solving a master equation (also called rate equation) for probabilities of each state’s occupation [9, 10]:
where P is the vector of the occupation probabilities of n levels Pi, i = 1..n and W
is a n × n matrix, whose elements Wi jare rates for transitions j → i for i 6= j , and
Wi i= −P j 6=i
Wj i. These rates can be written as
Wj i= 1 ħ(Γ S i jf S i j+ Γ D i jfi jD), (2.16) Wi j= 1 ħ(Γ S i j(1 − f S i j) + Γ D i j(1 − f D i j)). (2.17)
Here,ΓS(D)i j are the bare rates (determined by the thicknesses of the tunneling barri-ers) for electron tunneling from the molecule in state j into the source (drain), leav-ing the molecule in state i ,µi jare electrochemical potentials of the corresponding
transitions and fi jS(D)are the Fermi functions:
fi jS= 1 1 + exp((µi j+ V /2)/kBT ) , (2.18) fi jD= 1 1 + exp((µi j− V /2)/kBT ) . (2.19)
The steady-state current can then be found as I = eP
i , j
Wi jPi. In the simplest
case all the bare rates are equal to electronic couplings to the leads: ΓS,Di j = ΓS,D.
The current through one level then reads
I =e ħ
ΓSΓD
ΓS+ ΓD
. (2.20)
One consequence of the last equation is thatΓ can be estimated as the full width at half maximum (FWHM) of the conductance peak at the diamond edge when plotted as d I /dVbversus Vb.
2.1.3
I
NTERMEDIATE COUPLING:
EL ASTIC AND INEL ASTIC COTUN-NELING AND
K
ONDO EFFECTThe resonant sequential tunnelling, described in the previous section, is a first-order process inΓ. When the electronic coupling is comparable to the levels spac-ing or the temperature (Γ ∼ ∆,kBT ), not only the SET current rises as can be
ex-pected from Eq. (2.20), but also higher-order tunneling processes become impor-tant. In particular, they lead to a detectable current inside the Coulomb blockade regions. In this section we will discuss three different higher-order processes: elas-tic cotunneling, inelaselas-tic cotunneling and the Kondo effect.
2.1. THREE-TERMINAL TRANSPORT THROUGH MOLECULAR JUNCTIONS 25
S
D
S
D
FIGURE2.3: Elastic cotunneling.
Figure 2.3 shows schematically an elastic cotunneling process. There are no states availanble in the bias window and transport is forbidden in the first order. However, Heisenberg’s energy-time uncertainty principle allows an electron to tun-nel on the molecule from the source, as long as it tuntun-nels off to the drain or back to the source within a very short time period∆t.ħ/(EC+ ∆). This virtual process
conserves an energy and the molecule is in the same state at the end of the process as it was in the beginning. Since this process is elastic, it can occur at arbitrarily low bias voltage and leads to a constant nonzero background conductance in the Coulomb blockade regions.
To estimate the rate of elastic cotunnelingΓelnote first that tunneling from the
molecule to the drain is energetically favourable. The chance for it to occur within the time ħ/(EC+ ∆) is given by ΓDħ/(EC+ ∆). Taking into account the tunneling
rate from source to moleculeΓSwe get
Γel' ΓSΓDħ/(EC+ ∆). (2.21)
ForΓ << EC+ ∆, which is the requirement of the weak coupling regime, Γel<< Γ
and elastic cotunneling is negligible. However, it gives a substantial contribution to the current when this condition is not fulfilled. Note that Eq. (2.21) is only a rough estimation of the elastic cotunneling rate. A rigorous derivation can be found, for instance, in Ref. [11].
Inelastic cotunneling is a virtual tunneling proccess leaving a molecule in an ex-cited state. It is schematically shown in Figure 2.4. An electron from the molecule tunnels to the drain, being immediately replaced by another electron from the source, which occupies the excited state. Relaxation to the ground state takes place subsequently. In contrast to elastic cotunneling this process is suppressed at low bias, since energy is transferred from the electron to the molecule. The onset of inelastic cotunneling occurs at Vb' ∆/e and in stability diagram it appears as a
dia-V
GV
ba b
S
D
S
D
FIGURE2.4: Single-electron transport in an intermediate coupling regime. (a) Typical stability diagram. The horizontal inelastic cotunneling lines inside the Coulomb blockade region connect to slanted SET lines. (b) A sketch of inelastic cotunneling transport via a virtual state.
mond edges. At these points the horizontal line should meet the corresponding SET excitation lines. The SET lines, however, can be indistinguishable due to level broadening if the coupling is too strong. In this case inelastic cotunneling can pro-vide valuable spectroscopic information. The technique basing on the same prin-ciple, called inelastic electron tunneling spectroscopy (IETS) has been widely used in a surface science [12, 13], being recently adopted for STM research of nanoscale objects [14, 15].
The line shape of the cotunneling trace (d I /dVb trace at fixed gate voltage)
can vary from a step [16] to a peak [17](see Figure 2.5). Neglecting the intrinsic linewidth the step shape can be approximated by the following formula [18]:
d I dVb = A e+ Ai · (F µ ∆ − eVb kBT ¶ + F µ ∆ + eVb kBT ¶¸ , (2.22)
where F (x) = [1 + (x − 1)ex] / [ex− 1]2, Aiis a constant prefactor and Aeis the
con-tribution from the elastic cotunneling. This expression gives two steps of width 5.4kBT centered at Vb= ±∆/e.
More frequently in experiments the cotunneling threshold is seen as a conduc-tance peak. This happens due to the fact that cotunneling current is different for the ground state and the excitated state that becomes available above the thresh-old: Iexc6= Ig r. The total inelastic cotunneling current reads I = Pexc(Iexc+ R) +
Pg r(Ig r+ Γi nel), where Pg r (exc)are the occupations of the ground (excited) states,
Γi nel is the rate of inelastic cotunneling and R is the relaxation rate from the
ex-cited to the ground state [11]. This relaxation is not due to the interaction with environment: it is another cotunneling process with a rate comparable toΓi nel.
2.1. THREE-TERMINAL TRANSPORT THROUGH MOLECULAR JUNCTIONS 27
Immediately above the threshold Pexcrises, thus lowering Pg r, which inhibits the
current and results in a decrease of d I /dV . It should be noted here that because of the relaxation even at zero temperature cotunneling steps would have a width of ħR.
Another important higher-order process is the Kondo effect. It was first ob-served in 1930’s in bulk metals where the inclusion of ferromagnetic impurities re-sulted in a conductance minimum at low temperatures [19]. Only thirty years later it was shown by Jun Kondo, that in second order perturbation theory the antiferro-magnetic interaction of conductance electrons with localized spins leads to a diver-gence of the scattering probability for electrons close to the Fermi energy EF [20].
Although such a divergence means inapplicability of the perturbation theory, it can be used to explain the conductance behaviour, and the effect is named after Kondo since then. In 1988 it was shown theoretically that a similar effect in localized states connected to reservoirs by tunnel barriers causes a maximum of the conductance at zero bias due to the increased density of states around EF[21]. Experimentally
the Kondo effect can be observed in semiconductor quantum dots [22, 23] and in single-molecule devices [24, 25] in the intermediate coupling regime.
The Kondo effect is a rich and complicated many-body phenomenon; its theo-retical description requires the use of numerical techniques like perturbative poor-man scaling [26] or non-perturbative numerical renormalization group (NRG) [27]. A review of these methods can be found, for instance, in Ref. [28, 29]. We will re-strict the discussion to the simplified picture that is sketched in Figure 2.6b. When the number of electrons residing on the molecule is odd, the molecule has a non-zero spin, and its ground state is spin-degenerate. At non-zero bias elastic cotunneling can bring an unpaired electron to the drain and replace it with an electron from the source with an opposite spin orientation. Such a spin-flip process is a conduc-tion mechanism responsible for the spin-1/2 Kondo effect. The stability diagram is shown in Figure 2.6a. The zero-bias conductance peak is only present in the diamonds with an odd occupation of the molecule: the ground states for even-occupied states are non-degenerate, all electrons are then paired and there is no
Vb dI/dVb Ai Ae Δ/e Vb dI/dVb Δ/e
a
b
FIGURE2.5: d I /dVb(Vb) trace for inelastic cotunneling. (a) Steps of d I /dV smeared with the temper-ature mark an onset of inelastic cotunneling. The shape of the curve is described by 4.2 (b) A d I /dVb peak due to higher-order processes.
S
D
S
D
V
G
V
ba b
even
odd
even
FIGURE2.6: Kondo effect. (a) A stability diagram. Kondo resonance is observed a zero-bias d I /dV peak in Coulomb diamonds with an odd population. (b) A sketch of Kondo spin-flip.
Kondo effect. Note, however, that this odd-even asymmetry can be violated in case of nontrivial orbital filling, where a high-spin Kondo effect can arise [30]. More-over, the Kondo effect can also occur in case of other types (not spin) of degenarate states like, for example, orbital degeneracy in carbon nanotubes [31].
The energy scale for formation of the Kondo state is given by the Kondo tem-perature TK, that depends on the charging energy and the coupling. For
single-molecule devices the coupling is typically stronger than the one that can be reached in semiconductor quantum dots. Consequently, the Kondo temperature is typically higher: TK∼ 10 − 50 K for molecules and TK.1 K for quantum dots. The
height of the Kondo peak (the zero-bias conductance) has a characteristic temper-ature dependence (Figure 2.7b) [32]:
G(T ) = Gel+G0
4ΓSΓD
ΓS+ ΓD[1 + (2
1/s
− 1(T /TK)2]−s, (2.23)
where Gel is the constant offset due to elastic cotunneling, G0= e2/ħ is the
con-ductance quantum and s is a parameter, depending on the spin S of the dot [33]: s = 0.22 for S = 1/2 and it is less for higher spin values. The maximum conductance is predicted to increase logarithmically with decreasing temperatures and to satu-rate at a value Gel+ 2e2/ħ in case of symmetric lead-dot coupling (unitary limit).
This model is empirical and does not always work well for high spins. There are dif-ferent models proposed [34], but up to now it is not clear, which model should be used for S > 1/2-Kondo effect. All of them, however, predict the same trend of loga-rithmic scaling and saturation at low temperatures T << TKwith less steep curves
and slower saturation for higher spins.
2.1. THREE-TERMINAL TRANSPORT THROUGH MOLECULAR JUNCTIONS 29 dI/dVb Vb 0 a b c dI/dVb log T T FWHM
FIGURE2.7: Characteristic dependencies of the Kondo effect. (a) The zero bias Kondo resonance. Dashed line shows the splitting at high magnetic field. (b) Temperature dependence of the maximum of the zero-bias conductance. (c) Temperature dependence of the full width at half maximum (FWHM).
Lorentzian with a temperature-dependent width (Figure 2.7c) [35]:
F W H M =2 e
q
(πkBT )2+ (2kBTK)2. (2.24)
At low temperatures F W H M ' 2kBTK/e. Note that close to the diamond edges the
molecule is in the so-called mixed-valence regime [32], and the Kondo resonance is broader than in the middle of the diamond.
In a high magnetic field B the Zeeman effect splits the Kondo peak in two com-ponents, separated by twice the Zeeman splitting: ∆Vb= 2g µBB , where g is the
Landé factor, andµBis the Bohr magneton [36]. This splitting is only visible above
some critical field Bc∼ kBTK/2gµB[37]. Such a Zeeman splitting and the
logarith-mic temperature dependence are hallmarks of the Kondo effect.
Finally, when the molecule is in the strong coupling regime (Γ >> EC,∆,kBT ),
the states in the leads and in the molecule are strongly hybridized and transport occurs via elastic coherent tunneling from the source to the drain without stopping on the molecule. The Coulomb blockade signatures are washed out and the current is independent of the gate voltage.
2.1.4
T
HE ROLE OF SPINMolecular excitations mentioned in the previous sections can be of different na-ture: vibrational [38], electronic [39] or magnetic [40]. The latter ones are of special importance for this thesis. As was already discussed, spin is crucial for the emer-gence of the Kondo effect, however its role in quantum dot devices is much broader. Probably the simplest example of spin excitations is a singlet-triplet transition (Figure 2.8), observed in GaAs heterostructures [41], carbon nanotubes [42] and single molecules [40]. In quantum dots with an even occupation the ground state spin configuration is determined by the exchange interaction with characteristic
