Dynamic correlations in coupled electron-electron and electron-hole quantum wire systems

We investigate theoretically the ground-state behavior of the coupled electron-electron and electron-hole quantum wire systems by incorporating dynamic correlation effects within the quantum version of Singwi, Tosi, Land, and Sjölander theory. The numerical results are presented for the pair-correlation function, the ground-state energy, the static density susceptibility, and the static and dynamic local-field correction factors over a wide range of system parameters, viz., linear particle number density $r_s$, wire size $b$, and interwire spacing $d$. The results reveal that the inclusion of the dynamical nature of particle correlations brings in quantitative as well as qualitative changes in the ground-state behavior of both the electron-electron and electron-hole wire systems. In particular, it is found that these (dynamic) correlations can cause the (homogeneous) liquid phase, in these quantum wire systems, to become unstable against a phase transition into a(n) (inhomogeneous) coupled Wigner crystal ground state at sufficiently low particle density and/or narrow wire size in the close approach of two wires. The interwire correlations are found to reduce the critical $r_s$ for the onset of Wigner crystallization with respect to an isolated quantum wire system, and at $b∕a_0^{*}=1$ the reduction in $r_s$ is about $15\%$ and $4\%$ in the electron-hole and electron-electron wire systems, respectively; $a_0^{*}$ is the effective Bohr atomic radius. Our prediction of Wigner crystallization for the electron-electron wire system agrees qualitatively with the recent results of Tanatar et al., which they have obtained on the basis of an approximate density functional theory calculation.

Figure 1

Pair-correlation functions $g_{11}\left(r\right)$ and $g_{12}\left(r\right)$, commencing, respectively, in the lower and upper halves of the figure, for the e-h wire system in the qSTLS (solid lines) approach at: (a) $r_s=1,2.5$, and $4$; (b) $d∕a_0^{*}=6$ and $4.15$; and (c) $b∕a_0^{*}=2,1$, and $0.28$ at indicated wire parameters. The STLS results (dashed lines) are at: (a) $r_s=4$; (b) $d∕a_0^{*}=4.15$; and (c) $b∕a_0^{*}=0.28$.

Figure 2

Pair-correlation functions $g_{11}\left(r\right)$ and $g_{12}\left(r\right)$, commencing, respectively, in the lower and upper halves of the figure, for the e-e wire system in the qSTLS (solid lines) approach at: (a) $r_s=1,2.5$, and $4$; (b) $d∕a_0^{*}=6$ and 3.7; and (c) $b∕a_0^{*}=2,1$, and 0.58 at indicated wire parameters. The STLS results (dashed lines) are at: (a) $r_s=4$; (b) $d∕a_0^{*}=3.7$; and (c) $b∕a_0^{*}=0.58$.

Figure 3

Comparison of ground-state energy (per particle) $E_{gs}$ between the qSTLS and RPA approaches for the coupled e-h [in panel (a)] and e-e [in panel (b)] wire systems at $d∕a_0^{*}=4$ and 2; legends indicate the values of $d∕a_0^{*}$.

Figure 4

(a)–(d) In-phase component of the static density susceptibility ${\chi }_{+}\left(q,0\right)$ for the coupled e-h wire system for different $d∕a_0^{*}$ at indicated $r_s$ and $b∕a_0^{*}$, according to the qSTLS theory. For comparison, the STLS results are given in (d) at $d∕a_0^{*}=2.5$ and 2.49. Legends indicate the values of $d∕a_0^{*}$.

Figure 5

In-phase component of the static density susceptibility ${\chi }_{+}\left(q,0\right)$ for the coupled e-h wire system for different $b∕a_0^{*}$ at indicated $r_s$ and $d∕a_0^{*}$, according to the qSTLS approach; legends indicate the values of $b∕a_0^{*}$.

Figure 6

Critical wire spacing $d_c∕a_0^{*}$ depicting the points of instability for the coupled e-h wire system as a function of $r_s$ at $b∕a_0^{*}=1$ and $0.5$. The lines are just a guide for the eye. For each case the arrows show the critical $r_s$ where crossover from the CDW instability to the Wigner crystal instability occurs.

Figure 7

Intrawire [in panel (a)] and interwire [in panel (b)] static local fields $G_{11}\left(q,0\right)$ and $G_{12}\left(q,0\right)$ for the coupled e-h wire system in the qSTLS approach at indicated $r_s$, $b$, and $d$. The STLS local fields are shown for comparison at $d∕a_0^{*}=6$ and $4.15$. Legends indicate the values of $d∕a_0^{*}$.

Figure 8

(a)–(c) Out-of-phase component of the static density susceptibility ${\chi }_{-}\left(q,0\right)$ for the coupled e-e wire system in the qSTLS approach for different $d∕a_0^{*}$ at indicated $r_s$ and $b∕a_0^{*}$; legends indicate the values of $d∕a_0^{*}$.

Figure 9

Out-of-phase component of the static density susceptibility ${\chi }_{-}\left(q,0\right)$ for the coupled e-e wire system for different $b∕a_0^{*}$ at indicated $r_s$ and $d∕a_0^{*}$; legends indicate the values of $b∕a_0^{*}$.

Figure 10

Intrawire [in panel (a)] and interwire [in panel (b)] static local fields $G_{11}\left(q,0\right)$ and $G_{12}\left(q,0\right)$ for the coupled e-e wire system in the qSTLS approach at indicated $r_s$, $b$, and $d$. The STLS local fields are shown for comparison at $d∕a_0^{*}=6$ and $3.7$. Legends indicate the values of $d∕a_0^{*}$.

Figure 11

Frequency dependence of intrawire [in panel (a)] and interwire [in panel (b)] local fields $G_{11}\left(q,\omega \right)$ and $G_{12}\left(q,\omega \right)$ for the coupled e-h wire system at $q∕q_F=3.4$, $r_s=4.25$, $b∕a_0^{*}=1$, and $d∕a_0^{*}=5$ and $2.91$. Thin and thick lines represent, respectively, the real and imaginary parts. Legends indicate the values of $d∕a_0^{*}$, and ${\Omega }_F$ is the Fermi frequency.

Figure 12

(a), (b) Dynamic local fields for the coupled e-e wire system at indicated parameters; description of the curves is exactly the same as in Fig. 11 except for the fact that $r_s=4.8$.