Dispersion interactions between semiconducting wires

The dispersion energy between extended molecular chains (or equivalently infinite wires) with nonzero band gaps is generally assumed to be expressible as a pair-wise sum of atom-atom terms which decay as $R^{-6}$. Using a model system of two parallel wires with a variable band gap, we show that this is not the case. The dispersion interaction scales as $z^{-5}$ for large interwire separations $z$, as expected for an insulator, but as the band gap decreases the interaction is greatly enhanced; while at shorter (but nonoverlapping) separations it approaches a power-law scaling given by $z^{-2}$, i.e., the dispersion interaction expected between metallic wires. We demonstrate that these effects can be understood from the increasing length scale of the plasmon modes (charge fluctuations), and their increasing contribution to the molecular dipole polarizability and the dispersion interaction, as the band gaps are reduced. This result calls into question methods which invoke locality assumptions in deriving dispersion interactions between extended small-gap systems.

Figure 1

(Color online) The second-order dispersion energy of parallel wires modeled using Hückel theory obtained from Eq. (1). The bonding interaction is of alternating strength and is controlled by the ratio ${\beta }^{\prime }/\beta$. The lines are numerical power-law fits to the small-$z$ and large-$z$ data points. In the metallic $\left(\beta ={\beta }^{\prime }\right)$ and perfect insulating $\left({\beta }^{\prime }=0\right)$ cases, a single power-law fits the whole range of $z$ with the expected exponents $-2$ and $-5$, respectively. The intermediate cases show a crossover between these two limiting regimes as $z$ is varied.

Figure 2

(Color online) Distorted ${\left({\text{H}}_2\right)}_n$ chains. The distortion parameter is defined as $\eta =y/x$. In all our chains we have chosen $x=1.4487\text{a}\text{.u}\text{.}$

Figure 3

(Color online) The second-order dispersion energy of pairs of parallel ${\left({\text{H}}_2\right)}_n$ chains. Energies have been normalized by the number of ${\text{H}}_2$ units $n$. The solid lines are dispersion energies for chains with $n=32$ and the dashed lines for chains with $n=16$ (only $\eta =1.0$ and 2.0).

Figure 4

Matrix representation of the charge-flow polarizability matrix ${\alpha }_{00,00}^{a{a}^{\prime }}$ for ${\left({\text{H}}_2\right)}_{32}$ chains with $\eta =2$,1.5,1.25, and 1 (clockwise, from top left). Sites $a$ and $a^{\prime }$ are represented along the $x$ and $y$ axes. Because these terms span a number of orders of magnitude of both signs we have plotted $\text{ln}\left(\left|{\alpha }_{00,00}^{a{a}^{\prime }}\right|\right)$. Color scheme: Black rectangles correspond to terms of order 1 a.u. and white rectangles to terms of order ${10}^{-3}\text{a}\text{.u}\text{.}$

Figure 5

(Color online) The second-order dispersion energy calculated using the polarizabilities of ${\left({\text{H}}_2\right)}_{32}$ chains with distortion parameters $\eta =1$ and 2. The contribution from the charge-flow polarizabilities, ${\alpha }_{00,00}^{a{a}^{\prime }}$, is displayed separately from the sum of contributions up to the dipole-dipole polarizabilities.