Thermal effects on the Wigner localization and Friedel oscillations in many-electron nanowires

Thermal effects on the total charge density are studied for a one-dimensional correlated quantum dot by means of the path integral Monte Carlo method. The competition between Friedel and Wigner oscillations at zero temperature is driven by the ratio between the interaction of electronic strength and the kinetic energy of electrons. At the onset of the formation of a Wigner molecule, we show that thermal enhancement of Wigner oscillations occurs in a range of temperatures, which can be observed in the electron density. We further show that low-temperature Friedel oscillations may change to Wigner oscillations upon an increase in the temperature.

Figure 1

(a) Schematic of the nanowire described by confined electrons in a harmonic confinement. (b) Electron density for $N=6$ with different values of the softening parameter $\tilde{\lambda }$ in the Coulomb potential [see Eq. (1)]. The other parameters are $\tilde{c}=0.6$ and $\tilde{T}=0.3$.

Figure 2

Electron densities for the $N=3$ nanowire as a function of the temperature. Four interaction strengths with $\tilde{\lambda }=0.02$ are considered: (a) $\tilde{c}=0.2$, (b) $\tilde{c}=0.4$, (c) $\tilde{c}=0.6$, and (d) $\tilde{c}=1.0$. The temperature $\tilde{T}$ goes from 0.02 to 1.2, and $\tilde{x}$ from $-1.2$ to 1.2.

Figure 3

Fourier transforms of the electron densities shown in Fig. 2. The second peak, at $k=2k_F$, is characteristic of Friedel oscillations and the third peak, at $k=4k_F$, corresponds to Wigner oscillations. The contour lines are plotted from $k=1.3k_F$ up to larger values in steps of $\sim 3.2\times {10}^{-4}$.

Figure 4

Electron density of a nanowire with (a) $N=4$, (b) $N=6$, and (c) $N=10$ electrons at three temperatures. Parameters $\tilde{c}$ and $\tilde{\lambda }$ are given in the figures. In all cases signatures of thermally enhanced Wigner localization are found.

Figure 5

(a) Electron density of a nanowire with $N=4$, $\tilde{c}=0.4$, and $\tilde{\lambda }=0.05$ at temperatures given in (b), where we give the Fourier transform of the electron densities. Note that $\tilde{T}=4.8$ is not visible for the chosen axis in (a), but its Fourier transform is in (b).

Figure 6

(a) Electron density of a nanowire with $N=20$, $\tilde{c}=1.0$, and $\tilde{\lambda }=0.05$ at temperatures given in (b), where we show the Fourier transform of the electron densities.