Spin transport properties in an organic molecule in the presence of Rashba spin-orbit interaction

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 ¹²C 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 method¹⁵ is used to describe the π bonds formed from 2p_z orbitals. The effect of Hydrogen atoms which make σ bonds with sp² 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 formalism¹⁶ and Hückel method, the Hamiltonian for this system can be written as

in the spin space, where

$Ĥ0$ is the effective Hamiltonian of π-electrons in the absence of Rashba spin-orbit interaction with matrix elements

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

$σ̂x$⁠, and

$\hat{\sigma }_{y}$

$σ̂y$ are the Pauli matrices. The Hamiltonian of Rashba spin-orbit interaction [Eq. (3)], in the spin space, can be written as

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

$p̂x$(≡ −iℏ∂/∂x) and

$\hat{p}_{y}$

$p̂y$(≡ −iℏ∂/∂y) are the x and y components of the electron momentum operator, respectively and

$i=\sqrt{-1}$

$i=−1$⁠. To obtain the matrix form of the operators

$\hat{H}_{Rashba}^{\uparrow \downarrow }$

$ĤRashba↑↓$ and

$\hat{H}_{Rashba}^{\downarrow \uparrow }$

$ĤRashba↓↑$⁠, we use the fact that the basis used in the Hückel formalism are actually 2p_z atomic orbitals of the carbon atoms. Using Eqs. (5) and (6), the matrix elements of operators

$\hat{H}_{Rashba}^{\uparrow \downarrow }$

$ĤRashba↑↓$ and

$\hat{H}_{Rashba}^{\downarrow \uparrow }$

$ĤRashba↓↑$ can be written as

and

where

is the wavefunction of the 2p_z orbital of the carbon atom i located at the position (x_i, y_i, 0) with

where a₀ is the Bohr radius of the hydrogen atom. The retarded self-energy of leads is given by¹⁶ 

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.

$Σ̂p,σR$ is the self energy of an electron with spin σ related to the lead that is connected to the carbon atom number p. Assuming the leads to be one-dimensional, the matrix elements of

$\hat{\Sigma }^{R}_{p,\sigma }$

$Σ̂p,σR$ are given by

and

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

$k≡2mE/ℏ$ represents the wavenumber of the electron in the leads, 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 as¹⁶ 

where

$1$ indicates the unit operator and

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

$ηR=4×10−10eV.m$ and an arbitrary lead coupling constant λ = 0.6. Here 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

$ηR=1.1×10−9eV.m$ as shown in Fig. 3. Here the lead coupling constant is the same as for Fig. 2. This is an important result because in this case, the electron can be transmitted through the molecule only with spin flip from up to down. At the peeks of 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

$\eta _{R}=1.1 \times 10^{-9} \it {eV.m}$

$ηR=1.1×10−9eV.m$⁠, the incoming electron with spin state up can transmit through the molecule with spin state down and vice versa. In this case, the system (molecule and leads) acts as spin inverter. By considering the up and down spin states as 0 and 1 bits, it can be said that the system can be used as a molecular NOT gate. The NOT gate introduced here is a perfect NOT gate because the probability of electron transmission through this gate without the spin inversion is zero (i.e., 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.