Optical conductivity of graphene in the presence of random lattice deformations
aa r X i v : . [ c ond - m a t . s t r- e l ] O c t Optical conductivity of graphene in the presence of random lattice deformations
A. Sinner , A. Sedrakyan , , and K. Ziegler Institute for Physics, Universit¨at Augsburg, Universit¨atsstr. 1, D-86159, Augsburg, Germany Yerevan Physics Institute, Br. Alikhanian 2, Yerevan 36, Armenia (Dated: October 20, 2010)We study the influence of lattice deformations on the optical conductivity of a two-dimensionalelectron gas. Lattice deformations are taken into account by introducing a non-abelian gauge fieldinto the Eucledian action of two-dimensional Dirac electrons. This is in analogy to the introductionof the gravitation in the four-dimensional quantum field theory. We examine the effect of thesedeformations on the averaged optical conductivity. Within the perturbative theory up to secondorder we show that corrections of the conductivity due to the deformations cancel each other exactly.We argue that these corrections vanish to any order in perturbative expansion.
PACS numbers: 73.22.Pr, 72.80.Vp
I. INTRODUCTION
Graphene, a two-dimensional sheet of carbon atomsforming a honeycomb lattice, has outstanding electronicproperties . This is due to the fact that there are twobands that touch each other at two Dirac nodes. More-over, the low-energy quasiparticles of undoped grapheneexperience a linear dispersion around two Dirac nodes.Transport properties, characterized by the longitudinalconductivity at the Dirac nodes, are quite robust and donot vary much from sample to sample. Exactly at theDirac point a minimal conductivity has been observedin a number of experiments . There are two impor-tant questions regarding this minimal conductivity: (I)is the value of the minimal conductivity “universal” (i.e.independent of additional modifications of the graphenesheet such as ripples or impurities) and (II) what is itsactual value in units of e /h ? A discrepancy between thecalculated conductivity of Dirac fermions and the exper-imentally observed minimal conductivity of graphene bya factor of roughly 1 /π has been the subject of a substan-tial number of publications. The central idea is that ei-ther disorder or electron-electron interaction mayaffect the value of the minimal conductivity. Moreover,the value of minimal conductivity at low temperaturesdepends on the order of varies limits (e.g. frequency ω → T →
0) and is related to thescaling property σ min ( ω, T ) = σ min ( ω/T ) . Below wewill employ the zero-temperature formalism which sug-gests T → ω →
0. This yields for the DC limit ofthe AC conductivity the value π/ .An additional problem in terms of disorder is that itis not clear what role is played by different types of dis-order. Since disorder, depending on its type, may breakdifferent internal symmetries of the Dirac Hamiltonian,a classification according of the different types is crucial.On the other hand, the origin of disorder in graphenecan be different. Besides impurities inside the graphenesheet and in the substrate, the deformation of the lat-tice (e.g. ripples) might be the main source of disor-der . In general, it is believed that surface corruga-tions may influence electronic transport properties of graphene. It is crucial to notice that lattice defor-mations do not break the chiral symmetry at the Diracpoint, in contrast to potential disorder or a random gapcaused by a random deposition of hydrogen . There-fore, it is expected that this type of disorder has a ratherweak effect on transport properties . This is supportedby calculations, where the lattice deformations are ap-proximated by an uncorrelated random vector potentialin the Dirac Hamiltonian . This type of disorder has noeffect on the minimal conductivity . More recently, how-ever, a more general theory of lattice deformations withlong-range correlations revealed a dramatic increase ofthe minimal conductivity for weak disorder . In thispaper we will study a similar model by an alternativeapproach to check whether or not this dramatic increaseof the minimal conductivity can be reproduced.First we consider the deformation of the graphene sheetin three dimensions and show that in the continuum limitthe dynamics of the electrons on the two-dimensionalsurface is defined by the so-called induced Dirac ac-tion presented in Ref. [22]. In our approach the inter-nal deformations of the graphene sheet and the defor-mations perpendicular to the sheet direction are unifiedinto one schema, while in the approach developed in pa-pers there are separate internal 2D gravity andadditional non-abelian gauge fields. The deformations ofthe sheet in three dimensions by local SO (3) rotationsof the basic vectors in our approach carry the degrees offreedom of the additional gauge field.Then we develop a replica-trick based field theory totake the random character of surfaces into account andto calculate the average optical conductivity by a pertur-bative expansion. Our main result is that the randomlattice deformations do not affect the robust character ofminimal conductivity, contrary to the result presented inRef. [17]. II. THE MODEL
We depart from a model of hopping fermions on theregular 2D honeycomb lattice. Honeycomb lattice has ne e e µ µ = FIG. 1: Hopping vectors on regular honeycomb lattice natural partition into two triangular sub-lattices and wemark electronic fields associated with sites of the sub-lattices as ( ¯ ψ ~n,α , ψ ~n,α ) , α = 1 ,
2. The action of electronshopping on a line with the lattice spacing | ~e | reads S [ ¯ ψ, ψ ] = i X t,~n ( ¯ ψ t,~n ∂ t ψ t,~n + ¯ ψ t,~n γ ψ t,~n + ~e ) , but when fermions change hopping direction in two di-mensional space they fields should also be rotated by acorresponding angle (Fig. 1). On the honeycomb lattice(Fig. 2) we have S [ ¯Ψ , Ψ] = i X t,~n,i (cid:0) ¯Ψ t,~n ∂ t Ψ t,~n + ¯Ψ t,~n γ Ψ ′ t,~n + ~e i (cid:1) , (1)Ψ t,~n = (cid:18) ψ t,~n, ψ t,~n, (cid:19) , i = 1 , , , where γ , γ j , j = 1 , γ = σ , γ = σ , and γ = σ and fieldsΨ ′ ~n + ~e = Ψ ~n + ~e = e ~e · ~∂ Ψ ~n , (2a)Ψ ′ ~n + ~e = e i π γ Ψ ~n + ~e = e i π γ e ~e · ~∂ Ψ ~n , (2b)Ψ ′ ~n + ~e = e − i π γ Ψ ~n + ~e = e − i π γ e ~e · ~∂ Ψ ~n , (2c)are rotated by ± π/ ~e , spinor repre-sentations of the rotation group SO (3). In the paper bySemenoff was shown that the spectrum of low-energyexcitations of the hopping fermions on honeycomb lattice(corresponding to the continuum limit of the model) co-incides with the spectrum of Dirac fermions in 3D space.Below we will show that the continuum limit of the ac-tion of fermions hopping on honeycomb lattice Eq. (1) isdefined by the Dirac action in three dimensional coordi-nate space. This will allow to construct the continuumlimit of the generalized hopping model on the randomlydeformed lattice. e e e FIG. 2: (Color online) Regular honeycomb lattice.
In order to find a continuum limit of the action Eq. (1)one expands translational operators e ~e i · ~∂ ≃ ~e i · ~∂ andsubstitute Eqs. (2a) for Ψ’s into the action Eq. (1). Thenafter some simple algebra one will obtain: S [ ¯Ψ , Ψ] = i X t,~n ¯Ψ t,~n (cid:18) ∂ t + 34 γ i ~µ i · [ ←− ∂ − −→ ∂ ] (cid:19) Ψ t,~n (3) → i Z d ξdt ¯Ψ (cid:18) ∂ t + 12 γ i ~µ i · [ ←− ∂ − −→ ∂ ] (cid:19) Ψ , (4)where we introduced orthonormalized vectors ~µ = ( ~e − ~e ) / √ ~µ = ~e . In the line (4) we have rescaledthe fields and coordinates as Ψ → / ξ → ξ/ t → t and µ i → µ i . It is clear from Eq. (4) that vec-tors ~µ a = µ ia ( ~ξ )ˆ e i , a = 1 ,
2, with ˆ e i , i = 1 , ξ i , i = 1 ,
2. Indeed, consider deforma-tion of the honeycomb lattice (cf. Fig. 3) and attachto the sites a new coordinates ξ ′ i . Then the vectors µ ia , a = 1 , , i = 1 , ξ i , i = 1 , µ ia ( ~ξ ) = ∂ξ ′ j ∂ξ i µ ja ( ~ξ ′ ) . (5)We regard now the vectors µ ai as vielbeins in a 2D planewhich obey the orthogonality relation µ ai µ a,j = δ ij anddefine the metric µ ai µ aj = g ij . After integration by partsin Eq. (3) and using the relation ˆ µ i ˆ µ j = g ij + i √ g ǫ ij γ with ˆ µ i = γ a µ ai and g = det[ g ij ] one will obtain S [ ¯Ψ , Ψ] = i Z d ξdt ¯Ψ (cid:18) ∂ t + γ a µ ja (cid:2) ∂ j − i γ Γ j (cid:3)(cid:19) Ψ , (6)where Γ j = i √ g ǫ ab µ ka ∇ j µ k,b is a standard spinor connec-tion corresponding to the vielbein µ ja and ∇ j denotes a e e e FIG. 3: (Color online) Random honeycomb lattice. covariant derivative. For a scalar function f it reduces toa usual partial derivative: ∇ i f = ∂ i f , while for a vectorvalued function f j it is ∇ i f j = ∂ i f j + Γ kij f k , where Γ kij represent Christoffel symbols.Let us now consider deformations of the honeycomblattice in a three dimensional space . This means thattwo γ a , a = 1 , γ matrices in a SO (3) rotated plane which is tangent tocurved surface at the point ξ i :ˆ x a ( ξ i ) = U ( ξ i ) − γ a U ( ξ i ) . (7)As it is shown in Ref. [22] local rotations by U ( ~ξ ) producea 2D surface embedded into 3D Euclidean space if U − ∂ µ U = 14 (cid:0) ˆ x a ∂ µ ˆ x a + ˆ n∂ µ ˆ n (cid:1) , (8)where ˆ x a = µ µa ∂ µ x α γ α and ˆ n = n α γ α , α = 0 , , ξ respectively. This will occur since Eq. (8)will fulfill the Gauss-Codazzi equations , which rep-resents the necessary conditions for the surface x α to beembedded into 3D Euclidean space.Then we should rotate also fermionic fields by the samematrices Ψ → U Ψ, after which the action becomes S = i Z d ξdt ¯Ψ U − (cid:18) ∂ t + γ a µ µa [ ∂ µ − i γ Γ µ ] (cid:19) U Ψ , (9)where Γ µ = i Tr(ˆ nU − ∂ µ U ) is the spinor connection onthe surface x α . By use of Eq. (7) this expression can besimplified essentially (see details in Refs. [22,26]) acquir-ing the form S [ ¯Ψ , Ψ] = i Z d ξdt ¯Ψ (cid:18) ∂ t + 12 √ g ˆ γ µ (cid:2) ←− ∂ µ − −→ ∂ µ (cid:3)(cid:19) Ψ , (10)where g νµ = ∂ ν x · ∂ µ x = ∂ ν x α ∂ µ x α (11) is the metric on the surface x α ( ~ξ ) induced by its embed-ding into 3D Euclidean space, g denotes its determinantand ˆ γ µ = ∂ µ x · γ = ∂ µ x a γ a (12)represent the induced Dirac matrices . In a flat space,i.e. for x ( ~ξ ) = x + ˆ e µ ξ µ the induced metric reduces to ausual diagonal matrix. One can call action Eq. (10) theinduced Dirac action since the matrices ˆ γ µ are induced byembedding. The expression in Eq. (10) is a generalizationof 2D action Eq. (4) to 3D space. III. EFFECTIVE ACTION FOR SMALLCORRUGATIONS
Performing integration by parts in Eq. (10) we arriveat S [ ¯Ψ , Ψ] = i Z d ξdt ¯Ψ (cid:0) ∂ t + √ g ˆ γ µ [ ∂ µ + Γ µ ] (cid:1) Ψ . (13)Here, the quantity Γ µ = 12 ˆ γ ν ∇ µ ˆ γ ν (14)plays the role of an induced spinor connection, where ∇ µ denotes the operator of covariant differentiation and isdefined as ∇ µ ( · · · ) = 1 √ g ∂ µ ( √ g · · · ) . Let us derive the asymptotic action for small corruga-tions of the graphene sheet. In this case the surface x can be asymptotically represented as x ( ξ , ξ ) ≈ x + ˆ e µ ξ µ + x ′ ( ξ , ξ ) , (15)Plugging Eq. (15) into Eq. (11) we obtain the asymptoticsof the metric tensor: g νµ ≈ δ νµ + ǫ νµ + ǫ µν , (16)where ǫ νµ = ˆ e ν · ∂ µ x ′ . (17)Thus the metric tensor is in general neither diagonal norsymmetric. Its determinant is found using common rela-tions g ≈ ǫ + 2 ǫ = 1 + 2 ǫ νν , (18)and correspondingly its square root: √ g ≈ ǫ νν . (19)Using Eqs. (12), (15 ) and (19) we arrive at the effectiveaction for small fluctuations ǫ : S [ ¯Ψ , Ψ; ǫ ] ≈ i Z d ξdt ¯Ψ (cid:0) γ ∂ t + [1 + ǫ νν ]˜ γ µ ∂ µ (cid:1) Ψ+ i Z d ξdt ¯Ψ˜ γ µ ∂ µ ǫ νν Ψ , (20)with induced γ − matrices ˜ γ µ = e µa γ a . For further pur-poses we associate the spatial fluctuations with a bosonicfield ǫ νν = Λ( ~ξ, t ) , and its gradient with a static vector-disorder like term: ∂ µ ǫ νν = ∂ µ Λ( ~ξ, t ) = B µ ( ~ξ, t ) . Hence the action formally becomes S [ ¯Ψ , Ψ; Λ , B ] ≈ i Z d ξdt ¯Ψ (cid:0) γ ∂ t + ˜ γ µ ∂ µ (cid:1) Ψ+ i Z d ξdt Λ ¯Ψ˜ γ µ ∂ µ Ψ+ i Z d ξdt ¯Ψ˜ γ µ B µ Ψ . (21)The action derived this way reproduces the ansatz actionconsidered in Ref. [17].Below we consider topologic defects in the flat space.Technically that means that we replace zweibeins e µa bya unity-matrix. We are ultimately interested in the effectof this sort of the disorder on the optical conductivity. Inorder to perform such calculations we have to make somesuggestions regarding the correlators of the introducedquantities. One usually requires the vector disorder fields B µ to be gaussian correlated, i.e. h B µ ( ~ξ, t ) i = 0 , (22) h B µ ( ~ξ, t ) B ν ( ~ξ ′ , t ′ ) i = g δ µν δ ( t ) δ ( t ′ ) δ ( ~ξ − ~ξ ′ ) , (23)which guarantees that the vector associated with the ran-dom disorder is static. In Fourier-space these expressionsread with the short-hand Q = ( q , q ): h B µ ( Q ) i = 0 , (24) h B µ ( Q ) B ν ( Q ′ ) i = g (2 π ) δ µν δ ( q ) δ ( Q + Q ′ ) . (25)From Eqs. (23) and (25) we are lead to the correlators ofthe scalars Λ, since we have an exact relationship h B µ ( Q ) B ν ( Q ′ ) i = i q µ q ′ ν h Λ( Q )Λ( Q ′ ) i , (26)which leads to h Λ( Q )Λ( Q ′ ) i = g q + µ (2 π ) δ ( q ) δ ( Q + Q ′ ) , (27)where we have introduced an infrared cutoff µ of the orderof the inverse lattice spacing in order to avoid long wave-length divergences. Inverse Fourier transform yields forthe h ΛΛ i− correlator h Λ( ~ξ, t )Λ( ~ξ ′ , t ′ ) i = g δ ( t ) δ ( t ′ ) log (cid:12)(cid:12)(cid:12) µ ( ~ξ − ~ξ ′ ) (cid:12)(cid:12)(cid:12) . (28)Furthermore we will always assume h Λ( ~ξ, t ) i = h Λ( Q ) i = 0 . (29) The Fourier transform of Eq. (20) expressed in terms ofthe scalar fields Λ only reads S [ ¯Ψ , Ψ , Λ] = − Z Q ¯Ψ Q ( q γ + γ · q )Ψ Q − Z Q Z P Λ P ¯Ψ P + Q Γ( P + Q, P, Q )Ψ Q , (30)where γ · q = γ µ q µ . The two-particle vertex is obtainedfrom Eq. (30) in limit Λ → P, Q ) = (2 π ) δ ( P − Q ) G − ( Q ) , (31)where G − ( Q ) = q γ + γ · q , (32)represents the inverse free propagator and correspond-ingly G ( Q ) = q γ + γ · q q + q (33)the free Dirac propagator. For the three-particle vertexfunction Γ( K ¯Ψ , K Λ , K Ψ ) follows from Eq. (30)Γ( K ¯Ψ ; K Λ , K Ψ ) = (2 π ) δ ( K ¯Ψ − K Λ − K Ψ ) × γ · ( k Λ + 2 k Ψ ) . (34)Furthermore we have to augment Eq. (30) by the in-teraction between fermions and the radiation field S opt [ ¯Ψ , Ψ , A ] = − Z P Z Q A P ¯Ψ P + Q γ Ψ Q , (35)which suggests the presence of an electric field appliedto the graphene sheet. Interaction Eq. (35) gives rise tothe optical conductivity due to polarization of the chargecarriers. The corresponding bare vertex is defined asΓ ( K ¯Ψ , K A , K Ψ ) = (2 π ) δ ( K ¯Ψ − K A − K Ψ ) γ . (36)The full action acquires the form¯ S [ ¯Ψ , Ψ , Λ , A ] = S [ ¯Ψ , Ψ , Λ] + S opt [ ¯Ψ , Ψ , A ] . (37)The electron-gauge boson interaction renormalizes elec-tronic spectrum and therefore should have an effect onthe response to the radiation field. IV. OPTICAL CONDUCTIVITY OFGRAPHENE
We consider first ideal graphene. The correspondingEucledian action is obtained from Eq. (37) if we assume
K−K P+KP
FIG. 4: Bare polarization bubble −K KP+KP
FIG. 5: Dressed polarization bubble
Λ = 0: S [ ¯Ψ , Ψ; A ] = − Z Q ¯Ψ Q ( q γ + γ · q )Ψ Q − Z P Z Q A ( P − Q ) ¯Ψ P γ Ψ Q , (38)The optical conductivity of 2D Dirac electron gas canbe calculated from the electronic polarization σ = 4 k lim k → ∂ ∂k Π( K ) , (39)where Π( K ) denotes the irreducible polarization. Fac-tor 4 in front of this expression arises from the takingboth spin and valley degeneracy into account. To theleading order it is given by the diagram shown in Fig. 4.Algebraically we have for the polarization bubble:Π( K ) = Z P Tr { γ G ( P ) γ G ( K + P ) } , (40)with the bare Dirac propagators G ( Q ) defined inEq. (33). We give some details of the calculation in theAppendix. The irreducible polarization is obtained asΠ( K ) = 116 k p k + k , (41)and the optical conductivity in SI-units is σ = 14 e ~ = π e h . (42)In what follows we calculate corrections of the conduc-tivity in Eq. (42) due to lattice deformations described in Sec. III. To the leading order in momenta the renor-malized inverse fermionic propagator can be written as G − ( Q ) = G − ( Q ) − Σ( Q ) ≈ Z − q γ + Z − γ · q , (43)with renormalization factors Z − , = 1 − γ ,µ ∂∂q ,µ Σ( Q ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Q =0 = γ ,µ ∂∂q ,µ G − ( Q ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Q =0 , (44)where Σ( Q ) denotes the fermionic self-energy. On an-other hand, the dressed electron-photon vertex can bewritten in the following from:˜Γ ( Q ) ≈ ˜ eγ + O ( Q ) , (45)where ˜ e denotes the renormalization of the elementarycharge due to lattice deformations, such that the effectiverenormalized action reads˜ S [ ¯Ψ , Ψ; A ] ≈ − Z Q ¯Ψ Q ( Z − q γ + Z − γ · q )Ψ Q − ˜ e Z P Z Q A ( P − Q ) ¯Ψ P γ Ψ Q . (46)The effect of the lattice defects on the optical con-ductivity can be calculated from the dressed polarizationshown in Fig. 5. The dressing effect of the lattice defectsis taken into account by replacing bare Greens functions G in Fig. 4 by the full propagators G defined in Eq. (43)and bare vertices γ by the dressed ones ˜Γ from Eq. (45).Algebraically we obtain˜Π( K ) = Z P Tr { ˜Γ G ( P )˜Γ G ( K + P ) } = 2˜ e Z − Z P p ( p + k ) − α p · ( p + k )[ p + α p ][( p + k ) + α ( k + p ) ] , (47)where α = Z − /Z − . The integration can be performedwith some effort, but we restrict our attempts to a moredirect task, i.e. we calculate only the modified opticalconductivity ˜ σ analogously to the optical conductivity ofideal graphene as we did before. We take the derivativewith respect to k under the integral and employ residuetheorem in order to integrate out loop frequency p . Atthe end of the calculation we arrive at˜ σ = ˜ e Z σ . (48)Surprisingly, apart from the vertex renormalization onlyrenormalization of the frequency contributes to the mod-ificated conductivity. Therefore our task reduces to thecalculation of the renormalization factors Z and ˜ e whichis performed perturbativelly below. Γ Γ
Q PP Q−P = + Σ , Σ ∼∼ FIG. 6: Dressed fermionic Greens function and leading orderdiagram of the fermionic self-energy. Dashed lines denote the h ΛΛ i -correlator and white circles the ¯ΨΛΨ-vertices. Γ Γ ΓΓ a) Q−Q P−P
Γ ΓΓ Γ b) Q −Q−P P
FIG. 7: Second order self-energy corrections at zero externalmomenta.
V. CALCULATION OF RENORMALIZATIONFACTORS
In order to set up perturbative calculations we have toaverage over the lattice deformations. There are two pos-sible ways to implement such averaging: The replica trickand supersymmetry approach. The calculation below isbased on the replica-trick approach.According to the replica trick we introduce N copiesof fermions Ψ α , α = 1 , , . . . N with the same action¯ S = − Z Q ¯Ψ αQ ( q γ + γ · q )Ψ αQ − Z Q Z P Λ P ¯Ψ αP + Q Γ( P + Q, P, Q )Ψ αQ − Z Q Z P A P ¯Ψ αP + Q γ Ψ αQ . (49)Then we will calculate the diagrams describing renormal-ization of the fermionic propagator and electron-photonvertex function and perform limit N → N will vanish. These include for instancecontributions arising from diagrams containing closedfermionic loops.The diagrams of the fermionic self-energy to the order1 in replica indices and order g in lattice deformationsstrength are shown in Figs. 6 and 7. Correspondingly, thesame order diagrams of vertex corrections are depicted inFigs. 8 and 9. Retaining only frequency dependence ( p =0) in the analytical expressions for this contributions weobtain for the leading self-energy contribution (Fig. 6)Σ (2) ( p ) = g Z q Γ G ( p , q )Γ F ( q ) , (50) where Γ = Γ = γ · q and F ( q ) = 1 q + µ , (51)denotes the momentum dependent part of the h ΛΛ i− correlator defined in Eq. (28). The diagramof the next order in g depicted in Fig. 7a readsΣ (2)1 ( p ) = g Z q F ( q ) Z p F ( p ) Γ G ( p , q )Γ × G ( p , q + p )Γ G ( p , q )Γ , where Γ = γ · q and Γ = γ · (2 q + p ), while the diagramshown in Fig. 7b writesΣ (2)2 ( p ) = g Z q F ( q ) Z p F ( p ) Γ G ( p , q )Γ × G ( p , q + p )Γ G ( p , p )Γ , with the vertices Γ = γ · q , Γ = γ · (2 q + p ), Γ = γ · (2 p + q ) and Γ = γ · p . Eventually we obtain forthe contributions to the wave-function renormalizationfactor ∂∂p Σ (2) ( p ) (cid:12)(cid:12)(cid:12)(cid:12) p =0 = − ˆ e (52) ∂∂p Σ (4)1 ( p ) (cid:12)(cid:12)(cid:12)(cid:12) p =0 = − e , (53) ∂∂p Σ (4)2 ( p ) (cid:12)(cid:12)(cid:12)(cid:12) p =0 = −
152 ˆ e , (54)where we define ˆ e = g π log λµ , (55)with λ denoting some upper momentum cutoff. There-fore we obtain for the wave-function renormalization tothe second order in g Z − ≈ e + 332 ˆ e + O (ˆ e ) . (56)Now we look at the renormalization of the electron-photon vertex function. Diagrammatically the leadingorder correction is given by the second term on the righthand side of the diagram shown in Fig. 8. According toEq. (48) the main corrections to the conductivity arisefrom the momentum independent part of the vertex func-tion. We obtain for the vertices Γ = Γ = γ · q . Thealgebraic expression for the diagram depicted in Fig. 8 isgiven by˜Γ (1) (0) = g Z q Γ G ( q )Γ G ( q )Γ F ( q ) , (57) γ + γ ~~ Γ ~ −QQ FIG. 8: Leading order perturbative contribution to the ampu-tated electron-phonon coupling vertex. Dashed lines denotethe h ΛΛ i -correlator and white circles the ¯ΨΛΨ-vertices. Γ Γ a) ΓΓ Q−Q −PP γ Γ Γ b) ΓΓ Q−P −QP γ Γ Γ c) Γ Γ Q −Q−P P γ Γ Γ d) ΓΓ −QQ −P P γ FIG. 9: Second order vertex correction diagrams at zero ex-ternal momenta. The diagrams in the second row should becounted twice due to the mirror symmetry. which yields after the evaluation˜Γ (1) (0) = ˆ eγ , (58)where ˆ e from Eq. (55) is introduced and acquires themeaning of the leading order elementary charge renor-malization.Second order vertex corrections can be calculated fromthe diagrams shown in Fig. 9. Due to the mirror sym-metry diagrams c) and d) depicted in the second rowin Fig. 9 should be counted twice. Let us first considerthe contribution that arises from the diagram depicted inFig. 9a with two parallel ladder rungs. At zero externalmomenta we obtain for the vertices Γ = Γ = γ · q andΓ = Γ = γ · (2 q + p ). We obtain for the correction˜Γ (2)1 (0) = g Z q F ( q ) Z p F ( p ) Γ G ( q )Γ G ( p + q ) × Γ G ( p + q )Γ G ( q )Γ , (59)which, after performing integrations, yields˜Γ (2)1 (0) = 2ˆ e γ . (60)For diagrams depicted in Figs. 9b, c and d we proceedsimilarly. In the case of diagram b) we have the followingexpressions for the vertices: Γ = γ · q , Γ = γ · (2 q + p ),Γ = γ · (2 p + q ) and Γ = γ · p . Therefore the expression for this correction reads˜Γ (2)2 (0) = g Z q F ( q ) Z p F ( p ) Γ G ( q )Γ G ( p + q ) × Γ G ( p + q )Γ G ( p )Γ , (61)with the result ˜Γ (2)2 (0) = 52 ˆ e γ . (62)For diagram c) we have the following vertices: Γ = γ · q ,Γ = γ · (2 q + p ), Γ = γ · (2 p + q ) and Γ = γ · p , whereasthe expression for the correction reads˜Γ (2)3 (0) = g Z q F ( q ) Z p F ( p ) Γ G ( q )Γ G ( q )Γ × G ( p + q )Γ G ( p )Γ , (63)which yields the result˜Γ (2)3 (0) = 52 ˆ e γ . (64)Finally, diagram d) from Fig. 9 can be written alge-braically as follows:˜Γ (2)4 (0) = g Z q F ( q ) Z p F ( p ) Γ G ( q )Γ G ( q )Γ × G ( p + q )Γ G ( q )Γ , (65)where the vertices Γ ··· are given by Γ = Γ = γ · q andΓ = Γ = γ · (2 q + p ). Evaluation of Eq. (65) yields˜Γ (2)4 (0) = 72 ˆ e γ . (66)Hence, the second order contribution to the vertexfunction becomes˜Γ (2) (0) = ˜Γ (2)1 (0) + ˜Γ (2)2 (0) + 2˜Γ (2)3 (0) + 2˜Γ (2)4 (0) , (67)such that the dressed vertex function can be written asa series in ˆ e :˜Γ = ˜ eγ ≈ e (cid:18) e + 332 ˆ e + O (ˆ e ) (cid:19) γ . (68)This expression reproduces exactly the result which wehave obtained for the dressed vertex function in Eq. (56).Therefore we obtain from Eqs. (48), (68) and (56) for themodified conductivity˜ σ ≈ (cid:0) O (ˆ e ) (cid:1) σ , (69)i.e. the leading correction is of the order g . However,we can show to every order in perturbative expansionthat corrections arising from the propagator renormaliza-tion are exactly canceled by their counterparts departingfrom the electron-photon vertex renormalization. Con-sider the definition of the quasi-particle weight Z − givenin Eq. (44). Since the lattice deformations are static, allpropagators inside the diagram depend only on the ex-ternal Matsubara-frequency which thus becomes an in-dependent parameter. Hence the derivative with respectto the Matsubara frequency should be applied to everypropagator. Taking such a derivative of an average freepropagator h G ( q , q + k ) i k = Z k q γ + ( q µ + k µ ) γ µ q + ( q + k ) , (70)at zero external momentum and frequency we obtain ∂∂q h G ( q , q + k ) i k (cid:12)(cid:12)(cid:12)(cid:12) Q =0 = −h G (0 , k ) γ G (0 , k ) i k . (71)Equation (71) suggests that the expressions under theintegrals must be equal up to an irrelevant constant.Therefore each derivative of the free propagator with re-spect to the external frequency generates upon sendingexternal momenta and frequency to zero a bare electron-photon vertex. An irreducible n − th order diagram of theelectronic self-energy contains 2 n − q to such a diagram, 2 n − n -th ( n >
1) term in the perturbative series of self-energyto the corresponding vertex function correction: ∂∂q Σ ( n ) ( Q ) (cid:12)(cid:12)(cid:12)(cid:12) Q =0 = − ˜Γ ( n )0 (0 , , . (72)Summing over all n and subtracting γ on both sides wethen can assemble all contributions arriving at ∂∂q G − ( Q ) (cid:12)(cid:12)(cid:12)(cid:12) Q =0 = ˜Γ (0 , , . (73)On the right-hand side we have the charge renormaliza-tion ˜ eγ defined in Eq. (45), while the left-hand side rep-resents the wave-function renormalization factor Z − γ due to Eq. (43). Therefore Eq. (73) postulates the equal-ity Z − = ˜ e, which leads to the exact result for the modified conduc-tivity: ˜ σ = σ . (74) Importantly Eq. (73) is obtained without special empha-size on a disorder type and is not restricted to the con-sidered type. The only requirement we need is that thecorresponding term should not violate the chiral sym-metry of the pure graphene Hamiltonian and must be aquenched disorder. VI. CONCLUSIONS
In the present paper we address the question of the ef-fect which random deformations may have on the trans-port in graphene. The common believe is that surfacecorrugations in graphene influence its electronic trans-port properties, mainly the optical conductivity. It ispossible to describe deformations in graphene by a gaugefield that couples to the fermions living on the two di-mensional sheet. We have performed perturbative calcu-lations of the corrections due to lattice deformations tothe optical conductivity. Our results contrast the sugges-tions made in Refs. [17,21] where a substantional effectof the defects on the conductivity is proposed. We havefound that the minimal conductivity is robust with re-spect to the surface corrugations.
ACKNOWLEDGEMENTS
We acknowledge financial support by the DPG-grantZI 305/5-1.
Appendix
Below be evaluate the irreducible polarization of idealgraphene starting with Eq. (40). Upon performing thetrace over the pseudo-spin space we arrive atΠ( K ) = 2 Z P p ( p + k ) − P · ( P + K ) P ( P + K ) , where P · K = p k + p · k . Employing the Feynmanparametrization1 AB = Z dx [ xA + (1 − x ) B ] , and shifting P → P − xK we symmetrize the denominatorwith respect to P . Therefore odd powers of P appearingin the numerator may be dropped. We arrive atΠ( K ) = 2 Z P Z dx p − P + x (1 − x )[ K − k ][ P + x (1 − x ) K ] . Exploiting the rotational invariance we replace p = P / Z d d k (2 π ) d k + ∆) n = 1(4 π ) d/ Γ( n − d )Γ( n )∆ n − d/ , Z d d k (2 π ) d k ( k + ∆) n = 1(4 π ) d/ d n − d − n )∆ n − d/ − , which yields after integrating out x the result of Eq. (41). K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M.I. Katsnelson, I. V. Grigorieva, S. V. Dubonos, and A. A.Firsov, Nature (London) , 197 (2005). K. S. Novoselov, E. McCann, S. V. Morozov, V. I. Fal’ko,M. I. Katsnelson, U. Zeitler, D. Jiang, F. Schedin and A.K. Geim, Nat. Phys. , 177 (2006). M. I. Katsnelson, K. S. Novoselov, and A. K. Geim, Nat.Phys. , 620 (2006). Y. Zhang, Y.-W. Tan, H.L. Stormer, and P. Kim, Nature(London) , 201 (2005). E. Fradkin, Phys. Rev. B , 3263 (1986). P. A. Lee, Phys. Rev. Lett. , 1887 (1993). A. W. W. Ludwig, M. P. A. Fisher, R. Shankar, and G.Grinstein, Phys. Rev. B , 7526 (1994). K. Ziegler, Phys. Rev. B , 10661, (1997); Phys. Rev.Lett. , 3113 (1998). E. G. Mishchenko, Phys. Rev. Lett. , 216801 (2007); E.G. Mishchenko, Europhys. Lett. , 17005 (2008). D. E. Sheehy and J. Schmalian, Phys. Rev. Lett. ,226803 (2007). I. F. Herbut, V. Juri˘ci´c, and O. Vafek, Phys. Rev. Lett. , 046403 (2008). F. de Juan, A. G. Grushin, M. A. H. Vozmediano, Phys.Rev. B , 125409 (2010). K. Ziegler, Phys. Rev. B , 233407 (2007). M. Ishigami, J. H. Chen, W. G. Cullen, M. S. Fuhrer, andE. D. Williams, Nano Lett. , 1643 (2007). J. C. Meyer, A. K. Geim, M. I. Katsnelson, K. S.Novoselov, T. J. Booth, and S. Roth, Nature (London) , 60 (2007). E. Stolyarova, K. T. Rim, S. Ryu, J. Maultzsch, P. Kim, L. E. Brus, T. F. Heinz, M. S. Hybertsen, and G. W. Flynn,Proc. Natl. Acad. Sci. U.S.A. , 9202 (2007). A. Cortijo and M. A. H. Vozmediano, Phys. Rev. B ,184205 (2009). D. V. Khveshchenko, Europhys. Lett. , 57008 (2008). K. Ziegler, Phys. Rev. Lett. , 126802 (2009); Phys.Rev. B , 195424 (2009); J. H. Bardarson, M. V.Medvedyeva, J. Tworzydlo, A. R. Akhmerov, C. W. J.Beenakker, Phys. Rev. B , 121414(R) (2010); K. Ziegler,A. Sinner, Phys. Rev. B , 241404(R) (2010). A random vector potential has a strong effect on the localdensity of states fluctuations, leading to long-range corre-lations [K. Ziegler, Phys. Rev. B , 125401 (2008)]. M. A. H. Vozmediano, M. I. Katsnelson, and F. Guinea,
Accepted for publishing at Phys. Rep. , arxiv: 1003:5179,(2010). A. R. Kavalov, I. K. Kostov, and A. G. Sedrakyan, Phys.Lett. B , 331 (1986). A. Cortijo and M. A. H. Vozmediano, EPL , 47002(2007); A. Cortijo and M. A. H. Vozmediano, Nucl. Phys. B ,293 (2007); F. de Juan, A. Cortijo and M. A. H. Vozmediano, Phys.Rev. B , 165409 (2007). A. Sedrakyan and R. Stora, Phys. Lett. B , 442 (1987). B. A. Dubrovin, A. T. Fomenko, and S. P. Novikov,
Mod-ern geometry - methods and applications , Pt. 1, Springer,New York (1984). G. Semenoff, Phys. Rev. Lett.53