Collective excitation of quantum wires and effect of spin-orbit coupling in the presence of a magnetic field along the wire

The band structure of a quantum wire with the Rashba spin-orbit coupling develops a pseudogap in the presence of a magnetic field along the wire. In such a system spin mixing at the Fermi wave vectors $-k_F$ and $k_F$ can be different. We have investigated, using bosonization techniques, the collective mode of this system, and found that the velocity of this collective excitation depends sensitively on the strength of the Rashba spin-orbit interaction and magnetic field. Our result suggests that the strength of the spin-orbit interaction can be determined from the measurement of the velocity.

Figure 1

The geometry of a quantum wire with a magnetic field along the wire.

Figure 2

Upper figure: Solid lines represent the lowest energy subband structure of the quantum wire in the absence of the Dresselhaus term. (Dashed lines are for zero magnetic field.) Note that the Fermi energy lies in the pseudogap. When a finite value of magnetic field is present bands do anticross (in the figure $B=3\mathrm{T}$). The input parameters are ${\eta }_R=2\times {10}^{-9}\mathrm{eV}\mathrm{cm}$, $m^{*}=0.024m_e$. In this case the numerical value of $g$ is approximately 0.7. Lower figure: The spin-up (solid line) and -down (dashed line) components ${\left(u_k^{-}\right)}^2$ and ${\left(v_k^{-}\right)}^2$ for the lower $E_{-}\left(k\right)$ band. The input parameters are identical with the above figure. Note that ${\left(v_k^{-}\right)}^2=1-{\left(u_k^{-}\right)}^2$.

Figure 3

A Feynman diagram of the $g_4$ process. All four momenta $k_1,k_2,k_2+q,k_1-q$ are located near the right Fermi point. The dotted line indicates the matrix element $V_q$. See text for details.

Figure 4

A Feynman diagrams of the $g_2$ process. See text for details. There is another $g_2$ Feynman diagram in which $R\leftrightarrow L$.

Figure 5

A Feynman diagram of the $g_{1,\parallel }$ processes. There exists one more diagram where $R\leftrightarrow L$. See text.