Nonequilibrium electron relaxation in graphene

(1)

Vol. 33, No. 17 (2019) 1950183 (14 pages) c

World Scientific Publishing Company DOI: 10.1142/S0217979219501832

Luxmi Rani∗

Department of Physics, Bilkent University, C¸ ankaya 06800, Ankara, Turkey

luxmiphyiitr@gmail.com

Pankaj Bhalla

Beijing Computational Science Research Center, Haidian District, Beijing 100193, P. R. China

Navinder Singh

Physical Research Laboratory, Navrangpura, Ahmedabad 380009, Gujarat, India

Received 16 October 2018 Revised 24 April 2019 Accepted 15 May 2019 Published 18 July 2019

We apply memory function formalism to investigate nonequilibrium electron relaxation in graphene. Within the premises of two-temperature model (TTM), explicit expressions of the imaginary part of the memory function or generalized Drude scattering rate (1/τ ) are obtained. In the DC limit and in equilibrium case where electron temperature (Te) is equal to phonon temperature (T ), we reproduce the known results (i.e., 1/τ ∝ T4 when T ΘBG and 1/τ ∝ T when T ΘBG, where ΘBG is the Bloch–Gr¨uneisen temperature). We report several new results for 1/τ where T 6= Te relevant in pump– probe spectroscopic experiments. In the finite-frequency regime we find that 1/τ ∝ ω2 when ω ωBG, and for ω = ωBGit is ω-independent. These results can be verified in a typical pump–probe experimental setting for graphene.

Keywords: Graphene; electronic transport in graphene; conductivity of specific material; memory function formalism; nonequilibrium electron relaxation.

PACS numbers: 81.05.ue, 72.80.Vp, 72.80.-r, 64.60.-i

1. Introduction

Graphene is a unique two-dimensional material consisting of a single-atom-thick layer of carbon atoms that are closely packed in honeycomb lattice structure. In recent times, the study of electronic transport of hot carriers in graphene has created ∗_{Corresponding author.}

Int. J. Mod. Phys. B 2019.33. Downloaded from www.worldscientific.com

(2)

an enormous research interest in both the experimental and theoretical aspects due to the potential applications in electronic devices.1–10 _{In graphene, relaxation} of hot (photoexcited) electrons has been investigated experimentally4,5,10–15 _and theoretically.16–26In simple metals, electron relaxation dynamics is well understood and the two-temperature model (TTM) is extensively used to analyze the relaxation dynamics.27–33While in graphene, due to Dirac physics and peculiar band structure, the hot electron relaxation is different from that metal, and a detailed theoretical study is lacking.

In simple metals, hot electron relaxation happens via electron–phonon inter-actions. The mechanism of hot electron relaxation is as follows. A femtosecond laser pulse excites the electrons from equilibrium Fermi–Dirac (FD) distribution to a nonequilibrium distribution. This nonequilibrium electron distribution inter-nally relaxes via electron–electron interactions to a hot FD distribution in a time scale τee. Then through electron–phonon interactions, this “hot” FD distribution relaxes to a state in which electron temperature becomes equal to the phonon tem-perature, i.e., an equilibrium state. This process happens in a time scale τe–ph. In simple metals the inequality τee τe–phis true. And phonons remain in equilibrium during the whole process of relaxation (it is called the Bloch assumption32). This motivates the TTM: one temperature for electron subsystem (Te) and another for the phonon subsystem (T ). The electron relaxation in metals is extensively studied within the TTM model using the Bloch–Boltzmann kinetic equation.30–33 _{In the} analysis an important energy scale is set by Debye temperature, and it turns out that when T ΘD, the relaxation rate from the Bloch–Boltzmann equation is given as 1/τ ∝ T . In the opposite limit, i.e., T ΘD, it turns out that 1/τ ∝ T5. In order to study the hot electron relaxation in graphene, several experiments like pump–probe spectroscopy and photoemission spectroscopy have been used re-cently.34–36 On the theoretical side, the hot electron relaxation has been studied in graphene using the Bloch–Boltzmann equation16,17,22,37_{and density-matrix} the-ory.21 _{But all these studies are restricted to the DC regime.}

A detailed study of the frequency- and temperature-dependent scattering rate in graphene has been lacking in the literature. In the present investigation, we solved this problem using the method of memory function formalism.38The memory function formalism known as the projection operator method was pioneered by Mori and Zwanzig.39,40 _{The basic idea of this method is to study the time correlation} function systematically in the case of many-body correlated systems. The merit of using memory function formalism is that it directly deals with the dynamical behavior of electronic transport.38–42 We calculate the scattering rate in various frequency and temperature limits. Our main results are as follows.

In the DC case, scattering rate shows the fourth power-law of both electron and phonon subsystem temperatures below the Bloch–Gr¨uneisen (BG) temperature. Above the BG temperature, scattering rate is linearly dependent only on the phonon temperature. On the other hand, at finite frequency (ω = ωBG) and at higher temperatures, scattering rate is linearly dependent on both the phonon and electron

(3)

temperatures. At ω = ωBG, scattering rate is independent of frequency. However, it is observed that there is ω2_{-dependence in the lower-frequency regime.}

This paper is organized as follows. In Sec. 2, we introduce the model and mem-ory function formalism. We then compute the memmem-ory function (generalized Drude scattering rate) using the W¨olfle–G¨otze perturbative method.38_{Then various} sub-cases are studied analytically. In Sec. 3, we present the numerical study of the general case. In Sec. 4, we discuss the validity of the present approach. Finally the summary of our results and the main conclusions are given in Sec. 5.

2. Theoretical Framework

To study the electron relaxation in graphene, we consider the total Hamiltonian having three parts such as free electron (He), free phonon (Hp) and the interacting part, i.e., electron–phonon (Hep):

H = He+ Hp+ Hep. (1)

The different parts of Hamiltonian mentioned in the above equation are defined as He= X kσ kc†kσckσ, (2) Hp= X q ωq b†_qbq+ 1 2 , (3) Hep= X k, k0_,σ D(k − k0_)c† kσck0_σb_k−k0+ h.c.. (4)

Here, c†_kσ (ckσ) and b†q (bq) are the electron and phonon creation (annihilation) operators, σ is a spin k and q = k − k0 are the electron and phonon momentums, respectively. k = ~vF|k| is the linear energy dispersion term in graphene. The h.c. denotes the Hermitian conjugate term. D(k − k0) is the electron–phonon matrix element which is defined as22,43,44

D(q) = −i ₁ 2ρmωq 1/2 D0× q " 1 − _q 2kF 2#1/2 . (5)

Here, D0 is the deformation potential coupling constant for graphene, ρm is the surface mass density, kF is the Fermi momentum and ωq is the phonon energy. In the present study, we considered only the intraband transition. Also, we set ~ = kB= 1 throughout the calculations.

2.1. Calculation for generalized Drude scattering rate

Our aim is to calculate the generalized Drude scattering rate or imaginary part of the memory function. For the calculation of generalized Drude scattering, we

(4)

use the G¨otze–W¨olfle formalism.33,38,41,42 _{In this formalism, memory function is} expressed as M (z, T, Te) = zχ(z) χ0− χ(z) ' zχ(z) χ0 1 + χ(z) χ0 + · · · ' zχ(z) χ0 , (6)

where χ0 represents the static limit of correlation function and z is the complex frequency. χ(z) is the Fourier transform of the current–current correlation function:

χ(z) = i Z ∞

0

eizth[j1, j1]idt. (7)

Here, j1 = Σkσ(k · ˆn)c†kσckσ is the current density. ˆn is the unit vector along the direction of current. Using the equation of motion (EOM) method38,42 _{it can be} shown that

M (z, T, Te) =

hh[j1, H]; [j1, H]iiz=0− hh[j1, H]; [j1, H]iiz zχ0

. (8)

Using Eq. (1) and the definition of current density operator into the above equation and on simplifying,a _{we obtain}

M (z, T, Te) = 1 χ0 X kk0 |D(k − k0_)|2_{[(k − k}0_{) · ˆ}_n]2 × [f (1 − f0)(1 + n) − f0(1 − f )n] 1 (k− k0− ω_q) × ₁ (k− k0− ω_q+ z) + 1 (k− k0− ω_q− z) . (10)

Here, f = f (k, βe) and f0 = f (k0, β_e) are the Fermi–Dirac distribution

func-tions at different energies such as k and k0, and electron temperature T_e= 1/β_e.

n = n(ωq, β) is the Bose–Einstein distribution function, and T = 1/β is the phonon temperature. Also z = ω + iδ and δ → 0+_{. Here we assume a} steady-state situation in which electron temperature stays constant at Te, and phonon temperature also stays constant at T . This situation can be experimentally cre-ated by a continuous laser excitation of graphene. The memory function has real and imaginary parts: M (z, T, Te) = M0(ω, T, Te) + iM00(ω, T, Te). We are inter-ested in the scattering rate which is the imaginary part of the memory function

a_{The current density operator commutes with the noninteracting parts of the Hamiltonian, the} interacting part gives

C = [j1, H] = X

k,k0

[(k − k0) · ˆn][D(k − k0)c†_kσck0_σbk−k0− h.c.]. (9)

(5)

[i.e., M00(ω, T, Te) = 1/τ (ω, T, Te)]. In that case Eq. (10) can be simplified to 1 τ (ω, T, Te) = π χ0 X kk0 |D(k − k0)|2[(k − k0) · ˆn]2 × [f (1 − f0)(1 + n) − f0(1 − f )n] ×1 ω " δ(k− k0 − ω_q+ ω) − δ(_k− _k0− ω_q− ω) # . (11)

Converting the sums over momentum indices into integrals using the linear energy dispersion relations k = vFk and 0k= vFk0 and after further simplifying the above equation, we get 1 τ (ω, T, Te) = 1 τ0 Z qBG 0 dq × q3p1 − (q/2kF)2 × ( 1 −ωq ω [n(β, ωq) − n(βe, ωq− ω)] + (terms with ω → −ω) + · · · ) . (12)

Here, 1/τ0 = N2D20/32π3χ0ρmkFvs and qBG being the Bloch–Grüneisen momen-tum, i.e., the maximum momentum for the phonon excitations (i.e., vsqBG = 2kFvs = ΘBG). In graphene, a new temperature crossover known as Bloch– Grüneisen temperature (ΘBG) is introduced due to small Fermi surface (kF) as compared to Debye surface (kD).45,46 Thus in this system when kF kD, below the Bloch–Grüneisen temperature, only small number of phonons with wave vec-tor (kph< 2kF) can take part in scattering. Various limiting cases of Eq. (12) are studied in the next subsection.

2.2. Limiting cases for the generalized Drude scattering rate 2.2.1. Case-I: DC limit

Within this limit, curly bracket in Eq. (12) reduces to

2 lim ω→0 " n(β, ωq) − ∞ X m=0 ω2m _∂2m ∂ω2m q n(βe, ωq) + ωq ∂2m+1 ∂ω2m+1q n(βe, ωq) # . (13)

Here, we consider only m = 0, i.e., the leading-order case: 1 τ (ω, T, Te) = 1 τ0 Z qBG 0 dq × q3 s 1 − q 2kF 2 (n(β, ωq) − n(βe, ωq) − ωqn0(βe, ωq)). (14)

(6)

Using the relations ωq = vsq, ωBG ' ΘBG= 2vskF and defining ω_Tq = x, ω_Tq e = y, Eq. (14) becomes 1 τ (ω, T, Te) = 1 τ0 2 v4 s T4 Z ΘBG_T 0 dx × x 3 ex_{− 1} s 1 − x 2_T2 Θ2 BG + T_e4 Z ΘBG_Te 0 dy × y3 s 1 − y 2_T2 e Θ2 BG × y − 1 ey_{− 1} + y (ey_{− 1)}2 . (15) Subcase (a). When T, Te ΘBG, i.e., both the phonon temperature and electron temperature are lower than the Bloch–Gr¨uneisen temperature, Eq. (15) gives

1 τ (T, Te) = 1 τ0 2 v4 s T4×π 4 15+ T 4 e × π4 5 = 1 τ0 2 v4 s A1T4+ B1Te4 . (16) Here A1= π 4 15 and B1= 3A1.

Subcase (b). In the high-temperature regime, T, Te ΘBG, so Eq. (15) reduces to 1 τ (T, Te) = 1 τ0 2 v4 s 7 30T Θ 3 BG+ 1 6Θ 4 BG = 1 τ0 2 v4 s A2T + B2 . (17)

Here A2=₃₀7Θ3BGand B2=1₆Θ4BG. It is notable here that the scattering rate is in-dependent of electron temperature, and it only depends on the phonon temperature. Subcase (c). Here T ΘBG and Te ΘBG. In this regime scattering rate can be written as 1 τ (T, Te) = 1 τ0 2 v4 s 7 30T Θ 3 BG+ T 4 e π4 5 = 1 τ0 2 v4 s A3T + B3Te4 . (18) Here A3 = ₃₀7Θ3BG and B3 = π 4

5 . It is important to note that 1/τ leads to the linear phonon temperature dependence in high-temperature regime and shows the Te4-dependence below the BG temperature.

Subcase (d). Here T ΘBGand Te ΘBG. In this regime Eq. (15) gives 1 τ (T, Te) = 1 τ0 2 v4 s T4×π 4 15+ 1 6Θ 4 BG = 1 τ0 2 v4 s A4T4+ B4 . (19)

(7)

Table 1. The results of electrical scattering rate due to the electron–phonon interactions in different limiting cases. Here, A7=₃₀7Θ3BG, B7= −₃₀7Θ3BG, C7=20₃ΘBG; A8=π 4 15, B8= π4 5 , C8= 5π 2_{− ζ(5); A} 9=₃₀7Θ3BG, B9=π 4 5, C9= C8= const. and A10= π 4 15, B10= − 7 30Θ 3 BG, C10= 20 3ΘBG. No. Regimes 1_τ 1 ω = 0; Te, T ΘBG A1T4+ B1Te4 ω = 0; Te, T ΘBG A2T + B2 ω = 0; T ΘBG, Te ΘBG A3T + B3Te4 ω = 0; T ΘBG, Te ΘBG A4T4+ B4 2 ω = ωBG; Te, T ΘBG A5T4+ B5Te4 ω = ωBG; Te, T ΘBG A6T + B6Te 3 ω ωBG; T, Te ΘBG A7T + B7Te+ C7ω2Te ω ωBG; T, Te ΘBG A8T4+ B8Te4+ C8ω2Te2 ω ωBG; T ΘBG, Te ΘBG A9T + B9Te4+ C9ω2Te2 ω ωBG; T ΘBG, Te ΘBG A10T4+ B10Te+ C10ω2Te Here A4= π 4 15 and B4= 1 6Θ 4

BG. Hence 1/τ (T, Te) has a T4-dependence. Scattering rate is independent of the electron temperature. On the other hand, when T = Te, the result of scattering rate is identical as obtained in an equilibrium electron– phonon interaction in graphene case,22,45 as expected. These results are tabulated in Table 1.

2.2.2. Case-II: Finite-frequency regimes

Subcase (1). Consider ω = ωBG, then Eq. (12) becomes 1 τ (ω, T, Te) = 2 τ0 Z qBG 0 dq × q3p1 − (q/2kF)2× ( n(β, ωq) − n(βe, 2ωq) ) . (20) This can be simplified by setting ωq

T = x, ωq Te = y, then we have 1 τ (ω, T, Te) = 1 τ0 2 v4 s T4 Z ΘBG_T 0 dx × x 3 ex_{− 1} s 1 − x 2_T2 Θ2 BG − T4 e Z ΘBG_Te 0 dy × y 3 e2y_{− 1} s 1 − y 2_T2 e Θ2 BG . (21)

To further simplify the last equation, below we study the low- and high-temperature regimes separately.

In the low-temperature regime T, Te ΘBG, therefore Eq. (21) becomes 1 τ (ω, T, Te) = 1 τ0 2 v4 s T4×π 4 15− T 4 e × π4 240 = 1 τ0 2 v4 s A5T4+ B5Te4 . (22) Here A5= π 4 15 and B5= − π4 240.

(8)

In the high-temperature regime T, Te ΘBG, therefore Eq. (21) takes the following form: 1 τ (ω, T, Te) = 1 τ0 2 v4 s 7 30Θ 3 BGT − 7 60Θ 3 BGTe = 1 τ0 2 v4 s A6T + B6Te . (23)

Here A6 = ₃₀7Θ3BG and B6= −₆₀7Θ3BG. It is also noticeable here that in both the cases _{τ (ω,T ,T}1

e) shows frequency-independent behavior.

Subcase (2). At finite but lower-frequency (ω ωBG) case, with the relation ωq = vsq Eq. (12) becomes 1 τ (ω, T, Te) = 1 τ0 Z ΘBG 0 dq × ω_q3 s 1 − _ω q ΘBG 2 ₁ ewqT − 1 − ∞ X m=0 ω2m × ∂2m ∂ω2m q 1 ewqTe − 1 + ωq ∂2m+1 ∂ωq2m+1 1 ewqTe − 1 . (24)

This is the general equation of the imaginary part of memory function when frequency is lower than the Bloch–Gr¨uneisen frequency. The above equation can be further simplified by setting the variables ωq

T = x, ωq

Te = y, and for m = 1, Eq. (24)

reduces to 1 τ (ω, T, Te) = 1 τ0 2 v4 s T4 Z ΘBG_T 0 dx × x3 s 1 − x 2_T2 Θ2 BG 1 ex_{− 1} + ω2T_e2 Z ΘBG_Te 0 dy × y3 s 1 − y 2_T2 e Θ2 BG × ny+ 3n2y+ 2n 3 y− y ny− 7n2y− 12n 3 y− 6n 4 y . (25) Here ny = 1/(ey− 1). Furthermore, we study the frequency-dependent scattering rate at low- and high-temperature regimes of both electron and phonon subsystems. We consider first two terms (m = 0 and m = 1) in the series of Eq. (24). The analytic results obtained in the present subcase (ω ωBG) are presented in Table 1. It is observed that there is a ω2-dependence multiplied by the electron temperature in the lower-frequency regime. In the general case, numerical computations of Eq. (12) are presented in the next section. And in the appropriate limiting cases, numerical results agree with the analytical results presented in Table 1.

3. Numerical Analysis

We have numerically computed Eq. (12) in different frequency and temperature regimes. In Fig. 1(a), we depict the phonon temperature dependence of scat-tering rate 1/τ (T, Te) normalized by 1/τ0(=N2D02/16π3χ0ρmkFv5s) at zero fre-quency and at different electron temperatures. From Fig. 1(a), we observe that

(9)

Fig. 1. (Color online) (a) Variations of the scattering rate with phonon temperature at zero frequency and different electron temperatures. (b) Variations of the scattering rate with electron temperature at zero frequency and different phonon temperatures. Here both the electron and phonon temperatures are scaled with the Bloch–Gr¨uneisen temperature and 1/τ (T, Te) is scaled with 1/τ0. Panels (c) and (d) depict the contour plots T versus Te for the scattering rate at zero frequency.

at high-temperatures (Te, T ΘBG), 1/τ (T, Te) ∝ T . This can also be seen in the corresponding case (Te, T ΘBG) in Table 1. At very low-temperatures (T, Te ΘBG), 1/τ ∝ T4 and Te4. Figure 1(b) shows the dependence of 1/τ on Tein the DC limit. It is observed that 1/τ is independent of Te when Te ΘBG. Contour plots [Figs. 1(c) and 1(d)] depict the constant value of 1/τ in the Te–T plane. The contours for higher values of T and Teare for higher 1/τ .

From the contour plots, we notice that they are not symmetric around T = Te line. The physical reason for this asymmetry is that the scattering rate is differ-ently affected by phonon temperature and electron temperature (the pre-factor A1 of T4term is not equal to the prefactor B1 of Te4 terms). At very low-temperature T4 _{behavior is due to Pauli blocking effect. We notice that at high-temperature} (Te, T ΘBG), 1/τ (T, Te) is proportional to T , not Te. The reason for this behavior

(10)

Fig. 2. (Color online) (a) Variations of the scattering rate with phonon temperature at finite but lower frequency and at different electron temperatures, and inset shows the lower phonon temperature range and (b) Variations of the scattering rate with electron temperature at finite frequency and at different phonon temperatures, and inset shows the lower electron temperature range.

is that at high-temperatures phonon modes scale as kBT (hnqi = _eβωq1₋₁ ∝ kBT ), thus scattering increases with increasing temperatures linearly. For Te ΘBG the electron distribution can be approximated as Boltzmann distributions because ΘBG ' TF (the Fermi temperature). The temperature effect is exponentially re-duced in this case as compared to phonons (hnqi ∝ T ). Thus at high-temperatures, the scattering rate is proportional to T .

In Fig. 2(a), we plot the phonon temperature dependency of scattering rate τ0/τ (ω, T, Te) at lower frequency and at different temperatures of electrons. It is observed that at lower phonon temperature range, the magnitude of scattering rate increases with increasing temperature as T4 behavior. At higher T it shows T -linear behavior. In Fig. 2(b), the variations of electron temperature dependence of τ0/τ (ω, T, Te) at different phonon temperatures scaled with BG temperature are shown. The insets of Figs. 2(a) and 2(b) show the low-temperature behavior (T, Te ΘBG). The low-frequency behavior is similar to the DC case.

We further analyzed the scattering rate at zero-temperature in which both elec-tron subsystem and phonon subsystem are at zero-temperature. In this regime 1/τ scales as ω4as depicted in Fig. 3.

To study the scattering rate for finite frequency, we plot the ω-dependence of the scattering rate 1/τ (ω, T, Te) at different temperatures of electron and phonon subsystems in Fig. 4. Figure 4(a) depicts the variations of scattering rate with frequency at different phonon temperatures and at fixed electron temperature. At lower frequency it shows ω2 behavior.

(11)

Fig. 3. (Color online) Variation of the scattering rate with frequency at zero electron and phonon temperatures.

Fig. 4. (Color online) Variations of the scattering rate with frequency at different electron and phonon temperatures.

In Fig. 4(b), we plot the variations of scattering rate with frequency at different electron temperatures and fixed phonon temperature. From Fig. 4(b), it is clear that on increasing the electron temperature, scattering rate increases in lower-frequency regime.

4. Regime of Validity

Our theoretical analysis is mainly valid in the low-frequency regime ω ωBG. In this regime intraband transitions play a major role as illustrated in the present work. However, interband transitions become dominating in the range of frequencies

(12)

greater than the Bloch–Gr¨uneisen scale which we have not considered in the present study. We have checked the upper bound criteria, i.e., ω = ωBG, beyond which our calculations are no longer valid. Our study is not applicable in the high-frequency regime when interband transitions become important.

5. Summary and Conclusions

We presented a theoretical study of nonequilibrium relaxation of electrons due to their coupling with phonons in graphene by using the memory function approach. In the present model, we have considered only the intraband transition. In our results at zero-frequency limit, it is observed that if the electron temperature does not coincide with the phonon temperature, i.e., Te 6= T , then DC scattering rate has a fourth power-law behavior of both the electron and phonon temperatures (A1T4+B1Te4) below the BG temperature. While at higher temperature, 1/τ shows the T -linear dependency only (it does not depend on Te). Further, it is important to notice here that AC scattering rate shows a linear dependence on both the electron and phonon temperatures above the BG temperature. In the DC case, at the low-temperature regime scattering rate shows a T4_{power-law and at high-temperature} it shows a T -linear behavior which are in qualitative agreement with the prior experimental results.23,24,26

In Table 2, we compare the results of scattering rates for the simple metals and the present case of graphene. We observed that T5-law of 1/τ in the case of metals (in regimes ω = 0, T ΘD) changes to T4-law in the corresponding case in graphene.23,24,42,47 _{However, in the case of high-temperatures and ω = ω}

BG, temperature dependence of 1/τ in both metals and graphene remains the same.

In the low-frequency case (ω ωD) and in lower-temperature regimes (T, Te ΘD), 1/τ in metals has three terms (a6T5 + b6Te5 + c6Te5ω

2₎ whereas in the corresponding case of graphene this dependence changes to

Table 2. Comparison of nonequilibrium electron relaxation in metals and in graphene. Graphene (1

τ) 2D

23,24 _{Metals (}1

τ) 3D

33

Bloch–Gr¨uneisen Debye

No. Regimes temperature (ΘBG) temperature (ΘD)

1 ω = 0; Te, T ΘBG, ΘD A1T4+ B1Te4 a1T5+ b1Te5 ω = 0; Te, T ΘBG, ΘD A2T + B2 a2T + b2 ω = 0; T ΘBG, Te ΘBG A3T + B3Te4 — ω = 0; T ΘBG, Te ΘBG A4T4+ B4 — 2 ω = ωBG, ωD; Te, T ΘBG, ΘD A5T4+ B5Te4 a3T5+ b3Te5 ω = ωBG, ωD; Te, T ΘBG, ΘD A6T + B6Te a4T + b4Te 3 ω ωBG, ωD; T , Te ΘBG, ΘD A7T + B7Te+ C7ω2Te a5T + b5ω2Te ω ωBG, ωD; T , Te ΘBG, ΘD A8T4+ B8Te4+ C8ω2Te2 a6T5+ b6Te5+ c6Te5ω2 ω ωBG; T ΘBG, Te ΘBG A9T + B9Te4+ C9ω2Te2 — ω ωBG; T ΘBG, Te ΘBG A10T4+ B10Te+ C10ω2Te —

(13)

(A8T4+ B8Te4+ C8ω2Te2). These results can be verified in the typical pump–probe spectroscopic experiments.8,29,36

Acknowledgments

One of the authors (Luxmi Rani) is supported by the Scientific and Techni-cal Research Council of Turkey (T ¨UBITAK) ARDEB-1001 Program, Project No. 117F125. Pankar Bhalla acknowledges the support from NSAF Grant No. U1530401.

References

1. K. S. Novoselov et al., Nature 438, 197 (2005).

2. A. H. Castro Neto et al., Rev. Mod. Phys. 81, 109 (2009).

3. M. J. Allen, V. C. Tung and R. B. Kaner, Chem. Rev. 110, 132 (2010). 4. S. D. Sarma et al., Rev. Mod. Phys. 83, 407 (2011).

5. Z. Q. Li et al., Nat. Phys. 4, 532 (2008).

6. N. M. R. Peres, T. Stauber and A. H. Castro Neto, Europhys. Lett. 84, 38002 (2008). 7. C. N. R. Rao et al., Angew. Chem., Int. Ed. 48, 7752 (2009).

8. J. Shah, Hot Carrier in Semiconductor Nanostructures (Academic Press, London, 1992).

9. C. Voisin and B. Pla¸cais, J. Phys.: Condens. Matter 27, 160301 (2015). 10. S. Wu et al., Nano Lett. 12, 5495 (2012).

11. T. Li et al., Phys. Rev. Lett. 108, 167401 (2012). 12. N. M. Gabor, Science 334, 648 (2011).

13. S. Winnerl and M. Orlita, Phys. Rev. Lett. 107, 237401 (2011). 14. K. J. Tielrooij et al., Nat. Nanotechnol. 10, 437 (2015). 15. K. F. Mak et al., Phys. Rev. Lett. 101, 196405 (2008).

16. V. K. Dugaev and M. I. Katsnelson, Phys. Rev. B 88, 235432 (2013). 17. R. Kim, V. Perebeinos and P. Avouris, Phys. Rev. B 84, 075449 (2011). 18. T. Low et al., Phys. Rev. B 86, 045413 (2012).

19. J. M. Iglesias et al., Appl. Phys. Lett. 108, 043105 (2016). 20. W.-K. Tse and S. D. Sarma, Phys. Rev. B 79, 235406 (2009). 21. S. Butscher and F. Milde, Appl. Phys. Lett. 91, 203103 (2007). 22. E. Mu˜noz, J. Phys.: Condens. Matter 24, 195302 (2012). 23. D. K. Efetov and P. Kim, Phys. Rev. Lett. 105, 256805 (2010). 24. E. H. Hwang and S. D. Sarma, Phys. Rev. B 77, 115449 (2008). 25. Y.-W. Tan et al., Eur. Phys. J. Spec. Top. 148, 15 (2007). 26. M. S. Fuhrer, Physics 3, 106 (2010).

27. R. W. Davies, Phys. Rev. 181, 1118 (1969).

28. B. T. Wong and M. P. Meng, Two-temperature model coupled with e-Beam transport, in Thermal Transport for Applications in Micro/Nanomachining, Microtechnology and MEMS (Springer, Berlin, 2008), pp. 135–145.

29. P. C. Verburg, G. R. B. E. R¨omer and A. J. Huis In ’T Veld, Appl. Phys. A 114, 1135 (2014).

30. E. Majchrzak and J. Dziatkiewicz, Sci. Res. Inst. Math. Comput. Sci. (Czest. Univ. Technol., Online) 11, 63 (2012).

31. J. K. Chen, D. Y. Tzou and J. E. Beraun, Int. J. Heat Mass Transf. 49, 307 (2006). 32. N. Singh, Int. J. Mod. Phys. B 24, 1141 (2010).

(14)

33. N. Das and N. Singh, Int. J. Mod. Phys. B 30, 1650071 (2016). 34. B. A. Ruzicka et al., Opt. Mater. Express 2, 708 (2012). 35. P. A. George et al., Nano Lett. 12, 4248 (2008). 36. N. Liaros et al., J. Phys. Chem. C 120, 4104 (2016). 37. A. Meng et al., Nano Lett. 17, 5805 (2017).

38. W. G¨otze and P. W¨olfle, Phys. Rev. B 6, 1226 (1972). 39. H. Mori, Prog. Theor. Phys. 33, 423 (1965).

40. R. Zwanzig, Phys. Rev. 124, 983 (1961). 41. R. Kubo, J. Phys. Soc. Jpn. 12, 570 (1957).

42. N. Singh, Electronic Transport Theories: From Weakly to Strongly Correlated Materials (CRC Press, 2016).

43. G. D. Mahan, Many-Particle Physics, Physics of Solids and Liquids, 3rd edn. (Kluwer Academic/Plenum Publishers, New York, 2000).

44. J. M. Ziman, Electrons and Phonons: The Theory of Transport Phenomena in Solids, (Oxford Classic Texts in the Physical Sciences, Oxford University Press, New York, 2001).

45. L. Rani and N. Singh, J. Phys.: Condens. Matter 29, 255602 (2017). 46. P. Bhalla, Phys. Lett. A 381, 924 (2017).

47. K. S. Bhargavi and S. S. Kubakaddi, Phys. Status Solidi-Rapid Res. Lett. 3, 248 (2016).