A BGK model for charge transport in graphene. Armando Majorana Department of Mathematics and Computer Science, University of Catania, Italy

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1 A BGK model for charge transport in graphene Armando Majorana Department of Mathematics and Computer Science, University of Catania, Italy December 6, 28 arxiv:82.862v math-ph 5 Dec 28 Abstract The Boltzmann euation describes the detailed microscopic behaviour of a dilute gas, and represents the basis of the kinetic theory of gases. In order to reduce the difficulties in solving the Boltzmann euation, simple expressions of a collision operator have been proposed to replace the true Boltzmann integral term. These new euations are called kinetic models. The most popular and widely used kinetic model is the Bhatnagar-Gross-Krook BGK model. In this work we propose and analyse a BGK model for charge transport in graphene. MSC-class: 82C4-82C7 Primary 82D37 Secondary Introduction Graphene is a gapless semiconductor made of a sheet composed of a single layer of carbon atoms arranged into a honeycomb hexagonal lattice. In view of application in graphene-based electron devices, it is crucial to understand the basic transport properties of this material. In a semiclassical kinetic setting, the charge transport in graphene is described by four Boltzmann euations, one for electrons in the valence π band and one for electrons in the conductions π band, that in turn can belong to the K or K valley. In this paper we study the case of a single distribution function f for electrons belonging to a conduction band. This corresponds to a physical case, where a n-type doping or euivalently a high value of the Fermi potential is considered, and the electrons, belonging to a conduction band, do not move to the valence band. Moreover K and K are considered euivalent. Under these assumptions Boltzmann euation writes f t + kε x f e E kf = df, dt coll where t is the time, x and k are the position and the wave-vector of a charge particle, respectively. We denote by x and k the gradients with respect to the position and wave-vector, respectively. With a very good approximation the energy bands is given by εk = v F k, where v F is the constant Fermi velocity, and is the Planck constant divided by 2π. The elementary positive charge is denoted by e, and E is the electric field. The collision operator of Euation describes the interaction of electrons with acoustic, optical and K phonons. If the electric field E is not constant, then it must be self-consistently evaluated by coupling the Boltzmann euation with the Poisson euation for the electrostatic potential.

2 More recently accurate numerical solutions, based on the discontinuous Galerkin method, to the Boltzmann euation have been shown 2, 3. A comparison with results, obtained by using a direct simulation Monte Carlo approach, shows an excellent agreement, which gives a further validation of the numerical scheme. In this work the collision operator of Euation is replaced by a relaxation BGK collision operator. We propose an operator, which retains the fundamental properties of the Boltzmann euation, such the mass conservation, the same euilibrium distribution functions and it properly deals with Pauli s exclusion principle in the degenerate case. 2 The BGK model The collision operator of Euation writes 4 Sk,kft,x,k ft,x,kdk Sk,k ft,x,k ft,x,k dk, 2 where the transition rate Sk,k, related to electron-phonon scatterings, is described in detail in the following. The collision operator 2 vanishes if f is the Fermi-Dirac distribution Fk,µ =, εk µ +exp where k B is the Boltzmann constant, T the constant graphene lattice temperature, and µ is the chemical potential. If the electric field E is null, then Fermi-Dirac distributions are solutions, which do not depend on time t and space coordinates x, of the Boltzmann euation. In this paper we propose the following BGK collision operator Qft,x,k = Sk,kFk,µ ft,x,kdk Sk,k ft,x,k Fk,µ dk. 3 It is derived from Euation 2, replacing the distribution ft,x,k, inside the integrals, with a Fermi-Dirac distribution. Now the chemical potential µ becomes a new unknown. Therefore we must add an euation to the kinetic model. It is Qft,x,kdk =. 4 This euation guarantees the mass conservation, exactly as the same for the Boltzmann euation. If we define Φ k,µ = Sk,kFk,µdk and Φ k,µ = Sk,k Fk,µ dk, 5 then 3 becomes Qft,x,k = Φ k,µ ft,x,k Φ k,µft,x,k. 6 2

3 It is evident that, for every k and µ, Φ k,µ Fk,µ Φ k,µfk,µ = Sk,kFk,µ Fk,µdk Sk,k Fk,µ Fk,µ dk =. Using this identity, we have Qft,x,k = Φ k,µ ft,x,k Φ k,µ Fk,µ ft,x,k Fk, µ = Φ k,µ Fk,µ {Fk,µ ft,x,k Fk,µft,x,k} = Φ k,µ Fk,µ By defining the collision freuency of the BGK model κk,µ = Φ k,µ Fk,µ = Φ k,µ +exp the kinetic model writes Fk,µ ft,x,k. 7 εk µ, 8 f t t,x,k+ kε x ft,x,k e E kft,x,k = κk,µfk,µ ft,x,k, 9 κk,µfk,µ ft,x,kdk =. A solution of system 9- consists of the two functions ft,x,k and µt,x, which must satisfy suitable regularity conditions. The common definitions of the kernel S guarantees that the function κk,µ is a no negative continuous function in the set R. 2. The Pauli exclusion principle The distribution function f must be not negative and less or eual to one, according to the Pauli s exclusion principle. We prove that, if the electric field is null, then every spatial homogeneous solution of Euations 9- satisfies these conditions. Theorem Let be ft,k a function defined in,+, differentiable with respect to t for every k, and integrable, with respect to k, over the domain for each time t. Moreover the function µt is continuous in,+. If f and µ satisfy the euations f t,k = κk,µfk,µ ft,k and t κk,µfk,µ ft,kdk =, and f,k, for all k, then ft,k for t,k,+. Proof. Let be Therefore we have expkk,µ f t Kk, µ such that K t = κk,µ t. = expkk,µκk,µfk,µ expkk,µκk,µft,k 3

4 that is and then expkk,µft,k = expkk,µκk,µfk,µ. t expkk,µft,k expkk,µft,k. t= Hence the distribution function ft,k is always not negative, because f,k for every k. Since f,k for every k, we define gt,k = ft,k, and we note that g,k. It is evident that g satisfies the euation g t = κk,µ Fk,µ g and then, as before, taking into account that Fk,µ <, from expkk,µgt,k expkk, µ gt, k, t= it follows gt,k ft,k. 2.2 The kernel of the Boltzmann collision operator The kernel Sk,k is given by the sum of terms of the kind G ν k,k 2 n ν + δ εk εk+ ω ν +n ν δ εk εk ω ν. The index ν labels the νth phonon mode, G ν k,k is the scattering rate, which describes the scattering mechanism between phonons ν and electrons. The symbol δ denotes the Dirac distribution function, ω ν the νth phonon freuency, and n ν is the Bose-Einstein distribution for the phonon of type ν n ν =. e ων / For acoustic phonons, usually one considers the elastic approximation, and 2n ac G ac k,k 2 = πdack 2 B T 2π 2 2 σ m vp 2 +cosϑk,k, where D ac is the acoustic phonon coupling constant, v p is the sound speed in graphene, σ m the graphene areal density, and ϑ k,k is the convex angle between k and k. There are three relevant optical phonon scatterings: the longitudinal optical LO, the transversal optical TO and the K K phonons. The scattering rates are G LO k,k 2 = G TO k,k 2 = G K k,k 2 = 2π 2 πd 2 O σ m ω O cosϑk,k k +ϑ k,k k πd 2 O +cosϑk,k 2π 2 σ m ω k +ϑ k,k k O 2πD 2 K cosϑk,k 2π 2, σ m ω K where D O is the optical phonon coupling constant, ω O the optical phonon freuency, D K is the K- phonon coupling constant and ω K the K-phonon freuency. The angles ϑ k,k k and ϑ k,k k denote 4

5 the convex angles between k and k k and between k and k k, respectively. We used the same physical parameters of the Table of Ref. 3. Since the phonon freuency of longitudinal and transversal optical phonons coincide, then, as we sum the corresponding terms, the function cosϑ k,k k + ϑ k,k k can be eliminated, easily. Therefore, in this case, S is a sum of terms where now G ν k,k 2 = G ν cosϑ k,k. 2.3 The collision operator of the BGK model Taking into account Euation 9 and the definition 8, the collision operator of the BGK model writes { } εk µ Φ k,µ ft,x,k +exp. 2 This expression can be simplified. To this aim, we consider the function Φ k,µ, which is given by the sum of the following integrals n ν + δ +n ν δ G ν cosϑ k,k εk εk ω ν n ν + = εk+ ω ν G ν cosϑ k,k δ µ R +exp 2 n ν + G ν cosϑ k,k δ εk ω ν µ R + 2 +exp εk εk+ ω ν εk εk ω ν dk εk εk+ ω ν dk, Fk,µdk where z + = max{z,}. Introducing polar coordinates k = rcosϑ,sinϑ and k = r cosϑ,sinϑ, we have G ν cosϑ k,k δ = = = + 2π 2π dr 2π εk εk ω ν dϑ G ν cosϑ ϑδ G ν cosϑ ϑdϑ δ v F R G ν cosϑ dϑ r + ων v F v F dk where H is the Heaviside function. Analogously G ν cosϑ k,k δ v F r v F r ω ν r r ων H εk εk+ ω ν v F r+ ων v F r r Hr dr = 2π 2π dk = G ν cosϑ dϑ ν εk+ ω G ν cosϑ dϑ v F 2, εk ω ν + v F 2. 5

6 Hence Φ k,µ = ν v F 2 2π n ν + +exp G ν cosϑ dϑ εk+ ω ν εk+ ω ν n ν + µ +exp εk ω ν εk ω ν + µ. + In order to simplify the notation, we define So Φ k,µ = ν ψk,ξ;a,b = a+εk+b aεk b εk+b εk b+ +ξexp +ξexp 2π µ v F 2 G ν cosϑ dϑ ψ k,exp ;n ν, ω ν. 4 Then the BGK collision operator 2 is a linear combination, with constant positive coefficients, of the terms { } µ ψ k,exp ;n ν, ω ν εk µ ft,x,k +exp. 5 Since n ν and ω ν, then we can consider the function ψ, defined in 3, only for a, b, and ξ >. Lemma If ft,x,k, then the collision operator 2 is a strictly increasing function of µ on R. Proof. Since Euation 2 is a linear combination, with constant positive coefficients, of the functions 5, it is sufficient to prove that every function 5 is strictly increasing on R with respect to the variable µ. Again to simplify the notation, we consider the general application λk,ξ;a,b = ψk,ξ;a,b { ϕk +ξexp εk }, 6 where ϕk for all k. Euation 6 gives the expression 5, choosing the parameters a, b, ξ appropriately, and ϕk = ft,x,k, for fixed t and x. If we define εk b wk = exp and β = exp, then ψk,ξ;a,b = a+εk+b +ξβwk + aεk b + +ξβ wk, 6

7 where we have taken into account the identity aεk b + εk b+ +ξexp aεk b +. εk b +ξexp Hence we can write a+εk+b λk,ξ;a,b = + aεk b + +ξβwk +ξβ wk β ϕk+ ϕk = a+εk+b +ξβwk βϕk+ ϕk +aεk b + +ξβ wk ϕk ξwkϕk β ϕk βϕk which is strictly decreasing with respect to the positive variable ξ, because ϕk. µ Now the result follows for ξ = exp. It is useful, for the following, to establish some ineualities. From β β ϕk+ ϕk and βϕk+ ϕk β, we obtain β λk,ξ;a,b a+εk+b +ξβwk β ϕk +aεk b + +ξβ wk βϕk β a+εk+b ξβwk β ϕk +aεk b + ξβ wk βϕk, 7 and β λk,ξ;a,b a+εk+b +ξβwk β ϕk +aεk b + +ξβ wk βϕk a+εk+b a+εk+b β +βaεk b +ξβwk β + ϕk. 8 We remark that the variable ξ has not been involved in ineualities. 3 The mass conservation Euation, that guarantees the conservation of mass, can be written as { } εk µ Φ k,µ ft,x,k +exp dk =, i.e. ν 2π G ν cosϑ dϑ µ λ k,exp ;n ν, ω ν dk =. 9 ϕk=ft,x,k, 7

8 Theorem 2 If ft,x,k and εkft,x,k is integrable with respect to k over, for all t and x, then there exists a uniue µ satisfying Euation 9. Proof. If there exists a solution µ of Euation 9, then it must be uniue due to the Lemma. To prove the theorem, we show that every integrals of Euation 9 is negative for µ, and positive for µ +. We do not consider the meaningless case ft,x,k = almost everywhere. Taking into account the ineuality 7, we have λk,ξ;a,b ϕk=ft,x,k dk { a+εk+b ξβwk β ft,x,k +aεk b + = a+εk+b + aεk b + β ξ βwk β dk wk a+εk+bβ +aεk b + β ft,x,kdk, β ξβ wk βft,x,k where the last integral is finite due to hypotheses of the theorem. If µ, then ξ +, and lim λk,ξ;a,b µ ϕk=ft,x,k dk R 2 = a+εk+bβ +aεk b + β ft,x,kdk <. This implies that the l.h.s. of Euation 9 is negative for µ. Now we consider the ineuality 8. We have λk,ξ;a,b ϕk=ft,x,k dk β a+ εk+b +ξβwk dk where both integrals exist. Since, for µ >, we have εk+b + v F r +b dk = 2π r dr +ξβwk vf r µ +β exp = 2π + s+b v F 2 sds 2π µ v F 2 Therefore 2π µ v F 2 +β exp s µ s+b +β sds = 2π v F 2 +β } dk a+εk+b +βaεk b β + ft,x,kdk, 3 µ3 + 2 bµ2 lim λk,ξ;a,b µ + ϕk=ft,x,k dk = +. s+b sds s µ +β exp. This implies that the l.h.s. of Euation 9 is positive for µ +, and it concludes the proof. This theorem establishes that the kinetic model 9- is correctly-set. 8

9 4 Conclusions In this paper we propose a BGK model for charge transport in graphene. The collision operator of this model replaces the true non linear Boltzmann integral operator. A further euation, for a new unknown, is added to the kinetic euation in order to guarantee the mass conservation. The model would furnish good results when the electric field is not strong, so that the distribution function remains near an euilibrium Fermi-Dirac distribution. Moreover the simple collision term allows analytical investigations, which may be prohibitive for the Boltzmann euation. A numerical scheme, based on a discontinuous Galerkin method, for finding approximate solutions to the model, does not seem more simple than the scheme used for solving the Boltzmann euation in Ref. 5, due to the non-linearity of the integral euation for the mass conservation. Acknowledgements The author acknowledges the financial support provided by the project Modellistica, simulazione e ottimizzazione del trasporto di cariche in strutture a bassa dimensionalit, University of Catania. References A. H. C. Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim, The electronic properties of graphene, Reviews of Modern Physics, vol. 8, no., pp. 9 62, A. Majorana, G. Mascali, and V. Romano, Charge transport and mobility in monolayer graphene, Journal of Mathematics in Industry, vol. 7, no. 4, V. Romano, A. Majorana, and M. Coco, DSMC method consistent with the Pauli exclusion principle and comparison with deterministic solutions for charge transport in graphene, Journal of Computational Physics, vol. 32, pp , A. Tomadin, D. Brida, G. Cerullo, A. C. Ferrari, and M. Polini, Noneuilibrium dynamics of photoexcited electrons in graphene: Collinear scattering, auger processes, and the impact of screening, Physical Review B, vol. 88, no. 3543, pp. 8, M. Coco, A. Majorana, and V. Romano, Cross validation of discontinuous Galerkin method and Monte Carlo simulations of charge transport in graphene on substrate, Ricerche di Matematica, vol. 66, no., pp. 2 22, 27. 9