Many-body interactions in a quantum wire in the integer quantum Hall regime: Suppression of the exchange-enhanced $g$ factor

The collapse of Hall gaps in the integer quantum Hall liquid in a quantum wire is investigated. Motivated by recent experiment [Pallecchi et al., Phys. Rev. B 65, 125303 (2002)] previous approaches are extended to treat confinement effects and the exchange-enhanced $g$ factor in quantum wires. Two scenarios for the collapse of the $\nu =1$ state are discussed. In the first one the $\nu =1$ state becomes unstable at $B_{\mathrm{cr}}^{\left(1\right)}$, due to the exchange interaction and correlation effects, coming from the edge-state screening. In the second scenario, a transition to the $\nu =2$ state occurs at $B_{\mathrm{cr}}^{\left(2\right)}$, with a smaller effective channel width, caused by the redistribution of the charge density. This effect turns the Hartree interaction essential in calculating the total energy and changes $B_{\mathrm{cr}}^{\left(2\right)}$ drastically. In both scenarios, the exchange enhanced $g$ factor is suppressed for magnetic fields lower than $B_{\mathrm{cr}}$. Phase diagrams for the Hall gap collapse are determined. The critical fields, activation energy, and optical $g$ factor obtained are compared with experiments. Within the accuracy of the available data, the first scenario is most probable to be realized.

Figure 1

(Color online.) Energy dispersion curves for the lowest levels of the quantum wire (units of $\hslash {\omega }_c$) as a function of ${\tilde{k}}_x=k_x{l}_0$. The bottom (top) red solid line represents $E_{0,k_x,1}$ $\left(E_{0,k_x,-1}\right)$ and the blue dot horizontal line gives the exact position of $E_F$, when exchange-correlation effects are taken into account in the BS approach. The green dashed curve shows $E_{0,k_x,1}$ obtained within the HFA. The horizontal green dashed line indicates the position of the Fermi level $E_F$ within the HFA. The used parameters are $B=10\mathrm{T}$, $\hslash \Omega =0.65\mathrm{meV}$, and $k_F{l}_0=18.0$ $(W\approx 0.29\mu \mathrm{m})$, which correspond to parameters of sample 1 in Ref. 10. Here $r_0\approx 0.82$, ${\omega }_c∕\Omega \approx 26.6$, $\delta ={10}^{-3}$, and $v_g∕v_g^H\approx 8.5$. As for these conditions the activation gap is negative $(G_a\approx -2.9)$, there is no stable $\nu =1$ QHE state, in agreement with experiment (Ref. 10) and, respectively, here the blue dot line actually indicates the quasi-Fermi level.

Figure 2

(Color online.) Same as in Fig. 1 for the parameters pertinent to sample 2 of Ref. 10: $B=7.3\mathrm{T}$, $\hslash \Omega =0.46\mathrm{meV}$, and $k_F{l}_0=18.0$ $(W\approx 0.34\mu \mathrm{m})$. Now $r_0\approx 0.96$, ${\omega }_c∕\Omega \approx 27.4$, $v_g∕v_g^H\approx 9.45$ and the activation gap is positive $(G_a\approx 1.53>0)$ leading to a stable $\nu =1$ QHE state in the QW, in agreement the experimental result (Ref. 10) in which $G_a\approx 1.0$.

Figure 3

(Color online.) Same as in Fig. 2 for the confining frequency $\hslash \Omega =0.26\mathrm{meV}$, which corresponds to the estimated threshold for sample 2 in Ref. 10. Here ${\omega }_c∕\Omega \approx 48.5$, where $v_g∕v_g^H\approx 16.2$ and $G_a\approx 12.9$.

Figure 4

(Color online.) Dimensionless activation gap $G_a$, or activation gap $G$ in units of $\mid g_0\mid {\mu }_{B}B∕2$, for the $\nu =1$ state, as a function of the HA group velocity ${\tilde{v}}_g^H$, calculated within the BS scheme for $k_F{l}_0=18.0$ and $\delta ={10}^{-3}$. The red solid and green dashed curves correspond to $B=10.0$ and $7.3\mathrm{T}$, respectively. The square mark ($G_a\approx -2.94$, indicating the collapse of the $\nu =1$ state) corresponds to Fig. 1 parameters. The point marked by the circle (triangle) corresponds to Figs. 2 (3) and it $G_a\approx 1.53$ (12.9) clearly predicts the existence of the $\nu =1$ QHE state in this QW sample.

Figure 5

(Color online.) Effective “optical” $g$ factor $g_{\mathrm{op}}^{*}$ as a function of ${\tilde{k}}_x$, for the experimental conditions of Ref. 13: $B=14.0\mathrm{T}$, $\hslash \Omega =4.75\mathrm{meV}$. The red solid, green dashed, blue dotted, and black dash-dotted curves are depicted for $k_F{l}_0=1.83$, 3.0, 3.5, and 4.0 corresponding to linear density $n_L=8.5\times {10}^5$, $1.39\times {10}^6$, $1.63\times {10}^6$, and $1.87\times {10}^6{\mathrm{cm}}^{-1}$, respectively. The values of the $g_{\mathrm{op}}^{*}\approx 18.3$, 23.2 and 23.9 at ${\tilde{k}}_x=0$ for the solid, dashed and dotted curves are very close to the measured value $g_{\mathrm{op}}^{*}\approx 21$. The red solid curve corresponds to effective QW width $W\approx 250\mathrm{\AA }$.

Figure 6

(Color online.) Dimensionless activation gap $G_a$ as a function of ${\tilde{v}}_g^H$ (or ${\Omega }^2∕{\omega }_c^2$) calculated within the BS scheme for $B=14.0\mathrm{T}$ and $\nu =1$. The red solid, green dashed, blue dotted, and black dash-dotted curves were calculated for the same values of $k_F{l}_0$ and $W$ as given in Fig. 5. The square, circle, triangle, and inverse-triangle symbols indicated the values of $G_a$ for $\hslash \Omega =4.75\mathrm{meV}$ (or ${\omega }_c∕\Omega \approx 5.1$), used in Fig. 5. It is seen that the inverse-triangle mark $(G_a=-5.35)$ implies the collapse of the $\nu =1$ state.

Figure 7

(Color online.) Critical curves $\Omega$ vs $B_{\mathrm{cr}}$ for the collapse $(G_a=0)$ of the $\nu =1$ state in the GaAs-based QWs, for two linear densities. The first scenario $\Omega =\Omega \left(B_{\mathrm{cr}}^{\left(1\right)}\right)$, shown by the red solid curves, is calculated within the BS approach. The second scenario $\Omega =\Omega \left(B_{\mathrm{cr}}^{\left(2\right)}\right)$, indicated by the green dashed curves, is evaluated in the HFA from Eq. (17).

Figure 8

(Color online.) Comparison of the second scenario result for the critical curve $\Omega =\Omega \left(B_{\mathrm{cr}}^{\left(2\right)}\right)$ plotted by the red solid curve [from Eq. (17)] with the result from the Kinaret and Lee scenario $\Omega =\Omega \left(B_{\mathrm{cr}}^{\left(2\right),\mathrm{KL}}\right)$ plotted by the green dashed curve [from their Eqs. (12, 13, 14) in which the Hartree interaction is not taken into account (Ref. 7)]. The result of the second scenario with the Hartree interaction discarded, $\Omega =\Omega \left(B_{\mathrm{cr}}^{\left(2\right),F}\right)$, is plotted by the red dotted curve. The linear density for a QW is $n_L=6\times {10}^5{\mathrm{cm}}^{-1}$.

Figure 9

(Color online.) Linear scale plot of the critical curves for the experimental conditions of Ref. 13, with $n_L=8.5\times {10}^5{\mathrm{cm}}^{-1}$. The first scenario result, $\Omega =\Omega \left(B_{\mathrm{cr}}^{\left(1\right)}\right)$, calculated in the BS scheme, is given by the red solid curve and the red dot-dashed curve plots, $\Omega =\Omega \left(B_{\mathrm{cr}}^{\left(1\right),\mathrm{HF}}\right)$, the same curve, however, with neglected correlations, i.e., in the HFA. One can see clearly, by comparing these curves, the role of electron correlations. The green dashed line represents the second scenario result, $\Omega =\Omega \left(B_{\mathrm{cr}}^{\left(2\right)}\right)$, in the HFA. The result of the Ref. 7 model, $\Omega =\Omega \left(B_{\mathrm{cr}}^{\left(2\right),\mathrm{KL}}\right)$, is denoted by the blue dotted curve; in Fig. 9 it will practically coincide with our result, $\Omega =\Omega \left(B_{\mathrm{cr}}^{\left(2\right),F}\right)$, for omitted direct interaction in the second scenario. It can be seen that only the red solid curve, obtained within the first scenario, can explain the observed critical magnetic fields in which the collapse of the $\nu =1$ state occurs for different values of[13] $\Omega$ as the blue dotted curve should be discarded.

Figure 10

(Color online.) Same critical curves, as in Fig. 9, for the sample 1 (sample 2) quantum wire of Ref. 10 with $n_L=7.0\left(6.0\right)\times {10}^6{\mathrm{cm}}^{-1}$ plotted by the lower (upper) the red solid line, $\Omega =\Omega \left(B_{\mathrm{cr}}^{\left(1\right)}\right)$, the green dashed curve $\Omega =\Omega \left(B_{\mathrm{cr}}^{\left(2\right)}\right)$ and the blue dotted line $\Omega =\Omega \left(B_{\mathrm{cr}}^{\left(2\right),\mathrm{KL}}\right)$. Only the first scenario results explain well the experimental observations[10] (see the text).