# Theoretical Background

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## Abstract

The quantum conductance is one of the most famous novel properties of the single-molecule junction as a one-dimensional conductor.

## Keywords

Transmission Probability Tunneling Electron Differential Conductance Vibration Coupling Ballistic RegimeThe quantum conductance is one of the most famous novel properties of the single-molecule junction as a one-dimensional conductor. The d*I*/d*V* spectra of the single-molecule junction give basic information about the interaction between the transported electron and vibration of the molecule. This chapter presents the theoretical background of these novel physical phenomena of single-molecule junctions.

## 2.1 Electric States in One-Dimensional Conductor

*x*and

*y*directions but infinite in the

*z*direction. The Schrödinger equation can be expressed as in Eq. (2.1).

*z*direction is zero, the system can be described by a free electron model. The three-dimensional potential

*V*(

*R*) can be described as

*V*(

*x, y*). The wave function in the

*z*direction can be described as

*φ*=

*φ*(

*x, y*)exp(

*ik*

_{z}z), and the Schrödinger equation is expressed as in Eqs. (2.2) and (2.3).

*z*direction. On the other hand, the wave function is satisfied by Eq. (2.5) in the

*x*and

*y*directions.

*i*,

*j*is called an electric subband. The density of states of the sub b and

*D*

_{ ij }is expressed as follows, and the density of states can be described as in Fig. 2.1b [1].

*v*

_{ i,j }is the velocity of the electron in the

*i*,

*j*subband. When there are different subbands, the subbands can be called “modes” and “channels.”

## 2.2 Electron Transport in Nano-Sized Conductor

*S*and electrical conductivity

*σ*, as shown in Eq. (2.11).

*l*) and phase-coherence length (

*l*

_{ φ }). The mean free path, which represents the distance traveled by an electron between collisions, is about 100 nm, whereas

*l*

_{ φ }represents the distance within which the phase information is preserved. For gold,

*l*

_{ φ }is on the scale of 1 μm. Therefore, the single-atom and single-molecule junctions discussed in this thesis are smaller than

*l*and

*l*

_{ φ }. In this region, electrons travel ballistically though the conductor, as shown in Fig. 2.2b.

## 2.3 Landauer–Büttiker Formula

*μ*

_{ α }and

*T*

_{ α }represent the chemical potential and temperature, respectively, of the left and right electrodes, and

*k*

_{ B }is the Boltzmann constant. As discussed in the previous section, the wave function can be expressed separately in the longitudinal and transverse directions, and its energy is expressed by Eq. (2.6). The electron transport can be separated into incoming and outgoing states depending on the sample for each electrode. In order to express the incoming and outgoing states, we introduced creation and annihilation operators. For the outgoing states, we expressed the creation and annihilation operators as \(\hat{\text{a}}_{{\upalpha{\text{n}}}} ({\text{E}})\) and \(\hat{\text{a}}_{{\upalpha{\text{n}}}}^{\dag } ({\text{E}})\), respectively. For the incoming states, we expressed the operators as \(\hat{\text{b}}_{{\upalpha{\text{n}}}} ({\text{E}})\), \(\hat{\text{b}}_{{\upalpha{\text{n}}}}^{\dag } ({\text{E}})\), respectively. The index

*n*represents the quantum number to describe the quantum channels. The two operators were connected via the scattering matrix

*S*:

*S*is described using the transmitting (

*t*) and reflecting states (

*r*) as follows:

_{L}× N

_{L,}N

_{R}× N

_{R}, N

_{R}× N

_{L}, and N

_{L}× N

_{R}, for

*r, r’, t,*and

*t’,*respectively. In total, the dimensions of the scattering matrix are (N

_{L}+ N

_{R}) × (N

_{L}+ N

_{R}). The flux conservation and time-reverse symmetry lead to the unique unitary and symmetric character of the scattering matrix. The current operator in the left electrodes can be described using the field operator

*Ψ*

_{L.}

**r**_{ T }is the transverse coordination, and

*z*is the coordination along the leads. \(\upchi_{\text{Ln}} (\varvec{r}_{{\mathbf{T}}} )\) are the transverse wave functions. Because the observable values do not change drastically, especially near the Fermi energy, we can neglect the energy dependence of the quantities. For the condition where the physical quantity does not depend on the energy, the current operator is expressed as follows:

*L*in channel

*n*, and \({\hat{\text{n}}}_{\text{Ln}}^{ + } \left( E \right)\) and \({\hat{\text{n}}}_{\text{Ln}}^{ - } \left( E \right)\) are expressed by Eqs. (2.19) and (2.20), respectively.

*E*′ =

*ħ*Ω + E, and integrating over Ω yields:

*S*,

*S*

_{Lα;mk}(E) is the element of the scattering matrix that relates \({\hat{\text{b}}}_{\text{Lm}} \left( {\text{E}} \right)\) to \({\hat{\text{a}}}_{{\upalpha{\text{k}}}} \left( {\text{E}} \right)\). At equilibrium, the mean square of the annihilation operator is

*t*is the off-diagonal block of the scattering matrix, and

*t*

_{mn}=

*S*

_{RL;mn}. At the limit of 0 K and a small voltage, the electrical conductance

*G*is

*T*

_{ n }, and the mean current and conductance are expressed as follows:

*e*

^{2}/

*h*is referred to as the quantum unit and is expressed as

*G*

_{0,}which has a conductance of about 77 μS or 13 kΩ. If the transmission probability of all transmission channels is one, the conductance is an integer multiple value of

*G*

_{0}.

## 2.4 Electron Transport in a Single-Molecule Junction

*S*

_{1}and

*S*

_{2}are unitary, and the amplitude

*A*of the wave function is given by Eq. (2.31).

*W*is the distance between two barriers, and

*k*is the momentum of the electron. The transmission probability

*T*

_{0}is given by the square of the amplitude.

*T*

_{γ}and

*R*

_{γ}are the transmission and reflection at barriers 1 and 2, respectively, and

*δ*is the declination of

*r*

_{2}and

*r*′.

*T*

_{0}reaches a maximum at

*T*

_{0}reaches a maximum at the resonance position. Now,

*T*

_{1}and

*T*

_{2}are small, and the resonant position shifts from

*φ*=

*φ*

_{0}to

*φ*=

*φ*

_{0}+ δ. The momentum

*k*is expressed by

*k*= [2

*m*(

*E*+

*εφ*)/

*ħ*

^{2}]

^{1/2}, and

*k*shifts to

*k*+ (

*ε/v*

_{F}

*ħ*). The transmission probability

*T*

_{0}is expressed by

*Г*

_{ i }is the probability of the transition when an electron travels from the discrete level to the electron reservoir. The resonance is expressed by the Lorenz curve. The width

*Г*is expressed as the summation of the transition probabilities,

*Г*

_{1}and

*Г*

_{2}. Considering the single-transmission channel, the conductance value is given by

*Г*

_{1}is described by the relaxation time as

*Γ*represents the line width of the energy level after interacting with metals [4], as shown in Fig. 2.5. When the electric coupling increases, the relaxation time decreases, leading to a higher

*Г*, which means that

*Г*represents the electric coupling between the metal and molecule [4]. The Fermi energy is located between the HOMO and LUMO. Because the energy level closest to the Fermi level plays an important role in electron transport, the discrete level that we refer to is either HOMO or LUMO. When the energy difference between the HOMO and

*E*

_{F}is smaller than that between the LUMO and

*E*

_{F}, the carrier is a hole. Otherwise, the carrier is an electron. Equation (2.37) indicates that the smaller energy difference between the molecule’s orbital and

*E*

_{F}and the larger coupling Γ result in a large conductance value in the single-molecule junction.

*G*

^{ R/A }.

*I*and

*H*are the unit matrix and Hamiltonian matrix of the molecule, respectively.

*Γ*is expressed by the self-energy Σ

_{ L(R)}

^{ R/A }.

*Γ*

_{ L(R)}represents the broadening function. For a weak coupling system, Green’s function can be obtained from the zeroth Green’s functions that are calculated separately for the electrodes and the molecule.

*C*

_{ R,k },

*ε*

_{ k }, and

*η*represent the

*k*th molecule’s orbital coefficient at site

*R*, the

*k*th molecule’s orbital energy, and the infinitesimal value, which is determined by the relationship between Green’s function and the density of states (DOS). The term

*CC**represents the orbital distribution of a molecule that is bridged between metal electrodes. The NEGF method has explained the conductance of graphene, single-molecule junctions, and many nano-sized conductors [6]. Equations (2.39)–(2.42) express the physical effects expressed by scattering models. The transmission probability derived by the Green’s function explained the effects of the orbital alignments and the DOS of the metal electrodes.

## 2.5 Coupling Between Electron Transport and Vibration

### 2.5.1 Inelastic Electron Tunneling Spectroscopy

*V*<

*V*

_{k}, the electron tunnels from the left electrode (L) to the right electrode (R). When the electron obtains sufficient energy to excite the vibration (

*eV*

_{k}>

*hω*), the inelastic tunneling channel opens. The opening of this additional channel slightly increases the conductance of the junction. Therefore, the peak structure is observed in the

*d*

^{2}

*I/dV*

^{2}spectrum in the tunneling regime.

*Ψ*

_{ in }and

*Ψ*

_{ fm }are the initial and final states of inelastic scattering, respectively, and

*M*represents the electron–vibration coupling matrix.

### 2.5.2 Point-Contact Spectroscopy

*I*/d

*V*spectrum is called point-contact spectroscopy (PCS). In the ballistic transport regime, energy dispersion is described using a free electron model, as shown in Fig. 2.7a. In the ballistic regime, an electron occupies the states having momentum from –

*k*

_{z}to +

*k*

_{z}without a bias voltage. Note that

*k*

_{z}is the momentum in the direction parallel to the atomic junction. At a bias voltage of

*V*

_{k}, the electron obtains sufficient energy to excite the vibrational mode. Then, the electron transfers from the occupied states with a momentum of

*k*

_{z}to the unoccupied states with a momentum of –

*k*

_{z}, which indicates that electrons are backscattered, leading to a reduction in the conductance value. Therefore, a dip is observed at the vibrational energy in the d

^{2}

*I*/d

*V*

^{2}spectrum in the ballistic regime, as shown in Fig. 2.7b.

### 2.5.3 d^{2} *I*/d*V* ^{2} Spectrum Behavior

*T*,

*w*here we assume a single channel. The scattering approach also explains the limited understanding of the d

*I*/d

*V*spectrum behavior. In the ballistic regime, the ongoing states are fully occupied because almost all electrons travel from the left electrode to the right electrode. At

*V*

_{ k }, the only states to which the electrons can transfer are the unoccupied incoming states, leading to a reduction of the conductance. On the other hand, in the tunneling regime, almost all electrons are reflected, and the incoming states are occupied. At

*V*

_{k}, the only states to which electrons can transfer are unoccupied outgoing states, leading to an increase in the conductance value. Paulsson et al. explained the d

^{2}

*I*/d

*V*

^{2}spectrum behavior depending on the symmetricity (α) of the junction and transmission [9, 10], as shown in Fig. 2.8. When the junction is symmetric (α = 1), the d

*I*/d

*V*response behavior changes from a peak to a dip at

*T*= 0.5. This theory was experimentally verified with H

_{2}O and 1,4-benzenedithiol molecules, both of which have single channels [11, 12].

The d*I*/d*V* spectrum also shows a characteristic shape when the tunneling electron induces the elementary process of a reaction triggered by the vibration excitation [13, 14, 15, 16]. When the vibration coordination is the same as the reaction coordination, the reaction is induced by vibration with a higher energy than the reaction barrier. The hydrogen dissociation reaction of the trans-2-butene molecule on Pd(110) has been observed [14]. When the vibration coordination is not equal to the reaction coordination, the reaction is induced by the coupling between tunneling electrons and vibrational modes along the reaction coordination. In this mechanism, the CO molecule diffuses on the Cu substrate by the injection of the tunneling electrodes [15]. In a single-molecule junction, the transition between two meta-stable structures is observed by the injection of the tunneling electrons, thus exhibiting a peak structure in the d*I*/d*V* spectrum.

When a magnetic impurity exists on the nano-contacts, a peak or dip has been reported at *V* = 0, which is due to the Kondo effect. The Kondo effect is explained in terms of the many-body scattering problem. In a metallic atomic contact, this zero bias anomaly is also observed for Pt atomic junctions.

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