Localization transition in a ballistic quantum wire

The many-body wave function of localized states in one dimension is probed by measuring the tunneling conductance between two parallel wires, fabricated in a $\mathrm{Ga}\mathrm{As}∕\mathrm{Al}\mathrm{Ga}\mathrm{As}$ heterostructure. Tunneling conductance in the presence of a magnetic field perpendicular to the plane of the wires serves as probe of the momentum space wave function of the wires. One of the two wires is driven into the localized regime using a density tuning gate, whereas the other wire, still in the regime of extended electronic states, serves as a momentum spectrometer. As the electron density is lowered to a critical value, the state at the Fermi level abruptly changes from an extended state with a well-defined momentum to a localized state with a wide range of momentum components. The signature of the localized states appears as discrete tunneling features at resonant gate voltages, corresponding to the depletion of single electrons and showing Coulomb-Blockade behavior. Typically 5–10 such features appear, where the one-electron state has a single-lobed momentum distribution, and the few-electron states have double-lobed distributions with peaks at $\pm k_F$. A theoretical model suggests that for a small number of particles $(N<6)$, the observed state is a mixture of ground and thermally excited spin states.

Figure 1

(Color online) A schematic of the measurement setup with cleave plane front, perpendicular to $B$. Depicted: $2\mu \mathrm{m}$ wide top gates ($G_1$, $G_2$, and $G_3$), $20\mathrm{nm}$ thick upper wire at the edge of the 2DEG, $30\mathrm{nm}$ thick lower wire and $6\mathrm{nm}$ insulating AlGaAs barrier. $U_S\left(x\right)\left[U_M\left(x\right)\right]$: Schematic of UW gate-induced potentials for single-mode (multimode) wires, $E_F^U$ is the Fermi energy of the upper wire. Ohmic contact $O_1$ serves as source, $O_{2,3}$ as drains. Density is controlled by gate voltage $V_G$. Two-terminal current is marked by $I$, tunneling current by $I_T$.

Figure 2

(a) Plot of $d{G}_T∕d{V}_G$ vs $V_G$ and $B$ for a single-mode wire. Applied bias $V_{SD}=100\mu \mathrm{V}$ selected to avoid the zero-bias anomaly (Ref. 10), but is small enough to consider tunneling between the Fermi points of both wires. The upper and lower curves are momentum conserving tunneling features $B^{\pm }\left(V_G\right)$. Each curve is accompanied by finite size features marked by “FS”. At low densities LFs appear instead of these curves. $V_G^{*}$ marks the localization transition. (b) UW and LW densities extracted using Eq. (2).

Figure 3

(a) High resolution measurement of $d{G}_T∕d{V}_G$ of localization features for a single-mode wire, $V_{SD}=50\mu \mathrm{V}$, $d{V}_G=300\mu \mathrm{V}$. (b) Same as (a), taken for the second subband in a multimode wire. (c) $\Gamma \left(B\right)\propto {\mid \Psi \left(k\right)\mid }^2$ Extracted from panel (a). (d) $\Gamma \left(B\right)$ of panel (b). (e) $T\left(N\right)$ of panel (a), $N$ is the number of the added electron for each LF. (f) $T\left(N\right)$ of panel (b). $\Gamma$ and $T$ are defined in the text.

Figure 4

$d{I}_T∕d{V}_G$ as a function of gate voltage $V_G$ and source-drain bias $V_{SD}$, taken for the localization features of subband 1 in a multimode wire, at $B=2\mathrm{T}$. The tilted diamond features are due to resonant tunneling through the localized states. A line scan of $d{I}_T∕d{V}_G$ at a bias of $20\mu \mathrm{V}$ is superimposed on the plot. Inset: Second derivative of a Fermi function (line) and data (dots) of the second LF.