Order amidst disorder for two-dimensional nanoribbons with various boundary conditions

We show quantum systems with disordered Hamiltonians may exhibit order in commonly measured quantities. This counter-intuitive situation is demonstrated using a conventional tight binding model for two-dimensional nanoribbons with various boundary conditions. The analysis uses the traditional non-equilibrium Green's function (NEGF) methodology for electron transport. We study quantum dragon nanodevices that exhibit order amidst disorder. Each disordered Hamiltonian nanodevice shows order in both the bond currents and the local density of states (LDOS) of the propagating electrons.

Contrary to conventional standard intuition, by exact calculations we provide examples of disordered quantum nanodevices that amidst the disorder in the Hamiltonian exhibit order in commonly measured electron transport quantities.
Electrical resistance of nanodevices is directly related to the electron transmission probability T (E) at energy E of electrons injected via leads into the nanodevice [1][2][3][4][5]. Disorder is disruptive to electron transport as it causes coherent electron scattering, thereby reducing the transmission. Anderson [6] showed in one-dimensional (1D) models with short-range interactions that any amount of uncorrelated randomness gives a wave function which is exponentially localized. In the 1D case, long-range correlations in the disorder or long-range interactions in the Hamiltonian may produce extended states [7][8][9][10]. One of the authors showed in 2D models with cylindrical symmetry that short-range correlated disorder can lead to T (E) = 1 for all electrons that propagate through the leads [11], whimsically called quantum dragon nanodevices [11]. Here we show that even for different boundary conditions (BCs) 2D nanodevices can have T (E) = 1 for all energies E of electrons that propagate through leads. Furthermore, we show for all BCs quantum dragons show (a) E-mail: man40@msstate.edu (corresponding author) "order amidst disorder" in that the strongly disordered nanodevice has for the propagating electrons both the local density of states (LDOS) and the bond currents I k,k ordered, as in fig. 1.

Model and Hamiltonian. -
We study the standard single-orbital tight binding model. The model is on a graph [12] of N vertices with edges which can be embedded in 2D. We assume the graph vertices can be partitioned into slices each with m vertices. We index the vertices within a slice by i = 1, 2, · · · , m and the slices by j = 1, 2, · · · , . We number the N atoms by k = (j − 1)m + i. We here restrict ourselves only to graphs with intra-slice edges (between vertices with the same value of j) and inter-slice edges confined to being between vertices in slice j and adjacent slices j ± 1. As a model for materials, the graph vertices represent the locations of atoms, and the graph edges are hopping paths, often called bonds, for electrons between the atoms due to the electron wave function overlap of the two atoms labeled k and k . The tight binding Hamiltonian has the form The graph has N = m vertices each with an associated on site energy k . The creation (annihilation) operators for site k areĉ † k (ĉ k ). Every bond has a hopping term of strength t k;k , and the second sum is over all bonds (edges) of the graph. We study only graphs which may represent actual 2D nanomaterials by limiting ourselves to shortrange bonds, namely only nearest-neighbor (nn) or nextnearest neighbor (nnn). We connect the nanodevice to 1D tight-binding semi-infinite uniform incoming and outgoing leads, and choose the on-site energy of the lead atoms to be zero (setting our zero of energy) and nn hopping between all lead atoms to be −1 (setting our unit for energy). A Bloch wave function analysis of the tight binding leads gives the dispersion relation E = −2 cos q Lead a , with a the distance between lead atoms and q Lead the electron wave vector in the leads. Hence the leads have propagating modes for all electron energies in the range −2 < E < 2.
Because of the nn and nnn restrictions in eq. (1) in the physical-space basis the Hamiltonian is a block-tridiagonal N × N matrix. The diagonal block for slice j we denote by A j , which are m × m symmetric matrices. The superdiagonal blocks of H we denote by the m × m matrices B j,j+1 which need not be symmetric matrices, and the sub-diagonal blocks are B † j,j+1 . If there are no inter-slice nnn hopping terms, then all B j,j+1 = −tI m with the same hopping strength t as in the attached leads and where I m is the m × m identity matrix.
The tight binding model in eq. (1) is the traditional model to study conductance through nanodevices. Although the model is a low-level approximation for actual materials, it nevertheless has been well studied and applied to understand properties of materials. For example, with only nn hopping terms the tight binding model of eq. (1) (without disorder) has been used to study π-orbital graphene nanoribbons [13], even though an ab initio method is preferable [14]. For the tight binding model of eq. (1) we here predict disordered graphene nanoribbons can be quantum dragons that exhibit order amidst disorder, with an example the left-hand part of the plots in fig. 1. In this article we take the hopping to be a real number and hence study only zero external magnetic fields.
Our restriction to nn inter-slice bonds gives all A j to be symmetric tri-diagonal matrices, with diagonal elements the on-site energies i,j . The A j have the super-and subdiagonal elements the nn hopping −t i,j;i+1,j . We study quantum dragon nanodevices, and hence as described below insist all A j have a common eigenvector v Drak , with all m elements taken to be positive (which is always possible by the Perron-Frobenius theorem).
For ordered nanodevices all A j will be identical, and all hopping terms will be of strength t. All diagonal elements of A j will be the same value o , with the exception of the first ( 1,j ) and last ( m,j ) diagonal elements which will depend on the boundary conditions (BC). Physically the BCs are due to the different effective electrical potentials at the edges of the 2D nanoribbon. Our analysis works for any BC (any v Drak ), but in the figures we concentrate on six analytically tractable BCs [15] listed in table 1 for ordered rectangular nanoribbons. These have 26005-p2 Table 1: Listed are six different BCs from [15] for ordered rectangular nanodevices with largest eigenvalue λ Drak . The associated eigenvector v Drak has the index η for η = 1, 2, · · · , m.
We previously studied the case of free boundary conditions (k 1 = k 2 = 0) and demonstrated quantum dragon nanoribbons [16] for both uniform and dimerized leads, but did not analyze order amidst disorder in the LDOS and did not calculate the bond currents.
Nanoribbon length scales. -Any nanodevice has five important length scales. One is the bond length a (which we take to be the same as for the leads), and another is the device length L. For periodic (ordered) nanoribbons L ∝ a. Another length scale is the Anderson localization length ξ A [6,17]. Yet another is the "typical" distance a Dis in a periodic graph to have the graph and/or Hamiltonian parameters of eq. (1) disordered. For any nanodevice the length scales a, a Dis , and L are fixed after fabrication is complete. Introduce the additional energydependent length scale T (E) = L T (E)/(1 −T (E)). This length was used in [18] for the regime with a a Dis ≈ L as the effective elastic mean free path, the typical distance an electron travels before it scatters from the disorder. Strongly disordered devices have a Dis ≈ a L, so the electron mean free path is ill defined, but we find the length T (E) still useful. There are three different well-studied regimes of length scales, as in table 2. The definition of ballistic is "relating to projectiles and their flight", and for the regime with a L a Dis one can have T = ∞ and loosely think of projectile motion of an electron moving in an ordered background of atoms because there is no disorder to scatter the electrons. As in [18], the diffusive regime is when T L, with the transmission very small due to the disorder scattering. In both the diffusive and localized regimes there exist very narrow resonances with T (E necklace ) ≈ 1, and these necklace states have been observed in 1D [19]. The existence of long-range correlations in the disorder, and the interplay between diagonal and off diagonal disorder [20], can lead to effective mobility edges. The ballistic, diffusive and localized regimes of length scales are well studied [3][4][5]21], and all relate to typical disorder or the lack of any disorder.
Most researchers study typical disorder, and consequently models with locally correlated but arbitrarily strong disorder were expected to have T (E) 1 in 2D and 3D, as this is the diffusive or localized regimes. Nevertheless, we restrict our studies to quantum dragon devices, namely devices with strong atypical disorder but which have T (E) = 1 for all −2 < E < 2 and show order amidst disorder. Table 2 summarizes the four different regimes for electron transmission through 2D nanodevices. Nanodevices that are ordered (ballistic transport regime) or have appropriate atypical disorder (quantum dragon transport regime) have complete transmission, namely T (E) = 1 for all −2 < E < 2 in our units. Nanodevices with typical disorder, namely in the diffusive and localized regimes, have very small T (E) for almost all energies E.     fig. 2 at an energy E = π/3 is shown. The radius of the red sphere for atom k is proportional to LDOS k . The LDOS for slices with m = 1 are not shown, since they would overwhelm the figure. The LDOS are superimposed on the hopping terms, the cyan cylinders of fig. 2. Note both the ordered device and the quantum dragon disordered device have the same LDOS k for every atom k, thereby demonstrating order amidst disorder.
From the NEGF, the LDOS for each atom is calculated in the standard fashion using so only the diagonal elements of the imaginary part of the Green's function G enter. It is important to realize that the LDOS of eq. (2) contains a projection operator and so is only for the propagating electrons. To calculate the bond currents, introduce what is called the lesser Green's function G < (E) in the conventional fashion [21,22], with G < (E) ≈ GΓ L G † . The approximation is for zero temperature and very small differences between the chemical potentials of the macroscopic reservoirs at 26005-p4 ±∞. If there is a nonzero hopping term between sites at (i, j) and (i , j ), the local current which flows in this bond is [21] where we have chosen the injected current into the left lead at −∞ to be unity. Since G < (E) is a function of energy, the usual situation is that the bond currents are functions of the energy. Since H is disordered, one normally expects disordered bond currents, and this is seen in the diffusive and localized regimes of table 2.
Order amidst disorder in quantum dragon nanodevices. -In order to find strongly disordered quantum dragon nanodevices, as in table 2, one has to correlate the disorder. For every intra-slice A j submatrix of the Hamiltonian, we require A j v Drak = 0. For ordered nanodevices with different BCs from table 1, this means the eigenvalue λ Drak = 0. The eigenvalue zero of A j is because we have chosen the on-site energy of the lead atoms to be our zero of energy. For disordered quantum dragon nanodevices we choose independently at random the strength of each intra-slice bond t i,j;i+1,j . This procedure gives the intra-slice hopping portion of slice j as the Hermitian matrix T intra,j . The on-site energies for a quantum dragon nanodevice are then correlated by being set to so the device has T (E) = 1 for −2 < E < 2. For a hexagonal graph as on the left of fig. 1, eq. (4) allows independently choosing each intra-slice (vertical) hopping term with local correlations of only the two associated onsite energies. Appropriate disordered inter-slice matrices B j,j+1 can be found for any graph, but may be generalized since these matrices need not be symmetric. To have a quantum dragon, the singular value decomposition (SVD) of each B j,j+1 must have a singular value of −1 (the hopping energy between lead atoms) with corresponding left-singular and right-singular vectors v Drak [11]. We say a nanodevice exhibits order amidst disorder if it is disordered (the parameters i,j and t i,j;i ,j are not all the same and/or are not in a regular pattern) while the LDOS i,j is ordered for all −2 < E < 2. The different BCs for the ordered device depicted as the top figure in fig. 2 gives for the propagating electrons as displayed in the top figure of fig. 3. Since quantum dragon nanodevices have a basis where a uniform wire is connected to the two semi-infinite leads, the LDOS are also given by eq. (5), as seen in fig. 3. Ballistic nanodevices exhibit order in the propagating electron LDOS ( fig. 3) and the bond currents are independent of energy for −2 < E < 2. The supplementary Supplement cdf.cdf 1 contains a proof using NEGF of order amidst disorder for quantum dragon nanodevices and that the LDOS k calculated using eq. (2) satisfy eq. (5).
Using similar arguments, we have been able to prove for all quantum dragon nanodevices all the bond currents are independent of E. We have a proof (SM) for all quantum dragon nanodevices that all intra-slice bond currents are zero, independent of the types, locations, and size of the disorder, thereby showing order amidst disorder. We also have a proof (SM) for all quantum dragon nanoribbon devices the inter-slice bond currents depend only on the local hopping strength and v Drak . In particular, inter-slice, I i,j;i+1,j = 0 intra-slice.
For the special case B j,j+1 = −I m one has all I i,j;i,j+1 = v 2 Drak,i as shown in the bottom-left panel of fig. 1. The bond currents thus exhibit order amidst disorder.
Conclusions. -We have shown in standard tight binding Hamiltonians of eq. (1) using the standard NEGF formulation there are a large number of nanodevices with correlated but strong disorder that are quantum dragon nanodevices with T (E) = 1 for all −2 < E < 2. We have shown such nanoribbon devices have order amidst disorder, i.e., the nanodevice is disordered but the propagating electron LDOS and bond currents are ordered. See the supplementary material (CDF and SM) for additional nanodevices with order amidst disorder. Further research will consider 3D nanodevices [23], and devices that are nearly quantum dragon nanodevices. Our order amidst disorder predictions are based on exact calculations for the steady state quantum transport properties. Nevertheless the predictions need to be verified in real nanodevices (for example DNA nanodevices with m = 2 [24][25][26][27] or graphene nanoribbons [13]), discrete time crystals [28], or quantum emulators for the tight binding Hamiltonian [29,30].
Data availability statement: All data that support the findings of this study are included within the article (and any supplementary files).