Non-equilibrium electron relaxation in Graphene
NNon-equilibrium electron relaxation in Graphene
Luxmi Rani, ∗ Pankaj Bhalla, and Navinder Singh
Theoretical Physics Division, Physical Research Laboratory, Ahmedabad-380009, India. (Dated: May 8, 2019)We apply the powerful method of memory function formalism to investigate non-equilibrium electron relaxation in graphene. Within the premises of Two Temperature Model(TTM), explicit expressions of the imaginary part of the Memory Function or generalizedDrude scattering rate (1 /τ ) are obtained. In the DC limit and in equilibrium case whereelectron temperature ( T e ) is equal to phonon temperature (T), we reproduce the knownresults (i.e. 1 /τ ∝ T when T << Θ BG and 1 /τ ∝ T when T >> Θ BG , where Θ BG is theBloch-Gr¨uneisen temperature). We report several new results for 1 /τ where T (cid:54) = T e rele-vant in pump-probe spectroscopic experiments. In the finite frequency regime we find that1 /τ ∝ ω when ω << ω BG , and for ω >> ω BG it is ω independent and also electron tem-perature independent. These results can be verified in a typical pump-probe experimentalsetting for graphene. ∗ [email protected] a r X i v : . [ c ond - m a t . s t r- e l ] J a n I. INTRODUCTION
Graphene is a unique two dimensional material consisting of a single atom thick layer of car-bon atoms that are closely packed in honeycomb lattice structure. In recent times, the study ofelectronic transport of hot carriers in graphene has created an enormous research interest in boththe experimental and theoretical aspects due to the potential applications in electronic devices[1–8]. In graphene, relaxation of hot (photoexcited) electrons has been investigated experimen-tally in [4, 5, 9–12]and theoretically in [13–23]. In simple metals, electron relaxation dynamics iswell understood and the two temperature model (TTM) is extensively used to analyze the relax-ation dynamics [24–29]. While, in graphene due to Dirac physics and peculiar band structure, hotelectron relaxation is different from that metal, and a detailed theoretical study is lacking.In simple metals, hot electron relaxation happens via electron-phonon interactions. The mecha-nism of hot electron relaxation is as follows. A Femto-second laser pulse excites the electrons fromequilibrium Fermi-Dirac (FD) distribution to a non-equilibrium distribution. This non-equilibriumelectron distribution internally relaxes via electron-electron interactions to a hot FD-distributionin a time scale τ ee . Then through electron-phonon interactions, this “hot” FD-distribution re-laxes to a state in which electron temperature becomes equal to the phonon temperature i.e., anequilibrium state. This process happens in a time scale τ e − ph . In simple metals the inequality τ ee << τ e − ph is true. And phonons remain in equilibrium during the whole process of relaxation(it is called the Bloch assumption[28]). This motivates the two temperature model (TTM): onetemperature for electron sub-system ( T e ) and another for the phonon sub-system (T). The electronrelaxation in metals is extensively studied within TTM model using the Bloch-Boltzmann kineticequation[26–29]. In the analysis an important energy scale is set by Debye temperature, and itturns out that when T >> Θ D , the relaxation rate from the Bloch-Boltzmann equation is given as1 /τ ∝ T . In the opposite limit, i.e., T << Θ D it turns out that 1 /τ ∝ T .In order to study the hot electron relaxation in graphene, several experiments like pump - probespectroscopy and photo-emission spectroscopy has been used recently [30–32]. On the theoreticalside, the hot electron relaxation has been studied in graphene using the Bloch-Boltzmann equation[13, 14, 19]. But all these studies are restricted to the DC regime.A detailed study of frequency and temperature dependent scattering rate in graphene has beenlacking in the literature. In the present investigation, we solved this problem using the powerfulmethod of memory function formalism[33–35]. We calculate the scattering rate in various frequencyand temperature limits. Our main results are ;In the DC case, scattering rate shows the fourth power law of both electron and phonon sub-system temperatures below the BG temperature. Above the BG temperature, scattering rate islinearly dependent on phonon temperature only. On the other hand, at higher frequency and athigher temperature, scattering rate is independent on frequency and electron temperature. It isobserved that there is ω -dependence in the lower frequency regime.This paper is organized as follows. In section II, we introduce the model and memory functionformalism. We then compute the memory function (generalised Drude scattering rate) using theW¨olfle-G¨otze perturbative method[33]. Then various sub-cases are studied analytically. In sectionIII, we present the numerical study of the general case. Finally, we summarize our results andpresent our conclusions. II. THEORETICAL FRAMEWORK
To study the electron relaxation in graphene, we consider total Hamiltonian having three partssuch as free electron ( H e ), free phonon ( H p ) and interacting part i.e electron-phonon ( H ep ): H = H e + H p + H ep . (1)The different parts of Hamiltonian mentioned in the above equation are defined as H e = (cid:88) k σ (cid:15) k c † k σ c k σ , (2) H p = (cid:88) q ω q (cid:18) b † q b q + 12 (cid:19) , (3) H ep = (cid:88) k , k (cid:48) ,σ (cid:104) D ( k − k (cid:48) ) c † k σ c k (cid:48) σ b k − k (cid:48) + H.c. (cid:105) . (4)Here, c † k σ ( c k σ ) and b † q ( b q ) are electron and phonon creation (annihilation) operators, σ is a spin, k and q = k − k (cid:48) are electron and phonon momentum respectively. (cid:15) k = (cid:126) v F | (cid:126)k | is the linear energydispersion term in graphene. D ( k − k (cid:48) ) is the electron-phonon matrix element which is definedas[19, 36, 37] D ( q ) = − i (cid:18) ρ m ω q (cid:19) / D × q (cid:34) − (cid:18) q k F (cid:19) (cid:35) / . (5)Here, D is the deformation potential coupling constant for graphene, ρ m is surface mass densityand k F is the Fermi momentum and ω q is the phonon energy. Here, we set (cid:126) = k B = 1 throughoutthe calculations. A. Calculation for generalized Drude scattering rate
Our aim is to calculate the generalized Drude scattering rate or imaginary part of the memoryfunction. In a typical experimental set-up, reflectivity from a graphene sample is measured atvarious frequencies; and it is written as [35, 36]: R ( ω ) = ( n ( ω ) − + ( k ( ω )) ( n ( ω ) + 1) + ( k ( ω )) , (6)Where, n ( ω ) = 1 √ (cid:114)(cid:113) (cid:15) ( ω ) + (cid:15) ( ω ) + (cid:15) ( ω ) , (7) k ( ω ) = 1 √ (cid:114)(cid:113) (cid:15) ( ω ) + (cid:15) ( ω ) − (cid:15) ( ω ) . (8) (cid:15) ( ω ) and (cid:15) ( ω ) are the real and imaginary parts of the dielectric function which are related toreal and imaginary parts of the conductivity ( σ ( ω )). Thus, from the reflectivity data, frequencydependent conductivity can be obtained [35]. From conductivity data, by Kramers-Kronig (KK)analysis, real and imaginary parts of the memory function are obtained as the conductivity can bewritten as [33]: σ ( ω ) = − i ω + M ( ω ) . (9)For the calculation of generalized Drude scattering, we use the G¨otze-W¨olfle formalism [29, 35, 38].In this formalism, memory function is expressed as M ( z, T, T e ) = zχ ( z ) χ − χ ( z ) (cid:39) zχ ( z ) χ (cid:18) χ ( z ) χ + .... (cid:19) (cid:39) zχ ( z ) χ , (10)where, χ represents the static limit of correlation function (i.e. χ = N e/m ) and χ ( z ) is theFourier transform of the current-current correlation function: χ ( z ) = i (cid:90) ∞ e izt (cid:104) [ j , j ] (cid:105) dt. (11)Here, j = Σ kσ ( (cid:126)k. ˆ n ) c † k σ c k σ is the current density. ˆ n is the unit vector along the direction of current.Using the equation of motion (EOM) method [33, 35] it can be shown that M ( z, T, T e ) = (cid:104)(cid:104) [ j , H ]; [ j , H ] (cid:105)(cid:105) z =0 − (cid:104)(cid:104) [ j , H ]; [ j , H ] (cid:105)(cid:105) z zχ . (12)Substituting equation (1) and the definition of current density operator into the above equationand on simplifying , we obtain: M ( z, T, T e ) = 1 χ (cid:88) kk (cid:48) (cid:12)(cid:12) D ( k − k (cid:48) ) (cid:12)(cid:12) [( (cid:126)k − (cid:126)k (cid:48) ) . ˆ n ] × [ f (1 − f (cid:48) )(1 + n )) − f (cid:48) (1 − f ) n ] × (cid:15) k − (cid:15) k (cid:48) − ω q ) (cid:20) (cid:15) k − (cid:15) k (cid:48) − ω q + z ) + 1( (cid:15) k − (cid:15) k (cid:48) − ω q − z ) (cid:21) . (14)Here, f = f ( (cid:15) k , β e ) and f (cid:48) = f ( (cid:15) k (cid:48) , β e ) are the Fermi-Dirac distribution functions at differentenergies such as (cid:15) k and (cid:15) k (cid:48) and, electron temperature T e = β e . n = n ( ω q , β ) is the Bose-Einsteindistribution function, T = β is the phonon temperature. z = ω + iδ and δ → + . Here weassume a steady-state situation in which electron temperature stays constant at T e , and phonontemperature also stays constant at T. This situation can be experimentally created by a continuouslaser excitation of graphene. The memory function has real and imaginary parts: M ( z, T, T e ) = M (cid:48) ( ω, T, T e ) + M (cid:48)(cid:48) ( ω, T, T e ). We are interested in the scattering rate which is the imaginary partof the memory function (i.e. M (cid:48)(cid:48) ( ω, T, T e ) = 1 /τ ( ω, T, T e )). In that case equation (14) can besimplified to 1 τ ( ω, T, T e ) = πχ (cid:88) kk (cid:48) (cid:12)(cid:12) D ( k − k (cid:48) ) (cid:12)(cid:12) [( (cid:126)k − (cid:126)k (cid:48) ) . ˆ n ] × [ f (1 − f (cid:48) )(1 + n )) − f (cid:48) (1 − f ) n ] × ω (cid:34) δ ( (cid:15) k − (cid:15) k (cid:48) − ω q + ω ) − δ ( (cid:15) k − (cid:15) k (cid:48) − ω q − ω ) (cid:35) . (15)Converting the sums over momentum indices into integrals using the linear energy dispersionrelation (cid:15) k = v F k and (cid:15) (cid:48) k = v F k (cid:48) and after further simplifying the above equation, we get,1 τ ( ω, T, T e ) = 1 τ (cid:90) q BG dq × q (cid:113) − ( q/ k f ) × (cid:40) (1 − ω q ω ) [ n ( β, ω q ) − n ( β e , ω q − ω )]+(terms with ω → − ω ) .. (cid:41) . (16)Here, 1 /τ = N D π χ ρ m k F v s and q BG being the Bloch-Gr¨uneisen momentum i.e. the maximummomentum for the phonon excitations (i.e. v s q BG = 2 k F v s = Θ BG ). In graphene, a new temper-ature crossover known as Bloch-Gr¨uneisen temperature (Θ BG ) is introduced due to small Fermi The current density operator commutes with the non-interacting parts of the Hamiltonian, the interacting partgives C = (cid:88) k,k (cid:48) [( (cid:126)k − (cid:126)k (cid:48) ) . ˆ n ][ D ( k − k (cid:48) ) c † k σ c k (cid:48) σ b k − k (cid:48) − H.c. ] . (13) surface( k F ) as compared to Debye surface( k D )[38]. Thus in this system when k F << k D , belowthe Bloch-Gr¨uneisen temperature, only small number of phonons with wave vector ( k ph < k F )can take part in scattering. Various limiting cases of equation (16) are studied in the next section. B. Limiting cases for the generalised Drude scattering rate
Case-I : DC limitWithin this limit, curly bracket in equation (16) reduces to2 lim ω → (cid:34) n ( β, ω q ) − ∞ (cid:88) m =0 ω m (cid:26) ∂ m ∂ω mq n ( β e , ω q ) + ω q ∂ m +1 ∂ω m +1 q n ( β e , ω q ) (cid:27)(cid:35) , (17)Here we consider only m=0 i.e. the leading order case,1 τ ( ω, T, T e ) = 1 τ (cid:90) q BG dq × q (cid:115) − (cid:18) q k f (cid:19) (cid:18) n ( β, ω q ) − n ( β e , ω q ) − ω q n (cid:48) ( β e , ω q ) (cid:19) . (18)Using relations ω q = v s q , ω BG (cid:39) Θ BG = 2 v s k F and defining ω q T = x , ω q T e = y , the above equationbecomes, 1 τ ( ω, T, T e ) = 1 τ v s (cid:20) T (cid:90) Θ BGT dx × x e x − (cid:115) − (cid:18) x T Θ BG (cid:19) + T e (cid:90) Θ BGTe dy × y (cid:115) − (cid:18) y T e Θ BG (cid:19) × (cid:18) y − e y − y ( e y − (cid:19)(cid:21) (19)Subcase (a): T, T e << Θ BG , i.e., when both the phonon temperature and electron temperatureare lower than the Bloch-Gr¨uneisen temperature. Equation (19) gives1 τ ( T, T e ) = 1 τ v s (cid:20) T × π
15 + T e × π τ v s (cid:20) A T + B T e (cid:21) . (20)Here A = π and B = 3 A .Subcase (b) In high temperature case, T, T e >> Θ BG , equation (19) reduces to1 τ ( T, T e ) = 1 τ v s (cid:20) T Θ BG + 16 Θ BG (cid:21) = 1 τ v s (cid:20) A T + B (cid:21) (21)Here A = Θ BG and B = Θ BG . It is notable here that the scattering rate is independent ofelectron temperature, and it only depends on the phonon temperature.Subcase (c) T >> Θ BG , T e << Θ BG . In this case scattering rate can be written as1 τ ( T, T e ) = 1 τ v s (cid:20) T Θ BG + T e (cid:18) π (cid:19) (cid:21) = 1 τ v s (cid:20) A T + B T e (cid:21) (22)Here A = Θ BG and B = π . In this case 1 /τ leads to the linear phonon temperature depen-dence in high temperature regime and shows the T e - dependence below the BG temperature.Subcase (d) T << Θ BG , T e >> Θ BG . τ ( T,T e ) has T - dependence. Scattering rate is independentof the electron temperature. On the other hand, when T = T e , the result of scattering rate isidentical as obtained in an equilibrium electron-phonon interaction in graphene case [19, 38] asexpected. These results are tabulated in Table I. Case-II : Finite frequency regimesSubcase (1): Consider ω >> ω BG , then equation (16) becomes1 τ ( ω, T, T e ) = 1 τ (cid:90) q BG dq × q (cid:113) − ( q/ k f ) × (cid:40) n ( β, ω q ) − n ( β e , − ω ) − n ( β e , ω ) (cid:41) . (23)This can be simplified by setting ω q T = x , ωT e = ξ , then we have1 τ ( ω, T, T e ) = 1 τ v s T (cid:90) Θ BGT dx × x (cid:115) − (cid:18) x T Θ BG (cid:19)(cid:18) e x − − e − ξ − − e ξ − (cid:19) . (24)After simplifying the above equation, it is observed that there is only the phonon contribution athigher frequency. To further simplify the above equation, we study the following subcases:In the low temperature regime T << Θ BG , equation (24) becomes1 τ ( ω, T ) = 1 τ v s T (cid:20) π −
14 Θ BG T (cid:21) = 1 τ v s (cid:20) A T + B (cid:21) (25)Here, A = π and B = − Θ BG .In the high temperature regime T >> Θ BG , equation (24) takes the following form1 τ ( ω, T ) = 1 τ v s (cid:20)
715 Θ BG T −
14 Θ BG T (cid:21) = 1 τ v s (cid:20) A T + B (cid:21) (26)Here A = Θ BG . It is also noticeable here that in both the cases τ ( ω,T ) shows the frequencyindependent behavior. At T → τ ( ω ) shows saturation.Subcase (2): At finite but lower frequency ω << ω BG case, with relation ω q = v s q the equation(16) becomes 1 τ ( ω, T, T e ) = 1 τ (cid:90) Θ BG dq × ω q (cid:114) − ( ω q Θ BG ) (cid:20) e wqT − − ∞ (cid:88) m =0 ω m (cid:32) ∂ m ∂ω mq e wqTe − ω q ∂ m +1 ∂ω m +1 q e wqTe − (cid:33) (cid:21) (27)This is the general equation of the imaginary part of memory function when frequency is lowerthan the Bloch-Gr¨uneisen frequency. The above equation can be further simplified by setting thevariables ω q T = x , ω q T e = y , and for m=1, the equation (27) reduces1 τ ( ω, T, T e ) = 1 τ v s (cid:20) T (cid:90) Θ BGT dx × x (cid:115) − (cid:18) x T Θ BG (cid:19) e x − ω T e (cid:90) Θ BGTe dy × y (cid:115) − (cid:18) y T e Θ BG (cid:19) × (cid:0) n y + 3 n y + 2 n y − y (cid:0) n y − n y − n y − n y (cid:1)(cid:1) (cid:21) (28)Here, n y = e y − . Further we study the frequency dependent scattering rate at low and hightemperature regimes of both electron and phonon sub-systems. We consider first two terms (m=0and m=1) in the series of the equation (27). The analytic results obtained in the present subcase( ω << ω BG ) are presented in Table I. It is observed that there is ω -dependence multiplied by theelectron temperature in the lower frequency regime. In the general case, numerical computationsof equation (16) is presented in the next section. And in the appropriate limiting cases, numericalresults agree with analytical results presented in Table I. III. NUMERICAL ANALYSIS
We have numerically computed the equation (16) in different frequency and temperatureregimes. In Fig.1(a), we depict the phonon temperature dependence of scattering rate 1 /τ ( T, T e ) No Regimes τ ω = 0; T e , T << Θ BG A T + B T e .ω = 0; T e , T >> Θ BG A T + B ω = 0; T >> Θ BG , T e << Θ BG A T + B T e . ω = 0; T << Θ BG , T e >> Θ BG A T + constant.2 ω >> ω BG ; T << Θ BG A T . ω >> ω BG ; T >> Θ BG A T .3 ω << ω BG ; T, T e >> Θ BG A T + B T e + C ω T e . ω << ω BG ; T, T e << Θ BG A T + B T e + C ω T e . ω << ω BG ; T >> Θ BG , T e << Θ BG A T + B T e + C ω T e . ω << ω BG ; T << Θ BG , T e >> Θ BG A T + B T e + C ω T e . TABLE I: The results of electrical scattering rate due to the electron-phonon interactions indifferent limiting cases. Here, A = Θ BG , B = − Θ BG , C = Θ BG , and A = π , B = π , C = 5 π − ζ (5) and A = Θ BG , B = π , C = C = constant , and A = π , B = − Θ BG and C = Θ BG .normalized by 1 /τ (= N D π χ ρ m k F v s ) at zero frequency and at different electron temperatures.From Fig.1(a), we observe that at high temperatures ( T e , T >> Θ BG ), 1 /τ ( T, T e ) ∝ T . Thiscan also be seen in the corresponding case ( T e , T >> Θ BG ) in Table I. At very low temperature( T, T e << Θ BG ), 1 /τ ∝ T and T e . Fig.1(b) shows the dependence of 1 /τ on T e in the DC limit. Itis observed that 1 /τ is independent of T e when T e >> Θ BG . Contour plots (Fig.1(c) and Fig.1(d))depict the constant value of 1 /τ in T e and T plane. The contour for higher values of T and T e arefor higher 1 /τ .From the contour plots, we notice that they are not symmetric around T = T e line. The physicalreason for this asymmetry is that the scattering rate is differently effected by phonon temperatureand electron temperature (the pre-factor A of T term is not equal to the prefactor B of T e terms). At very low temperature T behavior is due to Pauli blocking effect. We notice that athigh temperature ( T e , T >> Θ BG ), 1 /τ ( T, T e ) is proportional to T , not T e . The reason for thisbehavior is that at high temperatures phonon modes scale as k B T ( < n q > = e βωq − ∝ k B T ), thusscattering increases with increasing temperatures linearly. For T e >> Θ BG the electron distributioncan be approximated as Boltzmann distributions because Θ BG (cid:39) T F (the Fermi temperature). Thetemperature effect is exponentially reduced in this case as compared to phonons ( < n q > ∝ T ).Thus at high temperatures, the scattering rate is proportional to T.0 T e (cid:61)(cid:81) BG T e (cid:61) (cid:81) BG T e (cid:61) (cid:81) BG T (cid:144) (cid:81) BG Τ (cid:144) Τ (cid:72) T , T e (cid:76) (a) T (cid:61)(cid:81) BG T (cid:61) (cid:81) BG T (cid:61) (cid:81) BG T e (cid:144) (cid:81) BG Τ (cid:144) Τ (cid:72) T , T e (cid:76) (b) T (cid:144) (cid:81) BG T e (cid:144) (cid:81) B G (c) T (cid:144) (cid:81) BG T e (cid:144) (cid:81) B G (d) FIG. 1: (a) Variation of the scattering rate with phonon temperature at zero frequency anddifferent electron temperatures. (b) Variation of the scattering rate with electron temperature atzero frequency and different phonon temperatures. Here both the electron and phonontemperatures are scaled with the Bloch-Gr¨uneisen temperature and 1 /τ ( T, T e ) is scaled with1 /τ . Figures (c) and (d) depict contour plots T vs T e for the scattering rate at zero frequency.In Fig.2(a), we plot the phonon temperature dependency of scattering rate τ /τ ( ω, T, T e ) atlower frequency and at different temperatures of electrons. It is observed that at lower phonontemperature range, the magnitude of scattering rate increases with increasing temperature as T behavior. At higher T it shows T-linear behavior. In Fig.2(b), the variation of electron temperature1dependence of τ /τ ( ω, T, T e ) at different phonon temperature scaled with BG temperature is shown.The insets of both the figures show low temperature behavior ( T, T e << Θ BG ). The low frequencybehavior is similar to the DC case. Ω (cid:144) Ω BG (cid:61) T e (cid:61) (cid:81) BG T e (cid:61)(cid:81) BG T e (cid:61) (cid:81) BG T (cid:144) (cid:81) BG Τ (cid:144) Τ (cid:72) Ω , T , T e (cid:76) Τ (cid:144) Τ (cid:72) Ω , T , T e (cid:76) (a) Ω (cid:144) Ω BG (cid:61) T (cid:61) (cid:81) BG T (cid:61)(cid:81) BG T (cid:61) (cid:81) BG T e (cid:144) (cid:81) BG Τ (cid:144) Τ (cid:72) Ω , T , T e (cid:76) Τ (cid:144) Τ (cid:72) Ω , T , T e (cid:76) (b) FIG. 2: (a)Variation of the scattering rate with phonon temperature at finite but lower frequencyand at different electron temperatures, and inset shows the lower phonon temperature range.(b)Variation of the scattering rate with electron temperature at finite frequency and at differentphonon temperatures, and inset shows the lower electron temperature range.In order to study the higher frequency regime, we plot the variation of the scattering rate withphonon temperature at higher frequency ( ω/ω BG = 100) and at different electron temperaturesin Fig.3(a). It is observed that at higher frequency, scattering rate is independent of the electrontemperature (compare with the corresponding entry given in Table I). Plot shows the T-linearbehavior above BG temperature and T behavior below lower BG temperature. These resultsagree with the result of [20, 21]. At higher frequency, the scattering rate is controlled by phonontemperature. The independence of 1 /τ from T e is also shown in the contour plot (Fig.3(b)).We further analyzed the scattering rate at zero temperature in which both electron subsystemand phonon subsystem are at zero temperature. In this regime 1 /τ scales as ω as depicted inFig.4.To order to study the scattering rate with frequency, we plot the frequency dependence behavior2 Ω (cid:144) Ω BG (cid:61) T e (cid:61) (cid:81) BG T e (cid:61)(cid:81) BG T e (cid:61) (cid:81) BG T (cid:144) (cid:81) BG Τ (cid:144) Τ (cid:72) Ω , T , T e (cid:76) Τ (cid:144) Τ (cid:72) Ω , T , T e (cid:76) (a) T (cid:144) (cid:81) BG T e (cid:144) (cid:81) B G (b) FIG. 3: (a)Variation of the scattering rate with phonon temperature at higher frequency( ω/ω BG = 100) and at different electron temperatures, and inset shows the same plot with lowerphonon temperature range over Θ BG . The inset also shows finite scattering rate even at zerophonon temperature which is nothing but the non-equilibrium generalization of the Holsteinmechanism [29, 35, 38]. (b) The contour plot depicts the behavior of scattering rate at higherfrequency ( ω/ω BG = 100). T e (cid:144) (cid:81) BG (cid:126) (cid:144) (cid:81) BG (cid:126) Ω (cid:144) Ω BG Τ (cid:144) Τ (cid:72) Ω (cid:76) Τ (cid:144) Τ (cid:72) Ω (cid:76) FIG. 4: (a)Variation of the scattering rate with frequency at zero electron and phonontemperatures, and inset shows the same plot at lower frequency over Bloch-Gr¨uneisen frequency.of the scattering rate 1 /τ ( ω, T, T e ) at different temperatures of electron and phonon subsystemsin Fig.5. Fig.5(a) depicts the variation of scattering rate with frequency at different phonon3 T e (cid:61) (cid:81) BG (cid:72) a (cid:76) T (cid:61) (cid:81) BG T (cid:61) (cid:81) BG T (cid:61) (cid:81) BG Ω (cid:144) Ω BG Τ (cid:144) Τ (cid:72) Ω , T , T e (cid:76) T (cid:61) (cid:81) BG (cid:72) b (cid:76) T e (cid:61) (cid:81) BG T e (cid:61) (cid:81) BG T e (cid:61) (cid:81) BG Ω (cid:144) Ω BG Τ (cid:144) Τ (cid:72) Ω , T , T e (cid:76) FIG. 5: Variation of the scattering rate with frequency at different electron and phonontemperatures.temperatures and at fixed electron temperature. At higher frequency, 1 /τ saturates and at lowerfrequency it shows ω behavior.In Fig.5(b), we plot the variation of scattering rate with frequency at different electron tem-peratures and fixed phonon temperature. From Fig.5(b), it is clear that on increasing the electrontemperature, scattering rate increases in lower frequency regime but scattering rate goes into satu-ration trend in the high frequency regimes, and become independent of electron temperature. Thiscan also be obtained from Table I (in the ω >> ω BG case). IV. CONCLUSION AND DISCUSSION
We presented a theoretical study of non-equilibrium relaxation of electrons due to their cou-pling with phonons in graphene by using the memory function approach. In our results at zerofrequency limit, it is observed that if both the electron and phonon temperature are not same, DCscattering rate has a fourth power law behavior of both the electron and phonon temperaures i.e.( A T + B T e ) below the BG temperature. While at higher temperature, 1 /τ shows the T-lineardependency only (it does not depend on T e ). Further, it is important to notice here that DCscattering rate and AC scattering rate shows the similar T-linear behavior at higher temperature.In Table II, we compare the results of scattering rates for the simple metals and the present case4 No Regimes Graphene (cid:18) τ (cid:19) Metals (cid:18) τ (cid:19) [29]2D 3DBloch Gr¨unisen DebyeTemperature (Θ BG ) Temperature (Θ D )1 ω = 0; T e , T << Θ BG , Θ D A T + B T e . a T + b T e .ω = 0; T e , T >> Θ BG , Θ D A T . a + b Tω = 0; T >> Θ BG , T e << Θ BG A T + B T e . - ω = 0; T << Θ BG , T e >> Θ BG A T . -2 ω >> ω BG , ω D ; T >> Θ BG , Θ D A T . a + b T . ω >> ω BG , ω D ; T << Θ BG , Θ D A T . a + b T ω << ω BG , ω D ; T, T e >> Θ BG , Θ D A T + B T e + C ω T e . a T + b ω T e ω << ω BG , ω D ; T, T e << Θ BG , Θ D A T + B T e + C ω T e . a T + b T e + c T e ω ω << ω BG ; T >> Θ BG , T e << Θ BG A T + B T e + C ω T e . - ω << ω BG ; T << Θ BG , T e >> Θ BG A T + B T e + C ω T e . - TABLE II: Comparison of non-equilibrium electron relaxation in metals and in grapheneof graphene. We observed that T -law of 1 /τ in the case of metals (in regimes ω = 0 , T << Θ D )changes to T -law in the corresponding case in graphene. However, in the case of high temperaturesand high frequencies, temperature dependence of 1 /τ in both metals and in graphene remains thesame.At higher frequency, the scattering rate is controlled by phonon temperature in both the cases(of metals and graphene). In the low frequency case ( ω << ω D ) and in lower temperature regimes( T, T e << Θ D ) 1 /τ in metals has three terms ( a T + b T e + c T e ω ) whereas in the correspondingcase of graphene this dependence changes to ( A T + B T e + C ω T e ). These results can be verifiedthat in a typical pump-probe experiments [8, 25, 32]. [1] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson, I. V. Grigorieva, S. V. Dubonosand A. A. Firsov (2005) Two-dimensional gas of massless Dirac fermions in graphene Nature
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