PHYSICAL REVIEW B 86, 125443 (2012)
Electron-phonon mediated heat flow in disordered graphene
Wei Chen and Aashish A. Clerk Department of Physics, McGill University, Montreal, Canada H3A 2T8 (Received 11 July 2012; published 26 September 2012) We calculate the heat flux and electron-phonon thermal conductance in a disordered graphene sheet, going beyond a Fermi’s golden rule approach to fully account for the modification of the electron-phonon interaction by disorder. Using the Keldysh technique combined with standard impurity averaging methods in the regime
kF l 1 (where kF is the Fermi wave vector and l is the mean free path), we consider both scalar potential (i.e., deformation potential) and vector-potential couplings between electrons and phonons. We also consider the effects of electronic screening at the Thomas-Fermi level. We find that the temperature dependence of the heat flux and thermal conductance is sensitive to the presence of disorder and screening, and reflects the underlying chiral nature of electrons in graphene and the corresponding modification of their diffusive behavior. In the case of weak screening, disorder enhances the low-temperature heat flux over the clean system (changing the associated power law from T 4 to T 3), and the deformation potential dominates. For strong screening, both the deformation potential and vector-potential couplings make comparable contributions, and the low-temperature heat flux obeys a T 5 power law.
DOI: 10.1103/PhysRevB.86.125443 PACS number(s): 65.80.Ck, 72.10.Di, 44.10.+i
I. INTRODUCTION In this work, we now ask how the above result is modified in the presence of electronic disorder. While great experimental The potential to exploit the exceptional thermal properties progress has been made in reducing disorder effects in of graphene in applications has recently generated con- graphene,12 the devices studied for bolometric applications in siderable activity:1–4 possible applications include sensitive Refs. 2–4 are sitting on a silicon substrate and have mean free bolometry and calorimetry for detecting infrared and terahertz paths that are 100 nm or less. In conventional metals, electronic radiation. Such detectors would ultimately be based on the sim- disorder can strongly affect the electron-phonon coupling at ple heating of electrons in a graphene sheet by the absorption temperatures low enough that the wavelength of a thermal of incident photons. Ideally, the detector electrons would be phonon is comparable to (or longer than) the electronic mean thermally decoupled from their surroundings, thus allowing free path.13–16 This defines a characteristic temperature scale any heating produced by the incident radiation to be long T below which disorder effects are important, lived. In this respect, graphene provides a potential advantage: dis Its low electron density and relatively weak electron-phonon kBTdis ≡ hs/l, (2) coupling implies that the expected low-temperature thermal decoupling between electrons and the lattice5,6 could occur where s is the speed of sound and l is the electronic mean free over a much wider temperature range than in a conventional path. As discussed in Refs. 13–16 (and below), the effects of metal.7,8 Further, one can effectively suppress the thermal link disorder are subtle: Depending on the nature of the disorder between graphene electrons and electrons in the contacts by and the electronic system, the power law δ in Eq. (1) can either 13–16 employing superconducting leads.9 be enhanced by disorder or be suppressed. Given the above, it is crucial to develop a rigorous and Here, we study how the additional complexity arising quantitative understanding of the thermal link between elec- from the unique electronic properties of graphene modify trons and phonons in graphene at low temperatures. Several the interplay of disorder and the electron-phonon interaction. recent theoretical works have addressed this problem in the The principle new ingredients arise from the effective chiral case of clean graphene (i.e., no electronic disorder).7,8,10,11 At nature of carriers in graphene, which both modifies electronic low temperatures, the heat flux (per volume) between electrons diffusion, and allows for a new kind of effective vector- 17–19 and longitudinal acoustic phonons takes the general form potential electron-phonon coupling. For simplicity, we will focus on disorder originating from charges in the substrate below the graphene flake, and thus take the disorder potential = − = δ − δ P (Te,Tph) F (Te) F (Tph) Te Tph , (1) to preserve the symmetries (valley and sublattice) of the low-energy graphene Hamiltonian.20,21 The impurity potential where F (T ) is called the energy control function, is a is thus also taken to be static, i.e., it does not move with coupling constant, and Te and Tph are the temperatures of the graphene sheet. We also consider the case where the the electrons and lattice (i.e., phonons), respectively. Previous graphene flake has been doped sufficiently that kF l 1(kF is theoretical works7,8,10,11 find that δ = 4 in the low-temperature the Fermi wave vector), meaning that the standard impurity- limit (assuming an unscreened deformation potential electron- averaged perturbation theory is appropriate.22,23 Combining phonon coupling); this has been confirmed in recent experi- this approach with the Keldysh technique then allows us ments on graphene in the clean limit.2–4 This power law is also to rigorously address the electron-phonon interaction in the identical to what would be expected for a clean conventional presence of disorder, in a manner analogous to the classic two-dimensional (2D) metal.11 works looking at this physics in a conventional disordered
1098-0121/2012/86(12)/125443(14)125443-1 ©2012 American Physical Society WEI CHEN AND AASHISH A. CLERK PHYSICAL REVIEW B 86, 125443 (2012) metal.13–16 We stress that properly addressing disorder effects Note that our results suggest that even though the bare involves going beyond the sort of golden rule calculation used VP coupling strength g2 is believed to be about an order of 11 18 to address the clean case. magnitude smaller than the bare DP coupling g1, if screening Note that since we work in the regime kF l 1, the tem- is strong, its contribution to the heat flux could be comparable perature scale Tdis below which disorder effects emerge will to or even larger than that from the DP coupling. This is necessarily be well below the Bloch-Gruneisen¨ temperature despite the relative enhancement of the DP coupling over the kBTBG = 2¯hskF ; we will thus explicitly focus on temperatures VP coupling by disorder. Further, our results suggest that the T 125443-2 ELECTRON-PHONON MEDIATED HEAT FLOW IN ... PHYSICAL REVIEW B 86, 125443 (2012) TABLE I. Energy control function [cf. Eq. (1)] of deformation potential and vector potential e-phonon couplings in graphene below the Bloch-Gruneisen¨ temperature TBG = 2¯hskF /kB. g1 and g2 are the deformation potential and vector-potential coupling constant, respectively, [cf. Eqs. (6)]. EF is the Fermi energy with respect to the Dirac point, vF is the Fermi velocity, l is the electronic mean free path, and ρM is the mass density of graphene per area. qTF is the Thomas-Fermi wave vector (as defined in Ref. 24), and ζ (n)istheζ function. Deformation potential T Weak screening Strong screening Weak screening Strong screening T 3 2 3 2 5 2 2 4 4 2 6 2 5 2 2 4 vF ρM 2ζ (3) g1 kB 3 24ζ (5) g1 kB 5 π g1 kB 4 8π g1 kB 6 30ζ (5) g2 lkB 5 π g2 kB 4 F (T ) E π2 4 2 T π2 6 4 2 T 15 5 3 T 63 7 5 2 T π2 6 4 T 15 5 3 T F h¯ ls h¯ s qTFl h¯ s h¯ s qTF h¯ s h¯ s 2. Electron-phonon interaction same for the LA and TA phonons and drop the superscript The electron-phonon interaction in graphene has been η in the frequency from now on. g1 (g2) is the deformation- studied extensively in Res. 24 and 26. For suspended graphene, potential (vector-potential) coupling constant. Previous works ∼ ∼ 18 there are both in-plane phonon modes and flexural phonon have estimated g1 20–30 eV and g2 1.5eV, though we note that even the value of the deformation-potential coupling modes (out of plane). In this work, we consider graphene 25,30,31 on a substrate (as in recent experiments probing thermal is subject to some debate. Our theory is thus not tied properties2–4), such that flexural motion is suppressed; we thus to specific values of these parameters, and we keep both the only focus on in-plane motion. Further, at low to moderate DP and VP couplings in our discussion. Note that transverse temperatures, the optical modes are barely excited and the phonons induce only a pure shear deformation, and hence dominant modes participating in cooling of hot electrons are couple only through the vector potential. acoustic modes; we thus focus exclusively on the coupling to these modes. B. Keldysh formalism for the kinetic equation Due to the Dirac Hamiltonian of electrons in graphene, there of electrons in graphene 17–19 are two distinct electron-phonon coupling mechanisms. Having established the basic electronic and electron- The first is a standard deformation-potential coupling, which phonon Hamiltonians [cf. Eqs. (3) and (5)], we now turn to corresponds to a local dilation of the lattice. In the Dirac our main goal of calculating the heat flux between electrons theory, it appears as a scalar potential (with respect to and phonons. We consider the standard situation where each pseudospin). The second mechanism is an effective gauge-field subsytem is independently in thermal equilibrium at its coupling or vector-potential coupling. This corresponds to own temperature (electrons at Te, phonons at Tph). In the the change in hopping matrix elements accompanying a pure disorder-free case, this heat flux can be conveniently calculated shear deformation, and enters the Dirac theory the same way by using Fermi’s golden rule to calculate electron-phonon as an external gauge field (the only proviso being that this scattering rates.7,8,10,11 Including disorder, we need a more phonon-induced vector potential is valley odd, and hence does general formalism, one that is capable of capturing the not break time-reversal symmetry). interference between electron-phonon and electron impurity Letting k denote a momentum-space electronic field scatterings [i.e., the vertex correction of the electron-phonon operator and bη,q a phonon annihilation operator, the total vertices Mˆ η(q) by disorder]. To that end, we make use of interaction between electrons and acoustic phonons can be the Keldysh technique,32,33 coupled with standard disorder- 26 written in the general form averaged perturbation theory. Such an approach was used by † † Kechedzhi et al. to study conductance fluctuations (in the H = Mˆ η(q) (b + b ). (5) ep k+q k η,q η,−q absence of any electron-phonon coupling).21 = η l,t k,q We start by noting that the heat flux of interest (i.e., energy Here η = l,t denote longitudinal (LA) and transverse (TA) lost/gained by the electrons) can be directly related to the × acoustic modes, respectively. The 2 2 coupling matrices collision integral I0(ε,p) appearing in a standard Boltzmann Mˆ η(q) take the form equation describing the dynamics of the electronic phase-space − 2iφq distribution function n[ε,p; t]: ˆ l = l g1 ig2e M (q) iqξq −2iφq , (6a) ig2e g1 dn[ε,p] I0(ε,p) ≡ . (8) 2iφq dt ˆ t = t 0 g2e e-ph scatt M (q) iqξq −2iφq , (6b) g2e 0 The collision integral I0(ε,p) tells us the rate of change where of n[ε,p; t] due to the emission and absorption of acoustic η η 1/2 phonons. ξ = (¯h/2ρM ω ) . (7) q q As electronic momentum relaxation is much faster than η Here ρM is the mass density of the graphene sheet, ωq is energy relaxation, to describe the latter process we can the phonon frequency for mode η, and φq is the angle of focus on times longer than the electron momentum relaxation the phonon wave vector q with respect to the x axis (which time; in addition, the distribution function will be sharply is taken to be along the armchair direction of the graphene peaked on shell (magnitude of momentum set by energy). lattice). For simplicity, we take the speed of sound to be the The relevant kinetics can thus be described by an electron 125443-3 WEI CHEN AND AASHISH A. CLERK PHYSICAL REVIEW B 86, 125443 (2012) distribution function n(ε; t) that depends only on energy, not on retarded, advanced, and Keldysh propagators. The retarded momentum. The kinetic equation for the electronic distribution and advanced phonon propagators appearing here are function n(ε; t) takes the form 2ω R/A(ω,q) = q , (13) ∂n(ε; t) 1 dp 2 − 2 ± = I¯ (ε) = I (ε,p)A(ε,p), (9) ω ωq iδ ∂t 0 πν (2π)2 0 with ωq = sq, while the Keldysh propagator is where K R A (ω,q) = [1 + 2N(ω,Tph)][ (ω,q) − (ω,q)], (14) 2 ε + i =− 2τ A(ε,p) Im (10) where N(ω,Tph) is the Bose-Einstein distribution evaluated at ε + i 2 − v2 p2 2τ F T = Tph. is the electron spectral function. The solid line in Fig. 1 represents an impurity-averaged ˆ β × The heat flux between electrons and lattice (for one valley electronic Green function G (ε,p); note that these are 2 and spin projection) is given by 2 matrices in pseudospin space. The retarded and advanced components are given by P (T ,T ) = ν dεεI¯ (ε). (11) i e ph 0 ε ± + vF σ · p Gˆ R/A(ε,p) = 2τ , (15) ± i 2 − 2 2 ε 2τ vF p The collision integral I0(ε,p) is obtained in the standard manner by calculating the electronic Keldysh self-energies ˆ while the Keldysh electron Green function is K R A arising from the electron-phonon interaction, to first order. One Gˆ (ε,q) = [1 − 2n(ε,Te)][Gˆ (ε,q) − Gˆ (ε,q)], (16) finds the general relation21 (see Appendix A) where n(ε,T ) is now the Fermi-Dirac distribution function i K A R evaluated at T = Te and chemical potential EF , where EF is I0(ε,p) =− Tr{ ˆ + [1 − 2n(ε,p; t)]( ˆ − ˆ )}. (12) 4 the Fermi energy (measured from the Dirac point). The self-energies ˆ j (j = K, R, and A)are2× 2 matrices Note that the electron Green function in Eq. (15) has an ε extremely simple form: It is just a free propagator with the (in pseudospin space), and are functions of both energy and → ± momentum p; they include the effects of disorder averaging. substitution ε ε i/2τ, corresponding to disorder-induced As we are considering a quasiequilibrium situation where both broadening of energy levels. This broadening represents the phonons and electrons are individually in thermal equilibrium, first mechanism by which the electron-phonon heat flux will n(ε,p; t)inEq.(12) can be replaced by a Fermi distribution be modified due to disorder; this broadening generally causes a suppression of the heat flux. The second key effect of disorder function at temperature Te, and the self-energies can be calculated assuming phonons are in thermal equilibrium at is via the vertex correction of the electron-phonon vertex temperature T . appearing in the self-energy in Fig. 1. Each vertex describes the ph emission or absorption of a phonon; heuristically, the disorder- In the regime of interest (kF l 1), the dominant self- energy diagram describing the leading-order electron-phonon induced vertex correction corresponds to the modification of contribution to the kinetic equation (in the presence of disor- the amplitude of such a process due to the diffusive motion der) is shown in Fig. 1. In this diagram, the wavy line represents of electrons. The full details of the renormalization of the a phonon propagator β (ω,q; η), where β = R,A,K denotes e-phonon vertices in a conventional (non-Dirac) metal by disorder using the Keldysh formalism are presented in Ref. 14. The Keldysh structure and renormalization of the e-phonon phonon vertices in graphene can be treated in a similar fashion. The × (a) only key difference comes from the 2 2 matrix structure associated with pseudospin; as we will see, this leads to interesting new physical consequences. e e e The electron-phonon vertices Mˆ (the η superscript is dropped here) in the Keldysh technique are represented by ,p + ω, p + q ,p γ the form Mˆ μ,ν, where the upper index is for phonons and (b) the lower for electrons. The indices γ,μ,ν each have two , p , p , p possible values, cl and q, due to the two-component structure ω, q 32 = + of the electron and phonon fields in the Keldysh formalism. The vertices with different indices are renormalized differently + ω, p + q + ω, p + q + ω, p + q by disorder in the Keldysh technique as shown in Ref. 14. Here, we only present the simplest case, the renormalization of the vertex Mˆ q . We stress here that our full calculation considers FIG. 1. (Color online) (a) Electron self-energy diagram. The cl,cl wavy line represents the phonon propagator, the solid line represents all relevant Keldysh-space vertices and their disorder-induced the electron propagator, and the square block represents the dressed renormalization. Upon summation of all self-energy diagrams electron phonon vertex by diffusion. (b) Vertex correction of the in the Keldysh formalism, one finds that the form of the renor- ˆ q electron-phonon interaction by impurity scattering. The dashed line malized vertex Mcl,cl appears directly in the final expression represents an impurity average. The square block represents dressed for the collision integral, Eq. (21).13,14 The renormalization of ˆ q vertex and the dot represents bare vertex. vertex Mcl,cl by disorder is depicted in Fig. 1(b). We focus on 125443-4 ELECTRON-PHONON MEDIATED HEAT FLOW IN ... PHYSICAL REVIEW B 86, 125443 (2012) this specific vertex in the remainder of this section and drop finally obtains the upper and lower indices from now on. 1 dq dω † In general, we may describe the renormalization of an I(ε,p) =− Tr R(ε,ω) Mˆ η(q) R(−ω; η) 3 0 electron-phonon interaction vertex by 2 = (2π) η l,t η Mˆ diff(q,ω) = Dˇ (q,ω) ◦ Mˆ 0(q), (17) × ˆ R + + ˆ − − + G (ε ω,p q)Mdiff( ω, q) H.c. , ˆ ˆ where M0 (Mdiff) is the bare (renormalized) vertex, and (21) Dˇ (q,ω) is a linear operator acting in the space of 2 × 2 matrices. It represents the diffusion propagator for electrons where R(ε,ω) is the expected combination of Bose-Einstein in graphene, with the nontrivial matrix structure reflecting the and Fermi-Dirac functions appropriate for phonon emission fact that charge and pseudospin diffusion are linked together. and absorption processes: It is convenient to write this expression using a basis of R(ε,ω) = N(ω,T )n(ε,T )[1 − n(ε + ω,T )] ph e e Pauli matrices. Defining the four-component vectors m,mdiff − + − + via [1 N(ω,Tph)][1 n(ε,Te)]n(ε ω,Te) = [n(ε,Te) − n(ε + ω,Te)][N(ω,Tph) − N(ω,Te)]. Mˆ = m · (1ˆ,σˆ ,σˆ ,σˆ ), (18a) 0 0 x y z (22) Mˆ = m · (1ˆ,σˆ ,σˆ ,σˆ ). (18b) diff diff x y z The first term of R(ε,ω) describes absorption of a phonon ω from energy state ε to ε + ω and the second term describes Equation (17) takes the form emission of a phonon ω from energy state ε + ω to state ε. As expected, R(ε,ω) vanishes if Te = Tph. We note that apart m diff(q,ω) = D(q,ω) · m0(q), (19) from the matrix structure of the electron-phonon vertices and D × electron propagators, the expression for the collision integral where is a 4 4 matrix. Using this representation, the has the same form as that found for conventional diffusive summation of ladder diagrams depicted in Fig. 1 results in metals.13,14,16 Nonetheless, we will see that the added matrix the form structure (which encodes the chiral nature of the graphene D(q,ω) = [1 − P(q,ω)]−1 , electronic excitations) gives rise to qualitatively new effects. 1 d2k P(q,ω) = σˆ α [Gˆ R(k + q,ω)] C. Diffusion propagator and renormalization [ ]αβ ij 2 ki 2πντ (2π) of the e-phonon vertex × ˆ A β [G (k,0)]jlσˆlk. (20) 1. Diffusion propagator Here, the indices α,β run from 0 to 3, and repeated indices It follows from Eqs. (11) and (21) that a key part of the are to be summed over; we also useσ ˆ 0 to denote the 2 × 2 disorder-induced modification of the electron-phonon heat flux unit matrix andσ ˆ i (i = 1,2,3) are the Pauli matrices. P is due to the modification of the effective electron-phonon describes a single “rung” in a standard diffusion ladder. Its interaction vertex. This modification is in turn directly related 4 × 4 matrix structure is now directly related to the fact to the chiral diffusive dynamics of electrons in graphene, as that each propagator carries an initial and final charge or described by Eq. (17). In this section, we discuss the diffusion pseudospin index. A similar structure is encountered when propagator in more detail, as well as the forms of the dressed considering diffusive dynamics in a system with strong spin- electron-phonon vertices. Note that we restrict our discussion orbit coupling or in 2D helical metals, as studied by Burkov here (as we do throughout the paper) on electrons in the K+ et al.34,35 valley. While the sign of the chirality will be different for holes, We will discuss the properties and physics encoded in or for the K− valley, this sign has no impact on the quantity of the matrix diffusion propagator D in more detail in the next interest, the e-phonon heat flux in the presence of disorder. section. For now, we only show how it enters in the final We focus in this section on the most interesting diffusive expression for the collision integral (and hence the heat flux). regime, where ql 1,ωτ 1 (i.e., we are interested in One finds that due to the causality structure of Keldysh length scales longer than l and time scales longer than τ). In Green functions, the two vertices in Fig. 1(a) cannot both this limit we can work to lowest nonvanishing order in ql and be simultaneously dressed by impurity scattering. Summing ωτ. The inverse diffusion propagator D−1(q,ω) = 1 − P(q,ω) up all the self-energy diagrams in the Keldysh formalism, one simplifies to ⎛ ⎞ ⎛ ⎞ − + 2 iω Dq 000 02 ivF qx 2ivF qy 0 1 ⎜ 0 1 1 − iω+Dq2 00⎟ 1 ⎜2iv q D q2 − q2 2Dq q 0 ⎟ D−1 =⎜ 2 τ ⎟+ ⎜ F x x y x y ⎟ (q,ω) ⎝ 1 1 − + 2 ⎠ ⎝ 2 − 2 ⎠ , τ 002 τ iω Dq 0 4 2ivF qy 2Dqx qy D qy qx 0 1 00 0τ 00 00 (23) in the diffusive limit, where D = vF l/2 is the usual diffusion constant in two dimensions. 125443-5 WEI CHEN AND AASHISH A. CLERK PHYSICAL REVIEW B 86, 125443 (2012) To gain intuition, it is useful to follow Ref. 34 and consider or the diffusion equations of Cooperons in graphene.36 The the real-space representation of the matrix diffusion propaga- interpretation here is similar to Ref. 35: The coupling between tor, which describes the coarse-grained evolution of charge charge and pseudospin dynamics in the diffusive limit is a and pseudospin densities [N(r,t) and Sj (r,t), respectively, result of the effective helicity of the electronic eigenstates. j = x,y,z]. The first term in Eq. (23) would simply lead to By helicity, we mean that in the ε>0 eigenstate of Eq. (3), uncoupled equations for each of these quantities: N would pseudospin will be aligned with momentum. Thus, a positive be described by a standard diffusion equation, while Sx gradient in, say, Sx in the x direction implies a corresponding and Sy would be described by diffusion equations with an positive gradient in the density of electronic x momentum. This additional decay term (rate 1/τ), corresponding to the fact that will then naturally cause the charge density N to decrease in pseudospin is not a conserved quantity. Sz has no dynamics in time: This is the third term in Eq. (24a). Alternatively, writing the limit we consider, as it precesses with frequency EF and Eq. (24a) in the form of a continuity equation, averages away on the time scale 1/τ. The second term in Eq. (23) complicates the above picture, ∂N =−∇· J, (25) as it now links the dynamics of charge and pseudospin ∂t densities. We thus obtain a set of coupled diffusion equations, one sees that the charge current density J has the form describing the dynamics of these quantities (note that we have taken into account the fact that the Pauli matrices are twice the J =−D∇N + vF (Sx xˆ + Sy yˆ). (26) pseudospin matrices): The first term is the usual diffusive current, while the second ∂N 2 ∂Sx ∂Sy = D∇ N − vF + , (24a) term corresponds to a “drift” current driven by the pseudospin ∂t ∂x ∂y density following from the Hamiltonian in Eq. (3) (i.e., the ∂S 3D ∂2 D ∂2 ∂2 S x = S + S + D S − x current operator is the pseudospin operator). ∂t 2 ∂x2 x 2 ∂y2 x ∂x ∂y y τ Turning to the dynamics of pseudospin densities, Eqs. (24b) v ∂N and (24c) again reflect the fact that pseudospin is not con- − F , (24b) 2 ∂x served, and effectively decays on a time scale τ due to elastic 2 2 2 impurity scattering. In addition, we see that the diffusion of ∂Sy D ∂ 3D ∂ ∂ Sy = Sx + Sx + D Sx − these densities is anisotropic: This is also a simple consequence ∂t 2 ∂x2 2 ∂y2 ∂x ∂y τ of helicity, as a net pseudospin density in a specific direction vF ∂N also implies a net momentum density in this direction which − . (24c) 2 ∂y reinforces the diffusion in this direction. Finally, inverting Eq. (23) [using as always the Pauli matrix These equations are analogous (but not identical) to the representation defined in Eq. (19)], one finds the diffusion diffusive dynamics for charge and spin in a 2D helical metal35 propagator in Eq. (20) in the diffusive limit to be ⎛ ⎞ 1 iqx l iqy l (−iω+2Dq2)τ (iω−2Dq2)τ (iω−2Dq2)τ 0 ⎜ iωτ+(q2−q2)l2 2 ⎟ ⎜ iqx l 3 + x y qx qy l 0 ⎟ D(q,ω) = ⎜ (iω−2Dq2)τ 2 2(iω−2Dq2)τ (iω−2Dq2)τ ⎟ . (27) ⎜ 2 − 2− 2 2 ⎟ ⎝ iqy l qx qy l 3 + iωτ (qx qy )l ⎠ (iω−2Dq2)τ (iω−2Dq2)τ 2 2(iω−2Dq2)τ 0 00 01 Note that the effective diffusion constant (i.e., the coefficient of interaction vertex due to diffusion, as given by Eqs. (19) q2 in the diffusion poles appearing above) is twice the value of and (27). Consider first the DP contribution to the vertex. The the standardly defined D appearing in Eqs. (24): Deff = 2D = bare DP vertex is just a diagonal matrix in the sublattice basis vF l. This effective doubling of the diffusion constant is a direct [cf. Eqs. (6)] proportional to the coupling constant g1.The consequence of the chiral nature of electrons in graphene, and impurity-dressed vertex in the diffusive limit takes the form is consistent with the results of previous transport studies.20,37 Also note that as expected from our discussion following −iφq 1 − iqle Eq. (23), only the charge-charge and charge-spin components ˆ = iqξq g1 (−iω+2Dq2) (−iω+2Dq2) MDP,diff(q,ω) iφ . of D [i.e., the top row and column] diverge in the limit of small τ − iqle q 1 (−iω+2Dq2) (−iω+2Dq2) ω and q. The lack of any corresponding large enhancement (28) of the spin components of D is directly tied to the fact that pseudospin is not a conserved quantity. The diagonal parts of the vertex acquire a diffusion pole 2. Disorder vertex correction of deformation-potential protected by charge conservation, analogous to the case of e-phonon vertex a normal metal. We will be interested in Eq. (28) with ω,q Having discussed the basic form of the matrix diffusion corresponding to a thermal phonon,hω ¯ = hsq¯ kBT .As propagator, we now turn to the renormalization of the e-phonon s vF in graphene, we thus have that over a wide range 125443-6 ELECTRON-PHONON MEDIATED HEAT FLOW IN ... PHYSICAL REVIEW B 86, 125443 (2012) of temperature vertex which is a 2 × 2 matrix in pseudospin space. The bare s vertices are T ⎛ ⎞ 2 2 − − iql − 3 + iωτ/2 2iφq − q l /2 4iφq 2 2 sin 3φq i 2 2 e i 2 2 e η=l l ⎝ −iωτ+q l 2 iωτ−q l −iωτ+q l ⎠ Mˆ (q,ω) = iqξ g2 . (34) VP,diff q − 2 2 3 + iωτ/2 2iφq + q l /2 4iφq − iql i 2 iωτ−q2l2 e i −iωτ+q2l2 e −iωτ+q2l2 sin 3φq The full interaction vertex for TA phonons can be obtained in we have all the necessary ingredients to evaluate Eq. (21) for a similar fashion. One finds that vector potentials arising from the electronic collision integral. From this, Eq. (11) directly TA and LA phonons make identical contributions to the heat yields the desired electron-phonon heat flux. As with the flux. disorder-free case, we again find that the DP and VP couplings contribute independently; we can thus meaningfully discuss the flux associated with each coupling. It is useful to express III. RESULTS AND DISCUSSIONS each of these heat fluxes in terms of an energy control function A. Heat flux without screening Fα(T )(α = DP,VP), defined via Having now determined both the renormalized electron- phonon vertices [cf. Eqs. (28) and (34)]aswellasthe = ¯ ≡ − disorder-averaged electronic Green functions [cf. Eq. (15)], Pα(Te,Tph) ν dεεIα,0(ε) Fα(Te) Fα(Tph). (35) 125443-7 WEI CHEN AND AASHISH A. CLERK PHYSICAL REVIEW B 86, 125443 (2012) ¯ Here, Iα,0(ε) is the momentum average of the collision integral 4 10 3 corresponding to the coupling mechanism α. l = 20 nm T We discuss each mechanism in turn, focusing as always on 100 s 2 the regime Tdis 125443-8 ELECTRON-PHONON MEDIATED HEAT FLOW IN ... PHYSICAL REVIEW B 86, 125443 (2012) integrand in Eq. (36) in the dirty limit l → 0 can ultimately be traced to the diffusive enhancement of the DP vertex [cf. 10 l = 20 nm Eq. (28)]. In contrast, the corresponding q dependence in Ref. 1 T3 can be traced to the energy of a virtual electronic state in a 2 m 0.01 second-order process involving both e-phonon and e-impurity K scattering events. W 10 5 A G 10 8 2. Vector-potential coupling heat flux T4 11 The energy control function FVP(T ) for vector-potential 10 s 0.01 0.1 1 10 100 coupling in the temperature regime Tdis VP-mediated heat transport. Note that as the bare VP coupling G 10 8 constant g2 has been estimated to be more than an order of magnitude smaller than the corresponding DP coupling 18 10 11 constant g1, it follows that in the absence of screening, 0.01 0.1 1 10 100 the heat flux associated with the VP coupling is expected to T K be negligible in comparison to that associated with the DP coupling. The thermal conductance associated with the VP FIG. 5. (Color online) Thermal conductance per unit area G/A coupling is shown in Figs. 4 and 5. associated with the vector-potential coupling versus temperature T , including the effects of disorder, showing the effects of varying the mean free path l as indicated; screening is neglected. Remaining B. Heat flux with electronic screening parameters are the same as Fig. 4. Both the suppression of the We now consider how the above results are altered if one in- low-temperature thermal conductance and shift of the crossover cludes the screening of the e-phonon interaction. As discussed temperature with increasing disorder are clearly evident. 125443-9 WEI CHEN AND AASHISH A. CLERK PHYSICAL REVIEW B 86, 125443 (2012) 12,25 5 where the Thomas-Fermi wave vector qTF is given by 10 e2 100 DP unscreened q = 4 k (43) TF κhv F DP screened ¯ 2 F 0.1 VP m and κ is an effective dielectric constant. Using the value of κ K 10 4 appropriate to graphene on a SiO2 substrate, one has qTF W 12 3.2kF . Note that as we focus on the regime kF l 1, the A 10 7 effects of screening will generally set in at a much higher G 10 temperature Tsc = sqTF/kB than the temperature Tdis below 10 which disorder effects become important. 10 13 One can now easily include the effects of screening 0.01 0.1 1 10 100 into our theory by making the substitution g1 → g1,sc(q)in T K Eq. (36) for the energy control function FDP(T ) determining the deformation-potential mediated heat flux. One finds FIG. 7. (Color online) Thermal conductance per unit area G/A ⎧ 8π 4g2E k6 versus temperature T , showing the effects of electronic screening. ⎨ 1 F B 6 7 3 5 2 T , if Tdis T Tsc, Parameters are identical to Fig. 6,exceptwehavenowtakenthe 63ρMh¯ vF s qTF FDP,sc(T ) = ⎩ 24 g2E mean free path to be l = 10 μm, meaning that we are effectively 1 F 5 2 6 3 4 2 ζ (5)T , if T Tdis,Tsc. π ρMh¯ vF s qTFl in the clean limit. Unlike the disordered case shown in Fig. 6,we (44) now see that at low temperatures, the contribution of the vector- potential coupling dominates that from the deformation-potential The suppression of the DP coupling by screening at low coupling. temperatures T Tdis implies that its associated heat flux can now become comparable or even smaller in magnitude to that associated with the VP coupling [cf. Eq. (41)]. In the versus temperature for both mechanisms is presented, for low-temperature limit, both mechanisms yield energy control both the strongly and weakly screened cases. The fact that 5 functions F (T ) ∝ T , with both mechanisms are comparable is markedly different from 2 what happens in the screened, disorder-free case, which is FDP,sc(T ) g 1 ∼ 1 . (45) realized when T T T . In this case, the VP heat flux F (T ) g2 q2 l2 dis sc VP 2 TF will dominate the DP heat flux at low temperatures, as the VP 4 We see that the largeness of qTFl can compensate for the heat flux scales like T , while the DP heat flux scales like 6 relative smallness of g2 with respect to g1, leading both T (see Fig. 7). This behavior in the clean limit is similar mechanisms to make comparable contributions. This behavior to expectations for e-phonon contribution to the electrical is demonstrated in Fig. 6, where the thermal conductance resistivity, where it has also been argued that the VP coupling can dominate at low temperatures.26 However, we note that a recent experiment measuring 104 the e-phonon contribution to the electrical resistivity of a DP unscreened suspended graphene flake suggests that screening does not 10 27 DP screened seem to be playing a role even when T m VP results are compatible with the predictions for an unscreened K 0.01 deformation-potential interaction (see also Ref. 25). The W 10 5 same conclusions could be drawn from recent experiments A measuring the hot electron heat flux in graphene.2–4 G 10 8 Finally, we note that with our theory, the only way to obtain a T 3 power law in the heat flux at low temperatures [i.e., 10 11 δ = 3inEq.(1)] is via an unscreened deformation potential. 0.01 0.1 1 10 100 Thus, measurements of the low-temperature heat flux could T K also serve as a diagnostic tool for assessing the importance of screening the deformation potential. FIG. 6. (Color online) Thermal conductance per unit area G/A versus temperature T , showing the effects of electronic screening. 12 2 g1 = 20 eV, g2 = 1.5 eV, a carrier density n = 10 /cm , and a mean free path l = 20 nm. The red short-dashed line corresponds to an IV. CONCLUSIONS unscreened deformation-potential coupling, while the blue dashed- dotted line corresponds to a screened deformation-potential coupling, We have presented a comprehensive theory showing how electronic disorder modifies the electron-phonon interaction in with a Thomas-Fermi wave vector qTF = 3.2kF as appropriate graphene (both the vector-potential and deformation-potential for graphene on SiO2 (Ref. 12). The dashed purple curve is the contribution from the vector-potential coupling. Despite its much couplings), and how this in turn has observable consequences smaller bare coupling constant, we see that at low temperatures, for the heat flux between the electrons and lattice (acoustic) both the deformation-potential and vector-potential couplings make phonons. We focused on the relatively simple situation where almost equal contributions when both screening and disorder effects the graphene is doped away from the Dirac point, and where are included. the impurity potential can be considered smooth on atomic 125443-10 ELECTRON-PHONON MEDIATED HEAT FLOW IN ... PHYSICAL REVIEW B 86, 125443 (2012) scales (implying that the disorder potential preserves the could be parametrized as Gˆ K = Gˆ R ◦ F − F ◦ Gˆ A, where F is pseudospin and valley symmetries of the graphene Hamil- the Hermitian distribution function matrix. The Green function tonian). We found that the unusual diffusion dynamics of obeys the following Dyson equation: electrons in graphene that results from their chirality also has implications for how disorder modifies electron-phonon −1 0 Gˆ A − ˆ A Gˆ K Gˆ R 0 ◦ = physics. We also found that this modification is quite different − 1, (A2) ˆ R 1 − ˆ R − ˆ K Gˆ A 0 for the deformation-potential coupling versus the effective G0 vector-potential coupling. In the absence of screening, the contribution to the heat flux in Eq. (1) from both couplings where the bare retarded and advanced Green functions in 4 R/A − ˆ 1 = − + ˆ · has temperature dependence of T in the clean limit T s/l, graphene are (G0 ) (x,x ) δ(x x )(i∂t iσ ∂r )in consistent with previous work. In the disorder limit T 125443-11 WEI CHEN AND AASHISH A. CLERK PHYSICAL REVIEW B 86, 125443 (2012) a measure of the pseudospin density induced by interaction, Eq. (19) in such basis becomes 0 while theσ ˆ component determines the collision integral for ⎛ √ ⎞T the charge density distribution function and is the one of + 1+q2l2 ⎜ 1 q2l2 ⎟ interest in this paper. From Eq. (A5), one gets the collision ⎜ − i ⎟ → = l ⎜ ql cos φq ⎟ integral for charge distribution function mDP,diff(ω 0,q) iqξq ⎜ ⎟ g1, (B2) ⎝ i ⎠ − sin φq ∂n(ε,p; t) 1 ∂f (ε,p; t) i ql I (ε,p) = =− 0 =− Tr{ ˆ K (ε,p) 0 0 ∂t 2 ∂t 4 T + [1 − 2n(ε,p; t)]( ˆ A − ˆ R)(ε,p)} (A6) where the superscript means transpose of the column vector to row vector and the same for m VP,diff below. Written in the as shown in Eq. (12). sublattice basis, the renormalized deformation-potential vertex becomes ⎛ √ ⎞ APPENDIX B: FULL FORM OF THE DIFFUSION 1+q2l2 −iφq 1 + − ie PROPAGATOR AND RENORMALIZED e-PHONON ˆ → = l ⎝ q2l2 √ql ⎠ MDP,diff(ω 0,q) iqξq g1. VERTEX − ieiφq + 1+q2l2 ql 1 q2l2 The full form of the diffusion propagator D(ω,q) crossing (B3) s the whole temperature regime Tdis ⎛ ⎞ i sin 3φ −4iφq − q −ie2iφq 1 + √ 1 − i √e ˆ → = l ⎝ ql 2 1+q2l2 2 1+q2l2 ⎠ MVP,diff(ω 0,q) iqξq − 4iφq i sin 3φ g2. (B6) ie 2iφq 1 + √ 1 + i √e − q 2 1+q2l2 2 1+q2l2 ql In the diffusive limit, it reduces to Eq. (34) (again dropping the frequency dependence there), while in the clean limit ql 1, it reduces to the bare vertex. APPENDIX C: DETAILS OF THE MATRIX COLLISION INTEGRAL 1. Deformation potential The renormalized deformation potential in the whole regime of ql is presented in Appendix B. Plugging in the renormalized deformation potential to Eq. (21) and integrating over the phonon frequency, one gets the collision integral for deformation- potential coupling as ∂n(ε,p; t) dq 1 i 1 + q2l2 = = 2 2 + + + IDP,0(ε,p) i g1 (qξq ) ε ωq 1 ∂t (2π)2 ε + ω + i 2 − v2 |p + q|2 2τ q2l2 q 2τ F v q p + q p + i F + iv l x x y y + h.c. R(ε,ω ) − (ω →−ω ,q →−q) , (C1) l F q2l2 q q q where ωq = sq. 125443-12 ELECTRON-PHONON MEDIATED HEAT FLOW IN ... PHYSICAL REVIEW B 86, 125443 (2012) The collision integral for deformation potential after average over the electron momentum becomes 1 dp dq 1 + q2l2 1 1 1 ¯ = = 2 2 2 + − − IDP,0(ε) A(ε,p)IDP,0(ε,p) τ g1 q ξq 1 1 πν (2π)2 (2π)2 q2l2 1 + q2l2 q2l2 1 + q2l2 × R(ε,ωq ) − (ωq →−ωq ,q →−q) . (C2) The heat flux is then PDP(Te,Tph) = ν dεεI¯DP,0(ε) = FDP(Tph) − FDP(Te), (C3) where the energy control function FDP(T ) is presented in Sec. III. 2. Vector potential Plugging in the renormalized vector potential in Appendix B to the collision integral Eq. (21) and separating the component for the charge distribution function, one gets the kinetic equation of the charge distribution function due to vector-potential coupling after averaging over the angle of electron momentum as ∂n(ε; t) dq 1 1 1 1 = ¯ = 2 2 2 + − − IVP,0(ε) τ g2 q ξq 1 1 R(ε,ωq ) ∂t (2π)2 2 1 + q2l2 1 + q2l2 2q2l2 1 + q2l2 − (ω →−ωq ,q →−q) . 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