# Generation-recombination processes via acoustic phonons in a disorded
graphene

F. T. Vasko<EMAIL_ADDRESS>V. V. Mitin  Department of Electrical
Engineering, University at Buffalo, Buffalo, NY 1460-1920, USA

###### Abstract

Generation-recombination interband transitions via acoustic phonons are
allowed in a disordered graphene because of violation of the energy-momentum
conservation requirements. The generation-recombination processes are analyzed
for the case of scattering by a short-range disorder and the deformation
interaction of carriers with in-plane acoustic modes. The generation-
recombination rates were calculated for the cases of intrinsic and heavily-
doped graphene at room temperature. The transient evolution of nonequilibrium
carriers is described by the exponential fit dependent on doping conditions
and disorder level. The characteristic relaxation times are estimated to be
about 150 - 400 ns for sample with the maximal sheet resistance $\sim$5
k$\Omega$. This rate is comparable with the generation-recombination processes
induced by the thermal radiation.

###### pacs:

72.80.Vp, 73.61.-b, 78.60.-b

##  I Introduction

Transport 1  and optical 2  properties of graphene as well as noise phenomena
in this material 2a  are not completely understood for the regime of nonlinear
response. The treatment of nonequilibrium carriers requires not only
verification the momentum and energy relaxation processes but also
understanding of the interband generation-recombination processes which
determine electron and hole concentrations far from equilibrium (similar
transport conditions take place for the bulk gapless materials, see review 3
and references therein). Effective interband transitions via optical phonons
of energy $\hbar\omega_{0}$ take place for the energy of carriers greater than
$\hbar\omega_{0}/2$, see Refs. 5 and 6 where the cases of optical excitation
and heating by dc current were analyzed. At lower energies, the generation-
recombination processes become ineffective because the Auger transitions are
forbidden due to the symmetry of electron-hole states 6  (c.f. with 7 ). Since
the carrier’s velocity $\upsilon\simeq 10^{8}$ cm/s exceeds significantly the
sound velocity $s$, the interband transitions via acoustic phonons are also
forbidden due to the momentum-energy conservation laws. Only slow generation-
recombination processes induced by the thermal radiation are allowed in a
perfect graphene. 8  To the best of our knowledge, consideration of a disorder
effect on the interband transitions via acoustic phonons in the low-energy
region, $\varepsilon<\hbar\omega_{0}/2$, is not performed yet. Thus, the
evaluation of the generation-recombinaton rate caused by the interaction of
carriers with the acoustic phonon thermostat under violation of the momentum-
energy laws in a disordered graphene (allowed electron-hole transitions are
depicted in Fig. 1) is timely now.

In this paper, the calculations are performed for the model of short-range
disorder whose parameters are taken from the mobility data, 1 ; 9  for samples
with the maximal resistance of 2 - 6 k$\Omega$ per square. The probability of
electron-hole transitions is expressed through the averaged spectral density
functions and is calculated taking into account the contribution of interband
interference. Due to slowness of the interband transitions, the
quasiequilibrium distributions of electrons and holes with the same
temperature are used for the description of a temporal evolution of
nonequilibrium concentrations of carriers. The electron and hole
concentrations are also connected through the electroneutrality condition with
the surface charge controlled by a gate voltage.

Figure 1: Interband generation-recombination transitions via acoustic phonons
with energies $\sim\hbar\omega_{ac}$ (thick arrows) between broadened
electron-hole (e - h, shown by grey) states in the low-energy region,
$\varepsilon<\hbar\omega_{0}/2$. Thin lines show the ideal dispersion law.

The results were obtained for the cases of intrinsic and heavily-doped
graphene at temperature $T$ and can be briefly summarized as follows. The
concentration balance equation is written through the chemical potential
normalized to $T$ and the characteristic rate, which is proportional to a
carrier-phonon coupling and increases with temperature as $T^{2}$. The
transient evolution of nonequilibruim population can be fitted by an
exponential decay with the relaxation time 150 - 400 ns at room temperature
and a typical disorder level corresponding to the maximal sheet resistance
$\sim$5 k$\Omega$. This time scale appears to be comparable to the
recombination rate via thermal radiation and the mechanism under consideration
can be verified by temperature and temporal measurements.

The paper is organized as follows. In the next section we present the basic
equations which describe the generation-recombination processes under
consideration. In Sec. III we evaluate the generation-recombination rates and
analyze their dependencies on temperature, disorder level, and doping
conditions. The last section includes the discussion of the approximations
used and conclusions. In Appendix we consider the generation-recombination
mechanism caused by the interaction with the thermal radiation.

##  II Basic Equations

Temporal evolution of carriers in a random potential, which are weakly
interacting with the acoustic phonon modes, is described by the distribution
$f_{\alpha t}$ over the states $|\alpha)$ with energies
$\varepsilon_{\alpha}$. An exact with respect to a disordere effect kinetic
equation takes the form 10

$\displaystyle\frac{\partial f_{\alpha t}}{\partial
t}=\sum\limits_{\alpha^{\prime}}\left[W_{\alpha^{\prime}\alpha}f_{\alpha^{\prime}t}\left(1-f_{\alpha
t}\right)\right.$ $\displaystyle\left.-W_{\alpha\alpha^{\prime}}f_{\alpha
t}\left(1-f_{\alpha^{\prime}t}\right)\right].$ (1)

The transition probability $W_{\alpha\alpha^{\prime}}$ is written within the
Born approximation with respect to the carrier-phonon interaction with $q$th
phonon mode of frequency $\omega_{q}$:

$\displaystyle
W_{\alpha\alpha^{\prime}}=\frac{2\pi}{\hbar}\sum\limits_{q}\left|(\alpha|\hat{\chi}_{q}|\alpha^{\prime})\right|^{2}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}$
(2)
$\displaystyle\times\left[(N_{q}+1)\delta\left(\varepsilon_{\alpha}-\varepsilon_{\alpha^{\prime}}-\hbar\omega_{q}\right)+N_{q}\delta\left(\varepsilon_{\alpha}-\varepsilon_{\alpha^{\prime}}+\hbar\omega_{q}\right)\right].$

Here the operator $\hat{\chi}_{q}$ determines the carrier-phonon interaction
$\sum_{q}\left(\hat{\chi}_{q}\hat{b}_{q}+H.c.\right)$ where $\hat{b}_{q}$ is
the annihilation operator of $q$th mode and $N_{q}$ is the Planck distribution
of phonons at the equilibrium temperature $T$. Note, that the transition
probabilities $W_{\alpha^{\prime}\alpha}$ and $W_{\alpha\alpha^{\prime}}$ are
connected by
$W_{\alpha^{\prime}\alpha}=\exp\left[-\left(\varepsilon_{\alpha}-\varepsilon_{\alpha^{\prime}}\right)/T\right]W_{\alpha\alpha^{\prime}}$
and

$\sum\limits_{\alpha\alpha^{\prime}}\left[W_{\alpha^{\prime}\alpha}f_{\alpha^{\prime}t}\left(1-f_{\alpha
t}\right)-W_{\alpha\alpha^{\prime}}f_{\alpha
t}\left(1-f_{\alpha^{\prime}t}\right)\right]=0$ (3)

due to the particle conservation law.

The concentrations of electrons and holes, $n_{t}$ and $\overline{n}_{t}$,
which are averaged over random disorder (such averaging is denoted as
$\langle\ldots\rangle$), are given by

$\left|\begin{array}[]{*{20}c}{n_{t}}\\\
\overline{n}_{t}\end{array}\right|=\frac{4}{L^{2}}\left\langle\sum\limits_{\alpha}\left|{\begin{array}[]{*{20}c}{\theta\left(\varepsilon_{\alpha}\right)f_{\alpha
t}}\\\ {\theta\left(-\varepsilon_{\alpha}\right)\left(1-f_{\alpha
t}\right)}\end{array}}\right|\right\rangle,$ (4)

where $L^{2}$ is the normalization area and the step function
$\theta(\pm\varepsilon)$ appears due to the symmetry of electron-hole spectrum
[see Eq. (10) below]. Since effective intraband scattering is caused by the
phonon thermostat and carrier-carrier interaction, the quasiequilibrium
distributions over the conduction and valence bands are imposed during a
short-time scales and below we use

$\tilde{f}_{\varepsilon
t}=\left\\{{\begin{array}[]{*{20}c}{\left[{\exp\left({\frac{{\varepsilon-\mu_{t}^{>}}}{T}}\right)+1}\right]^{-1},}&{\varepsilon>0}\\\
{\left[{\exp\left({\frac{{\varepsilon-\mu_{t}^{<}}}{T}}\right)+1}\right]^{-1},}&{\varepsilon<0}\\\
\end{array}}\right..$ (5)

Due to effective energy relaxation the same temperatures are established in
both bands. At the same time the electon and hole concentrations are
determined through the different chemical potentials, $\mu_{t}^{>}$ and
$\mu_{t}^{<}$, respectively. The chemical potentials are connected by the
electroneutrality condition $n_{t}-\overline{n}_{t}=n_{s}$, where the surface
charge $en_{s}$ is controlled by the gate voltage, $V_{g}$, according to
$n_{s}=aV_{g}$ with $a\simeq 7.2\times 10^{10}$ cm-2/V written for the SiO2
substrate of thickness 0.3 $\mu$m.

The concentration of electrons is governed by the balance equation
$dn_{t}/dt=(dn/dt)_{ac}$ with the generation-recombination rate

$\displaystyle\left(\frac{dn}{dt}\right)_{ac}=\int\limits_{0}^{\infty}d\varepsilon\int\limits_{-\infty}^{0}d\varepsilon^{\prime}W(\varepsilon,\varepsilon^{\prime})~{}~{}~{}~{}~{}$
(6)
$\displaystyle\times\left[{\exp\left({\frac{\varepsilon^{\prime}-\varepsilon}{T}}\right)\left(1-\tilde{f}_{\varepsilon
t}\right)\tilde{f}_{\varepsilon^{\prime}t}-\left({1-\tilde{f}_{\varepsilon^{\prime}t}}\right)\tilde{f}_{\varepsilon
t}}\right].$

We take into account that the intraband transitions (when
$\varepsilon,\varepsilon^{\prime}>0$) vanish in (6) and transform the
transition probability as follows

$W(\varepsilon,\varepsilon^{\prime})=\frac{4}{{L^{2}}}\left\langle{\sum\limits_{\alpha\alpha^{\prime}}{\delta\left({\varepsilon-\varepsilon_{\alpha}}\right)\delta\left({\varepsilon^{\prime}-\varepsilon_{\alpha^{\prime}}}\right)W_{\alpha\alpha^{\prime}}}}\right\rangle.$
(7)

Further, we introduce the exact spectral density function

$A_{\varepsilon}\left({l{\bf x},l^{\prime}{\bf
x}^{\prime}}\right)=\sum\limits_{\alpha}{\delta\left(\varepsilon-\varepsilon_{\alpha}\right)\Psi_{l{\bf
x}}^{(\alpha)}\Psi_{l^{\prime}{\bf x}^{\prime}}^{(\alpha)*}},$ (8)

which is determined through the double-row wave function $\Psi_{l{\bf
x}}^{(\alpha)}$ with $l=$1,2. The column $\Psi_{\bf x}^{(\alpha)}$ is a
solution of the eigenvalue problem $(\hat{h}+V_{\bf x})\Psi_{\bf
x}^{(\alpha)}=\varepsilon_{\alpha}\Psi_{\bf x}^{(\alpha)}$ written through the
single-particle Hamiltonian $\hat{h}$ and a random potential $V_{\bf x}$.
Using the definition (2) one obtains the probability
$W(\varepsilon,\varepsilon^{\prime})$ as follows

$\displaystyle W(\varepsilon,\varepsilon^{\prime})=\frac{8\pi}{\hbar
L^{2}}\sum\limits_{\bf
q}|C_{q}|^{2}\left(N_{q}+1\right)\delta\left(\varepsilon-\varepsilon^{\prime}-\hbar\omega_{q}\right)$
(9) $\displaystyle\times\int d{\bf x}\int d{\bf x}^{\prime}e^{i{\bf
q}\cdot({\bf x}-{\bf x}^{\prime})}{\rm
tr}\left\langle\hat{A}_{\varepsilon^{\prime}}\left({\bf x},{\bf
x}^{\prime}\right)\hat{A}_{\varepsilon}\left({\bf x}^{\prime},{\bf
x}\right)\right\rangle,$

where $\bf q$ is the in-plane wave vector and $|C_{q}|^{2}$ is the matrix
element of deformation interaction. 9  As a result, $(dn/dt)_{ac}$ is
expressed through the two-particle correlation function. Since the main
contributions to (6) appears from $\varepsilon\neq\varepsilon^{\prime}$, this
correlation function can be decoupled according to
$\left\langle{\hat{A}_{\varepsilon^{\prime}}\left({{\bf x},{\bf
x}^{\prime}}\right)\hat{A}_{\varepsilon}\left({{\bf x}^{\prime},{\bf
x}}\right)}\right\rangle\approx\hat{A}_{\varepsilon^{\prime},\Delta{\bf
x}}\hat{A}_{\varepsilon,-\Delta{\bf x}}$, where
$\hat{A}_{\varepsilon,\Delta{\bf
x}}=\left\langle{\hat{A}_{\varepsilon}\left({{\bf x},{\bf
x}^{\prime}}\right)}\right\rangle$ is the averaged spectral function given by
2$\times$2 matrix.

Below, we calculate the probability (9) using the model of the short-range
disorder described by the Gaussian correlator $\langle V_{\bf x}V_{\bf
x^{\prime}}\rangle=\bar{V}^{2}\exp[-({\bf x}-{\bf
x^{\prime}})^{2}/2l_{c}^{2}]$, where $\bar{V}$ is the averaged amplitude,
$l_{c}$ is the correlation length, and the cut-off energy
$E_{c}=\upsilon\hbar/l_{c}$ exceeds the energy scale under consideration.
According to Refs. 12 and 13 the retarded Green’s function in the momentum
representation takes form:

$\displaystyle\hat{G}_{\varepsilon,{\bf p}}^{R}=\hat{P}_{\bf
p}^{(+)}G_{\varepsilon,p}+\hat{P}_{\bf
p}^{(-)}G_{\varepsilon,-p},~{}~{}~{}~{}~{}~{}$ (10) $\displaystyle
G_{\varepsilon,p}\approx\left[\varepsilon(1+\Lambda_{\varepsilon}+ig)-\upsilon
p\right]^{-1},~{}~{}~{}\Lambda_{\varepsilon}=\frac{g}{\pi}\ln\left(\frac{E_{c}}{|\varepsilon|}\right)$

where $\hat{P}_{\bf
p}^{(\pm)}=\left[1\pm(\hat{\mbox{\boldmath$\sigma$}}\cdot{\bf p})/p\right]/2$
are the projection operators on the conduction ($+$) and valence ($-$) bands,
$\hat{\mbox{\boldmath$\sigma$}}$ is the isospin Pauli matrix, and
$g=(\bar{V}^{2}l_{c}/\hbar\upsilon)^{2}\pi/2$ is the coupling constant. Here
we restrict ourselves by the Born approximation when the self-energy
contribution $\varepsilon(\Lambda_{\varepsilon}+ig)$ is written through the
logarithmically-divergent real correction and the damping factor. Note, that
these corrections vanish at $\varepsilon\to 0$. From a comparision with the
mobility data 1 ; 9  one obtains that the parameters $g\simeq$0.45, 0.3, and
0.15 correspond to the sheet resistances $\sim$6, $\sim$4, and $\sim$2
k$\Omega$ per square, respectively. The density of states,
$\rho_{\varepsilon}=-4{\rm Im}\sum_{\bf p}{\rm tr}\hat{G}_{\varepsilon,{\bf
p}}^{R}$ is shown in Fig. 2a and $\rho_{\varepsilon}=\rho_{-\varepsilon}$, i.
e. the electron-hole symmetry is not violate due to disorder. Since
$\rho_{\varepsilon}$ increases in comparision to the ideal case,
$\overline{\rho}_{\varepsilon}=2|\varepsilon|/[(\hbar\upsilon)^{2}\pi]$, the
energy-dependent renormalized velocity, $\upsilon V_{\varepsilon}$ decreases
up to 10% if $g\leq$0.5 and the concentration of carriers in an intrinsic
graphene increases up to 2 times, see Figs. 2b and 2c, respectively (here
$n_{T}\simeq 8.1\times 10^{10}$ cm-2 is the equilibrium concentration at room
temperature and at $g\to$0).

Figure 2: (Color online) (a) Density of states $\rho_{\varepsilon}$ versus
energy at $g=$0 (1), 0.15 (2), 0.3 (3) and 0.45 (4). (b) Ratio
$V_{\varepsilon}=\sqrt{\overline{\rho}_{\varepsilon}/\rho_{\varepsilon}}$
versus $g$ for $\varepsilon=$20 meV (1), 30 meV (2), 40 meV (3), and 50 meV
(4). (c) Equilibrium concentration of non-doped graphene $n_{eq}$ versus $g$
normalized to $n_{T}\simeq 0.52(T/\hbar\upsilon)^{2}$.

Figure 3: Contour plots of dimensionless kernels
$w(\varepsilon/T,\varepsilon^{\prime}/T)$ for $g=$0.4 (a) and $g=$0.2 (b).

Further, we use the standard relation $\hat{A}_{\varepsilon,{\bf
p}}=i\left(\hat{G}_{\varepsilon,{\bf p}}^{R}-\hat{G}_{\varepsilon,{\bf
p}}^{R~{}+}\right)/2\pi$ and transform the probability (9) taking into account
the energy conservation law:

$\displaystyle
W(\varepsilon,\varepsilon^{\prime})\approx\left\\{|C_{q}|^{2}(N_{q}+1)\right\\}_{\hbar\omega_{q}=\varepsilon-\varepsilon^{\prime}}$
(11) $\displaystyle\times\frac{8\pi}{\hbar L^{2}}\sum\limits_{\bf
pp^{\prime}}\delta\left(\varepsilon-\varepsilon^{\prime}-s|{\bf p}-{\bf
p^{\prime}}|\right){\rm tr}\left(\hat{A}_{\varepsilon^{\prime},{\bf
p}^{\prime}}\hat{A}_{\varepsilon,{\bf p}}\right).$

The trace here should be taken using ${\rm tr}\left(\hat{P}_{\bf
p^{\prime}}^{(\pm)}\hat{P}_{\bf p}^{(\pm)}\right)=[1+({\bf p}\cdot{\bf
p^{\prime}})/pp^{\prime}]/2$ and ${\rm tr}\left(\hat{P}_{\bf
p^{\prime}}^{(\pm)}\hat{P}_{\bf p}^{(\mp)}\right)=[1-({\bf p}\cdot{\bf
p^{\prime}})/pp^{\prime}]/2$. This result differs from the standard
consideration, 13  because interference of electron and hole states gives an
essential contribution to $W(\varepsilon,\varepsilon^{\prime})$ due to the
matrix structure of the spectral density functions. After the integrations
over $\bf p$-plane, one transforms (11) into

$\displaystyle
W(\varepsilon,\varepsilon^{\prime})\equiv\Theta_{GR}w(\varepsilon/T,-\varepsilon^{\prime}/T)/T^{2},$
(12)
$\displaystyle\Theta_{GR}=\frac{\upsilon_{ac}s}{\upsilon^{2}}\frac{T}{\hbar}\left(\frac{T}{\pi\hbar\upsilon}\right)^{2},~{}~{}~{}~{}~{}~{}\upsilon_{ac}=\frac{D^{2}T}{4\hbar^{2}\rho_{s}\upsilon
s^{2}},$

where we separated the dimensionless kernel, $w(\xi,\xi^{\prime})$, and the
factor, $\Theta_{GR}$, which is written for the case of the deformation
interaction of carriers with the in-plane acoustic modes, see Refs. 10 and 15.
Here $D$ is the deformation potential, $s$ is the sound velocity, and
$\rho_{s}$ is the sheet density of graphene. At room temperature and typical
other parameters 9  we obtain $\upsilon_{ac}\simeq 0.96\times 10^{6}$ cm/s and
$\Theta_{GR}\simeq 5.06\times 10^{19}$ cm-2s-1 (notice, that
$\upsilon_{ac}\propto D^{2}$ and we used $D\simeq$ 12 eV). The dimensionless
kernel is plotted in Fig. 3 and the probability
$W(\varepsilon,\varepsilon^{\prime})$ is suppressed fast if
$(\varepsilon-\varepsilon^{\prime})/T\geq$0.15. This cut-off factor is
determined by the weak ratio $s/\upsilon\simeq$1/137 mainly while parameter
$g$ determines a peak value of $W(\varepsilon,\varepsilon^{\prime})$, c.f.
Figs. 3a and 3b.

The generation-recombination rate (6) is written through (11) and (12) with
the use of the dimensionless variables $\xi=\varepsilon/T$ and
$\xi^{\prime}=\varepsilon^{\prime}/T$:

$\displaystyle\left(\frac{dn}{dt}\right)_{ac}=\Theta_{GR}\int\limits_{0}^{\infty}d\xi\int\limits_{0}^{\infty}d\xi^{\prime}w(\xi,\xi^{\prime})$
(13)
$\displaystyle\times\frac{e^{-\xi^{\prime}-\mu_{t}^{>}/T}-e^{-\xi^{\prime}-\mu_{t}^{<}/T}}{\left[\exp\left(\xi-\frac{\mu_{t}^{>}}{T}\right)+1\right]\left[\exp\left(\xi^{\prime}-\frac{\mu_{t}^{<}}{T}\right)+1\right]}$

For typical concentrations of carriers, $\mu_{t}^{<}$ and $\mu_{t}^{>}$ exceed
0.15$T$ and one can simplify the rate as follows:

$\displaystyle\left(\frac{dn}{dt}\right)_{ac}\approx\Theta_{GR}\frac{w_{g}\left[e^{-\mu_{t}^{>}/T}-e^{-\mu_{t}^{<}/T}\right]}{\left[e^{-\mu_{t}^{>}/T}+1\right]\left[e^{-\mu_{t}^{<}/T}+1\right]},$
$\displaystyle
w_{g}=\int\limits_{0}^{\infty}d\xi\int\limits_{0}^{\infty}d\xi^{\prime}w(\xi,\xi^{\prime})~{}~{}~{}~{}~{}~{}~{}$
(14)

where the averaged over energies kernel $w_{g}$ is plotted versus $g$ in Fig.
4, together with a simple parabolic fit. For the disorder level corresponding
to the resistance $\sim$5 k$\Omega$ per square, one obtains $w_{g}\simeq$0.02.

Figure 4: (Color online) Averaged kernel $w_{g}$ versus coupling constant $g$.

##  III Results

In this section we analyze the concentration balance equation
$dn_{t}/dt=(dn/dt)_{ac}$, where the right-hand side of Eq. (14) is written
through $\psi_{t}^{>}=\mu_{t}^{>}/T$ and $\psi_{t}^{<}=\mu_{t}^{>}/T$,
together with the initial conditions $\psi_{t=0}^{>}=\psi_{0}^{>}$ and
$\psi_{t=0}^{<}=\psi_{0}^{<}$ determined through the initial concentrations
$n_{t=0}$ and $\overline{n}_{t=0}$ according to Eq. (4). Variables
$\psi_{t}^{>}$ and $\psi_{t}^{<}$ are connected through the electroneutrality
condition

$\int\limits_{0}^{\infty}d\varepsilon\rho_{\varepsilon}\left(\frac{1}{e^{\varepsilon/T-\psi_{t}^{>}}+1}-\frac{1}{e^{\varepsilon/T+\psi_{t}^{<}}+1}\right)=n_{s}$
(15)

and below we consider the cases of an intrinsic graphene ($n_{s}=0$) and a
$n$-type heavily-doped graphene ($n_{s}>0$).

###  III.1 Intrinsic graphene

For the case under consideration, $\psi_{t}^{>}=-\psi_{t}^{<}\equiv\psi_{t}$
and the concentration of electrons (or holes, because now
$n_{t}=\overline{n}_{t}$) is given by
$n_{t}=\int_{0}^{\infty}d\varepsilon\rho_{\varepsilon}\left[\exp(\varepsilon/T-\psi_{t})+1\right]^{-1}$,
so that $n_{t}$ and $\psi_{t}$ are connected through

$\frac{dn_{t}}{dt}=\frac{d\psi_{t}}{dt}\int\limits_{0}^{\infty}\frac{d\varepsilon\rho_{\varepsilon}}{1+\cosh\left(\varepsilon/T-\psi_{t}\right)}.$
(16)

As a result, the concentration balance equation is derived from Eq. (13) as

$\frac{dn_{t}}{dt}=-\Theta_{GR}w_{g}\tanh\left(\frac{\psi_{t}}{2}\right)$ (17)

and Eqs. (16) and (17) are transformed into the first-order differential
equation for $\psi_{t}$ with the initial condition $\psi_{t=0}=\psi_{0}$ where
$\psi_{0}$ is determined through the $n_{t=0}$. The implicit solution of this
equation takes form:

$\displaystyle\nu_{GR}t=\int\limits_{\psi_{t}}^{\psi_{0}}d\psi
F(\psi),~{}~{}~{}~{}\nu_{GR}=w_{g}\frac{\upsilon_{ac}s}{\pi\upsilon^{2}}\frac{T}{\hbar},$
(18) $\displaystyle
F(\psi)=\tanh\left(\frac{\psi}{2}\right)\int\limits_{0}^{\infty}\frac{d\xi
r_{\xi}}{1+\cosh(\xi-\psi)}.$

Here $r_{\xi}=\rho_{\xi T}/\overline{\rho}_{T}$ is the dimensionless density
of states and the temporal evolution of $n_{t}$ is described through the
characteristic rate $\nu_{GR}$ and the dimensionless function $F(\psi)$. At
room temperature and at $w_{g}\simeq$0.02, one obtains $\nu_{GR}\simeq
1.85\times 10^{7}$ s-1 for the parameters used.

Figure 5 shows the transient evolution of $n_{t}$ normalized to the
equilibrium concentration $n_{eq}$, for the cases of recombination or
generation of carriers, if $n_{t=0}>n_{eq}$ or $n_{t=0}<n_{eq}$, respectively.
The relaxation becomes suppressed if the disorder level decreases both due to
a slowness of dependency on $\nu_{GR}t$, c.f. curves for $g=$0.5 and 0.25 in
Fig. 5, and, mainly, due to the relation $\nu_{GR}\propto w_{g}$, see Fig. 4.
Within a 5% accuracy, the evolution of $n_{t}$ can be fitted by the
exponential dependencies

$n_{t}\approx n_{0}+n_{T}\left[1-\exp(-\alpha\nu_{GR}t)\right]$ (19)

with the parameter $\alpha$ varying between 0.16 and 0.28 depending on the
initial conditions. The corresponding times, $(\alpha\nu_{GR})^{-1}$, vary
between 310 and 180 ns for $g\simeq$0.5. This time scale is comparable to the
radiative recombination times, see Appendix.

Figure 5: (Color online) Transient evolution of concentration $n_{t}$ at
different initial conditions: $n_{t=0}=3n_{eq}$ (1,2), $n_{t=0}=2n_{eq}$,
(3,4) and $n_{t=0}=0.5n_{eq}$ (5,6) for coupling parameters $g=$0.25 (1, 3, 5)
and $g=$0.5 (2, 4, 6). Dotted curves correspond to exponential fits (19), with
$\alpha=$0.16 (1), 0.125 (2), 0.19 (3), 0.155 (4), 0.22 (5), and 0.275 (6).

###  III.2 Heavily-doped graphene

We turn now to the case of a heavily doped graphene, when $\psi_{t}^{>}\gg 1$
and it is convenient to introduce a weak variation
$\delta\psi_{t}=\psi_{t}^{>}-\psi_{s}$ where $\psi_{s}$ corresponds to the
equilibrium case. Neglecting a hole concentration and using the step function
in $c$-band, one obtains $n_{s}$ from Eq. (15):

$n_{s}\approx\int\limits_{0}^{\psi_{s}T}d\varepsilon\rho_{\varepsilon}.$ (20)

As a result, $\delta\psi_{t}$ and $\psi_{t}^{<}$ are connected by the
electroneutrality condition (15) as follows

$\delta\psi_{t}\approx\frac{\overline{\rho}_{T}}{\rho_{\varepsilon=\psi_{s}T}}\int\limits_{0}^{\infty}\frac{d\xi
r_{\xi}}{1+\exp\left(\xi+\psi_{t}^{<}\right)},$ (21)

where the ratio $\overline{\rho}_{T}/\rho_{\varepsilon=\psi_{s}T}$ can be
found from Fig. 2a. Using Eq. (14) we transform the concentration balance
equation into the form:

$\frac{d\delta\psi_{t}}{dt}=-\frac{\nu_{GR}\overline{\rho}_{T}}{2\rho_{\varepsilon=\psi_{s}T}[1+\exp\left(\psi_{t}^{<}\right)]}.$
(22)

Substituting the relation (21) into Eq. (22) one obtains the first-order
differential equation for $\psi_{t}^{<}$, with the implicit solution

$\nu_{GR}t=\int\limits_{\overline{\psi}_{0}}^{\psi_{t}^{<}}d\psi\int\limits_{0}^{\infty}\frac{d\xi
r_{\xi}\left(1+e^{\psi}\right)}{1+\cosh\left(\xi+\psi\right)},$ (23)

where $\overline{\psi}_{0}=\psi_{t=0}^{<}$ appears from the initial condition.
Notice, that the factor $\overline{\rho}_{T}/\rho_{\varepsilon=\psi_{s}T}$
drops out from the solution (23), i. e. the transient process under
consideration does not depend on the doping level because the only low-energy
states are involved in the interband transitions.

Figure 6: (Color online) Transient evolution of concentration $n_{t}-n_{s}$ at
different initial conditions: $n_{t=0}-n_{s}=3n_{T}$ (1),
$n_{t=0}-n_{s}=2n_{T}$, (2) and $n_{t=0}-n_{s}=0.5n_{T}$ (3). Solid and dashed
curves correspond to coupling parameters $g=$0.5 and $g=$0.25, respectively.
Dotted curves correspond to the exponential fits.

Further, we plot the transient evolution of the hole concentration
$\overline{n}_{t}=n_{t}-n_{s}$ determined through $\psi_{t}^{<}$ according to
$\overline{n}_{t}=\int_{0}^{\infty}d\varepsilon\rho_{\varepsilon}\left[\exp(\varepsilon/T+\psi_{t}^{<})+1\right]^{-1}$.
Figure 6 shows the concentration $\overline{n}_{t}$ versus dimensionless time,
$\nu_{GR}t$, for the initial conditions written through $n_{T}$. Similarly to
the undoped case, the exponential fits
$(n_{t}-n_{s})/(n_{t=0}-n_{s})\approx\exp(-\beta\nu_{GR}t)$ with
$\beta\simeq$0.12 (1), 0.15 (2), and 0.2 (3) describe the transient evolution
with an accuracy $\sim$10% if $\nu_{GR}t<$10\. An enhancement of recombination
takes place at tails of transient evolution, if $\nu_{GR}t>$10\. Since the
relaxation rate increases with the disorder level, $\nu_{GR}\propto w_{g}$,
the recombination process becomes faster in spite of an opposite dependency on
$\nu_{GR}t$ in Fig. 6. The relaxation times, $\sim(\beta\nu_{GR})^{-1}$, vary
between 410 and 240 ns for $g=$0.5 and different initial conditions. Once
again, the recombination scale is comparable to the radiative recombination
process shown in Fig. 7b, Appendix.

##  IV SUMMARY AND CONCLUSIONS

We have examined the new channel for interband generation-recombination
process of carriers in a disordered graphene via acoustic phonons. The
efficiency of transitions increases with the disorder level and concentration
of nonequilibrium carriers as well as with temperature. We have found that the
relaxation rate belongs to submicrosecond range for the samples with typical
disorder level at room temperature.

Let us discuss the assumptions used in the presented calculations. The main
restriction of the results is the description of the response in the framework
of the quasiequilibrium approach, with different chemical potentials in $c$\-
and $v$-bands but the same temperature due to the fast energy relaxation
caused by phonon and carrier-carrier scattering processes. We also restrict
ourselves by the simplest model of the short-range disorder. By analogy with
the description of transport phenomena, 1 ; 9 ; 11  more complicated
calculations for finite-range disorder should give similar results. But the
case of impurities with a low-energy resonant level, which was discussed
recently in Refs. 16, requires a special consideration. We considered the
deformation interaction of carriers with longitudinal acoustic modes 9 ; 14
neglecting scattering by surface phonons of the substrate in agreement with
the experimental data. 16  Such a contribution can only restrict the energies
under consideration because of the lower surface phonon energy ($\sim$55 meV
for the SiO2 substrate). Since the non-diagonal components give a weak
contribution to the concentration balance equation under consideration, 10  we
take into account only diagonal components of the density matrix $f_{\alpha
t}$ while evaluating of the generation-recombination rate. The simplifications
mentioned above do not change either the peculiarities of the generation-
recombination processes or the numerical estimates of relaxation times given
in Sec. III.

Next, we briefly consider some possibilities for experimental verification of
the mechanism of interband transitions suggested. It is clear from a
comparison of the results in Sec. III and in Appendix that interband
transitions via acoustic phonons and via thermal radiation can be separated
due to different temperature and concentration dependencies of damping. A
possible contribution of the disorder-induced Auger process is beyond of our
consideration and requires a special study. In contrast to the ultrafast
optical measurements applied for the study of the relaxation and recombination
of high-energy carriers, 2  a transient evolution of concentration over time
scales $\sim$100 ns can be measured directly (e.g. in Ref. 5 the transient
response under abrupt switching on of a dc field lasts up to hundreds of
nanoseconds). But under a verification of the slow process examined, a
possible contact injection or a trapping into substrate states should be
analyzed.

To conclude, we believe that the generation-recombination via acoustic phonons
can be verified experimentally and more detailed numerical calculations are
necessary in order to separate this mechanism from other contributions. The
results obtained will stimulate a further study of the generation-
recombination processes which are essential in many transport and optical
phenomena far from equilibrium.

## ACKNOWLEDGMENT

This paper is based upon work supported by the National Science Foundation
under Grant No DMR 0907126.

*

##  Appendix A Radiative transitions

Below we describe the generation-recombination processes which are associated
with the interband transitions induced by the thermal radiation and evaluate
the radiative relaxation rate for the weak disorder case, $g\ll 1$. The
corresponding collision integral was evaluated in Ref. 9 and the kinetic
equation for the electron distribution $f_{ept}$ takes the form

$\frac{\partial f_{ept}}{\partial t}=\nu_{p}^{(R)}\left[N_{2\upsilon
p/T}\left(1-f_{ept}-f_{hpt}\right)-f_{ept}f_{hpt}\right],$ (24)

where $N_{2\upsilon p/T}$ describes the Planck distribution of the thermal
photons at temperature $T$. The hole distribution can be obtained from the
condition $\partial(f_{ept}+f_{hpt})/\partial t=0$. The interband absorption
or emission of photons are described by the first or second terms in the
right-hand side of Eq. (A.1) and are responsible for the generation or
recombination processes. The rate of spontaneous radiative transitions is
given by $\nu_{p}^{(R)}=\upsilon_{R}p/\hbar$ where we have introduced the
characteristic velocity $\upsilon_{R}\simeq$41.6 cm/s for graphene surrounded
by SiO2 layers. Similar to Eq. (6) contribution of the radiative collision
integral from (A.1) into the concentration balance equation takes the form
$(dn/dt)_{ac}=(4/L^{2})\sum\nolimits_{\bf p}\nu_{p}^{(R)}\left[N_{2\upsilon
p/T}\left(1-f_{ept}-f_{hpt}\right)-f_{ept}f_{hpt}\right]$.

For the case of an intrinsic graphene, the balance equation is written by
analogy with Sect. III A through $\psi_{t}=\mu_{t}/T$:

$\frac{d\psi_{t}}{dt}=-\nu_{R}\frac{F_{i}(\psi_{t})}{N(\psi_{t})},~{}~{}~{}~{}\nu_{R}=\frac{2\upsilon_{R}}{\upsilon}\frac{T}{\hbar}$
(25)

where $\nu_{R}^{-1}\approx$30 ns is the radiative recombination time at room
temperature. The functions $F_{i}(\psi)$ and $N(\psi)$ are given by

$\displaystyle
F_{i}(\psi)=\int\limits_{0}^{\infty}\frac{d\xi\xi\left(1-e^{-2\psi}\right)}{\left(1-e^{-2\xi}\right)\left(e^{\xi-\psi}+1\right)^{2}}$
$\displaystyle
N(\psi)=\int\limits_{0}^{\infty}\frac{d\xi\xi}{1+\cosh(\xi-\psi)}$ (26)

and the implicit solution of Eqs. (A.2), (A.3) is given by the similar to Eq.
(18) formula:

$\nu_{R}t=\int_{\psi_{0}}^{\psi_{t}}d\psi\frac{N(\psi)}{F_{i}(\psi)}.$ (27)

In Fig. 7a we plot the transient evolution of concentration versus the
dimensionless time, $\nu_{R}t$ for the same initial conditions as in Fig. 5.
These transient dependencies are described by the exponential decay given by
Eq. (19) with $\alpha\approx$0.25 for all cases. Thus, one obtains the
radiative recombination time $\sim 4/\nu_{R}\approx$120 ns which does not
depend on an initial concentration.

Figure 7: (Color online) Transient evolution of concentration due to interband
radiative transitions for intrinsic (a) and heavily-doped (b) graphene at
different initial conditions: $n_{t=0}=3n_{T}$ (1), $n_{t=0}=2n_{T}$ (2) and
$n_{t=0}=0.5n_{T}$ (3). Dotted curves correspond to exponential fits.

For the case of doped graphene, the concentration balance equation (22) should
be replaced by

$\displaystyle\frac{d\delta\psi_{t}}{dt}=-\frac{\nu_{R}}{2}F_{d}(\psi_{t}^{<}),$
(28) $\displaystyle
F_{d}(\psi)=\int\limits_{0}^{\infty}\frac{d\xi\xi^{2}}{\left(1-e^{-2\xi}\right)\left(e^{\xi+\psi}+1\right)}$

while the relation between $\delta\psi_{t}$ and $\psi_{t}^{<}$ takes form
[c.f. Eq. (21)]

$\delta\psi_{t}\approx\frac{1}{\psi_{s}}\int\limits_{0}^{\infty}\frac{d\xi\xi}{1+\exp\left(\xi+\psi_{t}^{<}\right)}.$
(29)

As a result, the equation for $\psi_{t}^{<}$ has the only difference from Eq.
(2) due to the replacement $F_{i}(\psi)$ by $F_{d}(\psi)$. The implicit
solution of Eqs. (A.5) and (A.6) is given by (A.4) with the same replacement.
In Fig. 7b we plot the transient evolution of hole concentration,
$\overline{n}_{t}/n_{T}$, for the same initial conditions as in Fig. 6. The
corresponding exponential fits are determined by the coefficients
$\beta\simeq$0.15 (1), 0.125 (2), and 0.06 (3), i.e. the relaxation rate
depends on hole concentration. At room temperature the radiative recombination
time $(\beta\nu_{R})^{-1}$ corresponds to the time interval between 190 and
480 ns.

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