Universal Scaling of Electron Transmission for Nearly Ballistic and Quantum Dragon Nanodevices

TL;DR

This work addresses how electron transmission in nearly ballistic and quantum-dragon nanodevices scales under small uncorrelated disorder. Using tight-binding models and NEGF, it shows that two universal scaling regimes govern the disorder-averaged transmission ${\cal T}_{\rm ave}(E)$: a very small-$\delta$ regime with ${1-\cal T}_{\rm ave}(E) \propto \delta^2$ dependent on device geometry and energy, and a DOS-driven regime for larger $\delta$ that incorporates a DOS-dependent factor ${\Upsilon}$ and a scale length $L_{\rm scale}$. The DOS enters the second regime via a scaling relation validated across 2D hexagonal, rectangular, and square-octagonal quantum dragons, confirming robust transport behavior near ideal unit transmission ${\cal T}(E)=1$. The results illuminate how order amidst disorder can persist in higher-dimensional systems, with implications for nearly perfect nanoelectronic devices and potential routes to quantum information processing and cloaking. The framework connects universal perturbative scaling, Fano resonances, and DOS-based scaling into a cohesive picture of transport in complex quantum nanosystems.

Abstract

We predict two different universal scaling regimes for the quantum transmission of metallic nanodevices following the addition of a small amount of uncorrelated disorder. A nanodevice is connected to two thin semi-infinite uniform leads, and the Non-Equilibrium Green's Function (NEGF) methodology yields the electron transmission ${\cal T}(E)$ as a function of the injected electron energy $E$. Ballistic nanodevices have no disorder and have ${\cal T}(E)=1$ for all $E$ that allow electron propagation in the leads. Quantum dragon nanodevices can have extremely strong properly correlated disorder, and still have ${\cal T}(E)=1$ for all $E$. Additional uncorrelated site disorder leads to Fano resonances in ${\cal T}(E)$. Averaging over the uncorrelated disorder we predict using perturbation theory two universal scaling regimes for ${\cal T}_{\rm ave}(E)$. The functional form of both universal scaling regimes depend on the device length and width, energy, and variance of the uncorrelated disorder. The second scaling regime, valid for small but somewhat larger uncorrelated disorder than the first scaling regime, also has the form dependent on the density of states of the system. These two scaling regimes are demonstrated to be valid via large scale computer calculations.

Figures (10)

• Figure 1: [id=MAN] Two quantum nanosystems based on split armchair single-walled carbon nanotubes with $m$ atoms in each slice. With appropriate boundary conditions and input/output leads, both are quantum dragons with unit transmission, as in Eq. (\ref{['Eq:Landauer']}), based on our model and method. (A) A nanodevice with $m$$=$$10$. Compare directly with the graphical abstract of the experimental article He2023 of a sub-5 nm graphene-nanoribbon/carbon-nanotube which also has $m$$=$$10$. The black spheres represent a constant on site energy in a tight-binding model, with all on site energies the same. The yellow spheres, located on the boundary atoms and the atoms of the attached thin leads, have zero on site energy. The cyan cylinders represent the hopping energies, which are all the same value in the device and leads. The radius of the cyan cylinders for the nanosystem-lead connections are proportional to the hopping strength. (B) Shows a double-cut armchair nanotube, without showing the two attached leads. To see the underlying structure of the graph, the Mathematica Mathematica command GraphPlot has been used.
• Figure 2: Technique to show complete electron transmission for atypical strongly disordered nanosystems. See text in Sec. 2.3.2 for full description of this quantum dragon nanodevice[id=MAN], including the color coding. The top diagram is the quantum dragon nanodevice with $m$$=$$16$ and $\ell$$=$$73$ depicted in real space. In the basis after the unitary transformation, depicted in the bottom, the system is a uniform wire plus a disconnected part of the Hamiltonian. A unitary transformation does not affect the transport properties, hence the device is a quantum dragon with ${\cal T}(E)$$=$$1$ for all $-2$$<$$E$$<2$.
• Figure 3: The effect of the Hamiltonian of Eq. (\ref{['Eq:HamTB']}) almost being a quantum dragon. (A) A device based on a 2D hexagonal graph with $\ell=12$ and $m=4$, showing the effect of the unitary transformation ${\bf U}_{12}$ on the device depiction when the device Hamiltonian is close to being a quantum dragon. (B) Shows ${\cal T}(E)$ for an example of a similar quantum dragon nanodevice based on a 2D hexagonal graph with $\ell=80$ and $m=22$ for four different strengths of uncorrelated disorder, $\delta=0$ (green, ${\cal T}(E)=1$ for all $E$), $\delta=0.002$ (red), $\delta=0.004$ (black), and $\delta=0.006$ (blue). The dashed vertical orange line is the location of an eigenvalue of the $\delta=0$ device Hamiltonian. In agreement with Eq. (\ref{['Eq:Fano01']}), all three finite $\delta$ values have ${\cal T}(E)$ that plunge to values numerically indistinguishable from zero. Note the expanded scales for $E$ and ${\cal T}(E)$. See text in Sec. \ref{['sec:3:Fano']} for full description.
• Figure 4: The quantity $1-{\cal T}_{\rm ave}(E)$ vs $\delta$ is shown for a single quantum dragon based on a 2D hexagonal graph with $\ell=80$ and $m=20$ so $N=1600$. The three energies shown are $E=-1,\>0,\>1$ shown as black, green, and red (open circles, squares, disks), respectively. The averages are over $M$$=$$10^5$ different values of the uncorrelated on site disorder chosen from a Gaussian distribution with mean zero and width $\delta$, with the random deviate chosen independently for each vertex and added to the on site energies of the quantum dragon values of that vertex. The three curves, color coded to the data points, show the predictions of Eq. (\ref{['Eq:T-scaling']}) with no adjustable parameters. Asymptotically for small $\delta$ these solid curves are close to a straight line of slope 2, agreeing with Eq. (\ref{['Eq:scale:Sec4']}). We see $1-{\cal T}_{\rm ave}$ is proportional to $\delta^2$, but exhibits a cross-over from averages dominated by a single nearby Fano resonance for small $\delta$ to the Eq. (\ref{['Eq:T-scaling']}) prediction when the averages are due to many Fano resonances. Inset: The same data illustrating the universal scaling predicted for very small $\delta$ in Eq. (\ref{['Eq:deltaSmall:01']}), with the abscissa ${\rm log}_{10}(\delta)$. The ordinate of each point is the lhs of Eq. (\ref{['Eq:deltaSmall:01']}), while the rhs of Eq. (\ref{['Eq:deltaSmall:01']}) is the horizontal cyan line. See text in Sec. \ref{['sec:4:NearQuantumDragon']} for full description.
• Figure 5: A test of the small $\delta$ universal scaling of Eq. (\ref{['Eq:deltaSmall:01']}), for five different types of nanodevices depicted by the five different colors. The predicted scaling value of unity is shown by the horizontal cyan lines. The energies shown are (A)$E=1$ and (B)$E=-\sqrt{2}$. See Sec. \ref{['sec:5:SmallDelta']} and \ref{['CSaF_AppB']} for complete information.