In the present letter, we study the spin-dependent electron transport in a 1,4-2-phenyl-dithiolate molecule using the Hückel method and equilibrium Green's function formalism. The effect of the contacts on spin transport is taken into account. It is shown that spin inversion can occur in the presence of the Rashba spin-orbit interaction. The necessary conditions for complete spin inversion are obtained by using contour maps. This study can provide a new route to design molecular spintronic devices.

In the last few decades, there has been a lot of efforts, both theoretically and experimentally, to make the electronic devices as small as possible. An emerging field known as molecular electronics might prove itself as the best candidate for this purpose.^{1} In addition to charge transport, spin transport in molecules which is known as molecular spintronics has been under attention recently.^{2} Organic molecules seem to be suitable candidates for spintronics because of the long spin relaxation time and large spin coherency of electrons in them.^{3} It is due to the fact that the organic molecules are composed of light atoms with small intrinsic spin-orbit interactions. Furthermore, the main element of organic compounds, i.e. carbon atom, has zero nuclear spin for its main isotope ^{12}*C* which results in a very small intrinsic spin-orbit interaction.^{3}

One of the organic molecules used in molecular spintronics is 1,4-n-phenyl-dithiolate which is composed of a linear chain of *n* benzene rings with two sulfur atoms at its ends.^{2} The first measurement of electronic properties of a 1,4-benzene-dithiolate molecule (a 1,4-n-phenyl-dithiolate molecule with *n* = 1) has been performed by Reed *et al.*.^{4} They have shown that in a 1,4-benzene-dithiolate molecule, the current-voltage relation is non-ohmic.

To the best of our knowledge, in the previous works done on molecular spintronics, ferromagnetic leads or single-molecule magnets have been used to manipulate the spin of electrons.^{2,5–13}

In the present letter, we study the spin-dependent transmission properties in a 1,4-2-phenyl-dithiolate molecule consisted of two benzene rings and two sulfur atoms as shown in Fig. 1. Here we use an electrical method, i.e. Rashba spin-orbit interaction, instead of ferromagnetic leads or single-molecule magnets to manipulate the spin of electrons. The use of Rashba spin-orbit interaction has the advantages of being easy to control and maintaining the symmetry between the spin up and spin down, relative to the use of ferromagnetic leads and single-molecule magnets. The molecule is connected to two metallic leads (see Fig. 1) and is placed on an insulating substrate. A perpendicular electric field is applied to the molecule by placing a metallic gate at the top of the structure to control Rashba spin-orbit strength.^{14} The Hückel method^{15} is used to describe the π bonds formed from 2*p*_{z} orbitals. The effect of Hydrogen atoms which make σ bonds with *sp*^{2} orbitals of carbon atoms, is neglected. In addition, the effects of electron-electron and electron-phonon interactions are neglected by considering a single electron transmission at zero temperature.

Using equilibrium Green's function formalism^{16} and Hückel method, the Hamiltonian for this system can be written as

in the spin space, where

Here *i*, *j* = 1, 2, ..., 12 are the lattice site numbers of the carbon atoms as shown in Fig. 1 and α is the on-site energy and β is the hopping energy for π-electrons. In Eq. (1),

is the Hamiltonian of the Rashba spin-orbit interaction, where η_{R} is the Rashba spin-orbit strength which depends on the substrate properties and can also be controlled by the applied gate voltage, and

where

is the space part of the Hamiltonian of Rashba spin-orbit interaction that flips the spin of electron from up to down and

is the space part of the Hamiltonian of Rashba spin-orbit interaction which acts inversely, i.e., flips the spin of electron from down to up. Here

*i*ℏ∂/∂

*x*) and

*i*ℏ∂/∂

*y*) are the

*x*and

*y*components of the electron momentum operator, respectively and

*p*

_{z}atomic orbitals of the carbon atoms. Using Eqs. (5) and (6), the matrix elements of operators

and

where

is the wavefunction of the 2*p*_{z} orbital of the carbon atom *i* located at the position (*x*_{i}, *y*_{i}, 0) with

where *a*_{0} is the Bohr radius of the hydrogen atom. The retarded self-energy of leads is given by^{16}

where *p* = 1, 10 is the lattice site number of a carbon atom which a sulfur atom is connected to it (see Fig. 1) and σ = ↑, ↓ indicates spin direction of the electron.

*p*. Assuming the leads to be one-dimensional, the matrix elements of

and

Note that the indexes μ and ν contain both the lattice site and spin state of the electron. In Eqs. (12) and (13),

*m*is the electron mass,

*a*= 0.142

*nm*is the

*C*−

*C*bond longitude,

*E*is the energy of the electron relative to the bottom of the conduction band of the leads,

*h*≡ 2πℏ is the Planck constant, and λ is the lead coupling constant between the carbon atom located at the lattice site number 1 (10) and the left (right) sulfur atom and via it to the left (right) lead. The retarded Green's function and the spectral function can be written as

^{16}

where

where

is the advanced self-energy. Finally, the transmission coefficient of electron with incoming spin state σ to outgoing spin state σ′ can be determined by

Note that, as shown in Fig. 1, the incoming and outgoing leads are connected via the sulfur atoms to the carbon atoms with lattice site numbers 1 and 10, respectively.

For the numerical calculations, we use α = 0 and β = 2.8 *eV* for the on-site and hopping energies of electron respectively.^{17} Actually, the magnitude of α maybe not zero and it depends on the composition of the leads and cannot be determined exactly. But our calculation shows that the main results obtained in the present study do not show any important change by changing the value of α. Furthermore, the lead coupling constant λ which includes all of the coupling effects of leads and sulfur atoms with carbon atoms of the molecule, cannot be easily obtained.

In Fig. 2, the transmission probabilities *T*_{↑↑} and *T*_{↑↓} are plotted versus electron energy for an arbitrary Rashba coefficient

*T*

_{↑↑}≡

*T*

_{1↑ → 10↑}and

*T*

_{↑↓}≡

*T*

_{1↑ → 10↓}denote the transmission coefficients for an electron with incoming spin state up, without and with spin flip respectively. As expected from an quantum interferometer, in Fig. 2 the transmission is only allowed for certain energy regions. It is seen that although there is a possibility for the electron to undergo spin flip, the transmission coefficient for electron without spin flip is greater, which means that in this state we can not expect a very interesting result

By changing the Rashba coefficient η_{R} via changing the gate voltage, we can obtain the case in which *T*_{↑↑} becomes equal to zero for all electron energy and *T*_{↑↓} becomes equal to one for some electron energies. This can occur for Rashba coefficient

*T*

_{↑↓}(i.e.,

*T*

_{↑↓}= 1), the electron is transmitted without any reflection. Our calculations show that

*T*

_{↓↑}( ≡

*T*

_{1↓ → 10↑}) =

*T*

_{↑↓}and

*T*

_{↓↓}( ≡

*T*

_{1↓ → 10↓}) =

*T*

_{↑↑}. In other words, the transmission coefficient for incoming electron with up and down spin states are equal for both with and without spin flip transmissions. Therefore, at proper value of Rashba spin-orbit strength, that is

*T*

_{↑↑}=

*T*

_{↓↓}= 0). In quantum computation, the efficiency of a NOT gate is an important factor. The efficiency of a perfect NOT gate can be defined as η ≡

*T*

_{↑↓}=

*T*

_{↓↑}. As shown in Fig. 3, efficiency of the NOT gate is nearly 100% at the peaks.

As mentioned before, the value λ = 0.6 for the lead coupling constant is chosen arbitrarily. We show that the (main) results obtained above is independent of the value of the lead coupling constant λ. In Fig. 4, the transmission coefficients *T*_{↑↑} and *T*_{↑↓} are shown for two different lead coupling constants, e.g., (a) for low coupling constant λ = 0.3 and (b) for high coupling constant λ = 1. It is observed that although there are some differences between the graphs of *T*_{↑↓} in Fig. 3 and Fig. 4, the spin-inversion with probability near 1, can take place regardless of the value of the lead coupling constant λ.

In order to obtain proper values of Rashba coefficient η_{R} for spin-inverter with high efficiency, in Fig. (5) the contour map for transmission coefficient *T*_{↑↓} is plotted versus the Rashba coefficient η_{R} and the electron incident energy *E*. Again, we mention that the contour map of *T*_{↓↑} is the same as the contour map of *T*_{↑↓}. It is observed that, the spin inversion takes place at six regions I-VI.

The value of the Rashba coefficient used in the present work might seem to be rather large, but recently Rashba coefficient of this order are realized.^{18,19}

In the present letter, the spin transport properties in a 1,4-2-phenyl-dithiolate molecule have been studied. The Rashba spin-orbit interaction which can be controlled by applied electric field has been used to manipulate the spin of electron. It has been shown that, for proper values of Rashba coefficient, the electron can transmit through the molecule with perfect spin inversion. In this case the system (the molecule and leads) can be used as a perfect spin-inverter or NOT spin gate with high efficiency (about 100% near the transmission peaks). The above results may provide a new way to design molecular spintronics devices which can be used in quantum computation.