Resonant Franz-Keldysh exciton effect in the narrow biased quantum wire subject to a strong magnetic field

We present an analytical investigation of quasi-one-dimensional excitons in thin uniform (single) and double nanoscaled cylindrical quantum wires (UQWR and DQWR) surrounded by a barrier of infinite height and exposed to external electric and strong magnetic fields. The DQWR is formed by inserting an impenetrable longitudinal barrier in a single-quantum wire. Both external fields are directed parallel to the quantum wire (QWR) axis. The radius of the QWRs and the magnetic length are taken to be much less than the exciton Bohr radius. For the dependencies of the positions and widths of the complex quasidiscrete energy levels of the indirect exciton in the DQWR, in which the carriers are separated by the insertion on the confinement, electric field strength and width of the interwire barrier are derived. The confinement (insertion) leads to an increase (decrease) of the exciton binding energy. The impact of the electric field ionization of the exciton is less pronounced for strongly confined and weakly separated carriers. The coefficient of the exciton absorption in the UQWR as a function of the confinement and electric field is calculated in an explicit form. The effect of the confinement and electric field on the exciton peak closely resembles that on the quasidiscrete level of the indirect exciton in the DQWR. Electron-hole attraction increases remarkably the optical Franz-Keldysh electroabsorption in the frequency region below the edge and distant from the exciton peaks. The coefficient of absorption reflecting the electric field ionization and autoionization caused by the coupling between the discrete and continuous exciton states adjacent to the different size-quantized or Landau levels is obtained analytically. A comparison of our analytical results with numerical data is performed. Estimates of the expected experimental values for the parameters of GaAs/GaAlAs QWR show that the autoionized exciton magnetostates in thin biased QWRs are sufficiently stable to be observed.

Figure 1

The DQWR composed of two coaxial QWRs of radius $R$ and quantum disk of height $d$ incorporated in one of the QWRs: QWRs are separated by an insertion of width $D$.

Figure 2

(Color online) The dependence of the binding energy $E_b\left(1\right)$ scaled to the electron Rydberg constant ${\text{Ry}}^{\left(e\right)}$ of the ground exciton state $n=1$ in the DQWR on the confinement $a_B/a_e$ and relative width $D/a_e$ of the insertion for the narrow barrier $\left(D/a_e\le a_B/a_e\right)$. Magnetic regime: $a_B⪡R$.

Figure 3

(Color online) The isoenergetic contours of the binding energies $E_b\left(1\right)/{\text{Ry}}^{\left(e\right)}=0.51,$ 0.59, and 0.69 of the ground exciton state $n=1$ in the DQWR with the narrow barrier $\left(D/a_e\le a_B/a_e\right)$, where $D$ is the width of the insertion, $a_B$ is the magnetic length, and $a_e$ is the electron Bohr radius. Magnetic regime: $a_B⪡R$.

Figure 4

(Color online) The width ${\Gamma }_1$ of the ground exciton state $n=1$ in the DQWR as a function of the dimensionless electric field $s=e{a}_{e}F/{\text{Ry}}^{\left(e\right)}$ and confinement $a_B/a_e$ for the different relative width $\alpha =D/a_e=0.1,0.2$ of the insertion. Magnetic regime: $a_B⪡R$.

Figure 5

(Color online) The relative width ${\Gamma }_1$ of the ground exciton state $n=1$ in the DQWR versus the dimensionless electric field $s=e{a}_{e}F/{\text{Ry}}^{\left(e\right)}$ and insertion width $D/a_e$.

Figure 6

(Color online) The dependence of the width ${\Gamma }^{\left(0\right)}\left(0\right)$ (50) of the ground exciton peak $n=0$ of the ground $l=0$ subband in the UQWR scaled to the exciton Rydberg constant ${\text{Ry}}^{\left(\text{ex}\right)}$ on the dimensionless electric field $s=e{a}_{0}F/{\text{Ry}}^{\left(\text{ex}\right)}$ and confinement $R/a_0$. Quantum wire regime: $R⪡a_B$.

Figure 7

The isowidth contours ${\Gamma }^{\left(0\right)}\left(0\right)/{\text{Ry}}^{\left(\text{ex}\right)}=\alpha$ of the ground exciton peak $n=0$ of the ground $l=0$ subband in the UQWR. See Fig. 6 for units.

Figure 8

(Color online) The ratio of the coefficients of the exciton ${\alpha }^{\left(0\right)}\left(\omega \right)$ and Franz-Keldysh ${\alpha }_{F\text{−}K}^{\left(0\right)}\left(\omega \right)$ electroabsorption calculated from Eq. (55) versus the red energy shift relative to the ground zero intersubband edge scaled to the exciton Rydberg constant ${\text{Ry}}^{\left(\text{ex}\right)}$ for the different electric field strengths $F$, determined by the parameter $c=W_F/{\text{Ry}}^{\left(\text{ex}\right)}$. Confinement parameter is taken to be $R/a_0=0.3$. Quantum wire regime: $R⪡a_B$.

Figure 9

(Color online) The dependence of the relative width $\Gamma \left(0\right)/{\text{Ry}}^{\left(\text{ex}\right)}$ of the ground exciton resonance state $n=0$ caused by both the autoionization and the electric field ionization on the dimensionless electric field $s=e{a}_{0}F/{\text{Ry}}^{\left(\text{ex}\right)}$ and confinement parameter $a_B/a_0$ (${\text{Ry}}^{\left(\text{ex}\right)}$ is the exciton Rydberg constant). The width $\Gamma \left(0\right)$ is calculated from Eqs. (66, 67). The width $\Sigma \left(0\right)={\Gamma }_g\left(0\right)+{\Gamma }_F\left(0\right)$ is the sum of the widths induced by the autoionization $\left[{\Gamma }_g\left(0\right)\right]$ and electric field $\left[{\Gamma }_F\left(0\right)\right]$ ionization calculated from Eqs. (68, 45) for $l=1$, respectively. Magnetic regime: $a_B⪡R$.