# Kernel polynomial method to Anderson transition in disordered
$\beta$-graphyne

G.X. Wang

(Received February 11, 2020, in final form April 26, 2020)

###### Abstract

By means of variable moment kernel polynomial method, we analyze the
localization properties of $\beta$-graphyne sheet subjected to the Anderson
disorder. To detect the localization transition we focus on the scaling
behavior of the normalized typical density of states. We find that there takes
place a metal-insulator transition and the critical disorder strength is of
the order of the bandwidth, which is contrary to the one-parameter scaling
theory stating that for infinite two dimensional systems, all the electronic
states are localized for an arbitrary strength of the Anderson disorder. As
its particular localization properties, it is reasonable to predict there will
exist dc conductivity for $\beta$-graphyne at zero temperature.

Key words: $\beta$-graphyne, kernel polynomial method, Anderson disorder,
localization properties

###### Abstract

Ç äîïîìîãîþ ìåòîäó ïîëíîìíîãî ÿäðà ç çìííèì ìîìåíòîì ïðîàíàëçîâàíî ëîêàëçàöéí
âëàñòèâîñò ëèñòà $\beta$-ãðàôíà ïä âïëèâîì àíäåðñîíâñüêîãî áåçëàäó. Äëÿ
âèÿâëåííÿ ëîêàëçàöéíîãî ïåðåõîäó ìè çîñåðåäèëèñÿ íà ïîâåäíö ñêåéëíãó
íîðìàëçîâàíî¿ òèïîâî¿ ãóñòèíè ñòàíâ. Âñòàíîâëåíî ñíóâàííÿ ïåðåõîäó ìåòàë-
çîëÿòîð, ïðèîìó ñèëà êðèòèíîãî áåçëàäó ïîðÿäêó øèðèíè çîíè, ùî ïðîòèðèòü
îäíîïàðàìåòðèíé òåîð¿ ñêåéëíãó, ÿêà ñòâåðäæó, ùî äëÿ íåñêíåííèõ äâîâèìðíèõ
ñèñòåì óñ åëåêòðîíí ñòàíè ëîêàëçîâàíèìè ïðè äîâëüíé ñèë àíäåðñîíâñüêîãî
áåçëàäó. Ñòîñîâíî êîíêðåòíèõ ëîêàëçàöéíèõ âëàñòèâîñòåé, ìîæíà ïåðåäáàèòè, ùî
ñïîñòåðãàòèìåòüñÿ ïðîâäíñòü ïîñòéíîãî ñòðóìó äëÿ $\beta$-ãðàôíó ïðè íóëüîâé
òåìïåðàòóð.

Ключов слова: $\beta$-ãðàôí, ìåòîä ïîëíîìíîãî ÿäðà, àíäåðñîíâñüêèé áåçëàä,
ëîêàëçàöéí âëàñòèâîñò

##  1 Introduction

Graphyne, a single layer of carbon sheet, was first predicted by Baughman et
al. in 1987 as a layer consisting of $sp$ and $sp^{2}$ hybridized carbon atoms
[1, 2]. Graphyne is similar in structure to graphene but it is constructed by
inserting carbon triple bonds ($-$C$\equiv$C$-$) into single bonds (C$-$C) in
graphene. Although graphyne is topologically equivalent to graphene, the
presence of triple bonds introduces rich optical and electronic properties [3]
that are quite different from those of graphene, such as higher carrier
mobility [4], preferred chemical properties and strongly anisotropic optical
adsorption [6, 5]. Because of these unique properties, graphyne has inspired
researchers to explore its potential applications in electronics, catalysis,
energy storage and biomedical fields [7, 8, 9, 10, 11, 12].

Due to different distributions of $sp$ and $sp^{2}$ hybridized carbon atoms,
there were proposed various kinds of graphyne. Specially, four different types
of graphyne were identified, namely $\alpha$-, $\beta$-, $\gamma$\- and
6,6,12-graphyne [1, 13]. In reference [13], it is reported that $\alpha$-,
$\beta$\- and 6,6,12-graphyne possess Dirac cone-like band structure around
the Fermi level [13]. Unlike graphene, the Dirac points of $\beta$-graphyne
not only appear at high-symmetry points but also at low-symmetry points.
According to these results, it is reasonable to expect that these graphynes
can be considered as competitors to graphene and the study of graphyne
allotropes will open insights into designing new-generation electronics.

So far, bulk graphyne sample is experimentally unobtainable but can be
considered chemically stable. Tremendous efforts have been devoted to
synthesizing and assembling a precursor and subunit of graphyne, a class of
molecules known as dehydrobenzoannulenes (DBAs) [15, 14]. Graphdiyne [16],
which is a sub-structure of graphyne with benzene rings linked by diacetylene
($-$C$\equiv$C$-$C$\equiv$C$-$), was successfully synthesized on a copper
substrate via a cross-coupling reaction using hexaethynylbenzene [17].
Moreover, graphdiyne can be fabricated by adopting low-temperature chemical
vapor deposition [18] or through a carbon-carbon coupling reaction among
constitutive monomers at a liquid/liquid or gas/liquid interface [19]. These
exciting experimental progresses indicate that the synthesis of graphyne and
its allotropes can be expected in the near future.

Since in real materials disorder is unavoidably present, it is of particular
importance to understand how disorder affects the physics of electrons in the
fabricated samples. Disorder can arise for various reasons, such as impurities
(charged impurities, chemical impurities, etc.), topological defects
(vacancies, edge disorder, etc.), ripples or ad-atoms [20, 21, 22, 23, 24].
For graphene, as the quality of samples is improving and synthesis techniques
are developing, the dominant disorder varies from bulk disorder to edge
defects. So far, researchers are devoted to fabrication of graphyne families
and it is reasonable that bulk disorder will play a dominant role in these
materials. When modelling a disorder, the Anderson model has been widely used
in studying the basic localization features.

For graphene, due to the linear dispersion at the Dirac point, there was
debated the issue whether the one-parameter scaling theory can hold [25, 26,
27, 28]. Amini et al. study the effect of on-site uncorrelated disorder on the
electronic properties of graphene [25], and find that weak disorder can
decrease the velocities of Dirac fermions, with extended states remaining.
However, when the disorder strength increases to a large enough amount, there
will be localized states around the Fermi point, which is consistent with the
results of Naumis [29]. Subsequently, Schileede pointed out for an arbitrary
disorder strength that all the states are localized and there exists no
mobility edge [30]. Amini et al. acknowledged that the mobility edge might be
induced by the kernel [31], but they argued the comment that the localization
can occur throughout the whole spectrum because an arbitrary weak disorder
strength is unreasonable since there exists minimal conductivity in graphene
samples, and Anderson transition was observed by Bostwick et al. [32].
Recently, further study has been undertaken [33, 34, 35, 36, 37].

Of late there exist plenty of well established numerical methods, such as
exact diagonalization method, Monte Carlo simulations, Lanczos recursion
method and kernel polynomial method (KPM). Every method is suited to
particular problems and has severe limitations. For example, Lanczos algorithm
is a powerful method to calculate extremal eigenstates of sparse matrices but
it becomes unstable owing to the loss of orthogonality between the generated
basis vectors. However, KPM, which expands the spectral function in terms of a
complete set of orthogonal polynomials, can be stable and more convenient to
deal with spectral densities and correlation functions. The spectral
properties, such as LDOS, can be calculated to arbitrary precision without an
exact diagonalization of the Hamiltonian [38].

In this Letter, we consider $\beta$-graphyne sheet with the tight-binding
approximation and focus on the effect of the Anderson-type bulk disorder [39].
With the modified KPM-the variable moment kernel polynomial method [40]
(VMKPM), we calculate the local density of states (LDOS) and analyze the
scaling behavior of normalized typical density of states (DOS). To detect the
Anderson transition, the normalized typical DOS can serve as an order
parameter.

##  2 Model and method

$\beta$-graphyne consists of a honeycomb lattice of carbon atoms with eighteen
sublattices as shown in figure 1. The tight-binding Hamiltonian of
$\beta$-graphyne with Anderson disorder can be written as

${H}=\sum\limits_{\left\langle{ij}\right\rangle}{({t_{ij}c_{i}^{{\dagger}}{c_{j}}+H.c.})}+\sum\limits_{i}{{\epsilon_{i}}c_{i}^{{\dagger}}{c_{i}}}\,,$
(2.1)

where $c_{i}(c_{i}^{{\dagger}})$ is the annihilation (creation) operator
destructing (creating) an electron at lattice site $i$; $\langle i,j\rangle$
represents the nearest-neighbor hopping with the hopping amplitude $t_{ij}$,
which is specifically denoted in figure 1; $\epsilon_{i}$ is the random on-
site potential induced by Anderson disorder. Here, we assume that the random
on-site potential $\epsilon_{i}$ has a uniform probability distribution as

$p(\epsilon_{i})=\frac{1}{\gamma}\theta\left(\frac{\gamma}{2}-\left|\epsilon_{i}\right|\right),$
(2.2)

in which the parameter $\gamma$ is a measure of disorder strength and $\theta$
is Heaviside step function.

Figure 1: (Colour online) The schematic diagram of lattice structure of
$\beta$-graphyne. $t_{1}$, $t_{2}$ and $t_{3}$ denote the hopping amplitudes
between the nearest-neighbor sites. The dashed lines specify the unit cell of
$\beta$-graphyne.

Many quantities can be used to describe localization properties, such as the
averaged localization length, inverse participation number or the distribution
of LDOS. Among these quantities, LDOS is the most favorable to study Anderson
localization both theoretically and experimentally with the fact that it can
be applied to more complex situations and can be directly measured by scanning
tunneling microscopy. The LDOS at lattice site $i$ can be defined as

${\rho_{i}}\left(E\right)=\sum\limits_{n}{{{\left|{\left\langle
i\right|\left.n\right\rangle}\right|}^{2}}\delta\left({E-{E_{n}}}\right)},$
(2.3)

where $\left|n\right\rangle$ is an eigenstate of $H$ with energy $E_{n}$ and
$\left|i\right\rangle=c_{i}^{{\dagger}}\left|0\right\rangle$.

To distinguish a localized state from an extended one, it is convenient to
compare two characteristic quantities: the typical DOS and the mean DOS, which
are defined, respectively, as

$\displaystyle{\rho_{\rm{ty}}}\left(E\right)$
$\displaystyle=\exp\left\\{{\frac{1}{{{K_{r}}{K_{s}}}}\sum\limits_{k=1}^{{K_{r}}}{\sum\limits_{i=1}^{{K_{s}}}{\ln\left[{\rho_{i}^{k}\left(E\right)}\right]}}}\right\\},$
(2.4) $\displaystyle{\rho_{\rm{me}}}\left(E\right)$
$\displaystyle=\frac{1}{{{K_{r}}{K_{s}}}}\sum\limits_{k=1}^{{K_{r}}}{\sum\limits_{i=1}^{{K_{s}}}{\rho_{i}^{k}\left(E\right)}},$
(2.5)

where $K_{r}$ denotes the number of disorder realizations and $K_{s}$ is the
number of different lattice sites for a given realization. $\rho_{\rm{ty}}$
and $\rho_{\rm{me}}$ are both finite for extended states while for localized
states $\rho_{\rm{ty}}$ tends to zero but $\rho_{\rm{me}}$ remains finite.

To detect the localization transition only for a single finite size system is
insufficient [30]. It is reasonable to take into account the scaling behavior
of typical DOS, $\rho_{\rm{ty}}$. Consider the ratio of the typical and mean
DOS

$R\left(E\right)=\frac{\rho_{\rm{ty}}\left(E\right)}{\rho_{\rm{me}}\left(E\right)}.$
(2.6)

$R(E)$, the normalized typical DOS, is positive for extended states whereas it
is equal to zero for localized states. Therefore, $R(E)$ can serve as an order
parameter to reveal the position of the Anderson transition. By calculating
$R(E)$ for different disorder strength and lattice sizes, we can analyze the
scaling behavior of the typical DOS.

Theoretically, LDOS can be evaluated to arbitrary precision using the KPM
[38], which is based on the expansion of the spectral function, such as
$\rho_{i}(E)$, into a finite series of Chebyshev polynomials
$T_{n}(x)=\cos({n\arccos x})$. Since Chebyshev polynomials are defined in the
interval [-1,1], correspondingly, the Hamiltonian $H$ needs to be rescaled
before expansion with the rescaled Hamiltonian $\tilde{H}=(H-b)/a$ and
rescaled energies $\tilde{E}=(E-b)/a$, where
$a=(E_{\max}-E_{\min})/({2-\eta})$ and $b=(E_{\max}+E_{\min})/2$. $\eta$ is a
small positive number and it is not essential to have accurate bounds on the
spectrum.

In terms of Chebyshev polynomials, LDOS can be approximated as

${\rho_{i}}(\tilde{E})=\frac{1}{\piup\sqrt{1-{\tilde{E}^{2}}}}\left[g_{0}{\mu_{0}}+2\sum\limits_{m=1}^{M}{g_{m}{\mu_{m}}{T_{m}}(\tilde{E})}\right],$
(2.7)

where the Chebyshev moments

${\mu_{m}}=\int_{-1}^{1}{{\rho_{i}}(\tilde{E}){T_{m}}(\tilde{E})\mathrm{d}\tilde{E}}=\sum\limits_{n}{\left\langle
i\right|\left.n\right\rangle\left\langle
n\right|{T_{m}}({{\tilde{E}_{n}}})\left|i\right\rangle}=\left\langle
i\right|{T_{m}}(\tilde{H})\left|i\right\rangle.$ (2.8)

Using the recursion relations between the Chebyshev polynomials,
${T_{m+1}}(\tilde{E})=2\tilde{E}{T_{m}}(\tilde{E})-{T_{m-1}}(\tilde{E})$,
these moments can be efficiently obtained through a recursive procedure and
the expression for $\mu_{m}$ can be very simple: $\mu_{m}=\left\langle
i\right|\left.\alpha_{m}\right\rangle$ in which
$\alpha_{m}=T_{m}(\tilde{H})\left|i\right\rangle$. The most expensive
computation is the multiplication between a sparse Hamiltonian matrix and a
vector. If the matrix is computed on the fly, without being stored, dimensions
of the order can be carried out to $10^{9}$ or more. Since the Hamiltonian is
a sparse matrix, the calculation can be done even on a desktop computer. In
contrast to Lanczos, which is unstable because of loss of orthogonality and
spurious degeneracy, Chebyshev iteration is completely stable.

Since the number of Chebyshev moments $\mu_{m}$ in equation (2.7) is finite,
it is necessary to convolute the approximated function with kernel polynomials
in order to damp out the Gibbs oscillations and ensure properties such as
positivity and normalization. In this paper, we use the strictly positive
Jackson kernel [41]

${g_{{}_{m}}}=\left({1-\frac{{m\phi}}{\piup}}\right)\cos\left({m\phi}\right)+\frac{\phi}{\piup}\sin\left({m\phi}\right)\cot\left(\phi\right)$
(2.9)

with $\phi=\piup/\left(M+1\right)$. The broadened peak for an order $M$
expansion of a $\delta$ function at position $x_{0}$ is a good approximation
to a Gaussian of width $\sigma$ [38]

$\sigma=\sqrt{\frac{{M-x_{0}^{2}\left({M-1}\right)}}{{2\left({M+1}\right)}}\left({1-\cos
2\phi}\right)}\approx\frac{\piup}{M}\sqrt{1-x_{0}^{2}+\frac{{4x_{0}^{2}-3}}{M}}.$
(2.10)

In order to distinguish between resolution and localization [40, 42], it is
essential to ensure a uniform energy resolution, namely constant $\sigma$,
over the whole spectrum by restricting the number of moments in equation
(2.10).

##  3 Calculation and discussion

In our calculation, the number of lattice sites is $N=4.5\times
10^{4}-1.8\times 10^{5}$ with periodic boundary conditions. Following the
reference [43], each carbon atom of $\beta$-graphyne is described by one
2$p_{z}$ orbital, and the corresponding hopping energies can be obtained, with
$t_{1}=-2$ eV, $t_{2}=-2.7$ eV and $t_{3}=-4.3$ eV. In figure 2, we directly
calculate the band structure and its associated DOS of ordered
$\beta$-graphyne without expanding in Chebyshev polynomials, which is in
consistence with the results of the [13].

Figure 2:  Band structure and DOS of ordered $\beta$-graphyne without
expanding in Chebyshev polynomials.

Figure 3: (Colour online) Typical DOS (blue) and mean DOS (red) for
$\beta$-graphyne on a $1.8\times 10^{5}$ lattice sites with periodic boundary
conditions for different disorder strengths: (a) $\gamma=0$; (b) $\gamma=2t$;
(c) $\gamma=4t$; (d) $\gamma=8t$; (e) $\gamma=10t$; and (d) $\gamma=12t$.
Results are obtained using $N\sigma=45$ at $K_{\rm{s}}=32$ sites for
$K_{\rm{r}}=32$ realizations of disorder. Disorder strength and energy are in
units of $t$ ($t=1$ eV).

We calculate the typical DOS $\rho_{\rm ty}$ and the mean DOS $\rho_{\rm me}$
of a $\beta$-graphyne sheet with $1.8\times 10^{5}$ lattice sites for
different sites and disorder realizations $K=32\times 32$. The results are
shown in figure 3. As $\beta$-graphyne possesses the Dirac structure [13], the
valence and conduction bands meet at the Dirac points and form the Dirac cone
in the vicinity of the Dirac point. Consistently, the DOS is zero at the Fermi
level, corresponding to $E=0$ in figure 3 (a). For the ordered
$\beta$-graphyne, $\rho_{\rm{me}}(E)$ and $\rho_{\rm{ty}}(E)$ coincide with
each other, as shown in figure 3 (a). With disorder increasing, starting from
two boundaries of the spectrum, $\rho_{\rm{ty}}(E)$ is suppressed and when
$\gamma>12t$ it vanishes, namely, at the critical disorder strength, the
entire spectrum is localized. At the same time, the value of
$\rho_{\rm{me}}(E)$ is always finite, no matter how disorder strength varies.

As mentioned above, only a single finite size system is insufficient to study
the localization properties. We need to consider the normalized typical DOS,
$R(E)$. Figure 4 presents $R(E)$ of five different system sizes for the energy
$E=0$ as function of disorder strength. As Anderson transition can be
influenced by the finite size of the system, and the resolution of KPM, for
different system sizes $R(E)$, does not coincide very well but we can find
that the curves tend to converge. More importantly, with an increasing
disorder strength, $\rho_{\rm{ty}}\to 0$, which indicates Anderson
localization. Then, by calculating $R(E)$, it is possible to reliably
determine the critical disorder strength $\gamma_{c}$. For example, we obtain
the $W_{\rm{c}}\simeq 11t$ [$E=0$ and $R\left(E\right)=0.05$], which is of the
order of bandwidth. The existence of critical disorder means that the
localization behavior of $\beta$-graphyne is similar to 3D case, but contrary
to the predicted 2D case of one-parameter theory.

Figure 5 shows the contours of the normalized typical DOS $R(E)$ in the
energy-disorder plane. Since localized electrons do not contribute to the
charge transport, the energy that separates localized states from the extended
ones is called the mobility edge. From the contour map, we can find the
mobility edge [$\rho_{\rm{ty}}(E)/\rho_{\rm{me}}(E)=0.05$] and apparently all
states become localized for $\gamma>12t$. In figure 5, we can also find that
the value of $R(E)$ at two boundaries of the spectrum is finite. However, from
figure 5 we know that $\rho_{\rm{me}}(E)$ [which coincides with the standard
density of states $\rho(E)$] is close to zero, which indicates that the
probability of reaching these energy states is particularly small. Thus, these
finite values of $R(E)$ will make no sense and the corresponding curve is
known as the Lifshitz boundaries.

Figure 4: (Colour online) Normalized typical DOS, $R(E)$, as a function of
disorder, are calculated at $E=0$ for five different lattice sizes, with
$K_{\rm{r}}\times K_{\rm{s}}=32\times 32$ and $N\sigma=45$.

Figure 5:  Contour plot of $R\left(E\right)$ in the energy-disorder plane.
Mobility edge obtained for $\rho_{\rm{ty}}/\rho_{\rm{me}}=0.05$ as well as the
Lifshitz boundaries.

Figure 6: (Colour online) LDOS map for different disorder strengths (a)
$\gamma=0$; (b) $\gamma=4t$; (c) $\gamma=8t$; and (d) $\gamma=12t$ at energy
$E=0$. Here, the calculations were performed on $4.5\times 10^{4}$ lattice
sites with $N\sigma=45$.

For a given energy $E$, $\rho_{i}(E)$ can measure the local amplitude of the
wave function at site $i$. In order to understand the internal structure of
the extended and localized states, we calculate the LDOS in the band center
($E=0$) for one realization of disorder in figure 6. For the ordered
$\beta$-graphyne sheet, the distribution of LDOS is uniform [figure 6 (a)].
When the disorder strength $\gamma=4t$, from figure 6 (b) we can see that
although the distribution of LDOS becomes nonuniform, the states still remain
delocalized. As the disorder strength increases to $\gamma=8t$, the states
clump into some finite regions of clusters, as shown in figure 6 (c), which
indicates that the localization takes place. However, between the clusters,
there exist bridges where the LDOS is very small but nonvanishing. When the
disorder strength reaches $\gamma=12t$, the states are confined to many
isolated islands, which implies that these states are totally localized and
the metal turns into an insulator, that is to say, the Anderson transition
occurs.

##  4 Conclusion

In summary, we use typical DOS $\rho_{\rm{ty}}$ and the normalized typical DOS
$R(E)$, which can be efficiently and precisely calculated by means of variable
moment kernel polynomial method, in order to detect the localization
transition of disordered $\beta$-graphyne sheet. When modelling a disorder,
the Anderson model is widely used in studying the basic localization features.
One parameter scaling theory predicts that disorder can induce localization
transition in 3D systems while all states are localized for infinite 2D case.
However, the metal-insulator transition in hydrogenated graphene [32],
disordered GNRs [44], and Si-MOSFET inversion layers [45] have been reported,
which are out of reach for Anderson model. For a single layer of disordered
$\beta$-graphyne sheet, we have proved that it also exhibits metal-insulator
transition and the critical disorder strength is of the order of the
bandwidth, which might be explained by percolation-based approaches [44, 46].
In our work, we only numerically analyze the localization properties of
$\beta$-graphyne. More details, such as the localization length, conductivity
[28, 47], the contribution of Dirac nodes and the reduced back-scattering, and
“resilience” properties of the Dirac nodes against weak to moderate disorder
[25, 48], need further investigation. Furthermore, we doubt that other
allotropes, such as $\alpha$\- and 6,6,12-graphyne, may possess similar
localization properties, which will be investigated systematically later.

##  5 Acknowledgements

The work is supported by the Doctoral Program Foundation of Henan Institute of
Technology.

## References

  * [1] Baughman R.H., Eckhardt H., Kertesz M., J. Chem. Phys., 1987, 87, 6687–6699, doi:10.1063/1.453405.
  * [2] Kang J., Li J., Wu F., Li S.S., Xia J.B., J. Phys. Chem. C, 2011, 115, 20466–20470, doi:10.1021/jp206751m.
  * [3] Coluci V.R., Braga S.F., Legoas S.B., Galvão D.S., Baughman R.H., Nanotechnology., 2004, 15, S142–S149, doi:10.1088/0957-4484/15/4/006.
  * [4] Chen J.M., Xi J.Y., Wang D., Shuai Z.G., Phys. Chem. Lett., 2013, 4, 1443–1448, doi:10.1021/jz4005587.
  * [5] Zheng J.J., Zhao X., Zhao Y.L., Gao X.F., Sci. Rep., 2013, 3, 1271, doi:10.1038/srep01271.
  * [6] Wang X.M., Mo D.C., Lu S.S., J. Chem. Phys., 2013, 138, 204704, doi:10.1063/1.4806069.
  * [7] Pei Y., Physica B, 2012, 407, 4436–4439, doi:10.1016/j.physb.2012.07.026.
  * [8] Yue Q., Chang S.L., Kang J., Tan J.C., Qin S.Q., Li J.B., J. Chem. Phys., 2012, 136, 244702,
doi:10.1063/1.4730325.

  * [9] Pan Q.Y., Liu H., Zhao Y.J., Chen S.Q., Xue B., Kan X.N., Huang X.W., Liu J., Li Z.B., ACS Appl Mater. Interfaces, 2019, 11, 2740–2744, doi:10.1021/acsami.8b03311.
  * [10] Zhang X.S., Wang Q., Jin Z.W., Chen Y.H., Liu H.B., Wang J.Z., Li Y.L., Liu S.Z., Adv. Mater. Interfaces, 2018, 5, 1701117, doi:10.1002/admi.201701117.
  * [11] Wang N., He J.J., Wang K., Zhao Y.J., Jiu T.G., Huang C.S., Li Y.L., Adv. Mater., 2019, 31, 1803202,
doi:10.1002/adma.201803202.

  * [12] Liu J., Chen C., Zhao Y., Adv. Mater., 2019, 31, 1804386, doi:10.1002/adma.201804386.
  * [13] Malko D., Neiss C., Viñes F., Görling A., Phys. Rev. Lett., 2012, 108, 086804,
doi:10.1103/PhysRevLett.108.086804.

  * [14] Kehoe J.M., Kiley J.H., English J.J., Johnson C.A., Petersen R.C., Haley M.M., Org. Lett., 2000, 2, 969–972, doi:10.1021/ol005623w.
  * [15] Haley M.M., Pure Appl Chem, 2008, 80, 519–532, doi:10.1351/pac200880030519.
  * [16] Haley M.M., Brand S.C., Pak J.J., Angew. Chem. Int. Ed., 1997, 36, 836–838, doi:10.1002/anie.199708361.
  * [17] Li G.X., Li Y.L., Liu H.B., Guo Y.B., Li Y.J., Zhu D.B., Chem. Commun., 2010, 46, 3256–3258,
doi:10.1039/b922733d.

  * [18] Liu R., Gao X., Zhou J.Y., Xu H., Li Z.Z., Zhang S.Q., Xie Z.Q., Zhang J., Liu Z.F., Adv. Mater., 2017, 29, 1604665, doi:10.1002/adma.201604665.
  * [19] Matsuoka R., Sakamoto R., Hoshiko K., Sasaki S., Masunaga H., Nagashio K., Nishihara H., J. Am. Chem. Soc., 2017, 139, 3145–3152, doi:10.1021/jacs.6b12776.
  * [20] Querlioz D., Apertet Y., Valentin A., Huet K., Bournel A., Galdin-Retailleau S., Dollfus P., Appl. Phys. Lett., 2008, 92, 042108, doi:10.1063/1.2838354.
  * [21] Li T.C., Lu S.P., Phys. Rev. B, 2008, 77, 085408, doi:10.1103/PhysRevB.77.085408.
  * [22] Zare M.H., Amini M., Shahbazi F., Jafari S.A., J. Phys.: Condens. Matter, 2010, 22, 255503,
doi:10.1088/0953-8984/22/25/255503.

  * [23] Roche S., Leconte N., Ortmann F., Lherbier A., Soriano D., Charlier J.C., Solid State Commun., 2012, 152, 1404–1410, doi:10.1016/j.ssc.2012.04.030.
  * [24] Pogorelov Y.G., Loktev V.M., Condens. Matter Phys., 2018, 21, 33704, doi:10.5488/CMP.21.33704.
  * [25] Amini M., Jarari S.A., Shahbazi F., Eur. Phys. Lett., 2009, 87, 37002, doi:10.1209/0295-5075/87/37002.
  * [26] Song Y., Song H.K., Feng S.P., J. Phys.: Condens. Matter, 2011, 23, 205501,
doi:10.1088/0953-8984/23/20/205501.

  * [27] Bardarson J.H., Tworzydlo J., Brouwer P.W., Beenakker C.W.J., Phys. Rev. Lett., 2007, 99, 106801,
doi:10.1103/PhysRevLett.99.106801.

  * [28] Fan Z.Y., Uppstu A., Harju A., Phys. Rev. B, 2014, 89, 245422, doi:10.1103/PhysRevB.89.245422.
  * [29] Naumis G.G., Phys. Rev. B, 2007, 76, 153403, doi:10.1103/PhysRevB.76.153403.
  * [30] Schleede J., Schubert G., Fehske H., Eur. Phys. Lett., 2010, 90, 17002, doi:10.1209/0295-5075/90/17002.
  * [31] Amini M., Jarari S.A., Shahbazi F., Eur. Phys. Lett., 2010, 90, 17003, doi:10.1209/0295-5075/90/17003.
  * [32] Bostwick A., McChesney J.L., Emtsev K.V., Seyller T., Horn K., Kevan S.D., Rotenberg E., Phys. Rev. Lett., 2009, 103, 056404, doi:10.1103/PhysRevLett.103.056404.
  * [33] Ardakani A.G., Nejad M.S., Eur. Phys. J. B, 2019, 92, 9, doi:10.1140/epjb/e2018-90513-x.
  * [34] Ilic S., Meyer J.S., Houzet M., Phys. Rev. B, 2019, 99, 205407, doi:10.1103/PhysRevB.99.205407.
  * [35] Fujimoto A., Perini C.J., Terasawa D., Fukuda A., Harada Y., Sasa S., Yano M., Vogel E.M., Phys. Status. Solidi B, 2019, 256, 1800541, doi:10.1002/pssb.201800541.
  * [36] Khademi A., Kaasbjerg K., Dosanjh P., Stohr A., Forti S., Starke U., Folk J.A., Phys. Rev. B, 2019, 100, 161405, doi:10.1103/PhysRevB.100.161405.
  * [37] Lemut G., Pacholski M.J., Ovdat O., Grabsch A., Tworzydlo J., Beenakker C.W.J., Phys. Rev. B, 2020,101, 081405, doi:10.1103/PhysRevB.101.081405.
  * [38] Weisse A., Wellein G., Alvermann A., Fehske H., Rev. Mod. Phys., 2006, 78, 275–306,
doi:10.1103/RevModPhys.78.275.

  * [39] Anderson P.W., Phys. Rev., 1958, 109, 1492–1505, doi:10.1103/PhysRev.109.1492.
  * [40] Schubert G., Fehske H., Quantum Percolation in Disordered Structures, Lecture Notes in Physics, vol. 762. Springer, Berlin, Heidelberg, 2009, doi:https://doi.org/10.1007/978-3-540-85428-9_5
  * [41] Silver R.N., Roeder H., Voter A.F., Kress D.J., J. Comp. Phys., 1996, 124, 115–130, doi:10.1006/jcph.1996.0048.
  * [42] Schubert G., Schleede J., Byczuk K., Fehske H., Vollhardt D., Phys. Rev. B, 2010, 81, 155106,
doi:10.1103/PhysRevB.81.155106

  * [43] Liu Z., Yu G.D., Yao H.B., Liu L., Jiang L.W., Zheng Y.S., J. Phys., 2012, 14, 113007,
doi:10.1088/1367-2630/14/11/113007.

  * [44] Adam S., Cho S., Fuhrer M.S., Das Sarma S., Phys. Rev. Lett., 2008, 101, 046404,
doi:10.1103/PhysRevLett.101.046404.

  * [45] Kravchenko S.V., Kravchenko G.V., Furneaux J.E., Pudalov V.M., D’lorio M., Phys. Rev. B, 1994, 50, 8039–8042, doi:10.1103/PhysRevB.50.8039.
  * [46] Das Sarma S., Lilly M.P., Hwang E.H., Pfeiffer L.N., West K.W., Reno J.L., Phys. Rev. Lett., 2005, 94, 136401, doi:10.1103/PhysRevLett.94.136401.
  * [47] Fan Z.Y., Uppstu A., Siro T., Harju A., Comput. Phys. Commun., 2014, 185, 28–39,
doi:10.1016/j.cpc.2013.08.009.

  * [48] Castro Neto A.H., Guinea F., Peres N.M.R., Novoselov K.S., Geim A.K., Rev. Mod. Phys., 2009, 81, 109–162, doi:10.1103/RevModPhys.81.109

Ukrainian 0

Ìåòîä ïîëíîìíîãî ÿäðà äëÿ àíäåðñîíâñüêîãî ïåðåõîäó ó íåâïîðÿäêîâàíîìó
$\beta$-ãðàôí Ã.Ê. Âàíã

Øêîëà ïðèðîäíèèõ íàóê, Õåíàíñüêèé òåõíîëîãíèé íñòèòóò, 453003 Ñíüñÿí, Êèòàé