Roughness scattering induced insulator-metal-insulator transition in a quantum wire

We investigated theoretically the influence of interface roughness scattering on the low-temperature mobility of electrons in quantum wires when electrons fill one or many subbands. We find that the Drude conductance of the wire with length $\mathcal{L}$ first increases with increasing linear concentration of electrons $\eta$ and then decreases at larger concentrations. For small radius $R$ of the wire with length $\mathcal{L}$ the peak of the conductance $G_{\max }$ is below $e^2/h$ so that electrons are localized. The height of this peak grows as a large power of $R$, so that at large $R$ the conductance $G_{\max }$ exceeds $e^2/h$ and a window of concentrations with delocalized states (which we call the metallic window) opens around the peak. Thus, we predict an insulator-metal-insulator transition with increasing concentration for large enough $R$. Furthermore, we show that the metallic domain can be subdivided into three smaller domains: (1) single-subband ballistic conductor, (2) many-subband ballistic conductor, and (3) diffusive metal, and use our results to estimate the conductance in these domains. Finally, we estimate the critical value of $R_c\left(\mathcal{L}\right)$ at which the metallic window opens for a given length $\mathcal{L}$ and find it to be in reasonable agreement with experiment.

Figure 1

The scaling phase diagram of roughness limited electron Drude mobility of a long quantum wire plotted as a function of radius $R$ and linear electron concentration $\eta$ for $d<a_B$ on a log-log scale. Different “phases” or regions are denoted by capital letters. Drude mobility expressions corresponding to these regions are given in Table 1. Region boundaries are given by the equations next to them. The schematic self-consistent electron potential-energy profile along the wire diameter and subbands occupied by electrons are shown for each region. Small arrows show the direction of mobility decrease in each region. The colored areas illustrate where the wire of length $\mathcal{L}$ is metallic. The dark-red, light-red, and pink regions correspond to the single-subband ballistic conductor, many-subband ballistic conductor, and diffusive metal regions for $R_c\left(\mathcal{L}\right)<a_B$, respectively. Electrons are localized in all the colorless regions. The border between them and colored regions is determined by the length of the wire $\mathcal{L}$. We assumed that $\mathcal{L}\sim 1\mu \mathrm{m}$ as is used in quantum devices. For shorter wires $R_c\left(\mathcal{L}\right)$ decreases, and the colored metallic regions expand to cover most of the area of the phase diagram.

Figure 2

The scaling behavior of the dimensionless Drude conductance of a quantum wire with length $\mathcal{L}$ and radius $R$ as a function of the linear electron concentration $\eta$ at different wire radii for $d\ll a_B$ on a log-log scale (full lines). The upper curve corresponds to $R=a_B$ and the lower one is $R={\left(a_{B}d\right)}^{1/2}$. They are obtained from cross sections of the phase diagram in Fig. 1 and the mobilities in Table 1. The dashed line on the upper curve shows the metal-insulator crossover near $\eta a_B=0.5$ induced by electron-electron interactions. We see that for $R=a_B$ the metallic window is open, while for $R={\left(a_{B}d\right)}^{1/2}$ the window is closed. $\mathcal{L}=a_B^{7/2}{\mathrm{\Delta }}^{-2}{d}^{-1/2}$ was chosen.

Figure 3

The scaling phase diagram of roughness limited electron mobility of quantum well at different well width $L$ and 2D electron concentration $n$ for $d\ll a_B$ on a log-log scale. Different “phases” or regions are denoted by Roman numerals. Mobility expressions corresponding to these regions are given in Table 3. Region boundaries are given by the equations next to them. The schematic self-consistent electron potential-energy profile along the $z$ axis of wells and levels (subbands) occupied by electrons are shown for each region. Small arrows show the direction of mobility decrease in each region. Apparently the maximum mobility is achieved in region I. The dashed line indicates schematically the border of the metal-insulator transition (MIT) at small enough $n$. At large $n$ there is no reentrant MIT in spite of the decreasing mobility.

Figure 4

Electron concentration within the nanowire (top) and the corresponding level schematic along the diameter (bottom). Regions of higher concentration correspond to darker shading. (a) Regions B and C of the scaling phase diagram Fig. 1, where all subbands are confined electrostatically forming an accumulation layer of thickness $D$ near the surface. (b) Regions E and F, where the lowest subbands are confined electrostatically, while the top subbands are confined geometrically. (c) Regions H and I, where all the subbands are confined geometrically.

Figure 5

The scaling phase diagram of roughness limited electron mobility of a quantum wire for the variable-radius model (VRM) plotted as a function of radius $R$ and linear electron concentration $\eta$ for $d<a_B$ on a log-log scale. Different phases or regions are denoted by capital letters. Mobility expressions corresponding to these regions are given in Table 4. Region boundaries are given by the equations next to them. The schematic self-consistent electron potential-energy profile along the the wire diameter and subbands occupied by electrons are shown for each region. Small arrows show the direction of mobility decrease in each region. All regions and the borders have the same definitions as Fig. 1, with the exception of a new region ${\mathrm{J}}^{\prime }$ that was previously forbidden. The dark-red, light-red, and pink regions correspond respectively to the single-subband ballistic conductor, many-subband ballistic conductor, and diffusive metal defined by the same conditions as the isotropic model for $\mathcal{L}=a_B^{7/2}{\mathrm{\Delta }}^{-2}{d}^{-1/2}$. We see that, for the same $\mathcal{L}$, the metallic window in the VRM is much smaller than the isotropic model. Electrons are localized in all colorless regions at $T=0$.