Confined electrons in effective plane fractals

As an emerging complex two-dimensional structure, plane fractal has attracted much attention due to its novel dimension-related physical properties. In this paper, we check the feasibility to create an effective Sierpinski carpet (SC), a plane fractal with Hausdorff dimension intermediate between one and two, by applying an external electric field to a square or a honeycomb lattice. The electric field forms a fractal geometry but the atomic structure of the underlying lattice remains the same. By calculating and comparing various electronic properties, we find parts of the electrons can be confined effectively in a fractional dimension with a relatively small field, and representing properties are very close to these in a real fractal. In particular, compared to the square lattice, the external field required to effectively confine the electron is smaller in the honeycomb lattice, suggesting that a graphene-like system will be an ideal platform to construct an effective SC experimentally. Our work paves a way to build fractals from a top-down perspective and can motivate more studies of fractional dimensions in real systems.

Figure 1

Schematic representation of the square-lattice Sierpinski carpet (a) and the honeycomb-lattice Sierpinski carpet (c) with different iteration $I$. The width of the sample is $W$ lattice cells [e.g., (a) $I=3$, $W=54$; (c) $I=2$, $W=33$]. (b) and (d) illustrate the corresponding effective SC where we add opposite on-site potential in the different regions, namely, adding $-V$ in the uncolored region and $V$ in the colored (red) one. Here we label the uncolored region as Area I, and the colored region as Area II in context.

Figure 2

The DOS of SC-$\square$ (a) and SC-$⬡$ (c) in different iterations. (b) The DOS of the effective SC-$\square$ for $I=5$, $W=486$ and on-site potential $V=4t$. (d) The DOS of the effective SC-$⬡$ for $I=4$, $W=298$, and on-site potential $V=3t$.

Figure 3

(a) The comparison between converged DOS of SC-$\square$ with $I=5$, $W=486$ and corresponding effective ones with different on-site potential $V$. (b) The comparison between converged DOS of SC-$⬡$ with $I=4$, $W=486$, and corresponding effective ones with different on-site potential $V$.

Figure 4

(a), (c) The real-space distribution of the quasi-eigenstates of the SC-$\square$ with $I=5$, $W=486$ at the central peak $E/t=0$ and the edge peak $E/t=-3.79$. For comparison, we show the real-space distribution of the quasi-eigenstates of the effective SC-$\square$ with $V=3t$ at the corresponding energy in (b) and (d).

Figure 5

(a), (c) The real-space distribution of the quasi-eigenstates of the SC-$⬡$ with $I=4$, $W=298$ at the central peak $E/t=0$ and the edge peak $E/t=-2.95$. For comparison, we show the real-space distribution of the quasi-eigenstates of the effective SC-$⬡$ with $V=3t$ at the corresponding energy in (b) and (d).

Figure 6

Energy dependence of the conductance $G\left(E\right)$ (in units of $e^2/h$) of the (real and effective) SC-$\square$ (a) and SC-$⬡$ (b) with different $V$. $I=3$ and $W=54$ for the (real and effective) SC-$\square$ and $I=2$, $W=33$ for the (real and effective) SC-$⬡$. Left and right panels refer to central lead and diagonal lead configurations, respectively. Here, the convergences of the conductance fluctuations in different geometries and lead configurations are presented, and we see that for similar lead configurations, the coverage of the spectrum in SC-$⬡$ is faster than SC-$\square$. It is worth mentionng that we show here results with relatively smaller samples because the conductance fluctuation of a large system is too dense for visual comparison.

Figure 7

BC algorithm analysis of the CFs for SCs. (a), (b) BC dimension of the SC-$\square$ with $I=5$, $W=486$ and the corresponding effective ones with different $V$. (c), (d) BC dimension of the SC-$⬡$ with $I=4$, $W=298$ and the corresponding effective ones with different $V$. Data in (a) and (c) and (b) and (d) refer to central lead and diagonal lead positions, respectively.