# Topological conditions for impurity effects in graphene nanosystems

Yuriy G. Pogorelov<EMAIL_ADDRESS>IFIMUP-IN, Departamento de Física,
Universidade do Porto, Porto, Portugal,  Vadim M. Loktev<EMAIL_ADDRESS>N. N. Bogolyubov Institute of Theoretical Physics, NAS of Ukraine, Kyiv,
Ukraine,
&
Igor Sikorsky Kyiv Polytechnic Institute, Kyiv, Ukraine,

###### Abstract

We consider electronic spectra of graphene nanotubes and their perturbation by
impurity atoms absorbed at different positions on nanotube surfaces, within
the framework of Anderson hybrid model. A special attention is given to the
cases when Dirac-like 1D modes appear in the nanotube spectrum and their
hybridization with localized impurity states produces, at growing impurity
concentration $c$, onset of a mobility gap near the impurity level and even
opening, at yet higher $c$, of some narrow delocalized range within this
mobility gap. Such behaviors are compared with the similar effects in the
previously studied 2D graphene and armchair type graphene nanoribbons. Some
possible practical applications are discussed.

graphene, armchair and zigzag edges, nanotubes, impurity adatoms, spectrum
restructuring

##  I Introduction

From the discovery, almost 20 years ago, of single-layer graphene Geim2004 ,
an enormous interest has been attracted not only to its two-dimensionality
(2D) Geim2005  but mainly to its massless, that is Dirac-like, spectrum of
electronic excitations Geim2009 . Studies of its various physical properties
have a very broad nomenclature Tomanek_Book ; Inagaki2014  (see also
Katsnelson ) but we focus here on certain aspects of non-ideal graphene
structures, yet restricted to a single dimension (1D), namely, of graphene
nanoribbons Wakabayashi2010  and graphene nanotubes Iijima2002  in presence of
impurities. In more detail, we consider here the electron quasiparticle
spectra in principal topological types of graphene 1D nanosystems and their
restructuring under effects by impurity atoms absorbed at different positions
over carbon atoms Wehling2009 . In this course, the main attention is given to
the cases when Dirac-like 1D modes exist in the nanotube spectrum
Wakabayashi1996  and then the impurity disorder effects on these modes are
compared with the previously studied similar effects in 2D graphene YDV2021
and in armchair type nanoribbons (ANR’s) PL2022 .

The presentation is organized as follows. We begin from description of
quasiparticle spectra for the two nanotube topologies: armchair nanotubes
(ANT’s, Sec. II) and zigzag nanotubes (ZNT’s, Sec. III), in the forms adjusted
to describe the impurity induced restructuring of spectra. This description,
within the simplest T-matrix approximation for the quasi-particle self-energy,
begins from the technically simpler ZNT case (Sec. IV) and then extends to the
more involved ANT case (Sec. V). The next comparison with the previous results
for 2D graphene and ANR’s (Sec. VI) reveals both qualitative similarities and
some quantitative differences in their behaviors. The obtained results are
then verified with some T-matrix improvements (Sec. VII): the self-consistent
T-matrix method and the group expansion (GE) method, both of them confirming
validity of the simple T-matrix results. At least, a comparison of the ZNT and
ANT host structures with the more general topology of twisted nanotubes is
done in Sec. VIII, suggesting a qualitative difference between the twisted and
non-twisted nanotubes in their sensitivity to impurity disorder. The final
discussion of these theoretical results and of some perspectives for their
practical applications is given in Sec. IX.

##  II Armchair nanotubes

Let us consider nanoribbons cut out of 2D graphene. There are two basic
orientations of their edges: zigzag and armchair, and we begin from the
armchair edge type. Such an ANR can be seen as a composite of $n$ chains
(labeled by $j$ indices) of transversal period $a$ (the graphene lattice
constant), each chain containing $N\gg 1$ segments (labeled by $p$ indices) of
longitudinal period $a\sqrt{3}$, each segment including 4 atomic sites
(labeled by $s$ indices, see Fig. 1).

Figure 1: An armchair nanoribbon composed of $n$ atomic chains ($j$-labeled),
each chain consisting of segments ($p$-labeled) with 4 atomic sites: $s=1$
(red), $2$ (green), $3$ (white), $4$ (blue), in each segment.

Next, we consider the closure between the 1st and $n$th chains of an ANR,
transforming it into an ANT without its twisting, see Fig. 2).

Figure 2: An armchair nanotube formed by closing links between the 1st and
$n$th chains from the armchair nanoribbon in Fig. 1.

For the following consideration of electronic dynamics, it is suitable to
combine the local Fermi operators $a_{p,j,s}$ at 4 $s$-sites from $j$th chain
in $p$th segment into the 4-spinor:

$a_{p,j}=\left(\begin{array}[]{c}a_{p,j;1}\\\ a_{p,j;2}\\\ a_{p,j;3}\\\
a_{p,j;4}\\\ \end{array}\right).$ (1)

Then the longitudinal translation invariance (with the $a\sqrt{3}$ period, see
Fig. 1) and the discrete transversal rotation invariance of the obtained ANT
suggest the Fourier expansion of the local spinor components in quasi-
continuous longitudinal momentum $-\pi/\sqrt{3}<k<\pi/\sqrt{3}$ and in
discrete transversal wave number $q=0,\dots,n-1$ (both in $a^{-1}$ units):

$a_{p,j,s}=\frac{1}{2\sqrt{nN}}\sum_{k,q}\exp\left[i\left(\sqrt{3}kp_{s}+\frac{2\pi
q}{n}j_{s}\right)\right]\alpha_{k,q,s},$ (2)

its components $\alpha_{k,q,s}$ forming the wave spinor $\alpha_{k,q}$. Here
the longitudinal numbers for different $s$-sites are: $p_{1,2}=p\pm 1/6$,
$p_{3,4}=p\pm 1/3$ and their transversal numbers are: $j_{1,2}=j$,
$j_{3,4}=j+1/2$. Then the ANT Hamiltonian with only account of hopping between
nearest neighbor atoms (its parameter, $t\approx 2.8$ eV Geim2009 , taken as
the energy scale in what follows) is presented in terms of the wave spinors
as:

$H=\sum_{k,q}\alpha_{k,q}^{\dagger}\hat{H}_{k,q}\alpha_{k,q},$ (3)

where the $4\times 4$ matrix:

$\hat{H}_{k,q}=\left(\begin{array}[]{cccc}0&h_{k}&0&h_{k,q}^{\ast}\\\
h_{k}^{\ast}&0&h_{k,q}&0\\\ 0&h_{k,q}^{\ast}&0&h_{k}\\\
h_{k,q}&0&h_{k}^{\ast}&0\end{array}\right)$ (4)

has non-zero elements $h_{k}={\rm e}^{ik/\sqrt{3}}$ and $h_{k,q}=2{\rm
e}^{ik/2\sqrt{3}}\cos\tfrac{\pi q}{n}$. The ANT spectrum results from four
eigen-values of this matrix at given $k$ and $q$ as:

$\displaystyle\varepsilon_{k,q;1}=-\varepsilon_{k,q;2}=-\varepsilon_{k,q},$
$\displaystyle\qquad\varepsilon_{k,q;3}=-\varepsilon_{k,q;4}=-\varepsilon_{k,n-q},$
(5)

where the basic dispersion law:

$\varepsilon_{k,q}=\sqrt{1+4\cos\tfrac{\sqrt{3}k}{2}\cos\tfrac{\pi
q}{n}+4\cos^{2}\tfrac{\pi q}{n}}.$ (6)

This can be seen either as the standard graphene dispersion law Wallace  but
with discrete transversal momentum numbers $q$ or, otherwise, as a set of $4n$
1D $k$-bands $\varepsilon_{k,q;f}$ (for $n$ possible values of $q$ and 4
values of $f$). Notably, a double degeneracy of these bands follows from Eqs.
5, 6 as:

$\varepsilon_{k,q;1}\equiv\varepsilon_{k,n-q;3},\quad\varepsilon_{k,q;2}\equiv\varepsilon_{k,n-q;4}.$
(7)

Thus, if $n$ is even, the ANT spectrum has 4 non-degenerated (for $q=0$ and
$q=n/2$) modes and $2n-2$ doubly-degenerated ones. Otherwise, if $n$ is odd,
there are two non-degenerated modes (only for $q=0$) and $2n-1$ doubly-
degenerated ones. The eigen-operators $\psi_{k,q;f}$ of these modes enter the
diagonal ANT Hamiltonian:

$H=\sum_{k,q,f}\varepsilon_{k,q;f}\psi_{k,q;f}^{\dagger}\psi_{k,q;f}.$ (8)

These operators at given $k$ and $q$ can be also combined into the 4-spinor
$\psi_{k,q}$, related to the $\alpha$-spinor as:

$\psi_{k,q}=\hat{U}_{k,q}\alpha_{k,q},$ (9)

through the unitary matrix:

$\hat{U}_{k,q}=\frac{1}{2}\left(\begin{array}[]{cccc}-z_{k,q}&-1&z_{k,q}&1\\\
z_{k,q}&-1&-z_{k,q}&1\\\ -z_{k,n-q}&1&-z_{k,n-q}&1\\\
z_{k,n-q}&1&z_{k,n-q}&1\end{array}\right),$

with the complex phase factor

$z_{k,q}=\exp\left[i\left(\frac{k}{\sqrt{3}}+\arctan\frac{\sin\tfrac{k}{2\sqrt{3}}}{2\cos\tfrac{\pi
q}{n}-\cos\tfrac{k}{2\sqrt{3}}}\right)\right].$

Contrariwise, the $\alpha$-spinor follows from the $\psi$-spinor by the
inversion of Eq. 9:

$\alpha_{k,q}=\hat{U}_{k,q}^{\dagger}\psi_{k,q}.$ (10)

Another important features of the spectrum by Eq. 5 are:

i) presence of 2 flat (dispersionless) modes for even $n$ (then at $q=n/2$)
and

ii) presence of 4 gapless Dirac-like modes (DLM’s) for $n$ being a multiple of
3 (then at $q=n/3$ and $q=2n/3$).

The latter are just the 1D analogs to the 2D graphene Dirac modes with their
nodal points $K$ (here at $k=2\pi/\sqrt{3},\,q=n/3$) and $K^{\prime}$ (here at
$k=0,\,q=2n/3$), also they are fully analogous to DLM’s in ANR PL2022 .

Due to the DLM’s special sensitivity to local impurity perturbations, our
following treatment is mainly focused on them. In this course, the most
relevant energy range is the Dirac window (DW), exclusively occupied with
DLM’s and delimited by the inner edges of their nearest neighbor modes (Fig.
3). From Eq. 6, this window results of width:

$\Delta_{\rm DW}=2\left|1-\cos\frac{\pi}{n}-\sqrt{3}\sin\frac{\pi}{n}\right|,$
(11)

and with growing $n\gg 1$ it is narrowing as $\sim 2\sqrt{3}\pi/n$.

Figure 3: Dispersion laws for ANT of $n=6$ chains, with doubly-degenerated
(black) and non-degenerated (red) modes, the Dirac window of width
$\Delta_{\rm DW}=2(\sqrt{3}-1)$ is in between the shaded ranges of the resting
modes.

Now, considering the low-energy spectrum range, the expansion of local
operators by Eqs. 2, 10 can be restricted to 8 $K,K^{\prime}$ DLM’s which
share 4 eigen-energies:

$\displaystyle\varepsilon_{k,K,1}=-\varepsilon_{k,K,2}=-\varepsilon_{k,K^{\prime},1}$
$\displaystyle\qquad\qquad\qquad=\varepsilon_{k,K^{\prime},2}=2\sin\frac{\sqrt{3}k}{4},$
$\displaystyle\varepsilon_{k,K,3}=-\varepsilon_{k,K,4}=-\varepsilon_{k,K^{\prime},3}$
$\displaystyle\qquad\qquad\qquad=\varepsilon_{k,K^{\prime},4}=2\cos\frac{\sqrt{3}k}{4}.$
(12)

Thus, the restricted expansion of a local spinor in eigen-spinors is presented
in the form:

$\displaystyle a_{p,j}=\frac{1}{2\sqrt{nN}}\sum_{k}\,{\rm
e}^{i\sqrt{3}kp}\left({\rm
e}^{i\tfrac{2\pi}{3}j}U_{k,K}^{\dagger}\psi_{k,K}\right.$
$\displaystyle\qquad\qquad\qquad\qquad+\left.{\rm
e}^{i\tfrac{4\pi}{3}j}U_{k,K^{\prime}}^{\dagger}\psi_{k,K^{\prime}}\right),$
(13)

including the unitary matrices:

$\displaystyle\hat{U}_{k,K}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\begin{array}[]{cccc}-iz_{k}&-1&iz_{k}&1\\\
iz_{k}&-1&-iz_{k}&1\\\ z_{k}&1&z_{k}&1\\\
-z_{k}&1&-z_{k}&1\end{array}\right),$ (18)
$\displaystyle\hat{U}_{k,K^{\prime}}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\left(\begin{array}[]{cccc}-z_{k}&-1&z_{k}&1\\\
z_{k}&-1&-z_{k}&1\\\ -iz_{k}&1&-iz_{k}&1\\\
iz_{k}&1&iz_{k}&1\end{array}\right)$ (23)

with $z_{k}={\rm e}^{ik/2\sqrt{3}}$. This expansion is suitable for the
following construction of impurity perturbation Hamiltonian.

##  III Zigzag nanotubes

For the case of ZNT we also consider their untwisted structure obtained from
an $n$-chain zigzag nanoribbon (ZNR, Fig. 4) by closure between its 1st and
$n$th chains (Fig. 5).

Figure 4: A zigzag nanoribbon of $j=1,\dots,n$ chains, with $s=1$ (blue), 2
(green), 3 (red), and 4 (white) sites in each $p$th segment. Figure 5: An
$n$-chain zigzag nanotube formed by closure between the 1st and $n$th chains
of zigzag nanoribbon from Fig. 4.

Comparison of Fig. 4 with Fig. 1 readily shows that the ZNT elementary cell
results just from 90 degrees rotation of the ANT one, so the analysis of ZNT
spectra simply follows that for ANT but with the $\sqrt{3}k\longleftrightarrow
2\pi q/n$ interchange. Thus the ZNT Hamiltonian in terms of 4-spinor wave
operators results, in analogy with the ANT form by Eq. 2, as:

$H=\sum_{k,q}\alpha_{k,q}^{\dagger}\hat{H}_{q,k}\alpha_{k,q},$ (24)

where the $4\times 4$ matrix:

$\hat{H}_{q,k}=\left(\begin{array}[]{cccc}0&h_{q}&0&h_{q,k}^{\ast}\\\
h_{q}^{\ast}&0&h_{q,k}&0\\\ 0&h_{q,k}^{\ast}&0&h_{q}\\\
h_{q,k}&0&h_{q}^{\ast}&0\end{array}\right)$ (25)

has non-zero elements $h_{q}={\rm e}^{i2\pi q/3n}$ and $h_{q,k}=2{\rm e}^{i\pi
q/3n}\cos k/2$. The ZNT spectrum at given $k$ and $q$ results from the
indicated i nterchange in Eqs. 5, 6, as:

$\displaystyle\varepsilon_{q,k;1}=-\varepsilon_{q,k;2}=-\varepsilon_{q,k},$
$\displaystyle\qquad\varepsilon_{q,k;3}=-\varepsilon_{q,k;4}=-\varepsilon_{n-q,k},$
(26)

where:

$\varepsilon_{q,k}=\sqrt{1+4\cos\tfrac{\pi
q}{n}\cos\tfrac{k}{2}+4\cos^{2}\tfrac{k}{2}}.$ (27)

This spectrum includes the same numbers of non-degenerated and doubly
degenerated eigen-modes as in the above considered ANT case. But it differs
from that case by:

i) absence of flat modes and

ii) presence of two Dirac nodal points: $k=2\pi/3$ ($K$) and $k=-2\pi/3$
($K^{\prime}$) at the same $q=0$ and for any ZNT width $n$ value. Notably, the
related two DLM’s are non-degenerated (compare with Eq. 12):

$\varepsilon_{0,k;3}=-\varepsilon_{0,k,4}\equiv\varepsilon_{k}=1-2\cos\tfrac{k}{2}.$
(28)

For the ZNT case, the DW width results $\Delta_{\rm DW}=2\sin\pi/n$, that is
narrowing with $n\gg 1$ as $\Delta_{\rm DLM}\sim 2\pi/n$ (to be compared with
the ANT case by Eq. 11).

Figure 6: Dispersion laws for ZNT of $n=6$ chains, with doubly-degenerated
(black) and non-degenerated (red) modes. The Dirac window of width
$\Delta_{\rm DW}=1$ is more narrow than for the ANT in Fig. 3

It should be also noted that, unlike a complete similarity between the ANT and
ANR spectra, there is an important difference between those for ZNT and ZNR
(the latter having no DLM’s at all but presenting instead a special edge mode
Nakada1996 , PL2022 ).

Then the expansion of local operators (an analog to Eq. 13), reduced to only
DLM eigen-operators, results as:

$a_{p,j,s}=\frac{1}{2\sqrt{nN}}\sum_{k}{\rm
e}^{ikp_{s}}u_{s}^{\dagger}\psi_{k},$ (29)

with 2-spinors:

$\displaystyle\psi_{k}=\left(\begin{array}[]{c}\psi_{0,k;3}\\\
\psi_{0,k;4}\end{array}\right),\,u_{1}=\left(\begin{array}[]{c}-1\\\
1\end{array}\right)=-u_{3},$ (34) $\displaystyle\qquad\qquad\quad
u_{2}=\left(\begin{array}[]{c}-1\\\ -1\end{array}\right)=-u_{4}.$ (37)

Due to relative simplicity of expansions by Eq. 29 in only 2 DLM’s, compared
to the ANT case by Eq. 13 with up to 8 DLM’s, we begin the next consideration
of impurity effects just from the ZNT case.

##  IV Impurity effects on ZNT

Now we can consider impurity effects on the above described nanotubes. The
simplest Lifshitz isotopic perturbation model Lifshitz  is known not to
produce impurity resonance effects in nanoribbons, both in ANR and ZNR PL2022
, therefore we begin from the more effective Anderson hybrid model Anderson ,
presenting its perturbation Hamiltonian for the ZNT case (with use of
2-spinors by Eq. 29) as:

$\displaystyle H_{\rm AZ}$ $\displaystyle=$
$\displaystyle\sum_{\sigma}\left[\varepsilon_{res}b_{\sigma}^{\dagger}b_{\sigma}\right.$
(38) $\displaystyle+$
$\displaystyle\left.\frac{\gamma}{2\sqrt{nN}}\sum_{k}\left({\rm
e}^{ikp_{\sigma}}b_{\sigma}^{\dagger}u_{s_{\sigma}}^{\dagger}\psi_{k}+{\rm
h.c.}\right)\right].$

It describes impurity adatoms with their resonance energy $\varepsilon_{res}$
(laying inside the host DLM range) and corresponding local Fermi operators
$b_{\sigma}$ at random positions $\sigma$, linked through the hybridization
parameter $\gamma$ to its nearest neighbor host atom at $s_{\sigma}$ site in
$p_{\sigma}$ segment of $j_{\sigma}$ chain (see Fig. 7). The random
$p_{\sigma}$, $j_{\sigma}$, and $s_{\sigma}$ values are distributed uniformly
with a low overall concentration: $c=(4nN)^{-1}\sum_{\sigma}1\ll 1$.

Figure 7: A fragment of ZNT with $\sigma$-th impurity adatom (orange) linked
by hybridization $\gamma$ to its nearest neighbor host atom (green) at the
($p_{\sigma},j_{\sigma},s_{\sigma}$)-site.

The next consideration goes in terms of (advanced) Green’s functions (GF’s)
whose Fourier-transform in energy:

$\langle\langle
A|B\rangle\rangle_{\varepsilon}=\frac{i}{\pi}\int_{-\infty}^{0}{\rm
e}^{i(\varepsilon-i0)t}\langle\left\\{A(t),B(0)\right\\}\rangle dt.$ (39)

includes the grand-canonical statistical average: $\langle O\rangle={\rm
Tr}\,\left[{\rm e}^{-(H-\mu)/k_{\mathrm{B}}T}O_{H}(t)\right]\bigl{/}\,{\rm
Tr}\,\left[{\rm e}^{-(H-\mu)/k_{\mathrm{B}}T}\right]$ of a Heisenberg operator
$O(t)={\rm e}^{iHt}O{\rm e}^{-iHt}$ under a Hamiltonian $H$ with chemical
potential $\mu$ and the anticommutator $\\{.,.\\}$.

As known Zubarev1960 ; Economou1979 , GF’s satisfy the equation of motion:

$\varepsilon\langle\langle
A|B\rangle\rangle_{\varepsilon}=\langle\left\\{A(0),B(0)\right\\}\rangle+\langle\langle[A,H]|B\rangle\rangle_{\varepsilon}.$
(40)

In what follows the energy sub-index at GF’s is mostly omitted (or enters
directly as its argument).

Consider now the GF $2\times 2$ matrix
$\hat{G}(k,k^{\prime})=\langle\langle\psi_{k}|\psi_{k^{\prime}}^{\dagger}\rangle\rangle$
made of $\psi$-spinors by Eq. 29. In absence of impurities, with use of the
Hamiltonian $H$ by Eq. 24, the explicit solution for this GF turns
$k$-diagonal: $\hat{G}(k,k^{\prime})\to\delta_{k,k^{\prime}}\hat{G}^{(0)}(k)$,
where

$\hat{G}_{0}(k)=\left(\varepsilon-\varepsilon_{k}\hat{\tau}_{3}\right)^{-1}$
(41)

with the Pauli matrix $\hat{\tau}_{3}$.

When passing to the disordered system with its Hamiltonian extended to
$H+H_{\rm AZ}$, we get the equation of motion for the $k$-diagonal GF matrix,
$\hat{G}(k,k)\equiv\hat{G}(k)$:

$\hat{G}(k)=\hat{G}_{0}(k)+\frac{\gamma}{2\sqrt{nN}}\sum_{\sigma}{\rm
e}^{-ikp_{\sigma}}\hat{G}_{0}(k)u_{s_{\sigma}}\langle\langle
b_{\sigma}|\psi_{k}^{\dagger}\rangle\rangle,$ (42)

and then its solution is generally sought in the self-energy form:

$\hat{G}(k)=\left(\hat{G}_{0}^{-1}(k)-\hat{\Sigma}_{k}\right)^{-1},$ (43)

including the self-energy matrix $\hat{\Sigma}_{k}$. To find it, we continue
the chain of equations of motion, now for the mixed (impurity-DLM) row-vector
GF:

$\langle\langle
b_{\sigma}|\psi_{k}^{\dagger}\rangle\rangle(\varepsilon-\varepsilon_{res})=\frac{\gamma}{2\sqrt{nN}}\sum_{k^{\prime}}{\rm
e}^{ik^{\prime}p_{\sigma}}u_{s_{\sigma}}^{\dagger}\hat{G}(k^{\prime},k).$ (44)

This gives the first contribution to $\hat{\Sigma}_{k}$ from its term with
$k^{\prime}=k$ used in Eq. 42:

$\frac{\gamma^{2}}{4nN}\sum_{\sigma}\frac{u_{s_{\sigma}}u_{s_{\sigma}}^{\dagger}}{\varepsilon-\varepsilon_{res}}=\frac{c\gamma^{2}}{\varepsilon-\varepsilon_{res}}.$
(45)

It is then extended by writing down the equation of motion for the resting
terms with $k^{\prime}\neq k$ in the r.h.s. of Eq. 44:

$\displaystyle\hat{G}(k^{\prime},k)=\frac{\gamma}{2\sqrt{nN}}\sum_{\sigma^{\prime}}{\rm
e}^{-ik^{\prime}p_{\sigma^{\prime}}}$
$\displaystyle\qquad\qquad\qquad\times\,\hat{G}_{0}(k^{\prime})u_{s_{\sigma^{\prime}}}\langle\langle
b_{\sigma^{\prime}}|\psi_{k}^{\dagger}\rangle\rangle,$ (46)

and choosing the term with $\sigma^{\prime}=\sigma$ in its r.h.s. This
generates the (scalar) impurity self-energy $\Sigma_{0}=\gamma^{2}G_{0}$ with
the DLM locator GF:

$G_{0}=\frac{1}{4nN}\sum_{k}u_{s_{\sigma}}^{\dagger}\hat{G}_{0}(k)u_{s_{\sigma}}=\frac{1}{4nN}\sum_{k}{\rm
Tr}\hat{G}_{0}(k),$ (47)

which enters the modified factor $(\varepsilon-\varepsilon_{res}-\Sigma_{0})$
in Eq. 44. Then the solution for $\hat{G}(k)$ in the simplest T-matrix
approximation for the self-energy reads:

$\hat{G}(k)=\left[\varepsilon-
cT(\varepsilon)\hat{\tau}_{0}-\varepsilon_{k}\hat{\tau}_{3}\right]^{-1},$ (48)

with the scalar T-matrix:

$T(\varepsilon)=\frac{\gamma^{2}}{\varepsilon-\varepsilon_{res}-\Sigma_{0}}.$
(49)

The next important GF, the DLM locator, is calculated by usual passing from
$k$-summation to integration:

$G_{0}(\varepsilon)=\frac{\varepsilon}{4\pi
n}\int_{0}^{2\pi}\frac{dk}{\varepsilon^{2}-\varepsilon_{k}^{2}}.$ (50)

Its analytic expression (see Appendix) is:

$\displaystyle
G_{0}(\varepsilon)=\frac{i}{4n}\left\\{\frac{\theta\left[(1-\varepsilon)(3+\varepsilon)\right]}{\sqrt{(3+\varepsilon)(1-\varepsilon)}}\right.$
$\displaystyle\qquad\qquad\qquad\qquad+\left.\frac{\theta\left[(1+\varepsilon)(3-\varepsilon)\right]}{\sqrt{(3-\varepsilon)(1+\varepsilon)}}\right\\},$
(51)

approximated in the low-energy range as:

$G_{0}(\varepsilon)\approx\frac{i}{2n\sqrt{3}}\left(1+\frac{\varepsilon^{2}}{3}\right).$
(52)

More generally, the DLM locator is defined as

$G(\varepsilon)=\frac{1}{4\pi N}\sum_{k}{\rm Tr}\,\hat{G}(k),$ (53)

with the diagonal GF matrix $\hat{G}(k)$ by Eq. 43. Within the T-matrix
approximation by Eq. 48, it results simply as:

$G(\varepsilon)=G_{0}(\tilde{\varepsilon}),$ (54)

with $\tilde{\varepsilon}=\varepsilon-cT(\varepsilon)$ used instead of
$\varepsilon$ in Eqs. 51 or 52. The resulting $G(\varepsilon)$ real and
imaginary parts as shown in Fig. 8 for the choice of Cu impurities with
$\varepsilon_{res}=0.03$, $\gamma=0.3$ Irmer2018  and $c=0.08$ only slightly
differ from those for unperturbed $G_{0}(\varepsilon)$ within the
$|\varepsilon-\varepsilon_{res}|\lesssim\gamma^{2}|G_{0}|$ range.

Figure 8: Real (blue) and imaginary (black) parts of the locator function
$G(\varepsilon)$ compared to those by the unperturbed $G_{0}(\varepsilon)$
(dashed) for ZNT with $n=12$ and Cu impurity concentration $c=0.08$.

Another set of elementary excitations in the disordered system, that due to
impurity atoms, defines the impurity locator GF:
$G_{imp}=N^{-1}\sum_{\sigma}\langle\langle
b_{\sigma}|b_{\sigma}^{\dagger}\rangle\rangle$, and its solution in the same
approximation reads:

$G_{imp}(\varepsilon)=cT(\varepsilon)/\gamma^{2}.$ (55)

Together, the DLM and impurity locators, Eqs. 54, 55, define the low-energy
density of states (DOS) as
$\rho(\varepsilon)=\rho_{h}(\varepsilon)+\rho_{imp}(\varepsilon)$ with its
host and impurity parts:

$\rho_{h}(\varepsilon)=\frac{2}{\pi}\,{\rm
Im}\,G(\varepsilon),\quad\rho_{imp}(\varepsilon)=\frac{2}{\pi}\,{\rm
Im}\,G_{imp}(\varepsilon)$ (56)

(taking the account of 2 spin values).

Figure 9: DOS parts near the impurity resonance level for the same system as
in Fig. 8.

In a disordered ZNT, host DLM’s contribute with $1/n$ charge carriers per site
and impurities do with $c$ carriers per site, so defining the Fermi level
$\varepsilon_{\rm F}$ from the equation:

$\frac{1}{n}+c=\int_{-3}^{\varepsilon_{\rm F}}\rho(\varepsilon)d\varepsilon$
(57)

(integrated from the bottom of DLM range). Then, in the simplest approximation
of Im$G(\varepsilon)\approx{\rm Im}G_{0}(\varepsilon)\approx 2/(\sqrt{3}\pi
n)$ (dashed line in Fig. 9), the Fermi level dependence on impurity
concentration $c$ results:

$\varepsilon_{\rm F}(c)\approx\frac{c}{c+c_{\ast}}\,\varepsilon_{res},$ (58)

where $c_{\ast}=[\gamma G_{0}(\varepsilon_{res})]^{2}$. Its fast initial
growth: $\varepsilon_{\rm F}(c)\approx(c/c_{\ast})\varepsilon_{res}$ at
$c\lesssim c_{\ast}$, changes to a slow approaching $\varepsilon_{res}$:
$\varepsilon_{\rm F}(c)\approx\varepsilon_{res}-(c_{\ast}/c)\varepsilon_{res}$
at $c\gtrsim c_{\ast}$ (see Fig. 10).

Figure 10: Fermi level in function of impurity concentration for the system by
Figs. 8, 9.

The next analysis of low energy spectra in this disordered system follows the
lines of similar cases by Refs. ILP1987 ; Pogorelov2020 ; PL2022 . Thus, the
modified dispersion laws are obtained from the standard equation Bonch :

${\rm Re}\det\hat{G}^{-1}(k)=0,$ (59)

which splits in two scalar equations:

$\varepsilon-c\,{\rm Re}\,T(\varepsilon)=\pm\varepsilon_{k}.$ (60)

They appear as cubic equations for energy in function of momentum,
$\varepsilon(k)$, and their analytic solutions, though standard, are rather
cumbersome. But they can be greatly simplified within the relevant low energy
range by using the host dispersion law $\varepsilon_{k}\approx\sqrt{3}k/2$,
linearized in momentum $k$ (referred to a Dirac point), and solving Eq. 60 for
this momentum in function of energy PL2022 :

$\pm k_{\varepsilon}=\pm\,2\,\frac{\varepsilon-c\,{\rm
Re}\,T(\varepsilon)}{\sqrt{3}}.$ (61)

An example of such solutions for ZNT with $n=12$ and $c=0.08$ in Fig. 11
demonstrates how coupling of each $\pm\varepsilon_{k}$ mode to the impurity
$\varepsilon_{res}$ mode forms resonance splitting of hybridized
$\pm\varepsilon_{1k}$ and $\pm\varepsilon_{2k}$ modes at $k=0$ to the interval
of $\approx\sqrt{\varepsilon_{res}^{2}+4c\gamma^{2}}$ around
$\varepsilon_{res}/2$.

Figure 11: Modified dispersion laws $\varepsilon_{1,k}$ and
$\varepsilon_{2,k}$ by Eq. 61 for for the system as in Figs. 8-10 compared to
the unperturbed $\varepsilon_{k}$ and to the impurity level
$\varepsilon_{res}$ (the momentum $k$ being referred to the $K$-point). The
mobility gaps (shadowed), define the mobility edges $k_{1,min}$, $k_{2,min}$,
and $k_{2,max}$ for quasiparticle momenta.

Next, this $k_{\varepsilon}$-form is used in the important test of dispersion
law validity for a disordered system, the Ioffe-Regel-Mott (IRM) criterion
IoffeRegel ; Mott :

$k_{\varepsilon}v_{\varepsilon}\gtrsim\tau^{-1}(\varepsilon),$ (62)

with the quasiparticle group velocity $v_{\varepsilon}=(\partial
k_{\varepsilon}/\partial\varepsilon)^{-1}$ and its inverse lifetime
$\tau_{\varepsilon}^{-1}=c\,{\rm Im}\,T(\varepsilon)$, that is the
quasiparticle mean free path to be longer of its wavelength.

Each $\varepsilon$ value that converts $\gtrsim$ into $\approx$ in Eq. 62
gives an estimate for a mobility edge $\varepsilon_{mob}$, separating the
ranges of band-like and localized states in the spectrum. An important rule
for these states in a multi-mode system is that they can not coexist, that is,
if, for a certain energy, the IRM criterion does not hold for at least one
mode, all other modes at this energy should be also localized Mott .

With use of Eqs. 61, 49, such equation for mobility edges can be written
explicitly in the form:

$\frac{c\gamma^{2}\Gamma}{D^{2}(\varepsilon)}\left|1+c\gamma^{2}\frac{D^{2}(\varepsilon)-2\Gamma^{2}}{D^{4}(\varepsilon)}\right|\approx\left|\varepsilon-c\gamma^{2}\frac{\varepsilon-\varepsilon_{res}}{D^{2}(\varepsilon)}\right|$
(63)

with $D^{2}(\varepsilon)=(\varepsilon-\varepsilon_{res})^{2}+\Gamma^{2}$ and
$\Gamma=\gamma^{2}G_{0}(\varepsilon)$. Then, using the $G_{0}(\varepsilon)$
value by Eq. 51, this equation can be solved numerically to estimate all
possible $\varepsilon_{mob}$ values and so delimit the band-like and localized
energy ranges in ZNT with impurities at given parameters of disorder
($\varepsilon_{res}$, $\gamma$, $c$) and of nanotube structure ($n$) as shown
in Fig. 11.

Here one localized range is found at the lower limit of resonance splitting,
near the shifted Dirac energy $-\varepsilon_{s}$ at all $c>0$, being of width
$\approx c\gamma^{2}/\varepsilon_{res}$. Another localized range emerges above
it, around $\varepsilon_{res}$, when $c$ reaches a certain critical value
$c_{0}$. And at yet higher critical concentration, $c_{1}\gg c_{0}$, the
latter range gets split in two, due to a specific interplay (when going away
from the Dirac point) between the growing momentum $k_{\varepsilon}$,
decreasing group velocity $v_{k}$, and increasing inverse lifetime
$\tau_{\varepsilon}^{-1}$ of hybridized modes. A more detailed description of
these restructured spectra for different nanostructures follows below.

##  V Impurity effects on ANT

It is also of interest to extend the above approach to another nanotube
topology, namely, to the more involved ANT case. To simplify description of
low energy impurity resonances here, we again restrict the expansions of local
operators by Eqs. 2 and 9 in 4-spinors $\psi(k,q)$ with unitary matrices
$\hat{U}(k,q)$, to only DLM’s $q=K,K^{\prime}$. Then the ANT perturbation
Hamiltonian results, instead of Eq. 38 for ZNT, in the form:

$\displaystyle H_{\rm
AA}=\sum_{\sigma}\varepsilon_{res}b_{\sigma}^{\dagger}b_{\sigma}+\frac{\gamma}{2\sqrt{nN}}\sum_{k,\sigma}\left\\{{\rm
e}^{ikp_{\sigma}}b_{\sigma}^{\dagger}\right.$ $\displaystyle\times\left[{\rm
e}^{iKj_{\sigma}}[u^{\dagger}(k,K;\sigma)\psi(k,K)]_{j_{\sigma}}\right.$
$\displaystyle\left.\left.+\,{\rm
e}^{iK^{\prime}j_{\sigma}}[u^{\dagger}(k,K^{\prime};\sigma)\psi(k,K^{\prime})]_{j_{\sigma}}\right]+{\rm
h.c.}\right\\},$ (64)

where the row spinor $u^{\dagger}(k,q;\sigma)$ is just the $j_{\sigma}$-th row
of $\hat{U}^{\dagger}(k,q)$.

Next we consider the 4$\times$4 GF matrices
$\hat{G}(k,q;k^{\prime},q^{\prime})\equiv\langle\langle\psi_{k,q}|\psi_{k^{\prime},q^{\prime}}^{\dagger}\rangle\rangle$
and the related equation of motion with the Hamiltonian $H+H_{\rm AA}$ for the
choice of $K$-modes GF:

$\displaystyle\hat{G}(k,K;k^{\prime},K)=\delta_{k,k^{\prime}}\hat{G}^{(0)}(k,K)$
$\displaystyle\qquad+\frac{\gamma}{2\sqrt{nN}}\sum_{\sigma}{\rm
e}^{-i(kp_{\sigma}+Kj_{\sigma})}\hat{G}_{0}(k,K)$
$\displaystyle\qquad\qquad\qquad\times\,u(k,K;\sigma)\langle\langle
b_{\sigma}|\psi_{k^{\prime},K}^{\dagger}\rangle\rangle,$ (65)

where
$G_{f,f^{\prime}}^{(0)}(k,K)=\delta_{f,f^{\prime}}(\varepsilon-\varepsilon_{k,K,f})^{-1}$
and the column spinor $u(k,q;\sigma)$ is the $j_{\sigma}$-th column of
$\hat{U}(k,q)$.

Then the equation (similar to Eq. 44 for the ZNT case) for the mixed GF,
$\langle\langle b_{\sigma}|\psi_{k^{\prime},K}^{\dagger}\rangle\rangle$:

$\displaystyle\langle\langle
b_{\sigma}|\psi_{k^{\prime},K}^{\dagger}\rangle\rangle\left(\varepsilon-\varepsilon_{res}\right)=\frac{\gamma}{2\sqrt{nN}}\sum_{\kappa^{\prime\prime}}{\rm
e}^{i(k^{\prime\prime}p_{\sigma}+Kj_{\sigma})}$
$\displaystyle\qquad\qquad\qquad\qquad\times
u^{\dagger}(k^{\prime\prime},K;\sigma)\hat{G}(k^{\prime\prime},k^{\prime})$
(66)

leads (in the same way as to Eq. 48) to the T-matrix solution for momentum-
diagonal GF matrix $\hat{G}(k,K)\equiv\hat{G}(k,K;k,K)$:

$\hat{G}(k,K)=\left\\{\left[\hat{G}^{(0)}(k,K)\right]^{-1}-cT(\varepsilon)\right\\}^{-1}$
(67)

where the T-function for this case differs from that by Eq. 49 only by the
form of its locator $G_{0}(\varepsilon)$. Using Eq. 12, it results here as:

$G_{0}(\varepsilon)=\frac{4i}{n\sqrt{1-(\varepsilon/2)^{2}}}.$ (68)

Then comparison with Eq. 51 shows that the impurity level damping for this
case turns $\approx 4\sqrt{3}$ times stronger than for ZNT at the same $n$
number. This produces strongly different behaviors of IRM mobility edges and
qualitatively different structures of localized and band-like spectra in these
two nanosystems. Now we extend comparison of such structures to other carbon
nanosystems with the same sort of impurities.

##  VI Comparison with other carbon nanosystems

The low-energy spectrum restructuring under impurity disorder effect is
suitably illustrated by a diagram of mobility edges $\varepsilon_{mob}$
between the localized and band-like energy ranges in function of impurity
concentration $c$. Such diagrams in Fig. 12 permit to compare the effects of
Cu impurities on Dirac modes in the previously considered 2D graphene
Pogorelov2020  and ANRs PL2022  together with the above obtained results for
ZNT and ANT. Some general features, noted for the ZNT case, are observed in
all of them:

Figure 12: Development of localized ranges (shadowed) in quasiparticle spectra
of: 2D graphene, zigzag nanotube (with a narrow blue range of divergence for
GF group expansion, see Sec. VII), armchair nanoribbon, and armchair nanotube
with growing Cu impurity concentration .

i) formation of a localized range (mobility gap) around the resonance level
$\varepsilon_{res}$, at reaching a certain critical concentration $c_{0}$, ,

ii) presence of another localized range around the shifted down Dirac level
$\varepsilon_{D}$, generally much narrower but existing at all $c>0$,

iii) opening, at a higher critical concentration $c_{1}\gg c_{0}$, of a narrow
band range within the $\varepsilon_{res}$-related mobility gap.

But this comparison also reveals notably different sensitivity of the
corresponding DLM’s to the impurity resonance level, depending both on their
topological properties (absence or presence of edges and the edge types) and
on discrete transversal numbers of chains in a system. Within the IRM
formalism, for given impurity parameters $\varepsilon_{res}$ and $\gamma$, it
depends on the host system through its locator function $G_{0}(\varepsilon)$,
like those by Eqs. 51, 68. This can be further compared with the previously
found $G_{0}(\varepsilon)$ values for 2D graphene: $\varepsilon/\sqrt{3}$
YDV2021  and for ANR with $M$ carbon chains: $4/(M+1)$ PL2022 .

Thus, the impurity-induced localization first occurs at an energy very close
to $\varepsilon_{res}$ and the related critical concentration $c_{0}$ can be
estimated from Eq. 63 by setting $\varepsilon=\varepsilon_{res}$ there. It
results generally in:

$c_{0}=\frac{[\gamma
G_{0}(\varepsilon_{res})]^{2}}{2}\left[1+\sqrt{1+4\frac{\varepsilon_{res}}{\gamma^{2}G_{0}(\varepsilon_{res})}}\right],$
(69)

and for the instance of ZNT with $n=12$ it gives $c_{0}\approx 6.6\cdot
10^{-4}$ in a good agreement with the numerical calculation result shown in
Fig. 12. With further growth of $c>c_{0}$, a continuous range of localized
states (mobility gap) appears around $\varepsilon_{res}$, of width growing as
$\Delta_{mob}\sim\gamma\sqrt{c-c_{0}}$.

Then, at reaching another critical concentration $c_{1}\gg c_{0}$, a certain
window of band-like states opens inside the mobility gap, due to the before
discussed faster resonance splitting between the initial $\varepsilon_{k}$ and
$\varepsilon_{res}$ modes than these split modes damping. This $c_{1}$ value
is also estimated from the numerical solution of IRM Eq. 62. It can be
presented in function of the single $G_{0}$ parameter as shown in Fig. 13.
Strictly speaking, this is only possible for 1D nanosystems where the low-
energy locator $G_{0}(\varepsilon)$ is practically constant, defined by their
topology and discrete width numbers. This dependence can be reasonably fitted
by the formula:

$c_{1}\approx\sqrt{0.01G_{0}+20G_{0}^{3}}.$ (70)

Figure 13: Upper critical concentration $c_{1}$ in function of locator $G_{0}$
value (black line) and its particular values for Cu impurities in ZNT with
$n=24$ (red circle), in 2D graphene (blue circle), in ZNT with $n=12$ (green
circle), in ANR with $M=47$ (pink circle), and in ANT with ($n=50)$ (orange
circle) together with the fitting curve by Eq. 70 (dashed red line).

The IRM test also indicates a similar mobility window to open under the same
impurities in 2D graphene with linear $G_{0}(\varepsilon)$ behavior. The
resulting $c_{1}$ value qualitatively agrees with the approximation by Eq. 70
at the choice of $G_{0}=G_{0}(\varepsilon_{op})$, $\varepsilon_{op}$ being
just the energy where the mobility window first opens (as included into Fig.
13).

Notably, the $G_{0}$ parameter decreases with the nanotube width as $\sim
1/n$, producing respective decrease of $c_{1}$ and so making the system
spectrum more sensible to impurity resonances. Thus, the $c_{1}$ value for ZNT
with $n>16$ should turn already below of that for 2D graphene (despite the
latter could be formally thought as the $n\to\infty$ limit), underlying the
importance of topological factors in these effects.

However, as it was already noted above, such widening of a nanotube would
produce a similar narrowing of the Dirac window $\Delta_{DW}$ in its spectrum,
delimiting the range of possible impurity effects. Therefore, the optimal
conditions for them should be sought from a certain compromise between the
parameters of impurity (energy level $\varepsilon_{res}$, hybridization
$\gamma$, and concentration $c$) and of host nanotube (topological type and
width $n$). Thus, for the considered Cu impurities, we estimate admissible
width limits of ZNT: $n\lesssim 40$, ANT: $n\lesssim 60$, and ANR: $M\lesssim
35$. Their comparison with the $c_{1}$ estimates in Fig. 13 suggests
possibility for the narrow conductivity window above $\varepsilon_{res}$ in
ZNT, graphene and maybe in ANT, but hardly in ANR (though the latter can
provide a similar window below $\varepsilon_{res}$).

##  VII Beyond T-matrix approximation

Besides the most common approach to spectra of disordered systems through the
single-impurity scattering in terms of T-matrix, there are its certain
extensions. One of them uses the self-consistent approximation to this
T-matrix Freed , another is based on group expansions of self-energy ILP1987
in series of terms corresponding to wave scatterings by various clusters of
increasing number of impurities.

###  VII.1 Self-consistent approximation

Let us begin from the self-consistent approximation where the T-matrix is
written as:

$T_{s-c}(\varepsilon)=\frac{\gamma^{2}}{\varepsilon-\varepsilon_{res}-\gamma^{2}G_{s-c}(\varepsilon)},$
(71)

with the self-consistent locator $G_{s-c}(\varepsilon)=G_{0}[\varepsilon-
cT_{s-c}(\varepsilon)]$.

Then, using the above approximated expression by Eq. 52, we obtain the self-
consistency equation for $G_{s-c}(\varepsilon)$:

$\displaystyle i\sqrt{3}\,n\,G_{s-c}(\varepsilon)+1$
$\displaystyle\qquad+\frac{1}{3}\left[\varepsilon-\frac{c\gamma^{2}}{\varepsilon-\varepsilon_{res}-\gamma^{2}G_{s-c}(\varepsilon)}\right]^{2}=0.$
(72)

Figure 14: Imaginary parts of locator functions: self-consistent
$G_{s-c}(\varepsilon)$ (solid), simple T-matrix $G(\varepsilon)$ (dashed), and
unperturbed $G_{0}(\varepsilon)$ (dash-dotted) for the ZNT system as in Figs.
8-11, mostly differing near the resonance level $\varepsilon_{res}$. Figure
15: Practical coincidence of self-consistent (solid lines) and simple (dashed
lines) T-matrix functions for the same system as shown in Fig. 14.

Its numerical solution for the characteristic case of ZNT with $n=6$ and
$c=0.015$ provides the real and imaginary parts by $G_{s-c}(\varepsilon)$ as
shown in Fig. 14 in comparison with the same parts of the simple
$G_{0}(\varepsilon)$, Eq. 49. It demonstrates that the self-consistency
correction only slightly changes $G_{0}(\varepsilon)$ in a vicinity of
$\varepsilon_{res}$, and also such parts of $T_{s-c}$ and $T_{0}$ are almost
coincident (Fig. 15). So this change has practically no effect on the IRM
results obtained above with use of the simple $G_{0}(\varepsilon)$. So the
corresponding mobility diagrams as in Fig. 12 remain also valid in the self-
consistent approximation.

###  VII.2 Group expansion

Next, we look for a group expansion (GE) of the self-energy matrix
$\hat{\Sigma}_{k}(\varepsilon)$ in the form:

$\hat{\Sigma}_{k}(\varepsilon)=c\hat{T}(\varepsilon)\bigl{[}\hat{1}+c\hat{B}_{k}(\varepsilon)+\dots\bigr{]},$
(73)

where the sum:

$\hat{B}_{k}(\varepsilon)=\sum_{r}\bigl{[}{\rm
e}^{-ikr}\hat{A}_{r}(\varepsilon)+\hat{A}_{r}^{2}(\varepsilon)\bigr{]}\bigl{[}\hat{1}-\hat{A}_{r}^{2}(\varepsilon)\bigr{]}^{-1}$
(74)

describes the effects of multiple scatterings between pairs of impurities at
longitudinal distance $r$ between them through the related scattering matrix
$\hat{A}_{r}(\varepsilon)=\hat{T}(\varepsilon)\hat{G}_{r}(\varepsilon)$ with
the correlator matrix

$\hat{G}_{r}(\varepsilon)=\frac{1}{4nN}\sum_{k^{\prime}\neq k}{\rm
e}^{ik^{\prime}r}\hat{G}_{0}(k).$

The omitted terms in the r.h.s. of Eq. 73 correspond to contributions by
clusters of three and more impurities.

Notably, the important specifics of the disordered carbon nanotubes (and
nanoribbons), unlike the commonly studied disordered 3D or 2D crystals,
consists in that:

$\ast$ here the longitudinal distance $r$ between different impurities takes
here only discrete values, namely, $2r$ takes integer values (so the sum
$\Sigma_{r}$ can be done without usual passing to integral $\int dr$) and

$\ast$ this distance can be also zero.

Thus, it can be seen from Figs. 5, 7 that for any impurity position $\sigma$,
there are $2n-1$ other positions $\sigma^{\prime}$ with the same longitudinal
coordinate $p_{\sigma}=p_{\sigma^{\prime}}$. Such impurity pairs at zero
longitudinal distance contribute into $\hat{B}_{k}$ with

$\hat{B}_{0}=c\left(1-\frac{1}{n}\right)\bigl{[}\hat{A}_{0}(\varepsilon)+\hat{A}_{0}^{2}(\varepsilon)\bigr{]}\bigl{[}\hat{1}-\hat{A}_{0}^{2}(\varepsilon)\bigr{]}^{-1}$
(75)

and this contribution results dominant over the resting sum $\sum_{r\neq 0}$
in Eq. 74 (see in Appendix).

Consider it in more detail for the example of ZNT where all the matrices in
the self-energy $\hat{\Sigma}_{k}$ can be substituted by scalars:
$A_{0}(\varepsilon)=T(\varepsilon)G_{0}(\varepsilon)$, and $T(\varepsilon)$
can be taken in the form by Eq. 49. Then the $B_{0}(\varepsilon)$ contribution
to GE by Eq. 73 is estimated using the explicit form of
$A(\varepsilon)=\Gamma/(e-\varepsilon_{res}-i\Gamma)$ with
$\Gamma=\gamma^{2}/(\sqrt{3}n)$ to give:

$B_{0}(\varepsilon)=(1-\frac{1}{2n})\Gamma\frac{\varepsilon-\varepsilon_{res}-i\Gamma}{(e-\varepsilon_{res})^{2}}.$
(76)

For comparison, the lowest degree resting term in $B_{k}(\varepsilon)$ is
evaluated with use of the approximation for $A_{r}(\varepsilon)$ by Eq. 90,
as:

$\sum_{r\neq 0}{\rm
e}^{-ikr}A_{r}(\varepsilon)\approx\frac{i\Gamma\varepsilon}{4n\pi^{2}}\,{\rm
Li}_{2}\left(-{\rm e}^{-ik/2}\right),$ (77)

where the polylogarithmic function AbS  at $k\ll 1$ is close to ${\rm
Li}_{2}(-1)=-\pi^{2}/12$. Thus the magnitude of the term by Eq. 77 turns more
than 4 orders below of that by Eq. 76 within the whole low-energy range, while
the next terms in $B_{k}(\varepsilon)$ result yet much smaller.

Then, taking the GE convergence criterion as $c|B_{0}(\varepsilon)|<1$, the
T-matrix validity condition results from Eq. 76 well approximated by:

$|\varepsilon-\varepsilon_{res}|\gtrsim\Gamma\sqrt{c},$ (78)

and it can only fail in a very narrow vicinity of $\varepsilon_{res}$ (the
blue vicinity in Fig. 12 for ZNT) deeply within the localized range, thus
simply confirming localization of states there. In a similar way, this
conclusion can be reached for other nanosystems considered here, justifying
the above obtained pictures of spectrum restructuring in them.

##  VIII Twisted nanotubes

When passing to yet more general situation of twisted nanotubes which
constitute a set of intermediate structures between the above considered ANT
and ZNT, each of them has its unit cell built on a certain longitudinal
translation vector $\tilde{\mathbb{a}}_{1}=(a_{11},a_{12})$ (in the
${\mathbb{a}}_{1},\,{\mathbb{a}}_{2}$ basis with $a_{11}$, $a_{12}$ being
natural numbers) and its orthogonal transversal vector
$\tilde{\mathbb{a}}_{2}=(-2a_{12}-a_{11},2a_{11}+a_{12})$ Kane .

Figure 16: Graphene lattice unit cell (light grey) and its related Brillouin
zone (dark grey) in comparison with those (respectively, light and dark pink)
for a twisted nanotube at the choice of $\tilde{\mathbb{a}}_{1}=(2,1)$.

Then it is readily seen that the related Brillouin zone (BZ) for such a
nanotube, built on its basic vectors
$\tilde{\mathbb{b}}_{1}=\pi\tilde{\mathbb{a}}_{1}/|\tilde{\mathbb{a}}_{1}|^{2}$
and
$\tilde{\mathbb{b}}_{2}=2\pi\tilde{\mathbb{a}}_{2}/|\tilde{\mathbb{a}}_{2}|^{2}$,
is much smaller then the BZ for 2D graphene, built on the respective basic
vectors ${\mathbb{b}}_{1}=4\pi/3(2,-1)$ and ${\mathbb{b}}_{2}=4\pi/3(-1,2)$.

This is illustrated in Fig. 16 for the example of smallest possible unit cell
in a nanotube with the strongest twist angle: $a_{11}=2,a_{12}=1$, resulting
in its BZ vectors $\tilde{\mathbb{b}}_{1}=2\pi/7(2,1)$ and
$\tilde{\mathbb{b}}_{2}=2\pi/21(-4,5)$. Their absolute values
$|\tilde{\mathbb{b}}_{1}|=2\pi/\sqrt{7}$ and
$|\tilde{\mathbb{b}}_{2}|=2\pi\sqrt{31}/21$ are only $\approx 0.56$ and
$\approx 0.4$ of that for the Dirac point vector $|{\mathbb{K}}|=4\pi/3$, so
they cannot reach this point neither its partner $K^{\prime}$ (see Fig. 16).
This BZ reduction turns yet much stronger for smaller twisting angles
(producing yet bigger unit cells) and eliminates the possibility for DLM’s to
occur in any twisted nanotube and leaves all of them out of the field of low
energy resonance effects considered above for the straight nanostructures.

##  IX Discussion of results

The above obtained results on restructured low-energy quasiparticle spectra in
carbon nanosystems can be discussed in the general context of disordered 1D
crystalline systems known not to contain conducting states at any degree of
disorder Anderson1958 ; Hjort ; Delande ; Vosk ; YaqiTao . But their presence
in the disordered nanotubes and nanoribbons indicates again the principally
qualitative difference of these structures from the strictly 1D chains.
Besides the above noted possibility for zero longitudinal distance between
different impurity positions, it can be yet illustrated by the behaviors of
correlator functions with growing inter-impurity distance $r$: convergent as
$\sim 1/r^{2}$ by Eq. 90 for nanotubes and divergent as $\sim 1/r$ for really
1D chains (as that by single $g_{r}(\varepsilon)$ by Eq. 88), making the
related GE divergent at all energies. So we can conclude that it is just the
presence of additional transversal degrees of freedom in quasi-1D systems that
enables their conductivity under disorder Ando1998 ; Ando2000 ; Biel2008  .

The found intermittence of conducting and localized energy ranges in the
considered nanosystems can be then used for their various practical
applications. Thus, the most straightforward effects are expected in frequency
$\omega$\- and temperature $T$-dependent electric conductivity, following from
the general Kubo-Greenwood formula Kubo , Greenwood  presented here in the
form:

$\displaystyle\sigma(\omega,T)=\frac{e^{2}}{\pi}\int
d\varepsilon\,\frac{f(\varepsilon,T)-f(\varepsilon^{\prime},T)}{\omega}$
$\displaystyle\times\int_{cond}dk\,v_{k}(\varepsilon)v_{k}(\varepsilon^{\prime}){\rm
Im}G_{k}(\varepsilon)\,{\rm Im}G_{k}(\varepsilon^{\prime}).$ (79)

Here the Fermi function $f(\varepsilon,T)=[{\rm
e}^{(\varepsilon-\varepsilon_{F})/T}+1]^{-1}$, the group velocity
$v_{k}(\varepsilon)=[\partial k_{\varepsilon}/\partial\varepsilon]^{-1}$, the
a.c. shifted energy $\varepsilon^{\prime}=\varepsilon+\hbar\omega$, and the
integration $\int_{cond}$ avoids localized ranges (as those shadowed in Fig.
11).

First of all, consider the simplest d.c. limit:

$\lim_{\omega\to
0}\frac{f(\varepsilon,T)-f(\varepsilon^{\prime},T)}{\omega}\to\frac{1}{4T{\rm
Cosh}^{2}[(\varepsilon-\varepsilon_{F})/T]},$

which then goes to $\delta(\varepsilon-\varepsilon_{F})$ at $T\to 0$, defining

$\sigma(0,0)=\int_{cond}dk\,v_{k}^{2}(\varepsilon_{F})[{\rm
Im}G_{k}(\varepsilon_{F})]^{2},$

and this turns zero for $\varepsilon_{\rm F}$ laying within a localized range.
But such insulating state can be converted to conducting by applying quite a
small external gate voltage $V_{g}$. Thus, for the ZNT case of Fig. 11, the
initial $\varepsilon_{\rm F}\approx\varepsilon_{res}=0.03$ (in units of
$t\approx 2.8$ eV) could reach the nearest mobility edges with gating either
$\approx 100$ meV upwards or $\approx 80$ meV downwards, and the resulting
reversible insulator-metal transitions should stay well resolved up to room
temperatures.

Otherwise, a quite sharp threshold in optical conductivity can be reached by
applying IR radiation of $\sim 10$ THz (which may be also combined with a
slight gate tuning $V_{g}\sim 5$ meV). A more detailed description of the
$\sigma(\omega,T)$ behavior readily follows from the above given T-matrix
solutions for $v_{k}(\varepsilon)$ and ${\rm Im}G_{k}(\varepsilon)$. All these
effects are most diversified with formation of multiple mobility edges (above
the second critical concentration $c_{1}$).

The above quantitative results was delimited to a single choice of Cu impurity
in its top-position over a host carbon atom, but they can be readily extended
to other impurities in different positions, providing a variety of possible
values for the relevant $\varepsilon_{res}$ and $\gamma$ parameters and so a
much broader field of resulting electronic dynamics. Nevertheless, their
qualitative features indicated in the present study should stay proper for all
of them.

At least, yet another practical condition for validity of the above
conclusions consists in that the nanotube (or nanoribbon) should be long
enough compared to the localization length $l_{loc}$ of quasiparticle states
near the mobility edges. The latter can be estimated as $l_{loc}\sim
v_{k_{j}}\tau(\varepsilon_{j})$ using Eqs. 48, 49, 61 for $j$th mobility edge
which results in $l_{loc}\sim 1/\Gamma$. Thus for the same instance of ZNT
with $n=12$ we obtain numerically $l_{loc}\sim 400$ nm, therefore such
nanotube should extend to more than $\sim 5\,\mu$m in length.

##  X Acknowledgements

The authors are thankful to L.S. Brizhik, A.A. Eremko, V.P. Gusynin and S.G.
Sharapov for their attention to this work and its valuable discussion. V.L.
acknowledges the partial support of his work by the Simons Foundation.

##  Appendix A Locator and correlator

We calculate the integral that contributes to the locator GF in a 1D
nanosystem, for an example of $\varepsilon_{k}$ mode in ZNT:

$g(\varepsilon)=\frac{1}{4\pi}\int_{-2\pi}^{2\pi}\frac{dk}{\varepsilon-1+2\cos\tfrac{k}{2}},$
(80)

valid at $-3<\varepsilon<1$. By the common change of variable,
$t=\tan\tfrac{k}{4}$, this integral is rewritten as:

$g(\varepsilon)=\frac{1}{\pi}\int_{-\infty}^{\infty}\frac{dt}{1+\varepsilon-(3-\varepsilon)t^{2}},$
(81)

giving the explicit result:

$g(\varepsilon)=i\frac{\theta(1+\varepsilon)\theta(3-\varepsilon)}{\sqrt{(3-\varepsilon)(1+\varepsilon)}}$
(82)

with the standard $\theta$-functions delimiting the $\varepsilon_{k}$ energy
range.

Another contribution to $G_{0}(\varepsilon)$ from $-\varepsilon_{k}$ mode,
valid at $-1<\varepsilon<3$, is

$g(-\varepsilon)=i\frac{\theta(1-\varepsilon)\theta(3+\varepsilon)}{\sqrt{(3+\varepsilon)(1-\varepsilon)}}$
(83)

which then enters the full expression for locator GF:

$G_{0}(\varepsilon)=\frac{g(\varepsilon)-g(-\varepsilon)}{4n}.$ (84)

The next step is to calculate, for the same ZNT, the correlator between a pair
of impurities at distance $r$:

$G_{r}(\varepsilon)=\frac{g_{r}(\varepsilon)-g_{r}(-\varepsilon)}{4n},$ (85)

through the integral:

$g_{r}(\varepsilon)=\frac{1}{4\pi}\int_{-2\pi}^{2\pi}f_{r}(k,\varepsilon)dk,$
(86)

Figure 17: Integration contour for Eq. 87.

with its integrand:

$f_{r}(k,\varepsilon)=\frac{{\rm e}^{ikr}}{\varepsilon-1+2\cos\tfrac{k}{2}},$

especially considering long distances, $r\gg 1$. It can be done passing to the
contour integral:

$\displaystyle\int_{C}f_{r}(k,\varepsilon)dk=g_{r}(\varepsilon)+\int_{0}^{\infty}\left[f_{r}(2\pi+iy,\varepsilon)\right.$
$\displaystyle\qquad\qquad\qquad\left.-f_{r}(-2\pi+iy,\varepsilon)\right]dy=0.$
(87)

where the contour $C$ in the complex momentum plane (Fig. 17) includes the
$f_{r}(k,\varepsilon)$ poles:

$\pm k_{\varepsilon}=\pm 2\arccos\frac{1-\varepsilon}{2}.$

For integration along the imaginary axis we use the relations
$\cos(\pm\pi+iy/2)=-\cosh(y/2)$ and ${\rm e}^{i(\pm 2\pi+iy)r}=(-1)^{2r}{\rm
e}^{-yr}$ (noting that the longitudinal distance $r$ between impurities takes
here only integer or half-integer values) and the explicit formula:

$\displaystyle I_{2r}(b)=\int_{0}^{\infty}\frac{{\rm e}^{-2ry}\,dy}{\cosh
y+b}=\frac{1}{2r+1}\left[\left(\frac{b}{\sqrt{b^{2}-1}}+1\right)\right.$
$\displaystyle\times{{}_{2}F_{1}}\left(1,2r+1;2r+2;\frac{1}{\sqrt{b^{2}-1}-b}\right)+\left(\frac{b}{\sqrt{b^{2}-1}}-1\right)$
$\displaystyle\times\left.{{}_{2}F_{1}}\left(1,2r+1;2r+2;-\frac{1}{\sqrt{b^{2}-1}+b}\right)\right]$
(88)

where ${{}_{2}F_{1}}(n,m;p;q)$ is the hypergeometric function Andrews .

Figure 18: Exact discrete values of the correlator $G_{r}(\varepsilon)$ by Eq.
89 (solid points) and their approximation by Eq. 90 (dashed curves) for the
choice of $\varepsilon=\varepsilon_{res}$ and $n=12$.

Then the sought correlator follows from Eqs. 85, 86, 88 analytically as:

$\displaystyle G_{r}(\varepsilon)=\frac{(-1)^{2r}}{16\pi
n}\left[I_{2r-1}\left(b_{\varepsilon}\right)-I_{2r+1}\left(b_{\varepsilon}\right)\right.$
$\displaystyle\qquad\qquad\quad\left.-\,I_{2r-1}\left(b_{-\varepsilon}\right)+I_{2r+1}\left(b_{-\varepsilon}\right)\right]$
(89)

with the energy dependent parameter

$b_{\varepsilon}=\frac{1-\varepsilon}{2}.$

Notably, the full form by Eq. 89 admits a very simple approximation:

$G_{r}(\varepsilon)\approx i\frac{(-1)^{2r}\varepsilon}{4n(2\pi r)^{2}},$ (90)

however quite precise at all non-zero inter-impurity distances (see Fig. 18)
and suitable for detailed evaluations as in analysis of particular GE terms
(Sec. VIIB).

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