# Stochastization of long living spin-cyclotron excitations in a spin-
unpolarised quantum Hall system

S. Dickmann and B. D. Kaysin  Institute of Solid State Physics, Russian
Academy of Sciences, Chernogolovka, 142432 Russia

###### Abstract

In the present study we address the kinetics of long-lived excitations at zero
temperature in an electronic quantum Hall system with filling factor $\nu=2$.
The initial coherent state of spin-cyclotron magnetoexitons with two-
dimensional wave vector ${\bf q}=0$ resulting from laser pumping is
stochastized over time due to emission of acoustic phonons. The elementary
emission process requires participation of two magnetoexitons, so the
effective rate of phonon emission is proportional to the excitation density
squared, and the stochastization process occurs nonexponentially with time.
The final distribution of magnetoexitons over 2D momenta, established as a
result of stochastization, is compared with equilibrium distribution at finite
temperature

In recent years there has been a clear interest in studying long-lived exciton
systems in two-dimensional semiconductor structures, obviously in connection
with the prospect of observing a macroscopic quantum condensate in a dense
multi-exciton ensemble (similar, for instance, to the Bose-Einstein
condensate). A special place here is occupied by purely electronic spin
magnetoexitons formed in the conduction band of quantum Hall systems.UFN  Due
to their high symmetry (in particular, their ‘hole’, being just a vacancy at
the almost occupied electron Landau level, has a negative cyclotron mass in
magnitude exactly equal to the electron cyclotron mass), the number of
effective relaxation channels is substantially reduced compared to common
electron–valence-hole excitons. As a result, the lifetime of purely electronic
cyclotron spin-flip excitons (CSFEs) reaches a record magnitude, $1\,$ms, in a
spin-unpolarised quantum Hall system.UFN  This CSFE at filling factor $\nu=2$
(see Refs. ka84, ; di05, ; ku05, ) is the subject of our present theoretical
study. Now, however, we will not discuss the physics of CSFE
relaxation/annihilation due to transition of the system to the ground state.
We address CSFE stochastization, decay of an initial coherent multi-excitonic
state, where all excitations have equal 2D momenta ${\bf q}\\!=\\!0$, into a
diffusive incoherent state provided that the total number of excitations
remains constant.

We remind that generally the CSFE in a ‘clean’ unpolarized quantum Hall
system, being a collective excitation from the $S\\!=\\!0$ ground state, is a
$S\\!=\\!1$ triplet characterised by 2D momentum ${\bf q}$. The three
components of the triplet differ by the values of spin quantum number
$S_{z}=-1,0,1$, thus being energetically gapped by Zeeman energy
$|g\mu_{B}B|$.ka84 ; di05 ; ku05  The lowest-energy excitation in GaAs quantum
Hall systems corresponds to the $S_{z}=1$ component and conventionally
represents promotion of an electron from the fully occupied zeroth Landau
level upward to the first one with a simultaneous spin flip from $s_{z}=-1/2$
to $s_{z}=1/2$. The energy of this CSFE component can be calculated in terms
of expansion over ratio $r_{s}\\!=\\!E_{\rm C}/\hbar\omega_{c}$ of
characteristic Coulomb energy $E_{\rm C}$ to cyclotron energy
$\hbar\omega_{c}$ (Refs. ka84, ; di05, ). In the case of ideally two-
dimensional electrons the $E_{\rm C}$ value would be equal to $e^{2}/\kappa
l_{B}$ ($\kappa$ is the dielectric constant, $l_{B}$ the magnetic length),
however, in a modern GaAs wide-thickness quantum well $E_{\rm C}$ is
considerably smaller. The CSFE energy counted off the ground-state level is
determined by formula

${E}_{\bf q}=\delta^{(0)}+{\cal E}_{q},$ (1)

where ${\cal E}_{q}$ is the ${\bf q}$-momentum dispersion which, calculated to
the first order in $r_{s}$, vanishes if $q\to 0$ (see review UFN,  and the
works cited therein). The value $\delta^{(0)}$ is the $q\\!=\\!0$
magnetoexciton energy,

$\delta^{(0)}\\!\equiv\\!\hbar\omega_{c}\\!\\!-|g\mu_{B}B|\\!+\\!{\varepsilon}_{0},$
(2)

that is, it includes the cyclotron and Zeeman energies and negative Coulomb
shift ${\varepsilon}_{0}$ that remains nonzero even if $q\\!\to\\!0$. [The
${\varepsilon}_{0}\\!<\\!0$ value representing the second order Coulomb
correction ($\sim\\!\hbar\omega_{c}r_{s}^{2}$) was calculated and
experimentally measured in works di05, ] and ku05, , respectively].

The specificity of experimental excitation is that initially, as a result of
laser pumping, a long-lived CSFE ensemble appears in the vicinity of point
${\bf q}=0$ of the phase $K$-space corresponding not to the minimum but to the
maximum of energy ${\cal E}_{q}$ in the range of actual values
$q\\!<\\!1.4\hbar/l_{B}$ of the 2D momenta where ${\cal E}_{q}$ is negative
(see Fig. 1). This ‘zero momentum’ ensemble is subsequently stochastized, so
that the main mass of excitons in the $K$-space diffuses to the vicinity of
the ‘shallow’ energy minimum at $q\\!\sim\\!q_{m}\\!\approx\\!0.9\hbar/l_{B}$
values, and finally CSFEs completely relax/annihilate therefrom. The
stochastization occurs without any change of the spin state, thus, certainly,
it is much faster than the total CSFE-relaxation process. However, the
stochastization is also associated, like relaxation, with emission of phonons
and limited by the laws of conservation of energy and momentum. In particular,
the stochastization is a two-excitonic process and, therefore, in this case
its rate strongly depends on the CSFE concentration.

Note that, if the electronic system were completely conservative, the
diffusion associated with the departure of the CSFE ensemble from the $q=0$
maximum would be simply impossible due to impossibility to provide an energy
release. The conservation is violated when the interaction with the lattice is
taken into account. At the same time in the ‘clean’ (translationary invariant)
system, the one-exciton process associated with the emission of a phonon is
kinematically forbidden: the equality following from the energy preservation
condition, ${\cal E}_{q}+\epsilon_{\rm ph}\\!({\bf k})={\cal
E}_{0}\\!\equiv\\!0$, is never fulfilled.foot1  Here $\epsilon_{\rm ph}$ is
the phonon energy, and ${\bf k}$ is the phonon wave vector with the 2D
component equal to $-{\bf q}$. For example, for bulk acoustic phonons within
approximation of isotropic dispersion law, we have $\epsilon_{\rm
ph}\\!=\\!\hbar c_{s}\sqrt{k_{z}^{2}\\!+\\!q^{2}}$, where it is known that
$c_{s}\\!>\\!3\,10^{5}\,$cm$\\!/$s. Fig. 1 shows the $-\hbar c_{s}q$
dependence (the blue straight line) which does not intersect the ${\cal
E}_{q}$ dispersion curve.

Figure 1: In the main part: the CSFE dispersions ${\cal E}_{q}$, calculated at
$B\\!=\\!4.18\,$T, in accordance with the equation (4) (orange curve) and by
using the analytical approximation (22) (green curve). The blue straight line
corresponds to dependence $-\hbar c_{s}q$ with $c_{s}\\!=\\!3\,10^{5}\,$cm/s.
In the inset: dependencies of ${\cal E}_{m}$ and $q_{m}$ on the magnetic field
found on the base of equations (4) and (5), where the $\chi(z)$ function is
considered to be fixed.

So, the elementary quantum transition process from the $q=0$ state to the
$q>0$ one has at least to be two-excitonic. We use an approach of ‘excitonic
representation’ (for more details see Refs. di05,  and di19, ), which main
idea is to abandon the basis of Fermi one-electron states and switch to the
basis of so-called exciton states that diagonalize some essential part of the
Coulomb interaction. The exciton states in a purely electronic quantum Hall
system are generated by exciton operators originally defined via Dirac
electron operators, i.e., if $p$ is the ‘intrinsic’ quantum number of a
continually degenerated Landau level and $a_{p}$ and $b_{p}$ are the
annihilation operators corresponding to binary indexes $a$ and $b$ [each
designates both the Landau level number and the spin sublevel,
$a=(n_{a},\sigma_{a})$], then the exciton creation operator is foot2

$\displaystyle{{\cal Q}_{ab\,{\bf q}}^{{\dagger}}={{\cal
N}_{\phi}^{-1/2}}\sum_{p}\,e^{-iq_{x}\\!p}\,b_{p+\frac{q_{y}}{2}}^{{\dagger}}\,a_{p-\frac{q_{y}}{2}}}\vspace{-3.mm}$
(3)

($p$ and ${\bf q}$ are considered here and in the following to be measured in
$1/l_{B}$ units), where ${\cal N}_{\phi}$ is the number of magnetic flux
quanta in the system in question. $\left(\mbox{Note also that}\,{\cal
Q}_{ab\,{\bf q}}\\!\equiv{\cal Q}_{ba-\\!{\bf q}}^{{\dagger}}\right)$. These
${\cal Q}$-operators have a very important property: when acting on the state
of the quantum Hall system they add value $\hbar{\bf q}/l_{B}$ to the total
momentum of the system since there occurs commutator equality
$\left[{\hat{P}},{\cal Q}_{ab{\bf q}}^{{\dagger}}\right]={\bf q}{\cal
Q}_{ab{\bf q}}^{{\dagger}}$, where ${\hat{P}}$ describes the dimensionless
(with $\hbar=l_{B}=1$) ‘momentum’ operator (see Ref. di19,  and references
therein). In particular, if $|0\rangle$ is the ground state, then the exciton
state ${\cal Q}_{ab\,{\bf q}}^{{\dagger}}|0\rangle$, if not zero, is the
eigenstate of momentum operator ${\hat{P}}$ with eigen quantum number ${\bf
q}$. Thus, exciton states, in contrast to single electron states, possess a
natural quantum number, namely, the 2D momentum whose existence is the
consequence of the translational invariance. We study CSFE at $\nu\\!=\\!2$,
hence in our case $a\\!=\\!(0,\downarrow)$ and $b\\!=\\!(1,\uparrow)$. The
${\cal Q}_{{\bf q}}^{{\dagger}}|0\rangle$ state (we omit the ‘$ab$’ index,
besides, note that ${\cal Q}_{{\bf q}}|0\rangle\\!\equiv\\!0$) represents an
eigen state of our quantum Hall system yielding, in accordance with the
solution of the many-electron Schrödinger equation,ka84 ; di19  the CSFE
energy to the first order in terms of parameter $r_{s}$ with the
$q$-dispersion part:

${}\\!\\!\\!{\cal E}_{q}\\!\\!=\\!\frac{e^{2}}{\kappa
l_{B}}\\!\\!\int_{0}^{\infty}\\!\\!\\!{ds}\,e^{-s^{2}/2}{\cal
F}(s)\left(1\\!-\\!\frac{s^{2}}{2}\right)\left[1\\!-\\!J_{0}(sq)\right]\vspace{-1.mm},$
(4)

($J_{0}$ is the Bessel function). Here the formfactor is (cf. Ref. an82, )

${}\\!\\!{\cal
F}(q)\\!\displaystyle{=\\!\int\\!\\!\\!\int\\!dz_{1}dz_{2}e^{-q|z_{1}-z_{2}|/l_{B}}|\chi(z_{1})\chi(z_{2})|^{2}\\!,}\vspace{-1.mm}$
(5)

where $\chi(z)$ describes the electron size-quantized functions in the quantum
well.

We consider the situation where the CSFE ensemble represents a rarefied gas,
and thus we ignore any direct interaction among excitons originating from the
$e$-$e$ Coulomb coupling accounted ab initio.di19  This approach is certainly
valid if the number of excitons $N$ is much smaller than ${\cal N}_{\phi}$
(number of electrons $\\!=2{\cal N}_{\phi}$). However, there is an indirect
inter-excitonic coupling via the GaAs lattice, which enables us to study the
two-exciton process of quantum transition from the initial state

$|N\rangle=({\cal Q}^{\dagger}_{\bf 0})^{N}|0\rangle\vspace{-1.mm}$ (6)

describing the CSFE ensemble to the final one

$|N;{\bf q}_{1},{\bf q}_{2}\rangle=({\cal Q}^{\dagger}_{\bf
0})^{N\\!-\\!2}{\cal Q}^{\dagger}_{{\bf q}_{1}}{\cal Q}^{\dagger}_{{\bf
q}_{2}}|0\rangle\vspace{-1.mm}$ (7)

with nonzero momenta ${\bf q}_{1}$ and ${\bf q}_{2}$. If $-{\bf q}$ is the 2D
component of the emitted phonon, i.e. ${\bf k}\\!=\\!(-{\bf q},k_{z})$, then

${\bf q}_{1}\\!+{\bf q}_{2}\\!-{\bf q}\\!=\\!0.\vspace{-1.mm}$ (8)

In addition, the energy conservation condition must hold:

${\cal E}_{q_{1}}+{\cal E}_{q_{1}}+\epsilon_{\rm ph}({\bf
k})\\!=\\!0.\vspace{-1.5mm}$ (9)

Using the excitonic representation approach, one can express any interaction,
saving the number of electrons in the conduction band, in terms of ${\cal
Q}$-operators of one kind or another depending on the correct choice of
operators $a_{p}$ and $b_{p}$. Commutation algebra of $Q$-operators and
mathematical expectations (matrix elements) containing various excitonic and
multi-excitonic states are known and have already been repeatedly calculated
(see Refs. di05, ; di19,  and works cited therein). In particular, it is not
difficult to find that in our case, when $1\\!\ll\\!N\\!\ll\\!{\cal
N}_{\phi}$, the norms squared of the initial and final states are equal to

${}\\!{}\\!\begin{array}[]{l}\langle N|N\rangle\approx N!\quad{\rm
and\vspace{1.mm}}\\\ \langle{\bf q}_{2},{\bf q}_{1};N|N;{\bf q}_{1},{\bf
q}_{2}\rangle\approx(N\\!-\\!2)!(1+\delta_{{\bf q}_{1},{\bf
q}_{2}}),\end{array}$ (10)

where $\delta_{...}$ is the Kronecker delta.

Now we present the interaction of electrons with 3D acoustic phonons in terms
of the excitonic representation (see also Refs. di96, , di00,  and di12, ).
The Hamiltonian is written as (see, e.g., Ref. io89, ):

${\hat{H}}_{\rm
e-ph}\\!\\!=\\!\\!\frac{{\hbar}^{1/2}}{LL_{z}^{1/2}}\\!\\!\\!\sum_{{\bf
q},{k}_{z},s}\\!\\!{U^{\prime}}_{s}({\bf k}){\hat{\cal P}}_{{\bf k},s}{\cal
H}_{{\rm e-ph}}({\bf q})\\!+\mbox{H.~{}c.},\vspace{-1.mm}$ (11)

where $L^{2}\\!=\\!2\pi{\cal N}_{\phi}l_{B}^{2}$ is the 2D area, and $L_{z}$
the dimension of the sample along ${\hat{z}}$,

${\cal H}_{\rm e\\!-\\!ph}({\bf q})\\!=\\!\\!\int\\!e^{i{\bf
qr}}{\hat{\Psi}}^{{\dagger}}({\bf r}){\hat{\Psi}}({\bf r})\,d^{2}r,\quad{\bf
r}=(x,y);$ (12)

${\hat{\cal P}}_{{\bf k},s}$ is the phonon annihilation operator (index $s$
denotes possible phonon polarizations: longitudinal or one of two transverse
polarizations), and ${U^{\prime}}_{s}({\bf k})\\!=\\!U_{s}({\bf
k})\Phi(k_{z})$ is the renormalized the vertex which includes the fields of
deformation and piezoelectric couplings. Averaging over coordinate $z$ has
already being performed and is reduced to the appearance of formfactor

$\Phi(k_{z})=\int\chi^{*}(z)e^{ik_{z}z}\chi(z)\,dz.\vspace{-1.mm}$ (13)

The isotropic model for the phonon field gale87  enables us to take into
account deformation and piezoelectric couplings independently and use the
approximation where we take no difference between longitudinal and transverse
sound velocities. In this case for the three-dimensional vertex one needs only
an expression for the sum of squares equal to

$\sum_{s}|U_{s}|^{2}=\pi\varepsilon_{\rm ph}({\bf k})/{\cal
K}_{0}^{3}\tau_{\rm A}({\bf k}),\vspace{-3.mm}$ (14)

where the phonon energy is $\epsilon_{\rm ph}=\hbar
c_{s}\sqrt{k_{z}^{2}\\!+\\!q^{2}}/l_{B}$ (here and further both, $k_{z}$ and
${\bf q}$, are considered to be dimensionless), ${\cal K}_{0}=2.52\cdot
10^{6}\,$cm-1 is the material parameter of GaAs (see Ref. gale87, ), and
$\tau_{\rm A}({\bf k})$ is the characteristic time of 3D acoustic phonons,
namely:

$\frac{1}{\tau_{A}({\bf k})}=\frac{1}{\tau_{D}}+\frac{5({\cal
K}_{0}l_{B})^{2}}{k^{6}\tau_{P}}(q_{x}^{2}q_{y}^{2}+q^{2}k_{z}^{2}),$ (15)

calculated under the condition that ${\hat{x}},\;{\hat{y}},\;{\hat{z}}$ are
the directions of the principal crystal axes of the cubic lattice.di96 ; di00
Nominal times $\tau_{D}\simeq 0.8\,$ps and $\tau_{P}\simeq 35\,$ps
characterize respectively deformation and polarization electron–acoustic-
phonon scattering in the three-dimensional GaAs crystal for longitudinal sound
velocity $c_{s}\\!=\\!5.14\cdot 10^{5}\,$sm/c (see Ref. di96,  and cf. Ref.
gale87, ).

The dimensionless operator (12) can be presented in terms of the excitonic
representation in the usual way.di96 ; di00  In our case the relevant terms
are:

${\cal H}_{{\rm e\\!-\\!ph},{\bf q}}={\cal
N}_{\phi}e^{-q^{2}\\!/4}\\!\left[{\cal A}_{{\bf
q}}\\!+(1\\!-\\!q^{2}\\!/2){\cal B}_{{\bf q}}\right],$ (16)

where we use intra-sublevel operators

$\displaystyle{{\cal A}_{{\bf q}}^{{\dagger}}\\!={{\cal
N}_{\phi}}^{-1}\\!\sum_{p}\,e^{-iq_{x}\\!p}\,a_{p+\frac{q_{y}}{2}}^{{\dagger}}\,a_{p-\frac{q_{y}}{2}}}\vspace{-3.mm}$

(${\cal A}_{{\bf q}}\\!\\!\equiv\\!{\cal A}_{-{\bf q}}^{{\dagger}},\;$ ${\cal
B}_{\bf q}^{{\dagger}}$ means the $a\to b$ substitution; we note that ${\cal
B}_{\bf q}^{{\dagger}}|0\rangle\\!\equiv\\!0$ and ${\cal A}_{\bf
q}^{{\dagger}}|0\rangle\\!\equiv\\!\delta_{{\bf q},0}|0\rangle$). When so
doing, comparing to previous works,di96 ; di00  one has taken into account
that the $a_{p}$ and $b_{p}$ operators belong to different Landau levels.
Therefore, the factors in the square brackets at ${\cal A}_{{\bf q}}$ and
${\cal B}_{{\bf q}}$ in Eq. (16) turn out to be Laguerre polynomials
$L_{0}(q^{2}\\!/2)\\!\equiv\\!1$ and
$L_{1}(q^{2}\\!/2)\\!\equiv\\!(1\\!-\\!q^{2}\\!/2)$.

Our next task is to calculate the transition matrix element of operator (11)
between the initial state $|N\rangle$ and the final one ${\hat{\cal P}}_{{\bf
k},s}^{\dagger}|N;{\bf q}_{1},{\bf q}_{2}\rangle$,

${}\\!{\cal M}_{{\bf k},s,{\bf q}_{1},{\bf
q}_{2}}\\!=\\!\displaystyle{\frac{\langle{\bf q}_{2},{\bf
q}_{1};\\!N|{\hat{\cal P}}_{{\bf k},s}{\hat{H}}_{\rm
e-ph}|N\rangle}{\left(\langle N|N\rangle\langle{\bf q}_{2},{\bf
q}_{1};N|N;{\bf q}_{1},{\bf q}_{2}\rangle\right)^{1/2}}},\vspace{-1.mm}$ (17)

and, thus, finding the probability of transition per unit of time according to
the well-known formula

$W_{{\bf k},s,{\bf q}_{1},{\bf
q}_{2}}\\!\\!=\\!\frac{2\pi}{\hbar}\\!\left|{\cal M}_{{\bf k},s,{\bf
q}_{1},{\bf q}_{2}}\right|^{2}\\!\delta[{\cal E}_{q_{1}}\\!+{\cal
E}_{q_{2}}\\!+\epsilon_{\rm ph}({\bf k})]\vspace{-1.mm}$ (18)

($\delta[...]$ is the Dirac delta-function). If we perform the summation

$R_{\bf p}=\displaystyle{\sum_{{\bf k},s,{\bf
q}_{2}\vspace{-1.mm}}\\!\\!W_{{\bf k},s,{\bf p},{\bf q}_{2}}},\vspace{-2.mm}$
(19)

we obtain the total probability of transition to a state, where one of
‘nonzero’ magnetoexcitons has a fixed wave vector: ${\bf q}_{1}\\!=\\!{\bf p}$
(or ${\bf q}_{2}\\!=\\!{\bf p}$).

Calculation of the ${\cal M}_{{\bf k},s,{\bf q}_{1},{\bf q}_{2}}$ value is
reduced to calculating the expectation

$M_{{\bf q},{\bf q}_{1},{\bf q}_{2}}\\!\\!\\!=\\!\langle 0|{\cal Q}_{{\bf
q}_{1}}{\cal Q}_{{\bf q}_{2}}({\cal Q}_{\bf 0})\\!{}^{N\\!-\\!2}{\cal H}_{{\rm
e\\!-\\!ph},{\bf q}}^{\dagger}({\cal Q}_{\bf
0}^{\dagger})\\!{}^{N}|0\rangle,\vspace{-1.mm}$ (20)

which is based on the commutation algebra for the ${\cal A}_{...}$-, ${\cal
B}_{...}$\- and ${\cal Q}_{.}..$-operators many times used in previous
works.di05 ; di19 ; di96 ; di00  The matrix elements, where bra- and ket-
vectors are multi-exciton states, have also been numerously calculated (see,
e.g., Ref. di19, ). In this letter we omit the details of relevant algebraic
manipulations and present only the main key points of this procedure. First,
note that the sum ${\cal A}_{\bf q}^{\dagger}\\!+\\!{\cal B}_{\bf
q}^{\dagger}$ commutes with ${\cal Q}_{\bf 0}^{\dagger}$, hence, the only term
of operator (16) contributing to expectation (20) is $-{\cal
N}_{\phi}(q^{2}\\!/2)e^{-q^{2}\\!/4}{\cal B}_{\bf q}^{\dagger}$. By
calculating

$\displaystyle{{\cal B}_{\bf q}^{\dagger}({\cal Q}_{\bf
0}^{\dagger})^{N}|0\rangle\equiv({N}/{\cal N}_{\phi}){\cal Q}_{\bf
q}^{\dagger}({\cal Q}_{\bf 0}^{\dagger})^{N\\!-\\!1}|0\rangle}\vspace{-1.5mm}$

and after that calculating the action of ${\cal Q}_{\bf q}$ onto state (7), we
obtain:foot3

${}\\!{}\\!{}\\!{}\\!{}\\!{}\\!{}\\!\begin{array}[]{l}\>M_{{\bf q},{\bf
q}_{1},{\bf q}_{2}}{}\\!{}\\!\vspace{1.mm}{}\\\
\>=\displaystyle{\frac{N}{{\cal
N}_{\phi}}}\,q^{2}e^{-q^{2}\\!/4}\\!\cos{\\![{\bf q}_{1}\\!\\!\times\\!{\bf
q}_{2}]}{\vphantom{\left(\\!\frac{N}{{\cal N}_{\phi}}\\!\right)}}\langle
N\\!\\!-\\!1|N\\!\\!-\\!\\!1\rangle\delta_{{\bf q}_{1}\\!+\\!{\bf q}_{2},{\bf
q}}.\vspace{-1.mm}\end{array}\vspace{-.5mm}$ (21)

Then we find matrix element (17) and finally with the help of Eqs. (10), (11)
and (13) – (15) perform summation in accordance with formula (19). To
implement this, we model formfactor (13) by employing as $\chi(z)$ a function
numerically calculated with the help of a conventional routine procedure
(first appeared in Ref. pi92, ; see also, for instance, Ref. di19, ), and in
our case already used for calculation of the dispersion curve shown in Fig. 1.
The specific function $\chi$ employed in the present work is relevant to a
quantum well where the filling factor $\nu\\!=\\!2$ would correspond to
magnetic field $B\\!=\\!4.18\,$T (see Ref. [zh19, ]). The result,
$\Phi(k_{z})$, is demonstrated in the inset in Fig. 2. We also model the
excitation spectrum (4) with polynomial:

${\cal E}_{q}=(e^{2}\\!/\\!\kappa l_{B}\\!){\cal
E}_{m}\\!\left(\\!2q^{3}\\!/q_{m}^{3}-3q^{2}\\!/q_{m}^{2}\right),\vspace{-1.mm}$
(22)

where ${\cal E}_{m}$ and $q_{m}$ are measured in $e^{2}\\!/\\!\kappa l_{B}$
and $1/l_{B}$ units, respectively [see Fig. 1; the graph of function (22) is
shown by the green line; the $B$-dependencies of ${\cal E}_{m}$ and $q_{m}$,
found under the condition of fixed size-quantized function $\chi(z)$ and
filling factor $\nu\\!=\\!2$, are shown in the inset]. So, we obtain

$R_{p}=I(p)(N/{\cal N}_{\phi})^{2}\\!/({\cal K}_{0}l_{B})^{3},\vspace{-2.mm}$
(23)

where

${}\\!{}\\!{}\\!{}\\!\begin{array}[]{l}\displaystyle{{}I(p)\\!=\\!\int\\!\\!\int
dqd\varphi\frac{q^{5}e^{-q^{2}\\!/2\\!}}{2\pi\tau_{A}(q,k_{z})}\vspace{1.mm}}\\\
\qquad\quad\times\displaystyle{|\Phi(k_{z})|^{2}\\!\left(\frac{q^{2}}{k_{z}}\\!+\\!k_{z}\right)\cos^{2}{\\!(pq\sin{\varphi})}}\end{array}\vspace{-1.mm}$
(24)

(certainly, $R_{p}\\!\equiv\\!R_{\bf p})$. Here $k_{z}$ is determined by
equations (8) and (9),

${}\\!{}\\!k_{z}({\bf p},{\bf q})\\!=\\!\sqrt{(l_{B}/\hbar
c_{s})^{2}\left({\cal E}_{p}+{\cal E}_{|{\bf q}\\!-\\!{\bf
p}|}\right)^{2}\\!\\!-\\!q^{2}}\,,\vspace{-1.mm}$ (25)

and also $\tau_{A}$ is presented by formula (15) where averaging
$\overline{q_{x}^{2}q_{y}^{2}}=q^{4}\\!/8$ is performed. The area of the
integration in Eq. (24) is determined by two conditions: (i) the energy in the
final two-exciton state with nonzero momenta (7) must be smaller than that in
the initial state, and (ii) the root expression in formula (25) for $k_{z}$
must be positive. Both can be presented by the inequality:

$(l_{B}/\hbar c_{s})\left({\cal E}_{p}\\!\\!+\\!{\cal E}_{|{\bf q}\\!-\\!{\bf
p}|}\right)+q<0.\vspace{-1.mm}$ (26)

The physical meaning of the value $R_{p}$ (23) is that it represents the rate
of appearance of a magnetoexciton with momentum ${\bf p}$ due to the
considered process of direct transition from the initial coherent state
$|N\rangle$ (6) to any state $|N;{\bf p},{\bf q}\rangle$ with unfixed number
${\bf q}$. When studying the problem kinetically and neglecting any inter-
magnetoexcitonic coupling, $R_{p}$ will mean the rate of filling of a ‘one-
particle’ magnetoexcitonic state with specific momentum ${\bf p}$. It is
obvious that the total rate induced by phonon-emission, $R\\!=\\!\sum_{\bf
p}R_{p}$, is, on the one hand, the rate of the coherent state $|N\rangle$
decay/stochastization, and, on the other hand, the rate of appearance of
nonzero magnetoexcitons in the system. The physical meaning of the value R
allows us to consider the kinetic equation:

$dN\\!/dt=\displaystyle{-R\equiv-\frac{N^{2}}{{\cal N}_{\phi}{\cal
T}}},\vspace{-2.mm}$ (27)

where

$\frac{1}{\cal T}=\frac{1}{({\cal
K}_{0}l_{B})^{3}}\\!\\!\int_{0}^{\infty}\\!\\!I(p)pdp.\vspace{-2.mm}$ (28)

is calculated with the help of Eqs. (24) and (26) (see the result in Fig. 2;
the employed formfactor $\Phi(k_{z})$ is demonstrated in the inset). For our
specific case, when $B\\!=4.18\\!$T (c.f. Ref. zh19, ), we get numerical value
${\cal T}\\!\approx\\!0.88\,$ns.

Solving equation (27), we obtain the time law of change of the number of
$q\\!=\\!0$ excitations:

$n(t)=n(0)\\!\mbox{\Large/}\\!\left[1+t\,n(0)/\\!{\cal
T}\right].\vspace{-1.mm}$ (29)

Here $n(t)\\!=\\!N\\!/{\cal N}_{\phi}$ is the concentration of zero-momentum
CSFEs, while value

$1-n(t)\\!/n(0)\equiv n(t)t\\!/\\!{\cal T}\vspace{-1.5mm}$ (30)

is the relative concentration of nonzero magnetoexcitons with respect to the
given total CSFE concentration $n(0)\\!=\\!N(0)/{\cal N}_{\phi}$ in the
system. When dividing the ‘partial’ rate $R_{p}$ by the total one $R$ we
obviously obtain a ‘one-particle’ nonzero magnetoexciton distribution
function,

$f_{p}=R_{p}/R,\vspace{-2.mm}$ (31)

normalized by unit, $\left(\sum_{\bf p}f_{p}\\!=\\!1\right)$. As both values,
$R_{p}$ and $R$, have the same time dependance, $\propto\\!{N(t)}^{2}$, the
ratio $f_{p}$ turns out to be time-independent. However, normalization by the
relative concentration of nonzero magnetoexcitons (30) seems to be more
natural. Indeed, the change from $f_{p}$ to
$f^{\prime}_{p}(t)\\!=\\!n(t)tf_{p}/{\cal T}$ reveals the physical meaning of
the value (31), namely: this one actually represents the final distribution
function $f^{\prime}_{p}(\infty)$ at $t\\!\to\\!\infty$, when the
stochastization is completed and only non-coherent excitations with nonzero
momenta are present in the system.

We neglect any correlation among appearing nonzero magnetoexcitons and their
coupling with the zero ones. This concerns also any inter-excitonic
correlations with participation of nonzero magnetoexcitons, including
correlation induced by coupling with the lattice and so resulting in energy
release (phonon emission). Thus, it has been assumed that the coherent
ensemble of zero-momentum excitons (6) is the only generator of nonzero
magnetoexcitons with neglecting any subsequent evolution of the emerging
nonzero magnetoexcitonic ensemble. In principle, this approach should be
suitable if the relative nonzero-magnetoexciton concentration (30) is small
($\ll\\!1)$ and, in addition, the temperature is sufficiently low to ignore
any phonon-absorption processes. In this case thermalization in the studied
electron system should be a much longer process than the stochastization
considered.

However, it is still interesting to compare the distribution function (31)
established due to stochastization to a thermodynamically equilibrium
distribution corresponding to some temperature. The latter should be Boltzmann
due to the rarefaction of magnetoexiton gas [$N(0)\\!\ll\\!{\cal N}_{\phi}]$,
namely:

$F_{p}^{(T)}=\displaystyle{e^{-{\cal E}_{p}\\!/T}\\!\\!\\!\mbox{
\huge{/}}\\!\\!\\!\int_{0}^{\infty}\\!\\!\\!e^{-{\cal
E}_{p}\\!/T}pdp}\,.\vspace{-1.mm}$ (32)

In Fig. 2 we demonstrate both distributions equally normalized, that is:
$F_{p}^{(T)}$ and $F_{p}={\cal N}_{\phi}f_{p}$. In the $T\to 0$ limit
$F_{p}^{(T)}\\!\\!\propto\delta(p-q_{m})$. However, already at temperature
$T\sim 0.5\,$K, the stochastization distribution $F_{p}$ becomes qualitatively
similar to the thermodynamically equilibrium one $F_{p}^{(T)}$.

Figure 2: The result of calculating the distribution function $F_{p}$ of CSFEs
emerging due to the stochastization process [the black line; see Eq. (31) and
the text], and the thermodynamically equilibrium distribution functions
$F_{p}^{(T)}$ at different temperatures (32). All graphs correspond to
$B=4.18\,$T. In the upper left: formfactor (13) squared as function of the
dimensionless $k_{z}$ value is shown.

So, the presented model results in a nonexponential decay of the initial
coherent CSFE ensemble (6). The time dependence of the decay (29) is
parameterized by time ${\cal T}$ (Fig. 2.). The number of zero magnetoexcitons
decreases by half during time ${\cal T}\\!/n(0)$ inversely proportional to the
initial CSFE concentration. A tenfold decrease will take time
$\approx\\!10{\cal T}\\!/n(0)$, therefore, for $n(0)\\!\leqslant\\!0.01$ it
occurs during $\gtrsim\\!1\,\mu$s [cf. the 3D characteristic electron-phonon
scattering time $\tau_{A}$ (15) which is $\sim\\!0.1\,$ps if
$q\\!\sim\\!k_{z}\\!\sim 1$, and the CSFE lifetime which is $>\\!50\,\mu$s at
$B\\!=\\!4.18\,$T if estimated on the basis of the recent experimental data
zh19 ).

Generally, a single magnetoexciton is able also to drift from the ${\bf q}=0$
state to some state with $q\\!>\\!0$. This weak stochastization channel exists
only due to violation of the translational symmetry of the system. It is known
that in modern wide-thickness quantum wells only the ‘smooth random potential’
(SRP) represents a real reason for such violation. Actual parameters
characterizing the SRP are: amplitude $\Delta\\!\simeq 5-7\,$K, and
correlation length $\Lambda\\!\simeq 50-70\,$nm ($\gg\\!l_{B}$). The
elementary process effectively resulting in stochastization and energy
release, is conversion of a zero-momentum magnetoexciton to a nonzero one with
dispersion energy ${\cal E}_{q}\simeq-(e^{2}\\!/\\!\kappa l_{B}){\cal E}_{m}$
and an acoustic phonon with energy $\epsilon_{\rm ph}({\bf
q}^{\prime},k_{z})=-{\cal E}_{q}$, where it is obvious that $q^{\prime}$ is
substantially less than $q$. Such a transition is described within the
framework of the second order perturbation-theory approach, which represents
the first order by electron-phonon coupling and simultaneously the first one
by the SRP. The estimation of the characteristic time for this process gives a
result that is at least $\gtrsim 10^{3}$ larger than the parameter ${\cal T}$
value. Thus, the single-magnetoexciton stochastization definitely ceases to be
dominant even at a CSFE concentration $n(0)\\!\simeq\\!0.1\%$.

We note that our approach does not allow us to consider asymptotically large
or small magnetic fields even if we ignore some difficulties of keeping $\nu$
equal to 2. Indeed, we are limited, first, by the condition of the two-
dimensionality of the problem, i.e. the cyclotron energy should be less than
the distance between the size-quantization energy levels:
$\omega_{c}\\!<\\!\hbar/m^{*}_{e}d^{2}$, where
$d^{-2}\\!\\!=\\!\\!\int\\!dz|\chi(z)|^{4}$; and second, by the need to comply
with the $r_{s}\\!<\\!1$ condition. In fact, the situation is even more
complicated, because by changing the magnetic field while keeping the value of
$\nu$, we must be aware that such a change corresponds to a change in the
quantum heterostructure. In other words, different values of $B$ are
associated with different heterostructures, therefore function $\chi(z)$ must
be appropriately recalculated depending on $B$. Accordingly, the formfactors
$F$ and $\Phi$ and spectrum ${\cal E}_{q}$ must be recalculated, too. However,
in the present theoretical work, we of course, have a formal right, by keeping
$\chi(z)$ and $\nu$, to calculate, at different $B$, the dispersion determined
by Eqs. (4) and (5). Qualitatively, the form of the dispersion curve does not
change, but the ${\cal E}_{q}$ spectrum presented in dimensionless units
becomes smoother with increasing $B$; for instance, the change of parameters
${\cal E}_{m}$ and $q_{m}$ is shown in the inset in Figure 1. The
dimensionless $k_{z}$ (25) and the integration domain (26) depend on the field
indirectly through the dependencies ${\cal E}_{m}(B)$ and $q_{m}(B)$,
meanwhile, calculations show that integrals in Eqs. (24) and (28) turn out to
be extremely sensitive to a change in $B$ (actually to a change in the
integration domain). For instance, if $B$ grows from $4$ to $8\,$T the
integral in Eq. (28) decreases by about 20 times. We emphasize that this is a
formal and physically rather meaningless result, yet, this is a significant
sign of the sharp dependence of our calculation on the CSFE spectrum.

At the same time, there is a hypothetical situation, although unrealistic but
quite self-consistent, where there are no problems associated with the
dependence of calculation on the magnetic field. Namely, this is an ultra two-
dimensional case: when $d\\!\ll\\!l_{B}$, but the $r_{s}$ smallness is not
violated and the $\nu\\!=\\!2$ condition holds. In this limit, both
formfactors $\Phi(k_{z})$ and $F(q)$ are set equal to unit. Then the
dimensionless value $k_{z}$ (25) and the integration domain (26) are
$B$-independent. As a result, with the help of Eqs. (23)-(28) [in Eqs. (25)
and (26) we use just analytical expression (4) with $F\\!=\\!1$] for this
ideally 2D case we find:

$1/{\cal
T}\approx\left(49.4B^{3/2}+3.9B^{1/2}\right)\,\mbox{ns${}^{-1}$},\vspace{-1.9mm}$
(33)

where $B$ is in Tesla. The first term here comes from the deformation
electron-phonon interaction and the second one from the e-ph polarization
coupling [see. Eq. (15)]. This formula for $B\\!=\\!4.18\,$T gives value
${\cal T}$ about 400 times smaller than our calculation above performed for a
real physical system.

In conclusion, we note another feature of the stochastization channel under
consideration. We have seen already that integral (24) and therefore the
obtained result (28) is very sensitive to changes in the condition (26) for
the phase volume of the occurring stochastization processes and, hence, to any
change of the energy dispersion spectrum (Fig. 1). However, we do not have
experimental data on this issue, and theoretical considerations leading to the
employed dependence [Eqs. (4), (22) and Fig. 1] contain significant
simplifications. In particular, we neglected any $q$-dependence of the
negative Coulomb shift associated with the second-order Coulomb correction
$\sim\\!r_{s}^{2}\hbar\omega_{c}$ to the energy spectrum (see above). We
ignored also any inter-magnetoexcitonic corrections. In the coherent state,
these virial corrections to the CSFE spectrum can be negative and more
significant than in the incoherent ensemble.di19  Then, effectively, when
recalculated by one excitation, the dependence ${\cal E}_{q}$ will be more
smoothed. Both mentioned corrections can reduce the $({\bf p},{\bf q})$-domain
of the integration in Eqs. (24) and (28), and thus lead to increasing the
stochastisation time.

The authors are grateful to L.V. Kulik for useful discussions. The research
was supported by the Russian Science Foundation: grant RSF-21-12-00386.

## References

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# Pixel-wise classification in graphene-detection with tree-based machine
learning algorithms

Woon Hyung Cho  Department of Physics, University of Seoul, Seoul 02504, Korea
Department of Smart Cities, University of Seoul, Seoul 02504, Korea  Jiseon
Shin  Department of Physics, University of Seoul, Seoul 02504, Korea  Young
Duck Kim  Department of Physics, Department of Information Display, KHU-KIST
Department of Converging Science and Technology, Kyung Hee University, Seoul
02447, Korea  George J. Jung  Department of Physics, University of Seoul,
Seoul 02504, Korea  Department of Smart Cities, University of Seoul, Seoul
02504, Korea

###### Abstract

Mechanical exfoliation of graphene and its identification by optical
inspection is one of the milestones in condensed matter physics that sparked
the field of 2D materials. Finding regions of interest from the entire sample
space and identification of layer number is a routine task potentially
amenable to automatization. We propose supervised pixel-wise classification
methods showing a high performance even with a small number of training image
datasets that require short computational time without GPU. We introduce four
different tree-based machine learning algorithms – decision tree, random
forest, extreme gradient boost, and light gradient boosting machine. We train
them with five optical microscopy images of graphene, and evaluate their
performances with multiple metrics and indices. We also discuss combinatorial
machine learning models between the three single classifiers and assess their
performances in identification and reliability. The code developed in this
paper is open to the public and will be released at github.com/gjung-
group/Graphene_segmentation.

##  1 Introduction

The fact that a few layers of graphene sheet can be prepared by simple
mechanical exfoliation novoselov2004electric ; yi2015review  has facilitated a
rapid growth of graphene and other two-dimensional (2D) van der Waals (vdW)
materials research. In particular, graphene has been studied in a wide range
of applications in recent years due to its unique electrical, mechanical, and
optical properties neto2009electronic ; koppens2014photodetectors ;
geim2007rise ; lee2008measurement ; cao2020elastic ; bonaccorso2010graphene ;
geim2013van ; ajayan2016van . The recent discovery of a robust unconventional
superconductivity in twisted graphene systems bistritzer2011moire ;
cao2018unconventional ; cao2018correlated ; park2021tunable ; hao2021electric
has reinvigorated research in graphene and other two-dimensional (2D) van der
Waals (vdW) materials chhowalla2013chemistry ; novoselov2005two ;
zeng2010white ; watanabe2004direct ; reich2014phosphorene ; chen2017rising .

After the mechanical exfoliation of 2D materials, the number of layers of
graphene or other vdW materials can be identified by various techniques
including atomic force microscopy (AFM) huang2015reliable ;
shearer2016accurate , Raman spectroscopy saito2011raman ; huang2015reliable ,
or optical microscopy (OM) blake2007making ; li2013rapid ;
jessen2018quantitative ; huang2019optical . Among them, the most commonly used
method is based on the optical contrast between single and multilayer graphene
layers with different thicknesses in RGB color space of OM images of materials
placed on a substrate of specific thickness blake2007making ; li2013rapid ;
jessen2018quantitative ; huang2019optical . However, it is a time consuming
process to process more than $\sim 10^{3}$ scanned OM images to identify the
interesting few layers exfoliated flakes regions deposited on the substrate.
Here we propose a practical machine learning image recognition method that can
be used to quickly identify specific target few layers graphene regions.

Traditionally, machine learning (ML) based techniques have been applied
succesfully in many different fields of industrial applications or services
requiring repetitive human labor. Especially, image classification using deep
learning (DL) has emerged as a game changer technique that has allowed drastic
reduction of analysis time from hours to seconds. For example, the
convolutional neural network (CNN) has been used in biomedical fields, such as
abdominal CT scan, cell, hippocampus, and pancreas segmentations
ronneberger2015u ; oktay2018attention ; vizcaino2021pixel , and in analyzing
big image data obtained from satellites zhao2018building ; leon2020big
providing significant aid to error-prone human eyes.

Recent works have used ML techniques to identify the number of layers in a
thin film of materials. These researches have employed supervised learning
such as support vector machines (SVM) lin2018intelligent ; yang2020automated ,
deep neural network (DNN) masubuchi2019classifying ; greplova2020fully ;
shin2021fast , U-Net which belongs to CNN saito2019deep ; dong20213d ;
siao2021machine , or unsupervised learning such as clustering
masubuchi2019classifying . The image classifications in the earlier works,
however, require too many images for the training dataset that need to be
labeled accordingly with layer number, for example, $\sim 10^{3}-10^{5}$
labeled images for DNN masubuchi2019classifying ; greplova2020fully ;
shin2021fast , and $\sim 10^{2}$ siao2021machine ; dong20213d  or $10^{3}$
labeled images for U-Net which are augmented from less than 50 labeled images
saito2019deep , and more than a dozen GB of GPU memory to process a batch of
image data.

Figure 1:  (a) Schematic diagram for exfoliation of graphene from graphite
using the scotch tape on top of the substrate SiO2/Si. (b) Motorized
microscope for scanning the sample. (c) Schematic diagram for the entire
sample is denoted by the green panel, the path indicated by the black arrow
for scanning the sample to find the graphene, and the camera of the motorized
microscope illustrated by the gray square. (d) Scanned images by the motorized
microscope along with the black arrows in (c).

In this paper we suggest a handy classification tool for detecting a specific
number of graphene layers pixel by pixel using supervised ML algorithms
requiring only a few OM images for training. We compare the performance of
four different tree-based ML algorithms such as decision tree (DT), random
forest (RF), extreme gradient boosting (XGBoost), and light gradient boosting
machine (LightGBM) in terms of accuracy, precision, recall, F1 score, and
indices such as receiver operating characteristic (ROC) curve, area under
curve (AUC), intersection over union (IOU). We take account of several
combinations between RF, XGB, and LGBM entailing improvements in multiple
metrics compared to using a single algorithm. Our source code was built with
scikit-learn pedregosa2011scikit  ML library and is provided as open-source
(See Ref. source ).

This paper is organized as follows. In Sec. 2 we first describe the sample
preparation for OM images, feature extraction using several filters to pre-
treat the images for ML algorithms. We elaborate on the four different tree-
based ML algorithms that we employed in this work in Sec. 3. In Sec. 4, we
evaluate and discuss the performances of a single tree-based ML algorithm and
combinations for fusions of them using the several metrics. In Sec. 5 we
present the conclusions.

##  2 Data Preprocessing

In the following we describe the data preprocessing techniques used for this
work that consists in dataset preparation from the OM images obtained from the
experimental system and feature extraction from the images by using different
filters.

###  2.1 Dataset preparation

The graphene samples in the present work were exfoliated using scotch tapes
and placed on top of 285 nm SiO2/Si substrates as shown in Fig. 1 (a). We use
a motorized microscope as seen in Fig. 1 (b) to scan the $20\times$ magnified
images with $1832\times 1372$ resolution. We scan the entire sample along the
path indicated by the black arrow as shown in Fig. 1 (c). Afterwards, we get
scanned OM images like Fig. 1 (d). To improve the speed of image preprocessing
and classification, images were resized with $458\times 343$ pixels which are
moderately large for finding graphene. We only use 5 OM images for training
and 2 images for testing. However, in terms of the number of segmentation
resolved by pixel, we use in total 19,636,750 pixels for the training dataset
and 785,470 pixels for the testing dataset. The mask for labeling was made
manually using the APEER platform apeer .

Figure 2:  Flowchart for data preprocessing for ML feature extraction. (a)
Original images from motorized spectroscopy. (b) Map for the hue (H) component
of the original images. (c) Map for the saturation (S) component of the
original images. (d) Map for the value (V) component of the original images.
(e) Images in grayscale. (f) Resultant map of the Prewitt edge filter for the
grayscale images. (g) Resultant map of the Scharr edge filter. (h) Resultant
map of the Gaussian filter.

###  2.2 Feature Extraction

Feature extraction is one of the most important key elements in ML. Here we
use the HSV color space and several commonly used filters that help us extract
relevant features to recognize and classify the monolayer graphene from the
background as we describe in the following. There are several color bases for
generating arbitrary colors that we are familiar with, such as RGB (red,
green, blue) and CMYK (cyan, magenta, yellow, black). However, here we
introduce the HSV (hue, saturation, value) color space, which is a familiar
way for humans to perceive color. The HSV color space describes colors with
their hue (H) together with saturation (S), namely, amount of gray, and value
(V) of brightness or luminous intensity as illustrated in Fig. 2 (b-d). Each
pixel of the graphene images obtained from the motorized microscope which were
originally expressed in RGB color space are transformed to HSV wen2004color ;
chen2007identifying .

We use several filters, the Prewitt, Scharr, and Gaussian filters in order to
extract the relevant features. We use the grayscale images for this process as
shown in Fig. 2 (e) for the following two reasons. Firstly, the grayscale
images preserve the essential information such as edge, shape, and texture
information from their original RGB representation. Secondly, we can reduce
complexity and unnecessary computational cost bui2016using . An edge of any
object in images can be defined as a boundary where the value of brightness
changes discontinuously, leading to a very sharp change in the associated
intensity gradient. Hence, we can find abrupt changes in the intensity values
for each pixel over the entire image given in HSV space, and find spots where
its derivative is maximum. We utilize the $3\times 3$ Prewitt, and Scharr
operators for the $x$ (vertical) and $y$ (horizontal) direction as defined
below prewitt1970object ; jahne1999handbook

$\textrm{Prewitt}_{x}=\frac{1}{3}\begin{pmatrix}1&0&-1\\\ 1&0&-1\\\
1&0&-1\end{pmatrix},$

$\textrm{Prewitt}_{y}=\frac{1}{3}\begin{pmatrix}1&1&1\\\ 0&0&0\\\
-1&-1&-1\end{pmatrix},$ (1)

and

$\textrm{Scharr}_{x}=\frac{1}{32}\begin{pmatrix}3&0&-3\\\ 10&0&-10\\\
3&0&-3\end{pmatrix},$

$\textrm{Scharr}_{y}=\frac{1}{32}\begin{pmatrix}3&10&3\\\ 0&0&0\\\
-3&-10&-3\end{pmatrix}.$ (2)

As an example, we present the resultant map for an image processed through the
Prewitt and Scharr filters in Fig. 2 (f) and (g), respectively. After the
convolution of the two operators with our grayscale image, we get resultant
$F_{x}$ and $F_{y}$ matrices of the same size as our source image. Hence, the
magnitude for each pixel is given by

$\textrm{Magnitude}=\sqrt{F_{x}^{2}+F_{y}^{2}}.$ (3)

Afterwards, we use the Gaussian smoothing kernel based on the 2D Gaussian
function of Eq. (4).

$G(x,y)=\frac{1}{2\pi\sigma^{2}}e^{-\frac{x^{2}+y^{2}}{2\sigma^{2}}}$ (4)

The Gaussian kernel marr1980theory  raises the quality of our model by
blurring redundant noises as illustrated in Fig. 2 (h). For example, the
$5\times 5$ Gaussian kernel for $\sigma=1$ is

$G^{\sigma=1}_{5\times 5}=\frac{1}{273}\begin{pmatrix}1&4&7&4&1\\\
4&16&26&16&4\\\ 7&26&41&26&7\\\ 4&16&26&16&4\\\ 1&4&7&4&1\end{pmatrix}.$ (5)

Figure 3:  Training (blue filled circles) and validation (red solid line) loss
in XGB algorithm using the binary cross-entropy loss function for the learning
rates (a) lr = $0.05$, (b) $0.1$, and (c) $1.0$. (d) Validation losses for
four different learning rates, lr = $0.05$ (black squares), $0.10$ (green
circles), $0.5$ (blue diamonds), and $1.0$ (red triangles). XGB is over-fitted
with the training dataset when the learning rate is larger than lr $\approx
0.5$.

##  3 Theoretical frameworks

In the following, we briefly summarize the theoretical footing of the four
tree-based ML classification algorithms employed in the current work. The
simplicity and efficiency of these models are at the heart of the practical
applicability our model to reduce the number of training images and shorten
the classification time. This section is devoted mostly to assess the training
and validation of the more advanced extreme gradient boost and light gradient
boosting machine models.

###  3.1 Decision tree

The decision tree (DT) is one of the basic algorithms in ML, that as its name
indicates infers the predicted label following a tree-like decision framework.
Firstly, we have a root (a single leaf) to begin with and we assign a label to
this root according to a majority vote among all the labels over the training
set. For example, we can imagine that the label could be the length of the
petal in the famous classification problem for Iris flowers. Then, we can
split the root into two groups depending on whether or not the dataset
satisfies the label and evaluate the effect of splitting over the iterations
by calculating a measure which is called gain. The gain quantifies the
improvement of the performance of our model due to the splitting. Among the
possible splits, we can either choose the one that maximizes the gain or
choose not to split the leaf. The merit of this algorithm is that we can see
the procedures of decision-making in the algorithm. However, the major
disadvantage of DT is the high risk of over-fitting to the training set which
can be caused by an outlier that is chosen to be a single node even though it
is non-representative quinlan1996learning ; myles2004introduction ;
loh2011classification .

Figure 4:  Training (blue filled circles) and validation (red solid line) loss
in LGBM algorithm using the binary cross-entropy loss function for the
learning rates (a) lr = $0.001$, (b) $0.01$, and (c) $0.05$. (d) Validation
losses for four different learning rates, lr = $0.001$ (black squares),
$0.005$ (green circles), $0.01$ (blue diamonds), and $0.05$ (red triangles).
LGBM is over-fitted with the training dataset as the learning rate increases
more than lr $\approx 0.05$.

###  3.2 Random forest

The random forest (RF) is literally an ensemble of the decision trees with
randomness such that RF generates multiple decision trees that chooses a class
by majority vote among the trees and aggregates them by using their average
for a regression. Therefore, RF is less vulnerable to over-fitting than DT.
However, the decision-making could slow down as the size of the dataset
increases breiman2001random ; liu2012new ; akar2012classification .

###  3.3 XGBoost

The extreme gradient boost (XGB or XGBoost) is a scalable end-to-end ML
algorithm which was proposed by Chen and Guestrin in 2016 chen2015xgboost ;
chen2016xgboost . XGB is based on a gradient boosting decision tree (GBDT).
GBDT combines base-learners (e.g., DT) into a single strong learner over the
many iterations by fitting the base-learners using a loss function (e.g., the
mean squared error). The aim of gradient boosting is to train the model to
minimize the loss function using the functional gradient descent which leads
the next iteration toward the direction of negative gradient. While GBDT uses
only the first-order derivatives of the loss function, XGB utilizes the
Newton-Raphson method for the functional gradient. Namely, the second-order
derivatives of the loss function are used in the fitting procedure.
Consequently, XGB proposes a newly distributed algorithm for tree searching,
outperforming RF in general but XGB has the drawback that it is generally more
time-consuming to perform friedman2017elements .

Fig. 3 shows the training and validation losses in XGB using the binary cross-
entropy loss function

$L_{\log}=\frac{1}{N}\sum_{i=0}^{N-1}-(y_{i}\log(p_{i})+(1-y_{i})\log(1-p_{i}))$
(6)

as implemented in scikit-learn pedregosa2011scikit  as a function of the
epochs for the three different learning rates (lr) where $y\in\\{0,1\\}$ is
the true label and $p=\operatorname{Pr}(y=1)$ is the probability and the $i$
index runs over all sample points. The training loss decreases and approaches
zero for all cases as the epoch increases. For lr $=0.05$ and $0.10$ the
validation loss decreases monotonically, while the validation loss has a local
minimum at the epoch $=16$ and grows as the epoch increases for lr $=1.0$,
implying that the model is over-fitted with the training dataset. Our model
starts over-fitting when lr is larger than $\approx 0.5$ as shown in Fig. 3
(d). Thus, in this paper, we use lr $=0.1$ for XGB unless stated otherwise.

###  3.4 LightGBM

The light gradient boosting machine (LGBM or LightGBM) also originates from
GBDT and inherits many strengths from XGB. However, LGBM has two significant
technical differences from XGB in structuring trees which are called gradient-
based one side sampling (GOSS) and exclusive feature bundling (EFB)
ke2017lightgbm .

The GOSS is a sampling tool that keeps the instances with large gradients but
randomly drops out some portion of the ones with small gradients. In EFB two
sparse features which are nearly exclusive integrate into one to reduce the
number of instances. In short, GOSS and EFB reduce the number and the size of
data instances remarkably while at the same time keeping high accuracy.
Consequently, LGBM is faster and requires less memory, as its name suggests
“light”. However, LGBM is prone to over-fitting when the size of the dataset
is small zhao2019predicting .

Figure 5:  (a) Source images of the domain of interest. The flakes of
monolayer graphene are enclosed by red dashed lines. (b) The flakes of
monolayer graphene are labeled as 1 (white) and the rest is labeled as 0
(black). (c) Resultant images of pixel-wise classifications for the four
algorithms, DT, RF, XGBoost (XGB), and LightGBM (LGBM).

We show the training and validation losses for different learning rates lr
$=0.001$, $0.01$, and $0.05$ in Fig. 4 (a-c) using the binary cross-entropy
loss function. One can see that LGBM model for our graphene OM image datasets
requires to be stopped early at the epoch $=20$ to avoid over-fitting where
the validation loss starts increasing when lr $=0.05$. Compared to other
learning rates, the loss starts increasing upon the epochs when lr $\gtrsim
0.05$ as shown in 4 (d). Therefore, we choose lr $=0.005$ for LGBM throughout
this paper.

##  4 Results and Discussions

###  4.1 Single classifiers

We train the ML models for each pixel in an OM image with each one of the four
classification models and use the five-fold stratified cross-validation to
maintain objectivity kohavi1995study . Thus, we obtain the probability of
having a monolayer graphene sheet at each pixel, and round off the probability
from the first decimal place so that we assign $0$ to pixels predicted as
background, and $1$ to the ones predicted as monolayer graphene at each pixel.
In Fig. 5 we visualize our results for the four tree-based ML algorithms
together with the original OM images and the target images labeled for the
monolayer graphene. The graphene monolayers in the OM images in Fig. 5 (a) are
enclosed by the red dashed line. As shown in Fig. 5 (b), the pixels for
monolayers are labeled as 1 (white) and the others are labeled as 0 (black).
Fig. 5 (c) shows the classification results from DT, RF, XGB, and LGBM
algorithms, respectively. Each algorithm appears to well differentiate the
monolayer graphene from the OM image. Since background pixels are dominant, it
leads to an highly imbalanced pool of pixels. Therefore we introduce several
metrics and indices to evaluate performance of the model

Algorithm metric | Accuracy (%) | Precision (%) | Recall (%) | F1 Score (%)
---|---|---|---|---
DT | 99.68 | 73.70 | 77.92 | 75.36
RF | 99.79 | 89.61 | 76.54 | 82.09
XGB | 99.72 | 88.01 | 75.26 | 80.80
LGBM | 99.77 | 91.23 | 73.37 | 80.96
Table 1: Evaluations of the four different tree-based classification
algorithms, decision tree (DT), random forest (RF), XGBoost (XGB), and
lightGBM (LGBM) using the four metrics such as accuracy, precision, recall,
and F1 score defined in Eqs. (7-10).

First, the accuracy as defined in Eq. (7) is the most basic metric to evaluate
the performance of a classification ML model.

$\textrm{Accuracy}=\frac{\textrm{TP + TN}}{\textrm{TP + FN + TN + FP}},$ (7)

where TP (TN) stands for the case where our model correctly predicts a
graphene (background) pixel, while FP (FN) represents the opposite case where
our model incorrectly predicts a background (graphene) pixel as a graphene
(background) pixel. The precision which is defined as Eq. (8) is obtained by
evaluating the ratio of correctly predicted graphene pixels. The higher
precision score means that there are more actual graphene pixels among the
pixels predicted as graphene by our model. If the precision is lower, the
portion of background pixels classified as graphene is higher.

$\textrm{Precision}=\frac{\textrm{TP}}{\textrm{TP + FP}}$ (8)

Similarly, we define the recall in Eq. (9). If the recall value is high, a
graphene pixel is less likely to be classified as a background pixel.

$\textrm{Recall}=\frac{\textrm{TP}}{\textrm{TP + FN}}$ (9)

Lastly, the F1 score in the Eq. (10) is nothing more than the harmonic mean
between the precision and the recall.

$\textrm{F1
score}=\frac{2\times\textrm{Precision}\times\textrm{Recall}}{\textrm{Precision
+ Recall}}$ (10)

The evaluation of the four tree-based classification algorithms using the four
metrics are shown in Table 1. The accuracies for the four algorithms are more
or less the same but DT has the smallest accuracy compared to others. DT also
has far less precision around $\sim 74\%$ in contrast with the other methods
which are above $\sim 88\%$. LGBM has the highest precision $\sim 91\%$.
Regarding the recall values, on the other hand, DT scores above $\sim 78\%$
which is superior to others. The RF, XGB, and LGBM record around $\sim
81-82\%$ in F1 score, while DT is lower by around $6\%$ with respect to the
other methods.

We further employ several additional performance measures such as the receiver
operating characteristic (ROC) curves, their area under the curve (AUC), and
the intersection over union (IOU). We show the ROC curve in Fig. 6 as a
function of the true positive rate (TPR) which stands for the recall, and the
false positive rate (FPR) which is defined as

$\textrm{False Positive Rate (FPR)}=\frac{\textrm{FP}}{\textrm{FP + FN}}.$
(11)

We get these ROC curves by changing the criteria of rounding off the
probability between $0$ (background) and $1$ (graphene) for each pixel
obtained from each ML model, leading to different values of TPR and FPR. The
AUC for the ideal classifier whose ROC is denoted by the red dashed line is
$1$ using normalized units, whereas the AUC for the random classifier denoted
by the yellow dashed line is $1/2$ as shown in Fig. 6. The AUCs for the four
different classification algorithms which range between $1/2$ and $1$ are
presented in Table. 2. Their AUCs have an order such that DT $\ll$ RF $<$ LGBM
$<$ XGB. One can see in the inset of Fig. 6 in details that XGB has a smaller
TPR than RF and LGBM at nearly zero FPR but it starts to surpass the others
when FPR $\gtrsim 0.05$.

Algorithm | AUC
---|---
DT | 0.8731
RF | 0.9792
XGB | 0.9942
LGBM | 0.9884
Table 2: Area under the curve (AUC) score for the four tree-based ML
algorithms such as decision tree (DT), random forest (RF), XGBoost (XGB), and
LightGBM (LGBM).

To evaluate the performance of the pixel-wise segmentation method we use the
intersection over union (IOU) index defined as

$\textrm{IOU}=\frac{\textrm{Target Pixel}\cap\textrm{Predicted
Pixel}}{\textrm{Target Pixel}\cup\textrm{Predicted Pixel}}.$ (12)

This index indicates how many pixels overlap between the target pixels and the
predicted pixels and returns a value in the range between $0$ and $1$. The IOU
values for the four algorithms are presented in Table 3 and they have an order
of DT $\ll$ RF $\ll$ LGBM $<$ XGB which indicates that the XGB is the most
suitable algorithm for pixel-wise segmentation of our datasets.

The RF classifier has the highest accuracy and F1 score, but its pixel-
resolved segmentation shows a relatively low IOU score that is slightly higher
than the DT. On the other hand, the XGB has the highest IOU score but less
accuracy, precision, recall, and F1 score than RF. The LGBM has higher
precision and IOU score than RF. In practice we can choose the classifier
depending on which metric or index is more relevant for the classification
task at hand or mix more than one classifier as we will explain in the next
section.

Algorithm | IOU Score (%)
---|---
DT | 54.61
RF | 63.86
XGB | 71.18
LGBM | 70.73
Table 3: Evaluation of the intersection over union (IOU) score for the four tree-based ML algorithms such as decision tree (DT), random forest (RF), XGBoost (XGB), and LightGBM (LGBM). Algorithm | Inference time (s)
---|---
DT | 0.1185
RF | 0.1413
XGB | 0.2822
LGBM | 0.2006
Table 4: The inference time per image for the four tree-based ML algorithms
such as decision tree (DT), random forest (RF), XGBoost (XGB), and LightGBM
(LGBM).

Regarding the inference time, as shown in Table. 4, the DT algorithm is the
fastest model with a computation time of $\sim$0.1 seconds per image, while
the XGB is the slowest one at $\sim$0.3 seconds per image using Intel(R)
Core(TM) i5-10400 CPU. In practice, this means that we can analyze a thosand
images in a matter of minutes regardless of the method used.

Figure 6: The receiver operating characteristic (ROC) curves for random classifier (yellow dashed), ideal classifier (red dashed), DT (blue solid), RF (orange solid), XGB (green solid), and LGBM (pink solid). The inset shows the same ROC curve for the finer range of false positive range (FPR) in Eq. (11) near 0.0 and the true positive range (TPR) near 1.0. Algorithm metric | Accuracy (%) | Precision (%) | Recall (%) | F1 score (%) | IOU score (%)
---|---|---|---|---|---
| RF+XGB
---
(precision, recall, F1 score)
99.77 | 89.36 | 72.95 | 79.96 | 66.82
| RF+LGBM
---
(precision)
99.78 | 91.23 | 73.37 | 80.96 | 66.82
| RF+LGBM
---
(recall, F1 score)
99.77 | 89.36 | 72.95 | 79.96 | 66.82
| XGB+LGBM
---
(precision, F1 score)
99.78 | 91.23 | 73.37 | 80.96 | 68.22
| XGB+LGBM
---
(recall)
99.77 | 88.01 | 75.26 | 80.80 | 67.98
| RF+XGB+LGBM
---
(precision, recall, F1 score)
99.78 | 90.64 | 73.82 | 81.03 | 68.28
Table 5: Algorithm choice versus metric table for five different combinations
of the three single classification algorithms random forest (RF), XGBoost
(XGB), and lightGBM (LGBM) using the five metrics accuracy, precision, recall,
F1 score, and IOU score. In parentheses below each one of the fused-
classifiers, we specify the metrics used to optimize the weights of the
classifiers. We listed several metrics when their performances were the same
down to two decimal places.

###  4.2 Fusions between classifiers

In this section, we combine the different single classifiers RF, XGB, and LGBM
in an attempt to improve the performance of our model as proposed in Ref.
li2014classifier  that calculates different weights for each classifier
depending on metrics accuracy. We will show that it is possible to improve the
overall performance of different metrics of our choice at the expense of
slightly reducing the IOU. We exclude the DT as it scores far fewer points
particularly in the precision, F1 score, AUC and IOU. We predict to find a
monolayer graphene pixel with the probability $P$ defined in Eq. (13) for each
pixel in an OM image when we utilize the combined probability consisting of
$N$ single classifiers through

$P=\sum_{i}^{N}w_{i}P_{i},$ (13)

where $P_{i}$ stands for the probability of finding a monolayer graphene at
that pixel using the $i^{\rm th}$ single classifier, and $w_{i}$ is the
$i^{\rm th}$ classifier’s weight which is given as li2014classifier

$w_{i}=\frac{{\rm metric}(i)^{10}}{\sum_{j=1}^{N}{\rm metric}(j)^{10}},$ (14)

where metric($i$) represents the $i^{th}$ single classifier’s metric. In other
words, the probability $P$ of a joint classification model is defined such
that the probabilities of finding a monolayer graphene at that pixel using
single classifiers are expressed as a linear combination of each classifier’s
probability multiplied by the respective weight coefficients. Like in the
single classifiers, we round off the final probability to either 0
(background) and 1 (graphene). Then we use $P$ at each pixel to calculate the
metrics as presented in Table. 5. We take into account four combinations,
RF+XGB, RF+LGBM, XGB+LGBM, RF+XGB+LGBM. We calculate the weights $w_{i}$ for
the $i^{\rm th}$ single classifier as defined in Eq. (14) taking one of the
four metrics – accuracy, precision, recall, and F1 score. We indicated within
parentheses the reference metric used to calculate the weights of the
classifiers, and listed more than one when they gave the same results down to
two decimal places. We exclude the accuracy as a weight because the accuracies
of RF, XGB, and LGBM are all on the order of $\sim 99.7\%$.

The single classifier RF has a higher accuracy of $99.79\%$, recall of
$76.54\%$, and F1 score of $82.09\%$, but it has a smaller IOU score of
$63.86\%$ than XGB and LGBM. If we join XGB with RF, regardless of which
metric we use, we get an improved IOU score by $\sim 3\%$, but sacrificing
accuracy by $0.01\%$, precision by $0.25\%$, recall by $\sim 3.6\%$, and the
F1 score by $\sim 2.1\%$ compared to using the single RF classifier. On the
other hand, we get improved accuracy by $0.05\%$, precision by $\sim 1.4\%$,
sacrificing recall by $2.3\%$, F1 score by $\sim 0.8\%$, and the IOU score by
$\sim 4.4\%$ compared to using the single XGB classifier.

If we join LGBM with RF, the results are different depending on the metrics we
use for a weight. In the case of precision as a weight, it has $\sim 1.6\%$
higher precision, $\sim 3\%$ higher IOU score than a single RF classifier. If
we use either recall or F1 score as a metric, it has benefit in IOU score by
$\sim 3\%$ compared to the single RF. On the other hand, combining LGBM with
RF has no merit in comparison with the single LGBM.

If we combine XGB and LGBM, in the case where either precision or the F1 score
is chosen as a weight, the accuracy is improved to $99.78\%$, and the IOU hits
$68.22\%$ which is less than both XGB ($71.18\%$) and LGBM ($70.73\%$)
separately. The precision ($91.23\%$), recall ($73.37\%$), and F1 score
($80.96\%$) are the same as those of a single classifier of LGBM. When we use
the recall as a weight, the IOU hits $67.98\%$ which is again smaller than
that of single XGB and LGBM classifiers. The accuracy is the same as that of
LGBM ($99.77\%$), but the precision ($88.01\%$), recall ($75.26\%$), and F1
score ($80.80\%$) have the same value as those of XGB.

Lastly, if we combine all three classifiers, all metrics and indices are
overall averaged regardless of the metrics used for the weights. The accuracy
and F1 score are improved in comparison with single XGB and LGBM classifiers,
and the precision becomes higher than single RF and XGB. The recall becomes
higher than a single LGBM, and the IOU is improved compared to a single RF.

The resulting fused-classifier metrics presented in Table 5 indicates that we
can indeed improve the overall performance of our models if we target a
particular metric to optimize, and at times it is possible to achieve an
overall improvement in several metrics at the expense of a small decrease in
the IOU by a few percents with respect to the XGB and LGBM single classifier
scores.

##  5 Conclusions

The preparation of graphene electronic devices during almost the last two
decades has relied on human inspection during many hours of routine scanning
in the entire space of samples to identify the number of graphene layers
deposited on the substrate. Recently several research teams have attempted to
introduce ML in this process to reduce such time-consuming and error-prone
human labor. However, the existing proposals are mostly focused on developing
DNN or CNN based models which require quite a large number of labeled training
dataset, and this threshold makes practical application difficult. In this
paper, we have proposed a pixel-wise tree-based ML classification tool that
can be used for samples prepared under consistent scanning conditions such as
illumination or the thickness of the substrate. Our method can distinguish a
specific number of graphene layers from the background and performs precisely
even with a small number of training images, in contrast to existing works
utilizing DL tools that require hundreds to thousands of image data to train a
model.

We have trained our ML model with four different tree-based algorithms such as
DT, RF, XGB, and LGBM, and examined their outcomes using several metrics and
indices – accuracy, precision, recall, F1 score, ROC, AUC, and IOU. There is
no absolute best model among the single classifiers as they have different
strengths and weaknesses in metrics and indices, and those metrics vary
depending on the data that we deal with. We have shown that it is possible to
improve the performance by combine more than one classifier. Considering
overall performance, we propose that the LGBM or XGB are good choices when
using a single classifier, and for the fused classifiers model, the
RF+XGB+LGBM shows satisfactory performance and can be chosen in routine
applications.

In this work, we have sorted out the monolayer graphene from the OM images,
but the same process can be performed to find any other number of multilayer
graphene or for any other 2D vdW materials such as hexagonal boron nitride,
transition metal chalcogenides or black phosphorus. Furthermore, our model
requires only a few images and costs less computational time than a few
minutes in total to train a ML model even without using a GPU. Therefore, we
expect our work will be of immediate utility for researches on 2D materials,
and will greatly ease the repetitive and time-consuming tasks in experiments.

###### Acknowledgements.

This work was supported by Korean NRF through the Grants No. 2020R1A5A1016518
(W.H.C), No. 2021R1A6A3A01087281 (J.S.), No. 2020R1A2C3009142 (J.J.). YDK was
supported by the KIST Institutional Program (Project No.2E31781-22-108). We
acknowledge computational support from KISTI Grant No. KSC-2021-CRE-0389 and
by the computing resources of Urban Big data and AI Institute (UBAI) at UOS
and the network support from KREONET. We also acknowledge partial support for
W.H.C. from the Korean Ministry of Land, Infrastructure and Transport (MOLIT)
from the Innovative Talent Education Program for Smart Cities.

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