Effective spin-orbit couplings in an analytical tight-binding model of DNA: Spin filtering and chiral spin transport

We derive a detailed analytical tight-binding (TB) model for a double helix emulating DNA with one type of nucleotide pair and a single oriented $\pi$ orbital per base. The TB model incorporates both kinetic and intrinsic spin-orbit (ISO) contributions as well as Rashba-type interactions coupled to an external electric field along the axis of the double helix. The helical structure of the molecule renders the ISO first order in the interaction strength (in the meV range) as in carbon nanotubes. The coupling between the ISO and the chirality of the molecule is manifest in the effective coupling parameters while the Rashba coupling is only weakly dependent on structural chirality. A continuum model at half filling is derived where the dispersion is linear around the Fermi level. Spin transport can be completely solved in the case of ISO and the dominant Rashba type term. Spin selectivity is shown to exist for this minimal model (with features similar to recent experimental findings) when the double helix is biased and thus time reversal symmetry is broken. The model also display robustness toward scattering because of the chiral nature of the eigenstates.

Figure 1

A base pair in the $X\text{−}Y$ plane where $\mathrm{\Delta }\varphi$, is the angle between consecutive bases and $\phi$ is a reference angle to the $X$ axis.

Figure 2

Structure of the model molecule in the $XY$ plane, which shows the orientation of the $p_x$ (blue online) and $p_y$ (red online) orbitals on each base. The inset shows the orientation of the $p_z$ (green online) parallel to the helix axis. We also depict the local axes $xy$ in a coordinate system that rotates with the base plane orbitals $p_x$ and $p_y$.

Figure 3

Each set of orbitals at a base site $ı$ has two nearest neighbors on the same DNA strand labeled $ȷ=1$ y $ȷ=2$, and has a neighbor on the opposite strand, which is labeled $ȷ=3$.

Figure 4

The kinetic Hamiltonian is entirely due to $p_z\text{−}{p}_z$ overlap between nearest neighbors. Both intrastrand and interstrand processes are available.

Figure 5

An external electric field along the helix axis couples $p_z$ and $s$ orbitals on the same base, to first order in ${\xi }_{sp}$. The finite overlap $E_{sz}$, due to the chiral structure of the DNA molecule, enables nearest neighbor hopping to the $p_z$ orbital. The process is not spin active (spin indicated by short arrows) and there is no coupling is this kind between DNA strands.

Figure 6

The figure depicts the process involved in coupling $p_z$ orbitals between nearest neighbours, by way of the intrinsic SO interaction and $p_x\text{−}{p}_z$ orbital overlap $E_{xz}$. This process is spin active since the spin (depicted by short arrows) is flipped on transfer.

Figure 7

Lowest-order Rashba coupling as a combination of Stark and SO interactions and $s\text{−}{p}_{x,y}$ overlaps. The process is spin active (spin depicted by short arrows) and linear in ${\xi }_p$ and ${\xi }_{sp}$.

Figure 8

Kinetic energy band vs $k$ in the first Brillouin zone. The Fermi level, chosen at zero energy, corresponds to $k=\pm K$. We have used DNA consistent parameters, i.e., helix radius $a=22.39$ au, pitch $b=66.89$ au, and $\mathrm{\Delta }\varphi =0.6038$ radians. All quantities given in atomic units.

Figure 9

Spectrum in the vicinity of the Fermi level of a helix with $\mathcal{N}=1$ and $M=10$. Two sets of subbands are separated by a SO-dependent gap $\mathrm{\Delta }$ that separates positive (lower-energy) and negative (higher-energy) helicity $\left(s\nu \right)$ states. $\tilde{n}$ runs a full set of values for fixed spin $s$ and $\nu$ labels. Each of the subbands is doubly degenerate. Also indicated is the scale of the curvarure of the levels in units of the kinetic energy $\left|T\right|$. The gap is the source of spin filtering and protects from backscattering, as described in the text.

Figure 10

Lower and higher subbands (each of them Kramers doublets) with positive and negative helicities $s\nu$, respectively. The gap $\mathrm{\Delta }$ separates the highest subband $s\nu =+$ to the lowest subband $s\nu =-$. If time-reversal symmetry is broken by an external bias, one spin direction is chosen with the resulting spin filtering effect, protected by the gap.