Energy-level pinning and the 0.7 spin state in one dimension: GaAs quantum wires studied using finite-bias spectroscopy

We study the effects of electron-electron interactions on the energy levels of GaAs quantum wires (QWs) using finite-bias spectroscopy. We probe the energy spectrum at zero magnetic field, and at crossings of opposite-spin-levels in high in-plane magnetic field $B$. Our results constitute direct evidence that spin-up (higher energy) levels pin to the chemical potential $\mu$ as they populate. We also show that spin-up and spin-down levels abruptly rearrange at the crossing in a manner resembling the magnetic phase transitions predicted to occur at crossings of Landau levels. This rearranging and pinning of subbands provides a phenomenological explanation for the 0.7 structure, a one-dimensional (1D) nanomagnetic state, and its high-$B$ variants.

Figure 1

(Color online) Evolution and crossings of 1D subbands in $B$. (a) Grey-scale diagram of $dG∕d{V}_g$ as a function of $V_g$ for $B=0$ to $16\mathrm{T}$ in increments of $0.2\mathrm{T}$. White represents conductance plateaus, and dark lines correspond to a subband populating. Right-moving (left-moving) lines are higher energy spin-up (lower energy spin-down) subbands. We refer to the three symbols later in the paper. (b) A close up of the $1\uparrow$ and $2\downarrow$ crossing. The trajectory of $1\uparrow$ is discontinuous, with its two parts overlapping in $B$. A and C indicate the nonquantized analog and complement structure, respectively, shown in (c) Conductance traces from $5.8\mathrm{T}$ (left) to $13\mathrm{T}$, covering the range of figure (b). The analog A (a variant of the 0.7 structure) and the complement C are indicated.

Figure 2

(Color online) (a) Grey-scale of $dG∕d{V}_g$ data at $5\mathrm{T}$ as a function of dc bias $V_{ds}$ and $V_g$. Labels indicate whether a branch corresponds to a subband intercepting ${\mu }_s$ or ${\mu }_d$, as shown in (b). A left (right) branch in $+V_{ds}$ corresponds to a subband intercepting the source (drain) chemical potential ${\mu }_s=\mu +e{V}_{ds}∕2$ $({\mu }_d=\mu -e{V}_{ds}∕2)$.

Figure 3

(Color online) (a) Grey-scale of $dG∕d{V}_g$ at $B=0$ as a function of $V_{ds}$. White regions are plateaus. A close-up (b) shows that $\gamma$, separating the 0.7 structure from $2e^2∕h$, does not split in $V_{ds}$—the left branch is absent. This is illustrated schematically in (c), where the “missing left branches” are represented by dashed lines. (d) $V_{ds}$ data at the crossing at $B=9\mathrm{T}$. At $V_{ds}=0$, $a$, $b$, and $c$ correspond to the $1\uparrow$, $2\downarrow$ and $1\uparrow$ features marked by symbols in Fig. 1. $a$ has no right branch in $V_{ds}$ and $c$ has no left branch. $b$ and $c$ next to the analog A are equivalent to $\beta$ and $\gamma$ next to the 0.7 structure in (b). Inset: Conductance trace for $V_{ds}=0$ at $B=9\mathrm{T}$. (e) The “missing branches” for $a$ and $c$ are represented by dashed lines in this schematic diagram of the crossings. (f) A close-up of $a$ and $b$, in data from a similar sample at $8.6\mathrm{T}$, demonstrates that $a$ has no right branch. (g) A schematic of (d), showing the configurations of subbands. Missing branches indicate that $1\uparrow$ is pinned to $\mu$ in the complement and analog regions A and C (see text).

Figure 4

(Color) (a) Conductance traces from Fig. 1a with schematics illustrating how pinning of $1\uparrow$ explains the complement and analog. Again, symbols indicate rises in conductance that correspond to features in Figs. 1, 3d, 3g (b) $dG∕d{V}_g$ as a function of $G$ and $V_{ds}$ at the crossing at $B=7.8\mathrm{T}$. Blue indicates plateaus, red indicates abrupt changes in $G$ with $V_g$ and white indicates slowly changing $G$. (c) Similar data for $B=9\mathrm{T}$ [also the data used to make Fig. 3d]. From $7.8\mathrm{T}\text{to}9\mathrm{T}$, the analog A strengthens as $G_{\text{analog}}$ decreases, and the complement C weakens as $G_{\text{complement}}$ decreases. $G_{\text{complement}}$ and $G_{\text{analog}}$ change immediately in finite $V_{ds}$, unlike the quantized $1.5(2e^2∕h)$ plateau which remains at fixed $G$ until it disappears at $V_{ds}\sim \pm 0.2\mathrm{mV}$.