# Decoding the conductance of disordered nanostructures: a quantum inverse
problem

S. Mukim1,2, J. O’Brien1,2, M. Abarashi3, M. S. Ferreira1,2, and C. G.
Rocha3,4∗ 1 School of Physics, Trinity College Dublin, Dublin 2, Ireland  2
Advanced Materials and Bioengineering Research (AMBER) Centre, Trinity College
Dublin, Dublin 2, Ireland  3 Department of Physics and Astronomy, University
of Calgary, 2500 University Drive NW, Calgary, Alberta T2N 1N4, Canada  4
Hotchkiss Brain Institute, University of Calgary, 3330 Hospital Drive NW,
Calgary, Alberta T2N 4N1, Canada<EMAIL_ADDRESS>
###### Abstract

Obtaining conductance spectra for a concentration of disordered impurities
distributed over a nanoscale device with sensing capabilities is a well-
defined problem. However, to do this inversely, i.e., extracting information
about the scatters from the conductance spectrum alone, is not an easy task.
In the presence of impurities, even advanced techniques of inversion can
become particularly challenging. This article extends the applicability of a
methodology we proposed capable of extracting composition information about a
nanoscale sensing device using the conductance spectrum. The inversion tool
decodes the conductance spectrum to yield the concentration and nature of the
disorders responsible for conductance fluctuations in the spectra. We present
the method for simple one-dimensional systems like an electron gas with
randomly distributed delta functions and a linear chain of atoms. We prove the
generality and robustness of the method using materials with complex
electronic structures like hexagonal boron nitride, graphene nanoribbons, and
carbon nanotubes. We also go on to probe distribution of disorders on the
sublattice structure of the materials using the proposed inversion tool.

  * August 2021

##  1 Introduction

It is a simple Quantum Mechanics problem to find the electronic conductance of
a nanoscale device by directly solving the Schrödinger equation that governs
the wave function of the quantum system under study. Obviously, this is the
case if and only if the underlying Hamiltonian is known, i.e., if all
scattering sources and interactions are fully specified. Trying to perform the
same task in reverse is significantly more challenging. For example, assuming
that the conductance of the system is known, how can one infer about the
Hamiltonian components from that information alone? To make matters worse,
what if the system is made of a heavily disordered material? Questions of this
type are generally labeled as Inverse Problems (IP) and are classified as
those which attempt to obtain from a set of observations the causal factors
that generated them in the first place. IP are integral parts of classical
visualization tools [1, 2, 3, 4] but far less common in the quantum realm. The
literature on the field of Quantum IP (QIP) is primarily focused on the
fundamentals of inversion processes, e.g., whether a problem is ill-posed [5],
whether solutions are unique and how stable they are [6]. Following a
different scope, studies of nanomaterials and their physical properties could
benefit from applications of QIP since they involve investigating structures
for which the underlying Hamiltonians are not always known [7, 8, 9, 10, 11,
12, 13, 14, 15].

Identifying the precise Hamiltonian that generates a specific observable is a
difficult task. In general, it consists of solving the Schrödinger equation
with a Hamiltonian containing one (or more) parameter(s) that must be changed
until the solution closely matches the original observation. Because of such a
large variance in phase space, finding the optimum parameter set can be
computationally demanding. With the advance of high-performance computing and
various optimization algorithms available, distinct methods for probing the
parameter phase space in electronic structure problems can be used. Neural-
network-based search engines [16, 17], genetic algorithms [18, 19, 20], and
more recently, machine-learning strategies [21, 22, 23, 24, 25, 26, 27, 28,
29] have been proposed and, with different degrees of success, can speed up
the search for the “inverted” solution. As a result, simulations are starting
to have an impact in reducing the time and cost associated with materials
design, specially those involving high throughput studies of material groups
[30, 31, 32, 33, 34, 35, 36, 37, 38, 39]. These are large-scale simulations
that generate volumes of data with the intention of identifying optimal
combinations that can be subsequently used as candidates for an exploratory
search for new materials. These methods certainly serve their purposes of
determining the optimum parameter phase space that fits a given observable.
Nonetheless, they can be less intuitive to implement due to their data-science
scope and somehow “black-box” approach embedded in their algorithms [40]. A
significant portion of the electronic structure community studying low
dimensional materials is certainly familiar with the Hamiltonian construction
in the Schrödinger equation but also with its reciprocal form, involving the
use of equations of motion for the inverse of the Hamiltonian operator which
can be identified as the Green’s function. Green’s function-based methods[41]
have played a central role as a theoretical tool to study electronic and
transport properties of nanoscale systems. These are versatile methods that
can be used to obtain the conductance of complex low dimensional materials,
including disordered ones.

A mathematically transparent inversion technique capable of extracting
structural and composition information from a disordered quantum device by
looking at its conductance signatures has recently been proposed [42]. Using
energy-dependent two-terminal conductance as input functions, the reported
inversion method identifies the concentration of impurities disorderly
distributed within a graphene nanoribbon (GNR) host. Despite claims that the
method is general, robust, and stable, evidence has so far been limited to
nitrogen-doped GNRs [42]. In this manuscript, we put the generality of this
inversion methodology to the test by implementing it with a variety of
systems, some described by extremely simple electronic structure models,
others by more realistic ones. Furthermore, the inversion accuracy is also
tested and shown to be able to identify correctly the impurity concentration
of a number of systems studied in this work. This suggests that this method
can indeed be employed as a useful characterisation tool with a variety of
systems composed of different materials and dimensions. The method relies on
working with an objective function that can be built directly from conductance
calculations which we call the ‘misfit’ function. A remarkable advantage of
the method is its systematic approach in dealing with the misfit function,
without the need of extra adjustments or parameter tuning to perform the
optimization search of concentration and impurity characteristics. This
article starts by illustrating in the next section the inversion methodology
applied to a very simple one-dimensional atomic chain system. In subsequent
sections, we demonstrate how this inversion tool can be implemented to extract
valuable information about nanoscale systems acting as hosts of chemical
impurities; hosts include carbon nanotubes (CNTs), GNRs, and hexagonal boron
nitride (hBN) nanoribbons, on which foreign atoms are randomly distributed
over their atomic structure.

##  2 Methodology

To demonstrate the versatility of this inversion methodology, we separate the
cases we study into two groups: one with systems possessing very simple
electronic structures and another in which the materials are described by more
realistic band structures. The purpose of separating the case studies is
twofold: (i) it is a lot easier to introduce the inversion procedure with
simple systems since it helps preventing artifacts coming from complex
electronic structure models and our focus lies entirely on the inversion
method itself; (ii) showing that the inversion method works for systems
regardless of their electronic-structure details and of their dimensionality
is strong evidence of the robustness of the methodology. With that in mind, we
will present the inversion method with the first group consisting of three
different one-dimensional systems, all possessing rather simple electronic
structures. The first system consists of an electron gas in the presence of
short-range scatters represented by randomly distributed delta functions. In
this case, the delta-functions act as impurities in the presence of an
otherwise homogeneous potential produced by an infinitely-long host material.
Mathematically, the single-particle Hamiltonian is defined by the following
potential

$V_{1}(x)=\sum_{j=1}^{N}\lambda\,\delta(x-x_{j})\,\,,$ (1)

where $j$ is an integer running from $1$ up to the total number of scatters
$N$, $x$ is the spatial variable, $x_{j}$ gives the randomly-selected position
of the $j^{th}$ scatter, and $\lambda$ is a positive constant that reflects
the scattering strength of the impurities, assumed to be all identical for
simplicity. The $N$ random values of $\\{x_{j}\\}$ are constrained to be
within a region of length $L$ and fully determine the Hamiltonian as well as
the corresponding transmission coefficient $T(E)$, $E$ being the energy. Note
that the integer number of impurities $N$ or the non-integer concentration
$n=N/L$ are considered equivalent control parameters and will be used
interchangeably throughout this article.

In the second case study considered here, we relax the short-range aspect of
our scatters by replacing the delta-function potentials with identical square
barriers of width $d$ and height $V_{0}$. In other words, the new Hamiltonian
has now a potential described by

$V_{2}(x)=\sum_{j=1}^{N}V_{0}\,[\Theta(x-x_{j})-\Theta(x-x_{j}-d)]\,\,,$ (2)

where $\Theta$ is the Heaviside step function. Once again, the barrier
positions are determined by randomly selected values of $\\{x_{j}\\}$ with the
additional constraint that barriers must not overlap, i.e., no pair of values
$x_{j}$ and $x_{j^{\prime}}$ must be less than a distance $d$ apart.

The third and final case studied departs from a free-electron gas description
for the electronic structure of the host material. Still keeping a simplified
picture of the electronic structure, we move on to a lattice representation of
the Hamiltonian, in one-dimension, captured by the tight-binding picture of an
infinitely long chain of atoms. With no loss of generality, we adopt the
following Hamiltonian

${\hat{H}}_{3}=\sum_{j}(\,|j\rangle t\langle j+1|+|j\rangle t\langle
j-1|\,)+\sum_{j^{\prime}}|j^{\prime}\rangle\lambda_{j^{\prime}}\langle
j^{\prime}|\,\,,$ (3)

which contains an off-diagonal hopping term and a diagonal term describing the
potential on all sites of the system. In equation (3), $j$ and $j^{\prime}$
are integers that label the atomic sites, $|j\rangle$ represents an electronic
orbital centred at site $j$ and $t$ is the electronic hopping between nearest-
neighbour sites only. $\lambda_{j^{\prime}}$ represents the on-site potential
of substitutional impurities that are once again randomly distributed within a
region of size $L$. It is worth mentioning that impurities in the form of
adatoms (adsorption) can also be considered in the Hamiltonian in equation (3)
without fundamental changes to our approach.

All three cases introduced above can be characterised by an infinitely-long
host containing a number $N$ of randomly distributed impurities confined to
within a finite-sized region and described by the respective equations (1),
(2), and (3). With this general setup, one can obtain the transmission
coefficient across all three systems following a method of choice, e.g. Green
function[41, 43, 44, 45] or scattering matrix[46]. The top three panels of
Fig. 1 depict the electronic transmission $T(E)$ for a specific configuration
of each one of these simple cases. It is worth noting that in each one of the
three cases, the exact number of impurities and their respective locations
defining the parent Hamiltonian that generated the $T(E)$ curve of Fig. 1 are
set aside and will not be used in any part of the subsequent inversion
calculation. In other words, $T(E)$ serves as the only input function from
which the inversion will take place. The task at hand is a QIP that aims to
find structural and composition information about the scattered impurities
using the seemingly noisy curves shown on the top panels of Fig. 1 as the only
starting point. The transmission coefficient is a rather representative
quantity to use as input function because it has several parallels in real
materials that go beyond this simple illustrative model. For example, the
conductance of a metal is commonly expressed in terms of $T(E)$, following the
Landauer-Büttiker formalism [46]. Likewise, other quantities such as thermal
and optical conductivity can also be expressed in a spectral representation
that resembles the transmission coefficient[47]. In this work, we put emphasis
on extracting materials’ composition from electronic transport analysis
following the misfit function minimization scheme as illustrated in the case
studies above. It is important to emphasize a general aspect of our
methodology which is the possibility of decoding not only electronic
conductance curves but also response functions of other physical natures, e.g.
mechanical, optical, magnetic, etc. For instance, this methodology was
recently adapted to decode the optical conductivity of disordered MoS2 films
as presented in Duarte et al. [48]. Another promising applicability of the
methodology is to use in the context of mechanical response in line with the
approach proposed by Bao et al. [49]. Note also that our misfit function
methodology is not limited to conductance curves derived from numerical
methods; our method is applicable to any energy-dependent response function
that can serve as a “target” function for the misfit function minimization
scheme. Typical experimental data that could be used with this methodology
could come from Hall-bar experiments, in our case, conducted in 2D
nanostructures. As long as the main methodology parameters (i.e., bandwidth,
size of the ensemble, transport regime, and hamiltonian flavor) are set to
achieve the desired accuracy, a convergence to a minimum in the misfit
function can be expected for any fluctuating energy-dependent response
function obtained from experiments or numerical means.

As previously alluded to, a naive approach would be to attempt a comparison of
the input function with the transmission coefficient of every single possible
disorder configuration but the prohibitively large number of combinations
proves this strategy impracticable.

Figure 1: (first row) Transmission coefficient ($T$), (second row) misfit
function ($\chi$), and (third row) error ($\alpha$) as a function of
representative physical quantities in disordered systems: $E$ is the energy
expressed in certain units of energy $t$, $N$ is the number of scatters, $n$
is the impurity concentration expressed in $\%$, B.W. is the normalized
bandwidth used for integration in the inversion procedure, and $M$ is the
total number of disordered configurations in the studied ensemble. These
results were obtained for three case study systems: (first column) electron
gas with delta functions representing scatters, (second column) electron gas
with identical square barriers of width $d=1$ and height $V_{0}=1$ in
arbitrary units, (third column) one-dimensional (infinite) atomic chain. In
all cases, $t=1$. For the third case (atomic chain), the transmission is
scaled in terms of the conductance quantum, $\Gamma_{0}=2e^{2}/h$ with $e$
being the elementary charge, and $h$ the Planck constant. Panel g) also
contains the transmission as a function of energy for the pristine atomic
chain (red curve). Impurities in the doped case were modelled with on-site
energy of $\lambda=0.5t$ and they were spread within a length of $L=100$
atomic layers. Transmission results in the first row are for the target
systems in which the misfit function locates a minimum: (a,b) electron gas
with $N_{min}=5$ delta functions, (d,e) electron gas with $n_{min}=4\%$ square
barrier scatters, and (g,h) 1D atomic chain with $n_{min}=8\%$ impurities.

Fortunately, the inversion procedure introduced in [42] provides a far more
efficient technique which combines configurationally-averaged (CA) values of
the transmission coefficient with the ergodic principle. More specifically,
the ergodic hypothesis assumes that a running average over a continuous
parameter upon which the transmission coefficient depends is equivalent to
sampling different impurity configurations and this can be used to assist us
in the inversion procedure. In mathematical terms, this is done through the
so-called misfit function defined as [42]

$\chi(n)={1\over{\cal E}_{+}-{\cal E}_{-}}\,\int_{{\cal E}_{-}}^{{\cal
E}_{+}}dE\,(T(E)-\langle T(E,n)\rangle)^{2}\,\,,$ (4)

where the integration limits are arbitrary energy values. $\chi$ can be
interpreted as a functional that measures the deviation between the input
transmission of the parent configuration $T(E)$ and its CA counterpart
$\langle T(E)\rangle$, defined as

$\langle T(E,n)\rangle={1\over M}\sum_{m=1}^{M}T_{m}(E).$ (5)

where $M$ is the total number of different disordered configurations taken
into account and $T_{m}(E)$ is the corresponding transmission coefficient for
each one of these configurations, which are themselves labelled by the integer
$m$. While both $T(E)$ and $\langle T(E)\rangle$ in the integrand above are
functions of energy, the latter is also a function of the impurity
concentration $n$. When plotted as a function of $n$, the misfit function
displays a very distinctive minimum at a value that corresponds to the real
impurity concentration. This can be seen in the middle panels of Fig. 1. The
concentration $n_{min}$ that minimises the misfit function coincides with the
real concentration $n_{i}$ in all three cases, indicating that the inversion
is successful. Also included in the Figure (bottom-row panels), we plot the
relative error $\alpha=|n_{min}-n_{i}|/n_{i}$, where $n_{min}$ stands for the
concentration for which the misfit function is minimum. The error $\alpha$ is
calculated over several different parent configurations in order to achieve
statistical significance.

One question that naturally arises is how relevant the energy integral of
equation (4) is. On the one hand, the energy integration plays a significant
part in identifying the real impurity concentration because $T(E)$
fluctuations of a single sample versus energy is equivalent to sample-to-
sample fluctuations at a fixed energy. Therefore, this energy integration is
similar to vastly augmenting the number of disordered configurations taken
into account. In fact, if we were to use equation (4) with ${\cal E}_{-}={\cal
E}_{+}$, the misfit function would not have any distinctive feature and would
not lead to a successful inversion [42]. However, if the integration limits
span a small fraction of the relevant energy range, the misfit function
acquires a very distinctive shape with minima located at the correct
concentrations.

Some further insights we can learn from Fig. 1 are as follows. Fig. 1(c) plots
the inversion error $\alpha$ as a function of the normalized bandwidth defined
as B.W.$=({\cal E}_{+}-{\cal E}_{-})/Z$, with $Z$ being the total bandwidth of
the electron gas system. Note that $\alpha$ is large for very narrow energy
windows. When the energy window for integration is broadened beyond $20\%$ of
the bandwidth, $\alpha<0.1$ in the case of $N=5$ delta function potentials.
This indicates that larger energy windows accounted in the misfit function can
capture more of the transmission features, minimizing then the error of the
inversion procedure. The integration in equation (4) can be hence replaced
with a discrete sum without major impact to the accuracy of the inversion
procedure. By reducing the number of energy points ${\cal N}_{\epsilon}$
involved in generating the misfit function, there is a point below which the
inversion accuracy drops quite significantly but this threshold is normally at
very low levels (${\cal N}_{\epsilon}<50$), which enables calculations with
systems for which the numerical complexity is significantly higher [42], as we
will demonstrate in the next section.

Regarding the error study done in Fig. 1(f) and the number of configurations
needed in the CA values $M$, the backbone of our inversion procedure is that
fluctuations in the conductance ($\Gamma$) contain very little system specific
information but the average conductance depends smoothly on the variable of
interest. In this sense, the higher the value of disorder realizations $M$,
the smaller the fluctuations in $\langle\Gamma(E,n)\rangle$. Universal
conductance fluctuations (UCF) can be used to estimate values of $M$ required
for sufficiently accurate results in the inversion procedure. Both for
diffusive [50, 51] and chaotic ballistic systems [52], var($\Gamma)\approx 1$
is the main fingerprint of UCF. Hence, the CA relative statistical error is
expected to scale with $[\sqrt{M}\times\langle\Gamma(E,n)\rangle]^{-1}$. To
bring the degree of fluctuations to an acceptable level that yields
statistically significant results, we estimate that $M$ should be of the order
of $10^{3}$ [53, 54]. As seen in Fig. 1(f), the error is only guaranteed to
stabilize at $M\sim 1000$. Depending on the complexity of the system, higher
values of $M$ can be required to achieve statistical significance.

Finally, Fig. 1(i) shows the inversion error $\alpha$ for an infinite 1D
atomic chain as a function of impurity concentration $n$. Substitutional
impurities were spread randomly over $L=100$ unit cells in length. One can see
that $\alpha$ is relatively small for concentrations below 10%, clearly
indicating that our inversion method is very reliable for dilute regimes. To
test the inversion procedure, we defined hamiltonians with the knowledge of
concentration as well as positions. By increasing the concentration of
scatters spread over a finite length, one can start forming pairs or atomic
clusters of impurities, in which the method is not equipped to distinguish
from single scatters. For this reason, we see a reduction in the accuracy of
our method ($\alpha$ increases) as impurity concentration increases.

##  3 Results and discussion

Figure 2:  (first row) Electronic transmission ($T$), (second row) misfit
function ($\chi$), and (third row) error ($\alpha$) as a function of
representative physical quantities in disordered systems: $E$ is the energy
expressed in units of hopping $t$ or directly in eV, $n$ is the impurity
concentration expressed in $\%$, $M$ is the total number of disordered
configurations in the studied ensemble, and B.W. is the normalized bandwidth
used for integration in the inversion procedure. These results were obtained
for three quasi-1D systems: (first column) carbon nanotube (CNT), (second
column) graphene nanoribbon (GNR), and (third column) hexagonal boron nitride
(hBN) nanoribbon. The transmission results on the top panels are scaled in
terms of the conductance quantum, $\Gamma_{0}=2e^{2}/h$ with $e$ being the
elementary charge, and $h$ the Planck constant. Top panels (a,d,g) contain the
transmission as a function of energy for the respective pristine systems (red
curves) and for the doped target systems (blue curves) in which the misfit
function locates a minimum: (a,b) CNT at $n_{min}=2.5\%$, (d,e) GNR at
$n_{min}=3\%$, and (g,h) hBN at $n_{min}=2\%$ impurities. The CNT is an
armchair (5,5) hosting impurities modelled with an on-site energy of
$\lambda=0.5\,t$ with $t=1$ spread over L=50 unit cells in length. The GNR is
an armchair-edge nanoribbon with 7 atoms along its width and it is hosting
impurities with an on-site energy of $\lambda=0.5\,t$ with $t=1$ spread over
L=100 unit cells in length. The hBN host is an armchair-edge ribbon with 8
atoms along its width and infinite in length. Carbon atoms act as impurities
and are spread over $L=100$ layers long. The on-site energy values to model B,
N, and C atoms are: $\lambda_{B}=-6.64$ eV, $\lambda_{N}=-11.47$ eV, and
$\lambda_{C}=-8.97$ eV[55]. See main text for details on the hopping
characteristics used to model the hBN.

We now proceed to introduce three new case studies in the second group of
systems, which are less simplistic and more realistic. The first system in
this group consists of carbon nanotubes (CNTs); the second is made of graphene
nanoribbons (GNRs); the third and final case is made of hexagonal boron
nitride (hBN) nanoribbons. Each one of these materials will contain a finite
concentration of substitutional impurities that differ from their pristine
atomic composition. One obvious difference to the previous cases is that these
are no-longer one-dimensional systems and their electronic structures are a
lot more involved than the ones displayed by the cases shown in Fig. 1. Even
though these new cases possess band structures that have finer details than
the ones in the first group, it is still convenient to maintain the tight-
binding description of the electronic structure. Successful inversions have
already been reported with Density Functional Theory (DFT) calculations used
to describe the multi-orbital electronic structure of doped graphene
nanoribbons [42]. Despite its success, ab-initio calculations are far more
time-consuming than semi-empirical approaches such as tight-binding, thus
justifying our choice of electronic structure model for this manuscript.
Therefore, the Hamiltonian appearing in equation (3) is still applicable to
describe the cases in this group, the main difference being that the integer
$j$ now accounts for all sites of these quasi-2D structures. Particularly for
the hBN case, the Hamiltonian will carry three on-site contributions to
account for the nitrogen ($\lambda_{N}$), boron ($\lambda_{B}$), and impurity
atoms as we will detail later on.

Results found for these new cases are shown in Fig. 2 with CNT on the left
panels, GNR on the middle panels and hBN on the right panels. Following the
same pattern of Fig. 1, the top-row panels display the electronic transmission
of the parent configuration that serves as the input function in the inversion
procedure. Note that in this case, these functions can be interpreted as the
Fermi-energy-dependent conductance. Also on the same plots are the impurity-
free conductance plots shown for comparison purposes. The middle-row panels
depict the corresponding misfit function $\chi(n)$, always plotted as a
function of the concentration of impurities $n$. Once again for the sake of
comparison, the same plot contains a vertical dashed line that indicates the
real impurity concentration contained in the parent configuration. Finally,
the bottom-row panels indicate the accuracy of the inversion procedure by
plotting the error $\alpha$ as a function of certain control parameters that
will be discussed later.

For the CNT results on the left-column panels of Fig. 2, they contain
impurities represented by on-site potentials of $\lambda=0.5$ in units of
hopping (see equation (3)). While this value corresponds to an arbitrary
choice that does not represent any specific atomic species, the inversion
method correctly identifies the impurity concentration for whichever choice of
$\lambda$ values. Such a good agreement between the inverted result and the
real concentration is seen in the misfit function plot for the nanotube. The
concentration that minimizes $\chi$, $n_{min}$, coincides with the
$n_{i}=2.5\%$ chosen for the parent configuration. Having performed the
inversion procedure with numerous different parent configurations, we were
able to assess its success rate in the case of nanotubes serving as hosts.
Fig. 2(c) depicts the error $\alpha$ plotted as a function of the bandwidth,
revealing that the accuracy of the method improved as the integration window
of the misfit function increased. The central-column panels display equally
good results for the GNR with the same choice of $\lambda=0.5$ in units of
hopping to mimic substitutional impurity atoms. Fig. 2(d) shows the
conductance of a pristine armchair-edged GNR with 7 atoms wide plotted as a
function of energy. Note that only the conduction band ($E>0$) is shown to
simplify visualization. On the same panel, the conductance of the parent
configuration containing 3% of impurities is also shown. Fig. 2(e) displays
the $n$-dependent misfit function with a distinctive minimum that agrees with
the real concentration, $n_{i}=3\%$. The accuracy for the nanoribbon
inversions is tested by plotting $\alpha$ as a function of concentration in
Fig. 2(f). Once more, we observe that the inversion method performs better in
dilute regimes of doping, with the error $\alpha$ increasing relatively slow
with the concentration of impurities.

The right-column panels of Fig. 2 show results for an armchair-edge hBN
nanoribbon with 8 atoms along its width. The ribbon is infinite in length but,
in its doped form, impurities are spread over a section of $L=100$ layers
long. The tight-binding Hamiltonian for the hBN host uses three distinct on-
site energy values to model B and N atoms on the host and C atoms as
impurities of the system: $\lambda_{B}=-6.64$ eV, $\lambda_{N}=-11.47$ eV, and
$\lambda_{C}=-8.97$ eV[55]. Hopping terms were parameterized as
$t=-6.17/a^{2}$ [55, 56] with $a$ being the pair-wise bond length. For a
boron-nitrogen bond, $a=1.43$ Å and, for the sake of simplicity, we consider
that boron-carbon and nitrogen-carbon bond lengths do not change significantly
from that value. The noisy conductance curve in Fig. 2(g) corresponds to the
parent system of a hBN ribbon doped with $n_{i}=2\%$ of substitutional
impurities and the red curve is the conductance of the pristine hBN
nanoribbon. The misfit function shown in Fig. 2(h) acquires a minimum at
$n_{min}=2\%$, evidencing that the method predicted correctly the
concentration of the parent system. Finally, Fig. 2(i) confirms that the
accuracy of the method is significantly improved as the number of disordered
configurations in the ensemble, $M$, increases.

Figure 3: (Left panel) Misfit function ($\chi$) surface plot taken for an
armchair-edge hBN nanoribbon with 7 atoms along its width. The ribbon is
infinite but carbon impurities were spread randomly over $L=100$ unit cells
along its length. The bar color express values of $\chi$. The control
parameters along x- and y-axis are $N_{N}$ and $N_{B}$, respectively; those
account for the number of carbon impurities replacing nitrogen ($N_{N}$) or
boron ($N_{B}$) atoms. Dashed lines intersect at the characteristics of the
parent system with $N_{N}=50$ and $N_{B}=20$. The minimum of $\chi$ correctly
predicts the characteristics of the parent system as it lays within the region
where $N_{N}=50$ and $N_{B}=20$. (Right panel) Curve of $\chi$ versus $N_{N}$
taken from left panel by slicing the misfit surface plot horizontally at
$N_{B}=20$ (red horizontal dashed line). A distinctive minimum at $N_{N}=50$
(red vertical dashed line) correctly finds the occupation of carbon impurities
on nitrogen sites of the parent system. Bandwidth considered for inversion is
70% of the entire spectrum.

Another way to prove the generality of our methodology is to extend its use to
a multi-dimensional phase space. Till this point, the inversion procedure
looked at one degree of freedom which was the impurity concentration. However,
it is possible to extend this analysis to a two-dimensional phase space, e.g.,
as we are going to illustrate. This analysis turns particularly interesting if
applied to the hBN case because impurities can replace two types of atoms on
the host (N or B). This means we can decompose the concentration information
into the two distinct sublattices. Considering that a total of $N$ impurities
can be substitutionally dope a segment length of an hBN nanoribbon, we can
write that $N=N_{B}+N_{N}$ where $N_{B}$ ($N_{N}$) is the number of boron
(nitrogen) atoms replaced by an impurity. Therefore, we can write the misfit
function in terms of these two degrees of freedom to probe occupation of
impurities on both boron and nitrogen sites as

$\chi(N_{B},N_{N})=\int_{{\cal E}_{-}}^{{\cal E}_{+}}dE\,\left[\,T(E)-\langle
T(E,N_{B},N_{N})\rangle\,\right]^{2}\,\,$ (6)

A certain number of C atoms are then spread over the hBN host in such a way
that they can replace equal portions of B and N atoms or they can cause a
sublattice unbalance. In other words, we can define a parameter $\delta$ as
$\delta=N_{N}-N_{B}$ that can serve as a metric for this unbalance: if
$\delta>0$, C atoms majorly occupy the nitrogen sublattice, if $\delta<0$, C
atoms majorly occupy the boron sublattice, and if $\delta=0$, C atoms occupy
both sublattices equally. A 2D contour plot of $\chi$ as defined in equation
(6) is presented in Fig. 3 left panel. The global minimum correctly coincides
with the occupation of impurities placed on boron ($N_{B}=20$) and nitrogen
($N_{N}=50$) sublattice of the parent system highlighted by the intersection
of vertical and horizontal dashed red lines. The right panel in Fig. 3
corresponds to an horizontal slice taken from the surface plot at fixed
$N_{B}=20$ to visualize the dependency of $\chi$ with $N_{N}$ and to highlight
the minimum of $\chi$ at $N_{N}=50$. The fundamental difference between boron
and nitrogen sites allows to identify occupation of carbon impurities on
corresponding sublattices. However, identifying the exact position of
impurities still remains an elusive task. This example proves that extending
the inversion methodology to inspect more than one degree of freedom does not
affect its generality and robustness. Looking at more than one degree of
freedom in the misfit function certainly increases computational cost and may
generate multiple minima. Nonetheless, the method can serve as a first
approach to determine the exact features of a parent system that can be
described in a multi-dimensional phase space. In other words, results as the
one depicted in Fig. 3 narrows down considerably the range of search for the
optimum $N_{B}$ and $N_{N}$ values. A refinement in the range of values can be
achieved by re-applying the inversion procedure within the reduced parameter
range and with an increased number of CA samples, $M$, and/or by increasing
the bandwidth established for the inversion procedure.

It is worth mentioning that our methodology is not immune to the typical
limitations regarding the investigation of large-scale systems containing over
$10^{3}$ atoms. Yet, we tested our methodology with relatively large systems,
e.g. nanoribbon systems with $\sim 2000$ atoms along their length. A single
point calculation to determine the accuracy of the method took approximately
30-60 minutes to complete depending on the ensemble size. Depending on the
computational resources available, this computing time can be significantly
reduced. Certainly systems treated with more complex Hamiltonians, beyond the
nearest-neighbour single-orbital tight-binding approximation, can take longer
computational times. Nonetheless, our methodology can be paired with other
scalable electronic transport approaches, e.g. as the one proposed by Fan et
al. [57] in which an optimization for the linear-scaling Kubo-Greenwood
formula is presented, allowing large-scale calculations in systems of $\sim
10^{6}$ atoms.

##  4 Conclusions

In this manuscript, we illustrated the use of an inversion methodology capable
of extracting information out of disordered systems, more specifically host
media perturbed by local potentials representing impurities or dopants. For
situations in which we know the initial conditions of the system of study, the
method may sound unimpressive. However, if we do not have access to its
initial conditions, one needs to find a way of retrieving the system initial
setup by inspecting response functions of the system subjected to
perturbations. In this work, we studied a series of solid state systems
ranging from electron gas, 1D atomic chain, up to carbon nanotubes, graphene
and hBN nanoribbons that give the conductance or electronic transmission as
response function. Each of these (host) solid state systems were perturbed by
a certain concentration of impurities and their presence induce fluctuations
in the transmission response functions, turning them into quite noisy spectra.
It is not straightforward to deduce the concentration of impurities doping the
hosts by only looking at the noisy profile of the transmission curve. We
proved the generality of an inversion methodology proposed by Mukim et al.
[42] in which unknown quantities such as the impurity concentration in doped
nanoscale systems can be uncovered by computing the so called “misfit
function”, an objective function written in terms of the configurationally
averaged conductance and the target conductance. The misfit function reveals
minima at locations in the phase space parameter that correspond to the
unknown initial conditions of the system, in this case, uniquely characterized
by the concentration of impurities doping a host material. The method itself
can be built upon numerous control parameters that inform on its accuracy and
performance such as relative error to target, influence of bandwidth window,
number of configurationally averaged samples in the ensembles, to name but a
few. We observed that the accuracy of the method improves considerably when:
(i) sufficiently large bandwidths are selected for the integration of the
misfit function, (ii) the sampling of the ensemble is increased, (iii) target
systems are in dilute regime of sufficiently low concentration of dopants. In
summary, the method proved capable of “decoding” noisy transmission response
functions into a more meaningful mathematical representation, the misfit
function, that exhibits minima at the characteristics defining the unknown a
priori initial conditions of the studied systems.

## Acknowledgements

This publication has emanated from research supported in part by a research
grant from Science Foundation Ireland (SFI) under Grant Number
SFI/12/RC/2278-P2. This work was also supported by U of C start-up funding and
partially support by the Natural Sciences and Engineering Research Council of
Canada (NSERC), [Discovery Grant funding reference number xxxxxx]. We also
acknowledge the WestGrid (www.westgrid.ca), the Compute Canada Calcul Canada
(www.computecanada.ca), and the CMC Microsystems (www.cmc.ca) for
computational resources.

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