Dimensionality of metallic atomic wires on surfaces

We investigate the low-energy collective charge excitations (plasmons, holons) in metallic atomic wires deposited on semiconducting substrates. These systems are described by two-dimensional correlated models representing strongly anisotropic lattices or weakly coupled chains. Well-established theoretical approaches and results are used to study their properties: Random phase approximation for anisotropic Fermi liquids and bosonization for coupled Tomonaga-Luttinger liquids as well as Bethe ansatz and density-matrix renormalization group methods for ladder models. We show that the Fermi and Tomonaga-Luttinger liquid theories predict the same qualitative behavior for the dispersion of excitations at long wave lengths. Moreover, their scaling depends on the choice of the effective electron-electron interaction but does not characterize the dimensionality of the metallic state. Our results also suggest that such anisotropic correlated systems can exhibit two-dimensional dispersions due to the coupling between wires but remain quasi-one-dimensional strongly anisotropic conductors or retain typical features of Tomonaga-Luttinger liquids such as the power-law behavior of the density of states at the Fermi energy. Thus it is possible that atomic wire materials such as Au/Ge(100) exhibit a mixture of features associated with one- and two-dimensional metals.

Figure 1

Dispersions $E\left(\vec{q}\right)$ of the upper edge of the continuum ($\theta =0$) of collective charge excitations (holons) in coupled TLLs as a function of $q=|\vec{q}|$. The red dash-dotted curve shows the dispersion (21) for a screened Coulomb potential. The solid black curve indicates the dispersion (25) for a phenomenological gaussian potential. The blue dotted curve corresponds to uncoupled TLLs. The dashed green curve represents a fit to the experimental data for plasmons in Au/Ge(100) presented in Ref. [23].

Figure 2

Dynamical charge structure factor $S(\vec{q},\omega )$ (27) calculated with DMRG for a two-leg ladder with $V_y=8t_{\parallel }$, $V_x=V_{xy}=0$, and $q_y=0$ as a function of the excitation energy $\hslash \omega$ for several values of $q_x$ from $\pi /\left(33a\right)$ (bottom) to $16\pi /\left(33a\right)$ (top). The system length is $L_x=32$ and the broadening is $\eta =0.4t_{\parallel }$. The units are $t_{\parallel }=1$ and $\hslash =1$.

Figure 3

Holon dispersions $E\left(\vec{q}\right)$ in two-leg ladders as a function of $q_x$ for fixed $q_y$. Symbols show values determined from the charge structure factor calculated with DMRG for $V_y=8t_{\parallel }$, $V_x=V_{xy}=0$, $q_y=0$ (circles) and $q_y=\pi /a$ (squares) as well as for $V_x=V_y=V_{xy}=4t_{\parallel }$, $q_y=0$ (triangles) and $q_y=\pi /a$ (diamonds). The Bethe ansatz solutions for $V_y=8t_{\parallel }$, $V_x=V_{xy}=0$ are represented by a solid red line ($q_y=\pi /a$) and a blue dashed line ($q_y=0$), respectively. The units are $t_{\parallel }=1$ and $a=1$.