Kramers-Kronig relation of graphene conductivity
aa r X i v : . [ c ond - m a t . m e s - h a ll ] M a r Kramers-Kr¨onig relation of graphene conductivity
Daqing Liu and Shengli Zhang
Department of Applied Physics, Xi’an Jiaotong university, Xi’an, 710049, China
Abstract
Utilizing a complete Lorentz-covariant and local-gauge-invariant formulation, we discuss graphene responseto arbitrary external electric field. The relation, which is called as Kramers-Kr¨onig relation in the paper,between imaginary part and real part of ac conductivity is given. We point out there exists an ambiguity inthe conductivity computing, attributed to the wick behavior at ultraviolet vicinity. We argue that to studyelectrical response of graphene completely, non-perturbational contribution should be considered.PACS numbers: 71.10.-w,72.10.Bg,73.63.-b
Graphene, a flat monolayer of carbon atoms tightlypacked into a two-dimensional honeycomb lattice, hasspawned many theoretical and experimental focuses. Asstated in ref. [1], graphene plays the role of bridge be-tween condensed physics and high energy physics. Thisis attributed to massless Dirac fermion behavior of quasi-electron in graphene, i.e. we can treat the quasielectronsin graphene as ultimately relativity particles.Such behavior also arouses many unusual propertiesof graphene, such as ac and dc conductivity. Manyattentions [2–5] are on such topic. However, there ex-ist discrepancies in the problem, including the discrep-ancy between theories and experiments on dc conduc-tivity, the famous missing ” π factor”, and conflictionamong different theoretical calculations [6] . So far dif-ferent theories are almost based on the perturbationalapproximation, even calculations may be did by multi-loop diagrams [7] .We introduce a correlation function with respect toonly one variable, the invariant amplitude of spatial-timeposition, x , to study the graphene conductivity non-perturbationally. From the correlation function, we showthere is relation between imaginary part and real part ofac conductivity. The function is very close to spectralfunction and we find that the perturbational calcula-tions to conductivity only include contributions from freevalence-conduction electron pairs. Therefore, besidesthese contributions, to compute conductivity completelywe should also consider other ones, such as excitation orimpurity. To check the statement, we perform a pertur-bational calculation of dc conductivity using the quan-tum field theory. This technique guarantees that the for- mulation is Lorentz-covariant and local-gauge-invariant.We point out that there exists discrepancy among differ-ent theoretical calculations, attributed to bad behaviorsat ultraviolet vicinity of δ -functions.We organize the paper as following: In section 2,we discuss the electrical response to arbitrary externalelectric field. A discussion on obtaining dc conductivityutilizing Kubo theory [8] is also given. The relation be-tween imaginary part and real part of ac conductivity islisted in section 3. We show an explicit perturbationalcomputing for conductivity in section 4 and a brief dis-cussion in section 5. To perform the calculation we first give the secondquantization on graphene briefly. Lagrangian density is L = ¯ ψ ( iγ µ ∂ µ − m ) ψ, (1)where ∂ µ = ∂∂x µ , ¯ ψ = ψ † γ , m is the mass of quasiparti-cle (To clarify we here endow the quasielectrons with anonzero mass), and γ ’s are γ = β = τ , γ = βτ , γ = βτ , where τ , τ and τ are three Pauli matrices. In the pa-per, the repeated indices are generally summed, unlessotherwise indicated. Furthermore, ~ = v F = e = 1 arealways set.The Hamiltonian is then [5] H = Z d r ψ † ( r ) β ( − iγ i ∂ i + m ) ψ ( r ) . (2)1ere, Latin indices i, j, etc. generally run over two spa-tial coordinate labels,1,2. (While Greek indices µ, ν, andso on, three spatial-time coordinate labels 0, 1, 2 with x the time coordinate.)Under the second quantization, we have ψ ( x ) = Z d p (2 π ) √ p [ a p u ( p ) e − ipx + b † p v ( p ) e ipx ] , ¯ ψ ( x ) = Z d p (2 π ) √ p [ a † p ¯ u ( p ) e ipx + b p ¯ v ( p ) e − ipx ] , where p = p m + p >
0. Solutions to positive energy u ( p ) and to negative energy v ( p ) satisfy respectively u † u = 2 p , ¯ uu = 2 m, u ¯ u = p/ + m,v † v = 2 p , ¯ vv = − m, v ¯ v = p/ − m. (3)The explicit forms of u and v are irrelevant.For simplification, we call operator a † p ( a p ), whichcreates(annihilates) quasielectron in conductionband, as creation(annihilation) operator which cre-ates(annihilates) electron, at the same time, we calloperator b † p ( b p ), which annihilates(creates) electron invalance band, as creation(annihilation) operator whichcreates(annihilates) hole. Both electron and hole havepositive energy, p .Since in the perturbational ground state, the valanceband is completely filled while the conduction bandis empty, the energy of the ground state is nonzero, E gnd = − R d p p m + p . To obtain a Lorentz invari-ant ground state, we perform a substraction for all thestates, E → E − E gnd . Under such subtraction, eachphysics quantity, such as energy, current, etc. should bein normal form [9] .With the substitution i∂ µ → p µ , we read Hamilto-nian operator eventually H = Z d p p ( a † p a p + b † p b p ) . (4)When concerning of electromagnetic interactions weshould make a substitution of p µ → p µ − eA µ in Eq.(2). Denoting A µ = g µν A ν with metric matrix g = diag { , − , − } , the interacting Lagrangian density is L int = − e ¯ ψγ µ ψA µ = − J µ A µ and the correspondingHamiltonian is H int = − Z d x L int = Z d x J µ ( x ) A µ ( x ) , (5) where, just as pointed out above, J µ ( x ) is in norm form: J µ ( x ) =: ¯ ψγ µ ψ ( x ) := Z d p d p ′ (2 π ) p p p ′ { a † p a ′ p e i ( p − p ′ ) x ¯ u ( p ) γ µ u ( p ′ )+ a † p b † p ′ e i ( p + p ′ ) x ¯ u ( p ) γ µ v ( p ′ )+ b p a p ′ e − i ( p + p ′ ) x ¯ v ( p ) γ µ u ( p ′ ) − b † p ′ b p e − i ( p − p ′ ) x ¯ v ( p ) γ µ v ( p ′ ) } . Unlike some papers, we here introduce a factor √ p associated with momentum integration, which is at-tributed to the Lorentz covariant [9, 10] .Both the electron number and the hole number areconservative without interaction. However, the only con-servation quantity is their difference when interactionsare included, N = Z d x : ψ † ψ := Z d p (2 π ) ( a † p a p − b † p b p ) . (6)Generally, the interacting Hamiltonian of graphenein external field A µ ( x , x ) is described by Eq. (5). Thedensity operator is ρ = ρ + δρ , where ρ is the equilib-rium density operator and δρ is the leading order correc-tion with respect to the external field.In Heisenberg picture we have iδρ ( x ) = [ H int , ρ ] . (7)Therefore [2] , δρ ( x ) = − i Z x −∞ dx ′ d x ′ [ J µ ( x ′ x ′ ) , ρ ] A µ ( x ′ x ′ ) . (8)Due to the spatial and time translation invariant, atzero temperature, on fixed time x , the current densityat arbitrary position is < J µ ( x ) > = T r ( δρ ( x ) J µ ( x ))= Z −∞ dx d x T µν ( x x ) A ν ( x + x x ) , (9)where T µν ( x x ) = i < | [ J ν ( x x ) , J µ (0 )] | > (10)with the notation of the ground state | > .Noticing in Eq. (9) and (10), variable x is definedon ( −∞ , x onto ( −∞ , ∞ ) as T µν ( x ) = iθ ( x ) < | [ J µ (0) , J ν ( x )] | > + iθ ( − x ) < | [ J ν ( x ) , J µ (0)] | >, (11)where θ ( x ) is step function: θ ( x ) = 1 for x ≥ θ ( x ) = 0 for x ≤
0. Tensor T µν ( x ) is vanishing for2pace-like x , i.e. the support of tensor T µν is time-likethree-dimensional vector x .Our expansion is different to the one in Ref. [2], where T µν has only forward term or backward term.The conductivity should be independent on the gaugetransformation. This means that, under a local gaugetransformation A ν → A ′ ν = A ν − ∂ ν f , where f is ar-bitrary function with f ( x = x ) = f ( x = −∞ ) = 0,the current density in Eq. (9) should be invariant. Inte-grating Eq. (9) by part we find that this requirement issatisfied provided ∂ ν T µν = 0 everywhere. The statementcan be proven by the facts: 1)charge conversation, i.e. ∂ ν J ν ≡
0; 2) ∂∂x µ θ ( x ) = − ∂∂x µ θ ( − x ) = δ µ δ ( x ); 3) theequal time commutation relation [ J ( t x ) , J µ ( t y )] = 0. T µν ( x ) is written as T µν ( x ) = − ( ∂ µ ∂ ν − g µν (cid:3) )Π( x ) , (12)where (cid:3) ≡ ∂ µ ∂ µ = ∂ ∂x − ∂ ∂x − ∂ ∂x , Π( x ) is scalarfunction with respect to only one variable, invariant am-plitude of three-dimensional spatial-time vector x . Afterdefining the Fourier transformation of function f ( x ) as f ( q ) = R d x dtf ( x ) e i q x with q x ≡ q µ x µ , we have, then, T µν ( q ) = ( q µ q ν − q g µν )Π( q ) , (13)where Π( q ) is the only function with respect to invariantamplitude of q .From Eq. (9) it seems that J µ is time dependentin the time-invariant external electric field. But this isnot true. It is enough to illustrate it by a special gauge, A ν = (0 , E x ,
0) or A ν = (0 , − E x , A is three-dimensional potential and E is the external electric field, E ν = ( E, J µ ( x ) = − E Z −∞ dx dq π e − i q x g µ q Π( q )( x + x ) , (14)where Π( q ) ≡ Π( q , q = ). J (charge density) and J are both vanishing and only J = 0: J ( x ) = E Z dq π q Π( q ) Z x −∞ dx x e iq x + ǫx e iq x = E Z dq π Π( q ) q q + iq ǫ (1 + iq x ) ≡ Eσ ( x ) , (15)where the additional factor e ǫx ( ǫ is a positive infinitesi-mal) is to guarantee that the external field is introducedadiabatically. When ǫ → q q + iq ǫ →
1, we can re-place q q + iq ǫ by unitary. Furthermore, since Π( q ) iseven function of q , R dq π Π( q ) iq x ≡
0. We finally have a time-independent conductivity σ = Z dq π Π( q ) ≡ Π( x = 0 , q = ) , (16)where, to obtain a meaningful quantity, we should per-form a substraction of Π, i.e. we make a substitution:Π( x = 0 , q = ) → Π( x = 0 , q = ) − Π( x = −∞ , q = ). In Fourier space, this substraction is the substitu-tion Π( q , q ) → Π( q , q ) − Π( q = 0 , q ). In the paper,we always make such substraction for all the physicalquantities.As expected, we obtain a time-independent currentdensity for a steady external field.It is not difficult to deduce the response to arbitraryexternal fields. Supposing the external electric field is acwith frequency ω , A = (0 , E e iωx , e ǫx and substitut-ing the potential into expression (9), we find that onlyx-component of current density is nonvanishing, J ( x ) = E e iωx Z dq π Π( q , ) iq q − ω + iǫ . (17)Since this potential stands for external electric field( E , E ) = ( iω E e iωx , σ = R dq π Π( q , ) q ω ( q − ω + iǫ ) (18a)= ω R dq π Π( q , ) q + R dq π Π( q , ) q q − ω + iǫ . (18b)Obtaining the dc conductivity from Eq. (18a) and(18b) corresponds to the results obtained from famousKubo theory. However, it is not obvious whether wecan obtain dc conductivity (16) from the limit of Eq.(18a): Firstly, to obtain Eq. (18b) from (18a) we neednot only convergence of all the integrations, such as R dq π Π( q , ) q ω ( q − ω + iǫ ) , etc. but also proper subtrac-tion of physics quantities. Secondly, when one chooses ω = 0 directly in Eq. (18b), he will meet an uncom-fortable situation: the first term in Eq. (18b) is an am-biguous . To obtain right dc conductivity we shouldperform computation as follows: we calculate the ac con-ductivity in the course of nature from Eq. (18) at ω = 0,with proper subtraction. At last we read the dc conduc-tivity utilizing the limit of ω →
0. This is just pointedout by Kubo [8] , which implies that, compare to resultsin references [3], results in references [4] is just the rightresults. Of course, all the results in references [3, 4] areobtained by perturbational approach.After suitable subtraction, the ac conductivity is σ = Z dq π Π( q , ) q q − ω + iǫ . (19)Generally, Π( q , ) is not convergent or well defined, aswill be shown by perturbative calculation in section 4.This is relevant to the wick definition of δ − function.3herefore, in order to obtain meaningful physical re-sult, we need to perform subtraction to cancel diver-gence. For instance, in Ref. [6] the author has proposeda soft δ − function. The subtraction should meet somephysical criteria. For instance, as found in above para-graph, to get the result in Eq. (19) from Eq. (18), afterthe subtraction Π is still the function of q rather thanthe function of q in Fourier space. In section 4 we shallshow a explicit subtraction to Π( x , q ). ¨ onig relation ofgraphene conductivity In this section we show a relation between imaginarypart and real part of graphene conductivity.We first give a non-perturbational proof that Π( q )is real. Π( q ) is real at q < q ) is also real at q > q >
0, after inserting complete intermediatestates P Γ | Γ >< Γ | , we have for T ≡ T µµ ( q ), T = Z d xd pe iqx (2 π ) i ( θ ( x ) − θ ( − x ))( e ipx − e − ipx ) s ( p ) θ ( p ) , where the spectral function s ( p ) is defined as2 πs ( p ) = X Γ < | J µ (0) | Γ >< Γ | J µ (0) | > (2 π ) δ ( p − p Γ ) . (20)The spectral function s ( p ), which is very close tostate density, includes not only perturbational contri-butions, but also non-perturbational contributions. Tostudy non-perturbational contributions, one should con-sider, for instance, excitations.Inserting R ∞ dtδ ( p − t ) ≡ p >
0, one obtains T = Z ∞ dts ( t ) Z d xe iqx i ( θ ( x ) − θ ( − x )) I, (21)where I = Z d p (2 π ) θ ( p ) δ ( p − t )( e ipx − e − ipx )= Z d p (2 π ) e − ipx δ ( p − t )( θ ( − p ) − θ ( p )) . Utilizing i Z d pe − ipx (2 π ) [ θ ( x ) θ ( p ) + θ ( − x ) θ ( − p )] δ ( p − t )= − Z d p (2 π ) e − ipx p − t + iǫ , i Z d pe − ipx (2 π ) [ θ ( − x ) θ ( p ) + θ ( x ) θ ( − p )] δ ( p − t )= Z d p (2 π ) e − ipx p − t − iǫ , we finally get T = 2 P Z ∞ dts ( t ) 1 q − t (22)with identity f ± iǫ = P f ∓ iπδ ( f ). Since s ( t ) ≥ T ( q ) and therefore Π( q ) are both real. The spectral den-sity is in fact a very important function, which will bestudied elsewhere [11] .Thus, from Eq. (19), we obtain an important relationbetween real part and imaginary part of conductivity σ Im σ ( ω ) = − ω Π( ω , )2 , Re σ ( ω ) = P Z dsπ Im σ ( s ) ω − s . (23)This relationship between real part and imaginary partof conductivity is beyond the perturbation approachand can be considered as Kramers-Kr¨onig relation ofgraphene conductivity. We hope the advanced study ofgraphene may check this relation.Eq. (23) is one of the main results of the paper. Itpoints out that the electrical response of graphene cannever be considered as a pure resistance, but a resis-tance parallel connected with a capacitor with capacitiv-ity Π( ω , ) /
2. Furthermore, due to the obvious relationbetween Im σ and Π, Im σ reflects the state structure ofgraphene. Im σ is therefore a non-perturbational probeto detect the state structure of graphene. In this view-point, Im σ is a more basic quantity than Re σ . Further-more, Eq. (23) is irrelevant to the idiographic interac-tions, which means that, the equation holds under verygeneral conditions, such as the existence of impurities orexcitations in graphene.More present works reveal that the graphene is rarelyflat, i.e. there are always ripples in graphene. The non-vanishing curvature, raised by ripples, will lead two maineffects: altering group velocity of quasiparticle and in-troducing effective gauge fields. The first effect possiblymakes a global correction to conductivity, which may beabsorbed into the redefinition of spectral function, Π.Furthermore, since the holding of Eq. (23) is irrelevantto idiographic interactions, we conclude that Kramers-Kr¨onig relation is still valid for corrugated graphene.Eq. (23) supplies one possible way to study the dis-crepancy of dc conductivity between theories and experi-ments. One may first perform perturbational computingto imaginary part of ac conductivity and then comparesthe perturbational result to experiments at different fre-quency. The discrepancy between these results reveals4ontribution which can not be ascribed to perturbationaltheory. Furthermore, the complete contributions to dcconductivity are not only from the Dirac nodal point,but from the spectral structure of carriers. We here deduce the dc conductivity in perturbationalapproximation. After that we shall discuss an ambiguitybesides the one pointed out in Ref. [6].We begin the game by a perturbational calculationof T µν . Noticing < | b p f a p ′ f a † p i b † p ′ i | > = (2 π ) δ ( p f − p ′ i ) δ ( p ′ f − p i ) and the normal ordering of operatorsin current density, we have, for perturbational groundstate, < | J µ ( x ) J µ (0) | > = Z d p d p ′ (2 π ) p p ′ F µµ e − i ( p + p ′ ) x , (24)where F µµ = ¯ v ( p ′ ) γ µ u ( p )¯ u ( p ) γ µ v ( p ′ ). Taking advantageof Eq. (3), one finds, F µµ = − p · p ′ − m = − ( p + p ′ ) − m . (25) T µµ ≡ T is given by a direct computing T ( x ) = i ( θ ( x ) − θ ( − x ))( (cid:3) − m ) K ′ ( x ) , (26)where K ′ ( x ) = Z d p d p ′ (2 π ) p p ′ ( e i ( p + p ′ ) x − e − i ( p + p ′ ) x )= Z d p ( e ipx + e − ipx )(2 π ) p Z d p ( e ipx − e − ipx )(2 π ) p From Z d p ( e ipx + e − ipx )(2 π ) p = Z d pe − ipx (2 π ) δ ( p − m ) , Z d p ( e ipx − e − ipx )(2 π ) p = Z d pe − ipx (2 π ) δ ( p − m )sgn( p ) , Fourier transformation of K ′ is K ′ ( p ) = − Z d q π δ (( p − q ) − m ) δ ( q − m )sgn( q ) . (27)Here p does not need to be on mass shell, i.e. p = p p + m is not needed, if the integrating factor is d p . We focus on the case of p = 0(Or, p is a time-like vector). Letting K ( x ) = ( (cid:3) − m ) K ′ ( x ) and K ( q ) = K ( q , q = ), we get K ( q ) = q + 4 m q θ ( q − m ) . (28) The nonzero contribution to K ( q ) is | q | > m . To sim-plify we let m = 0. Thus T ( x , p = 0) = ( θ ( − x ) − θ ( x )) δ ′ ( x ). Since Π( q , q = 0) = − q T ( q , q = 0), wefind the dc conductivity of graphene σ = 18 Z −∞ dx Z x −∞ dx ( θ ( − x ) − θ ( x )) δ ′ ( x ) , (29)utilizing Eq. (16). Notice that in above equation wehave made a subtraction Π( x , p ) = ∂ Π( x , p ) ∂x = 0 at x → −∞ .However, the functions, such as δ ′ ( x ) and θ ( x ), arenot well defined. This means that there possibly existsambiguity in Eq. (29). This ambiguity is different to theone pointed out in Ref. [6].We consider dc conductivity here. First let δ ( x ) bethe simplest form, δ ( x ) = 0 for | x | > a and δ ( x ) = 1 /a for | x | < a . In this case we obtain σ = = π π ≃ . π utilizing Eq. (29). This is just the result obtained inRef. [4]. Meanwhile, we can also let δ ( x ) be a some-what complex form [6] , δ ( x ) = π ηx + η . At this timewe get σ = π π π ≃ . π . Finally, we can also set δ ( x ) = T cosh ( x T ). We find σ = π (1 / π ≃ . π ≃ π π , numeral value of which is in agreement withthat in ref. [3].To see the physics meaning of T in δ ( x ), we writeout explicitly: θ ( − x ) = e t/T . The role of T is somelike temperature, which means that T − symbolizes thedisorder. a − in δ ( x ) and η in δ ( x ) [6] play the similarrole. Since σ , σ and σ are a − , η − and T -independentrespectively, we conclude that the dc conductivity is al-most temperature-independent near zero temperature,although the conductivity value is ambiguous because ofthe wicked behavior of δ -function. This is verified byexperments [12] .This is a unexpected occasion that the conductiv-ity, a physical observable quantity, varies with differentdefinitions of δ − function. The ambiguity is associatedby the different definitions of δ − function at ultravioletregion. One may argue that we can eliminate the am-biguity by a standard renormalization schedule in quan-tum field theory [9, 10] , however, this elimination is stillcontributed to the special definition of δ − function at ul-traviolet region. We think that the ambiguity impliesthat the dc conductivity of graphene depends on the be-havior of quasielectrons at high energy as well as thebehavior at Dirac nodal point. This is also pointed byKramers-Kr¨onig relation in Eq. (23). Unfortunately,linear dispersion relation of quasielectron does not holdat high energy, which means that, different numeral val-ues based on linear dispersion and perturbational ap-proaches, need corrections. On the other hand, when5e study electrical response of graphene, we always per-form calculations utilizing diagrams composed by dif-ferent Green functions. To include higher corrections,we should use loop diagrams. However, since coupling g = 2 πe /ǫ ~ v F is not small, g ∼
1, comparing to leadingorder, the loop corrections can not be ignored.One possibly expects that the correction to conduc-tivity given above are not large. If this is the case, ourcomputations and others [3, 4] indicate that about 30% ofthe full conductivity is from the perturbational contri-bution. A question is raised, then, where other contribu-tions to conductivity come from. A generalized versionof Eq. (28) tells us that, from the definition of state den-sity s ( q ), perturbational contribution to state density is2 πs pt ( q ) = q + 4 m q θ ( q − m ) , (30)at q > θ function in this equation reveals that, s pt only includes the contribution from pairs of free quasi-electron and hole. However, since there are complex in-teractions between electron and hole, electron and holemay be combined into excitations [14, 15] , or in otherwords, it is questionable to consider quasielectrons ingraphene as 2-dimensional electron gas with no inter-acting. To study electrical responses completely, onemust also consider the contribution of excitations(andimpurities), attributed to Eq. (22). In standard fieldtheory it is difficult to study the contribution pertur-bationally. We often nominate the contribution as non-perturbational one, such as we did in Ref. [13]. Since thecoupling is large on graphene, such contribution can notbe ignored when one consider electrical responses. Ap-parently, if m is large enough, the nonzero contributionfrom exciton appears before q = 4 m . We shall discusssuch contribution elsewhere [11] . The relationship between imaginary part, Im σ , andreal part, Re σ , of ac conductivity is given in paper. Im σ depends directly on details of state structure and onecan study state structure from Im σ . We consider itas a non-perturbational probe to detect state structureof graphene and it is therefore a very important quan-tity. Our formulae are Lorenz-covariant and local-gauge-invariant.We also perform an explicit perturbational calcula-tion using quantum field theory. The computing showsthat the conductivity is mainly manipulated by themomentum-energy relation and there is little nexus be-tween the conductivity and state density near Diracnodal point. The computing reveals that, due to the wicked behavior of δ -function, there is ambiguity ingraphene conductivity calculations. We argue that thefull perturbational studies need two corrections: one isdue to the incorrectness of carrier linear dispersion athigh-energy and the other is higher order correction. Be-sides these corrections, however, there is a furthermorecorrection which is nominated as non-perturbationalcorrections in the paper. This correction comes fromthe contribution of excitations, which is attributed toelectron-electron interactions.Authors are very grateful to Dr. M.G. Xia and Dr.E.H. Zhang. This work is supported by the Ministry ofScience and Technology of China through 973 - projectunder grant No. 2002CB613307, the National NaturalScience Foundation of China under grant No. 50472052and No. 60528008. References [1] M.I. Katsnelson and K.S. Novoselov,cond-mat/0703742.[2] V.P. Gusynin, S.G. Sharapov, J.P. Carbotte, Intern.J. Mod. Phys.
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