Molecular orbital concept on spin-flip transport in molecular junctions

Abstract

The spin-dependent electron transport correlated with spin-flip dynamics in a molecular junction was investigated in the wave-packet and Green’s function approaches. The molecular junction adopted in this work is described by a simple one-dimensional tight-binding chain including a localized spin. The spin exchange coupling J between the localized and conduction electron spins was taken into account through the s-d Hamiltonian. The wave-packet simulations showed that the transmission probabilities in both the spin-flip and no-flip processes show large peaks at the eigenvalues of the spin singlet (−3J/4) and triplet (J/4) states, and that, different transmission properties appear at the mid-gap of the two eigenvalues: the spin-flip process shows a moderate decrease, whereas the no-flip process an abrupt drop. Dividing the s-d Hamiltonian into two submatrices and referring to the molecular orbital concept for the coherent electron transport, we found that the moderate decrease in the spin-flip process at the mid-gap is the result of a coherent-and-cooperative contribution from the singlet and triplet states of the conduction and localized electron spins, and that, the abrupt drop in the no-flip process at the mid-gap is mainly caused by the coherent cancellation from the singlet and triplet states. The molecular orbital concept available for the electron transport including spin-flip scattering processes is described in Green’s function method, in analogy to the one derived for the spinless electron transport.

Appendices

Appendix 1: s-d Hamiltonian

The one-dimensional system adopted in this study is composed of electrodes and a localized electron spin. This system, thus, can be regarded as a metal including a single impurity spin. The Anderson Hamiltonian is an appropriate model to investigate the magnetic property of the metal with impurities, and we use the Anderson Hamiltonian to derive the s-d Hamiltonian, which is the story introduced in the book by Shiba [53]. The Anderson Hamiltonian is written as

$$ H = \sum_{k \sigma} \varepsilon_{k} c^{\dag}_{k \sigma} c_{k \sigma} + \sum_{\sigma} \varepsilon_{d} n_{d \sigma} + U n_{d \uparrow} n_{d \downarrow} + \frac{1}{\sqrt {N_A}} \sum_{k \sigma}\left(V_{k} c^{\dag}_{k \sigma} d_{\sigma} + \hbox{h.c.}\right) . $$

(23)

The first term represents the Hamiltonian of conduction electrons in electrodes, and $\varepsilon_{d}$ in the second term is the impurity d-level. The on-site energy U in the third term represents the coulomb interaction when the impurity d-level is occupied by two electrons. The forth term is the interaction between the orbitals in electrodes and the impurity. $c^{\dag}_{\sigma}$ and ${d^{\dag}_{\sigma}} (c_{\sigma}$ and d _σ) are the creation (annihilation) operators of an electron with spin σ in the electrodes and impurity level, respectively; n _dσ is the number operator, $d^{\dag}_{\sigma} d_{\sigma},$ and N _A is the total number of atoms. To investigate physical properties of the system described with the Anderson Hamiltonian, there are two standard approaches: (1) the third term in Eq. 23 is considered as a perturbation, and (2) the fourth term in Eq. 23 is considered as a perturbation. The latter approach is more suitable for the present one-dimensional model including a single localized spin, that is, the unperturbed Hamiltonian H ₀ and perturbation term H _I are

$$ H_{0} = \sum_{k \sigma} \varepsilon_{k} c^{\dag}_{k \sigma} c_{k \sigma} + \sum_{\sigma} \varepsilon_{d} n_{d \sigma} + U n_{d \uparrow} n_{d \downarrow} , $$

(24)

and

$$ H_{I} = \frac{1}{\sqrt {N_A}} \sum_{k \sigma}\left(V_{k} c^{\dag}_{k \sigma} d_{\sigma} + \hbox{h.c.} \right) , $$

(25)

respectively. The unperturbed Green’s function G ₀(E) and perturbed Green’s function G(E) can be represented as (E − H ₀)⁻¹ and (E − H)⁻¹, respectively. Using the relation $(A-B)^{-1} = A^{-1} + A^{-1}B A^{-1} + A^{-1}B A^{-1}B A^{-1} + \cdots ,$ we straightforwardly obtain the perturbed Green’s function in terms of G ₀ and H _I as

$$ G = G_0 + G_0 H_I G_0 + G_0 H_I G_0 H_I G_0 + G_0 H_I G_0 H_I G_0 H_I G_0 + \cdots. $$

(26)

This is useful to define the effective Hamiltonian of the perturbation term.

Let us firstly consider the system in which the d-level is occupied by an electron and investigate the effects from the interactions between the d-level and conduction levels. The unperturbed states are represented as $d^{\dag}_{\uparrow} | F \rangle$ and $d^{\dag}_{\downarrow} | F \rangle,$ where $| F \rangle$ indicates conduction electrons occupying the levels $\varepsilon_{k}$ up to the Fermi level. When we consider the first-order process H _I (i.e., $V_{k} c^{\dag}_{k \sigma} d_{\sigma})$ with respect to the unperturbed states $d^{\dag}_{\uparrow} | F \rangle$ and $d^{\dag}_{\downarrow} | F \rangle ,$ we can easily find that the expectation values of the first-order process are vanished because of the following relations: $d_{\sigma} | F \rangle = 0$ and $\langle F | d^{\dag}_{\sigma}= 0.$ For example, the first-order term for $d^{\dag}_{\uparrow} | F \rangle$ has the form of $\langle F | d_{\uparrow} c^{\dag}_{k \sigma} d_{\sigma} d^{\dag}_{\uparrow} | F \rangle$ or $\langle F | d_{\uparrow} d^{\dag}_{\sigma} c_{k \sigma} d^{\dag}_{\uparrow} | F \rangle .$ The observation allows us to rewrite the perturbed Green’s function as

$$ G = G_0 + G_0 H_I G_0 H_I G_0 + G_0 H_I G_0 H_I G_0 H_I G_0 H_I G_0 + \cdots. $$

(27)

Defining the second-order term H _I G ₀ H _I as H ^eff _I , we obtain

$$ G = G_0 + G_0 H^{\rm eff}_I G_0 + G_0 H^{\rm eff}_I G_0 H^{\rm eff}_I G_0 + \cdots. $$

(28)

This is the perturbed Green’s function for the Hamiltonian of H ₀ + H ^eff _I . Thus, we consider the second-order processes H _I G ₀ H _I to obtain the explicit form of the effective Hamiltonian. It is to be noted that the unperturbed Green’s function G ₀ is a function of energy E, and the energy dependence in the effective Hamiltonian H _I G ₀ H _I is not convenient for general use. However, the perturbed states in which the energy is close to the unperturbed energy $E_0 (= \sum\nolimits_{k'}^{\rm occ} \varepsilon_{k'} + \varepsilon_d$) are well mixed with the unperturbed state, and the effective Hamiltonian derived from H _I G ₀(E ₀) H _I can be recognized as a reasonable one.

Let us next consider the explicit form of the second-order process. There are two types for the intermediate states in which zero/two electrons occupy the d-level. For the zero occupation case with the initial state of $d^{\dag}_{\uparrow} | F \rangle ,$ we have

$$ \frac{1}{E_0 - H_0} V c^{\dag}_{k \uparrow} d_{\uparrow} d^{\dag}_{\uparrow} | F \rangle = \frac{1}{\varepsilon_d - \varepsilon_k } V c^{\dag}_{k \uparrow} | F \rangle, $$

(29)

where we use the anticommutator relation of Fermions (e.g., $[c_{k}, c^{\dag}_{k'}]_{+} = \delta_{k,k'}$) and the fact $H_0 c^{\dag}_{k \uparrow} d_{\uparrow} d^{\dag}_{\uparrow} | F \rangle = (\sum\nolimits_{k'}^{\rm occ} \varepsilon_{k'} + \varepsilon_k ) c^{\dag}_{k \uparrow} d_{\uparrow} d^{\dag}_{\uparrow} | F \rangle .$ Operating H _I at the left-hand side of Eq. 29, we obtain

$$ \begin{aligned} & V \left(d^{\dag}_{\uparrow} c_{k" \uparrow} + d^{\dag}_{\downarrow} c_{k" \downarrow}\right) \frac{1}{\varepsilon_d - \varepsilon_k } V c^{\dag}_{k \uparrow} | F \rangle \\ &= \frac{V^2}{\varepsilon_d - \varepsilon_k } \left(d^{\dag}_{\uparrow} c_{k" \uparrow}c^{\dag}_{k \uparrow} + d^{\dag}_{\downarrow} c_{k" \downarrow}c^{\dag}_{k \uparrow}\right) | F \rangle \\ &= \frac{V^2}{\varepsilon_d - \varepsilon_k } \left(\delta_{k,k"}d^{\dag}_{\uparrow} -c^{\dag}_{k \uparrow}c_{k" \uparrow} d^{\dag}_{\uparrow} - c^{\dag}_{k \uparrow} c_{k" \downarrow}d^{\dag}_{\downarrow} \right) | F \rangle . \end{aligned} $$

(30)

Since the expectation value of the process is obtained by applying $\langle F | d_{\sigma}$ at the left-hand side of Eq. 30, we readily confirm the second-order processes have non-zero expectation values.

For the initial state of $d^{\dag}_{\downarrow} | F \rangle$ in the zero occupation case, we have

$$ \frac{1}{E_0 - H_0} V c^{\dag}_{k \downarrow} d_{\downarrow} d^{\dag}_{\downarrow} | F \rangle = \frac{1}{\varepsilon_d - \varepsilon_k } V c^{\dag}_{k \downarrow} | F \rangle, $$

(31)

and obtain the following expression by operating H _I at the left-hand side of Eq. 31 as

$$ \begin{aligned} & V \left(d^{\dag}_{\uparrow} c_{k" \uparrow} + d^{\dag}_{\downarrow} c_{k" \downarrow}\right) \frac{1}{\varepsilon_d - \varepsilon_k } V c^{\dag}_{k \downarrow} | F \rangle \\ &= \frac{V^2}{\varepsilon_d - \varepsilon_k } \left(d^{\dag}_{\uparrow} c_{k" \uparrow}c^{\dag}_{k \downarrow} + d^{\dag}_{\downarrow} c_{k" \downarrow}c^{\dag}_{k \downarrow}\right) | F \rangle \\ &= \frac{V^2}{\varepsilon_d - \varepsilon_k } \left(\delta_{k,k"}d^{\dag}_{\downarrow} -c^{\dag}_{k \downarrow}c_{k" \uparrow}d^{\dag}_{\uparrow} - c^{\dag}_{k \downarrow} c_{k" \downarrow}d^{\dag}_{\downarrow} \right) | F \rangle . \end{aligned} $$

(32)

For the doubly occupation case with the initial states of $d^{\dag}_{\uparrow} | F \rangle$ and $d^{\dag}_{\downarrow} | F \rangle,$ we, respectively, have

$$ \begin{gathered} V\left( {c_{{k^{\prime } \uparrow }}^{\dag } d_{ \uparrow } + c_{{k^{\prime } \downarrow }}^{\dag } d_{ \downarrow } } \right)\frac{1}{{\varepsilon _{k} - \varepsilon _{d} - U}}Vd_{ \downarrow }^{\dag } c_{{k \downarrow }} d_{ \uparrow }^{\dag } |F\rangle \hfill \\ = \frac{{V^{2} }}{{\varepsilon _{k} - \varepsilon _{d} - U}}\left( { - c_{{k^{\prime } \uparrow }}^{\dag } c_{{k \downarrow }} d_{ \downarrow }^{\dag } + c_{{k^{\prime } \downarrow }}^{\dag } c_{{k \downarrow }} d_{ \uparrow }^{\dag } } \right)|F\rangle , \hfill \\ \end{gathered} $$

(33)

and

$$ \begin{aligned} & V \left(c^{\dag}_{k" \uparrow}d_{\uparrow} + c^{\dag}_{k" \downarrow}d_{\downarrow} \right) \frac{1}{\varepsilon_k - \varepsilon_d - U} V d^{\dag}_{\uparrow} c_{k \uparrow}d^{\dag}_{\downarrow} | F \rangle \\ &= \frac{V^2}{\varepsilon_k - \varepsilon_d - U } \left( -c^{\dag}_{k" \downarrow}c_{k \uparrow} d^{\dag}_{\uparrow} + c^{\dag}_{k" \uparrow} c_{k \uparrow}d^{\dag}_{\downarrow} \right) | F \rangle. \end{aligned} $$

(34)

From Eqs. 30, 32–34, we obtain the expression of the effective Hamiltonian with respect to $| F \rangle$ as

$$ \begin{aligned} &\frac{V^2}{\varepsilon_d - \varepsilon_k } \left( \delta_{k,k"}d^{\dag}_{\uparrow} -c^{\dag}_{k \uparrow}c_{k" \uparrow} d^{\dag}_{\uparrow} - c^{\dag}_{k \uparrow} c_{k" \downarrow}d^{\dag}_{\downarrow} \right.\\ &\left.+\,\delta_{k,k"}d^{\dag}_{\downarrow} - c^{\dag}_{k \downarrow} c_{k" \downarrow}d^{\dag}_{\downarrow}-c^{\dag}_{k \downarrow}c_{k" \uparrow}d^{\dag}_{\uparrow} \right) \\ &+\,\frac{V^2}{\varepsilon_k - \varepsilon_d - U } \left( -c^{\dag}_{k" \uparrow}c_{k \downarrow} d^{\dag}_{\downarrow} + c^{\dag}_{k" \downarrow} c_{k \downarrow}d^{\dag}_{\uparrow} \right.\\ &\left.-\,c^{\dag}_{k" \downarrow}c_{k \uparrow} d^{\dag}_{\uparrow} + c^{\dag}_{k" \uparrow} c_{k \uparrow}d^{\dag}_{\downarrow} \right) . \end{aligned} $$

(35)

Since this Hamiltonian is derived using the singly occupied d-level (i.e., $d^{\dag}_{\uparrow} | F \rangle$ and $d^{\dag}_{\downarrow} | F \rangle$), the creation operators for the d-level must be replaced with the number operators for general use. The effective Hamiltonian, thus, can be represented as

$$ \begin{aligned} H^{\rm eff}_I &= \frac{1}{N_A}\sum_{k,k"} \left[ \frac{V^2}{\varepsilon_d - \varepsilon_k } \left(-c^{\dag}_{k \uparrow}c_{k" \uparrow} d^{\dag}_{\uparrow}d_{\uparrow} - c^{\dag}_{k \uparrow} c_{k"\downarrow}d^{\dag}_{\downarrow}d_{\uparrow} \right.\right.\\ &\left.\left.-\,c^{\dag}_{k \downarrow} c_{k" \downarrow}d^{\dag}_{\downarrow}d_{\downarrow} -c^{\dag}_{k \downarrow}c_{k" \uparrow}d^{\dag}_{\uparrow}d^{\dag}_{\downarrow} \right)\right.\\ &\left.+\,\frac{V^2}{\varepsilon_k - \varepsilon_d - U } \left( -c^{\dag}_{k" \uparrow}c_{k \downarrow} d^{\dag}_{\downarrow} d_{\uparrow} + c^{\dag}_{k" \downarrow} c_{k \downarrow}d^{\dag}_{\uparrow} d_{\uparrow} \right.\right.\\ &\left.\left.-\,c^{\dag}_{k" \downarrow}c_{k \uparrow}d^{\dag}_{\uparrow}d_{\downarrow} + c^{\dag}_{k" \uparrow} c_{k \uparrow}d^{\dag}_{\downarrow}d_{\downarrow} \right) \right], \end{aligned} $$

(36)

where we omitted several terms that do not include the operators of conduction electrons for simplicity. Using the relations for the number counting and spin-flip operators, $n_{d \sigma} = d^{\dag}_{\sigma}d_{\sigma}, S_{+} = d^{\dag}_{\uparrow}d_{\downarrow},$ and $S_{-} = d^{\dag}_{\downarrow}d_{\uparrow},$ we have

$$ \begin{aligned} H^{\rm eff}_I &= \frac{1}{N_A}\sum_{k,k"} \left[ \frac{V^2}{\varepsilon_d - \varepsilon_k } \left(-c^{\dag}_{k \uparrow}c_{k" \uparrow} n_{d \uparrow} - c^{\dag}_{k \uparrow} c_{k" \downarrow}S_{-} \right.\right.\\ &\left.\left.-\,c^{\dag}_{k \downarrow} c_{k" \downarrow} n_{d \downarrow} -c^{\dag}_{k \downarrow}c_{k" \uparrow}S_{+} \right)\right. \\ &\left.+\,\frac{V^2}{\varepsilon_k - \varepsilon_d - U } \left( -c^{\dag}_{k" \uparrow}c_{k \downarrow} S_{-} + c^{\dag}_{k" \downarrow} c_{k \downarrow} n_{d \uparrow} \right.\right.\\ &\left.\left.-\,c^{\dag}_{k" \downarrow}c_{k \uparrow} S_{+} + c^{\dag}_{k" \uparrow} c_{k \uparrow} n_{d \downarrow} \right) \right]. \end{aligned} $$

(37)

Since the energy difference between $\varepsilon_{k}$ and the Fermi level is smaller than $\varepsilon_{d}$ and $\varepsilon_{d} + U,$ the effective Hamiltonian can be written as

$$ \begin{aligned} H^{\rm eff}_I &= \frac{1}{N_A}\sum_{k,k"} \left[ V^2 \left(-\frac{1}{\varepsilon_d} - \frac{1}{\varepsilon_d + U}\right) \times \left(c^{\dag}_{k \uparrow}c_{k" \uparrow} + c^{\dag}_{k \downarrow}c_{k" \downarrow} \right)( n_{d \uparrow} + n_{d \downarrow} ) \right.\\ &\left.+ V^2 \left(-\frac{1}{\varepsilon_d} + \frac{1}{\varepsilon_d + U}\right) \left(c^{\dag}_{k \uparrow}c_{k" \downarrow} S_{-} + c^{\dag}_{k \downarrow}c_{k" \uparrow} S_{+} \right) \right]. \end{aligned} $$

(38)

Using the condition n _{d ↑} + n _{d ↓} = 1, we reasonably obtain the effective Hamiltonian as

$$ \begin{aligned} H_{I}^{{{\text{eff}}}} & = \frac{1}{{N_{A} }}\sum\limits_{{k,k^{\prime } ,\sigma }} {\frac{{V^{2} }}{2}} \left( { - \frac{1}{{\varepsilon _{d} }} - \frac{1}{{\varepsilon _{d} + U}}} \right)c_{{k\sigma }}^{\dag } c_{{k^{\prime } \sigma }} \\ & \quad + \frac{1}{{N_{A} }}\sum\limits_{{k,k^{\prime } ,\sigma ,\sigma ^{\prime } }} {V^{2} } {\text{ }}\left( { - \frac{1}{{\varepsilon _{d} }} + \frac{1}{{\varepsilon _{d} + U}}} \right)c_{{k\sigma }}^{\dag } {\text{ }}\sigma _{{\sigma ,\sigma ^{\prime } }} c_{{k^{\prime } \sigma ^{\prime } }} \cdot {\mathbf{S}}, \\ \end{aligned} $$

(39)

where $\varvec{\sigma}$ is the Pauli matrix,

$$ \varvec{\sigma}_{\bf x}= \bordermatrix{ & \uparrow & \downarrow \cr \uparrow & 0 & 1 \cr \downarrow & 1 & 0 \cr }, \varvec{\sigma}_{\rm y } = \bordermatrix{ & \uparrow & \downarrow \cr \uparrow & 0 & -i \cr \downarrow & i & 0 \cr}, \varvec{\sigma}_{\bf z} = \bordermatrix{ & \uparrow & \downarrow \cr \uparrow & 1 & 0 \cr \downarrow & 0 & -1 \cr}, $$

(40)

and we use the following relations, S ₊ = S _x  + i S _y and S ₋ = S _x  − i S _y . The first term represents the potential scattering process that are independent of the spin-direction. The second term represents the spin-dependent scattering processes including spin-flip processes. Regarding the term $- 2 V^2 \left(-\frac{1}{\varepsilon_d} + \frac{1}{\varepsilon_d + U}\right)$ as the spin-spin interaction J, we finally obtain the s-d Hamiltonian as

$$ H_{s-d} = -\frac{J}{2N_A}\sum_{k,k",\sigma, \sigma'} c^{\dag}_{k \sigma} \varvec{\sigma}_{\sigma,\sigma'} c_{k" \sigma'} \cdot {\bf S}. $$

(41)

From the s-d Hamiltonian, we easily obtain the matrix elements in the spin sub-space as follows:

$$ {\bf H}_{\bf sd} = \bordermatrix{& \uparrow \Uparrow & \downarrow \Downarrow & \uparrow \Downarrow & \downarrow \Uparrow \cr \uparrow \Uparrow & \frac{J}{4}& 0 & 0 & 0 \cr \downarrow \Downarrow & 0 & \frac{J}{4} & 0 & 0 \cr \uparrow \Downarrow & 0& 0 & -\frac{J}{4} & \frac{J}{2} \cr \downarrow \Uparrow & 0 & 0 & \frac{J}{2} & -\frac{J}{4} \cr}. $$

(42)

It is to be noted that the factor N _A is taken into account through the normalized amplitudes in the wave packets.

Appendix 2: The Crank-Nicholson scheme

The time-dependent Schrödinger equation is

$$ i \frac{\partial \psi}{\partial t} = H \psi , $$

(43)

and the formal solution of the equation is

$$ \psi (x,t) = e^{-iHt}\psi (x,0). $$

(44)

When we introduce the finite grid for time with the interval of $\Updelta t,$ the wave function after the time propagation of $\Updelta t$ can be written as

$$ \psi (x,t + \Updelta t ) = ( 1 - i H \Updelta t) \psi (x,t) . $$

(45)

This is accurate up to the first order in time. However, it is well known that the strategy in Eq. 45 is numerically unstable and the time propagation operator is not unitary. To avoid these difficulties, we consider the time propagation in the reverse direction, t + $\Updelta t \rightarrow t .$ In this case, we have the following relation:

$$ \psi (x, t) = ( 1 + i H \Updelta t) \psi (x,t + \Updelta t) . $$

(46)

Using Eqs. 45 and 46, the wave function at $t + \Updelta t / 2$ is written as

$$ \psi (x,t + \Updelta t / 2 ) = ( 1 - i H \Updelta t / 2) \psi (x,t) $$

(47)

and

$$ \psi (x,t + \Updelta t / 2) = ( 1 + i H \Updelta t / 2) \psi (x, t+ \Updelta t) . $$

(48)

Eliminating the term $\psi (x,t + \Updelta t / 2)$ using Eqs. 47 and 48, we obtain

$$ (1 + i H \Updelta t / 2) \psi (x, t+ \Updelta t) = ( 1 - i H \Updelta t / 2) \psi (x,t), $$

(49)

and thereby

$$ \psi (x, t+ \Updelta t) = \frac{( 1 - i H \Updelta t / 2)}{ ( 1 + i H \Updelta t / 2)} \psi (x,t). $$

(50)

We can easily confirm that the time propagation operator $\frac{( 1 - i H \Updelta t / 2)}{ ( 1 + i H \Updelta t / 2)}$ is unitary and that the time propagation is accurate up to the second order in time using the standard mathematical relations: $e^{A} = 1 + A + \frac{1}{2!} A^2 + \cdots$ and $(1+A)^{-1} = 1 - A + A^2 -A^3 \cdots.$

Appendix 3: The computational conditions for wave-packet simulations

There are two important computational conditions for wave-packet simulations: (1) the energy resolution in the wave-packet simulation must be quite fine in order to compare the results with those from Green’s function method and (2) we have to take care that an artificial reflection of the wave-packet at the left-/right-hand edge of the one-dimensional chain does not affect the true dynamics of the wave-packet scattered by the localized spin. As for the first point, the Gaussian type of wave-packet is adopted in this study, and thereby, the wave-packet is spatially well broadened when the energy of the wave-packet is finely focused on a certain value. This is very important for the calculations of the transmission functions from wave-packet dynamics as a function of energy. We tested several Gaussian wave-packets having different broadening width and consequently found that the wave-packet with broadening width of 1,200 sites or more is sufficient for the calculations of transmission functions and for the comparison with Green’s function results. As for the second point, we have to recognize that the firstly reflected wave-packet at the central spin site will arrive at the left-hand edge of the one-dimensional chain and in turn propagates again toward the spin site. If a portion of the wave-packet is trapped on the spin site (this will be probable for the weak interaction between the electrode and molecule), an artificial superposition between the trapped wave-packet and re-reflected wave-packet occurs. Since we calculate the transmission probabilities from the transmitted wave-packet at the right electrodes, the artificial superposition will cause the artificial errors in transmission functions. We of course take care about the reflection of the transmitted wave-packet at the right-hand edge, too. To avoid the artificial errors caused by the re-reflected wave-packets at the left-/right-hand edge, we simply adopted a large number of sites, 20,000 in this study, and confirmed the model size is sufficient for the present purpose.

As for the unit of simulation time, we determined the unit as follows: (1) we first assumed that each site in electrodes corresponds to a carbon atom, and second that the transfer integral t of 1 corresponds to 2.7 eV, which is a typical value in the carbon 2 p _π network; (2) we calculated the Fermi velocity from the energy band dispersion in the one-dimensional carbon chain, using the carbon--carbon bond distance of 1.4 Å; and (3) we determined the velocity of the right-moving wave packet at the Fermi level (i.e., the unit of simulation time) so as to be identical to the Fermi velocity calculated in Step 2.