PHYSICAL REVIEW B 85, 033403 (2012)

Layer-resolved conductivities in multilayer graphene

Takeo Wakutsu,1 Masaaki Nakamura,1 and Balazs Dora2

1Department of Physics, Tokyo Institute of Technology, Ookayama, Meguro-ku, Tokyo 152-8551, Japan2Department of Physics, Budapest University of Technology and Economics, Budafoki ut 8, 1111 Budapest, Hungary

(Received 13 December 2011; published 17 January 2012)

We study interlayer transport of multilayer graphenes in a magnetic field with various stacking structures(AB, ABC, and AA types) by calculating the Hall and longitudinal conductivities as functions of the Fermi energy.Their behavior depends strongly on the stacking structures and selection of the layers. The Hall conductivitybetween different layers is no longer quantized. Moreover, for AB stacking, the interlayer conductivity vanishesaround zero energy with increasing layer separation, and shows negative values in particular cases. The fact thatlongitudinal interlayer conductivity is suppressed by a magnetic field indicates that this system can be applied asa switching device.

DOI: 10.1103/PhysRevB.85.033403 PACS number(s): 72.80.Vp, 71.70.Di, 73.22.Pr, 81.05.ue

Introduction. Graphene has attracted increasing attentionin recent years.1 It consists of a two-dimensional hexagonallattice of carbon atoms, whose quasiparticles are governed bya massless Dirac equation. A variety of unusual phenomenaare observed in this system such as high mobility of chargecarriers, anomalous quantum Hall effect, and so on.2–4 Inaddition, bilayer graphene has also been studied intensely,5–10

which is characterized by intrinsic Landau level degeneracy atzero energy and a gate tunable band gap.

Due to the qualitative differences between mono- andbilayer graphene, attention has been focused on multilayergraphenes to determine how additional layers influence theirphysical response. One of the most intriguing property of thesesystems is variety of stacking structures such as AB (Bernal),ABC (rhombohedral), and AA (simple hexagonal) types.Graphene is usually produced by micromechanical cleavage ofgraphite, so that the common stacking structure is of AB type,since the natural graphite falls into this category. However,production of graphene with other stacking types is alsopossible by recent technology such as epitaxial methods.11,12 Inaddition, AA stacking can be realized by folding of a graphenesheet.13 In terms of band structure, multilayer graphenes withmore than ten layers are regarded as bulk graphites14 so thatfew layer graphenes have been considered to be importantsystems, interpolating between graphene and graphite. So far,diamagnetism,15 transport properties,16–23 band tunable bandgap,24,25 and energy spectra26–28 have been studied for thesesystems.

In this Brief Report, we investigate the transport propertiesbetween two different layers of multilayer graphenes in amagnetic field, using the Kubo formula. For example, thissituation is realized when we measure the voltage drop orinduced current in the top layer, while electric current isrunning through the bottom layer, as illustrated in Fig. 1. Inorder to calculate these “layer-resolved conductivities,” we es-tablish the formalisms to obtain eigenvalues and eigenstates ofmultilayer systems, using the block diagonalization technique.We find many interesting properties such as negative responseand switching effect by a magnetic field, absent in monolayergraphene.

Formalism. We consider three types of the stacking struc-tures of multilayer graphenes, AB, ABC, and AA types. Sincea monolayer graphene consists of two sublattices labeled by A

and B; carbon atoms in multilayer graphenes are specified by(Ai ,Bi) meaning the A and B sublattices in the i-th layer,respectively. Figure 2 shows the lattice structure of thesesystems with nearest neighbor intralayer (interlayer) couplingt (t⊥).

Among these three types, we discuss AB stackedgraphene in detail which is the most common structure ofgraphite. After taking the continuum limit of tight-bindingmodel, in a basis with atomic components for N layers,|A1〉,|B1〉, . . . ,|AN 〉,|BN 〉, the model Hamiltonian in the vicin-ity of K point (per spin and per valley) is

H =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣

H0 V 0 0 0

V † H0 V † 0 0

0 V H0 V 0

0 0 V † H0. . .

0 0 0. . .

. . .

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦

, (1)

with

H0 =[

0 vπ−vπ+ 0

], V =

[0 0t⊥ 0

], (2)

where π± ≡ πx ± iπy with π ≡ p + eA/c being the mo-mentum operator in a magnetic field ∇ × A = (0,0,B). v =(√

3/2)αt/h is the Fermi velocity with α being the latticeconstant. We have ignored long-range hopping terms except fort and t⊥ for simplicity. Since the commutation relation betweenthe momentum operators in Eq. (2) is [π±,π∓] = ∓2eBh/c,there are correspondences with the creation and annihilationoperators of the harmonic oscillator: π± → √

2 hla† and π∓ →√

2 hla for eB ≷ 0, where l ≡ √

ch/|eB|.In order to solve this model, we employ the matrix

decompositions of the Hamiltonian. It is already known thatthe Hamiltonians of AB and AA-stacked N -layer graphenescan be block diagonalized, considering Fourier modes ofthe wave function along the stacking direction.15,20,23 Thesame conclusion can also be obtained by factorization ofdeterminant of the Hamiltonians.16 According to these, theeffective Hamiltonian of AB-stacked N -layer graphenes canbe divided into isolated [N/2]G effective bilayer systems([x]G is the integer part of x), and one monolayer system

033403-11098-0121/2012/85(3)/033403(4) ©2012 American Physical Society

BRIEF REPORTS PHYSICAL REVIEW B 85, 033403 (2012)

induced current

SSubstrate

bias current

D

1st layer

N-th layer

FIG. 1. (Color online) Schematic illustration of the layer-resolvedmagneto transport between top and bottom layers in a multilayergraphene.

is added when N is odd. Similarly, the effective Hamiltonianof an AA-stacked N -layer graphene consists of N isolatedmonolayer systems with different potential energies.

Therefore, we can introduce a transformation matrixU for the AB stacked system which relates the wavefunction in the real space |A1〉,|B1〉, . . . ,|AN 〉,|BN 〉 andin the Fourier modes of the stacking direction |φ(A,even)

N−1 〉,|φ(B,even)

N−1 〉, |φ(A,odd)N−1 〉,|φ(B,odd)

N−1 〉,|φ(A,even)N−3 〉, . . . ,|�α〉,|�β〉. Here

we have used the notation defined in Refs. 15 and 27. Then theHamiltonian H is transformed into a block diagonalized form,

H′ = U †HU =

⎡⎢⎣Hsub(N − 1)

Hsub (N − 3). . .

⎤⎥⎦ . (3)

Here, Hsub(m) is a Hamiltonian of a bilayer system with aneffective hopping

Hsub(m) =

⎡⎢⎣

0 vπ− 0 t⊥λm

vπ+ 0 0 00 0 0 vπ−

t⊥λm 0 vπ+ 0

⎤⎥⎦ , (4)

with

λm = 2 sin

(mπ

2(N + 1)

), m = N − 1, N − 3, . . . > 0.

Using the above block diagonalized Hamiltonian, we easilyobtain eigenvalues of AB-stacked graphenes based on theknown results for monolayer and bilayer systems, replacing

FIG. 2. Lattice structures of (a) AB, (b) ABC, and (c) AA-stackedtrilayer graphenes, containing six sites in a unit cell. White and blackcircles denote carbon atoms which belong to A and B sublattices ineach layer.

the interlayer hopping t⊥ by the effective one t⊥λm as7,9

Eμm,n = s2

√2hv

l

{1

2(2n + 1 + (λmr)2

+ s1

√(λmr)4 + 2(2n + 1)(λmr)2 + 1)

} 12

, (5)

where r ≡ l√2hv

t⊥ and n denotes the Landau levels. The labelμ ≡ (s1,s2) specifies the outer and the inner bands (s1 = ±1),and positive and negative (s2 = ±1) energies, respectively.The eigenstates of AB-stacked graphenes in basis of the realspace |A1〉,|B1〉, . . . ,|AN 〉,|BN 〉 are obtained from those ofthe subsystems Hsub(m) and the transformation matrices U ,written as

|�n,μ〉 = [f 1

n,μ|n − 1〉 f 2n,μ|n〉 f 3

n,μ|n〉 f 4n,μ|n + 1〉 · · · ]T

,

(6)

where f 2k−1n,μ , f 2k

n,μ denote coefficients of the wave function forthe k-th layer, |n〉 is the number state of a,a†, and 1 � μ � 2N

is band indices of the multilayer system.The conductivity is given by the Kubo formula as

Re σij () = − Im ij ()

h, (7)

where ij with {i,j} ∈ {x,y} and ij is the Fourier transformof the current-current correlation function obtained afteranalytic continuation of the Matsubara form. The generalexpression of the conductivity of the multilayer graphenesis obtained by extending the result for bilayer graphene7 as

sgn(eB)σxy() + iσxx()

= −4e2v2

hl2

∑n

∑μ,ν

{X

(E

μ

n+1,Eνn ;

) − X(E

μ

n+1,Eνn ; −

)}

×(

N∑k=1

f 2k−1n+1,μf 2k

n,ν

)2

, (8)

with

X(A,B; ) ≡∑

n

[(iωn − A/h)−1(iωn+m − B/h)−1|iνm→

− ( → −)](βh)−1, (9)

where ωn is the fermionic Matsubara frequency including thechemical potential and effect of impurity scattering as iωn +[μ + i sgn(ωn) ]/h. νm is the bosonic Matsubara frequencyand ωn+m ≡ ωn + νm.

In the above formalism, the electric current operators aredefined by Ji = − δH

δAiwith Ai the vector potential in the

direction i = x,y. Now, we introduce current operators forparticular layers to calculate the conductivity between twodistinct layers as J k

i = − δHδAk

i

where k denotes layer number

(k = 1,2, . . . ,N), and Aki is the vector potential acting solely

in layer k in direction i. The current operator of the k-th layerhas only two matrix elements, (J k

i )2k−1,2k and (J ki )2k,2k−1, and

all other elements are vanishing. The general expression for thelayer-resolved conductivity between the k-th and l-th layers,

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BRIEF REPORTS PHYSICAL REVIEW B 85, 033403 (2012)

σ klij is given by Eq. (8) with the following replacement,

(N∑

k=1

f 2k−1n+1,μf 2k

n,ν

)2

→ f 2k−1n+1,μf 2k

n,νf2l−1n+1,μf 2l

n,ν . (10)

In the present model which includes only the nearest interlayercouplings as interlayer matrix elements, the layer currentoperator, J k

i does not have any momentum dependence.However, this situation can change in the presence of othermatrix elements, such as tilted interlayer hoppings. In suchcases, the layer current operators should be redefined appro-priately so that they become Hermitian and satisfy the relationJi = ∑N

k=1 J ki .

Numerical results. Based on the above discussions, wecalculate the layer-resolved Hall conductivity σ kl

xy of AB-stacked multilayer graphenes, as functions of the Fermi energy,for a strong magnetic field hωc � where ωc ∝ √

B is thecyclotron frequency. Our findings are summarized as: (i) σ kl

xy

is no longer quantized as integer times e2/h. (ii) around zeroenergy μ ∼ 0, σ kl

xy becomes almost zero when the source andthe drain are separated by more than three layers |k − l| � 3and the biased layer k is odd, and two layers |k − l| � 2 and k

is even. (iii) the layer-resolved Hall response can be negative(σ kl

xy < 0) around zero energy for |k − l| = 2 when the biasedlayer k is odd. Layers indices are growing in the direction ofa magnetic field. These features are summarized in Figs. 3(a)and 3(b). Similar behavior can also be seen in the longitudinalconductivity σ kl

xx which behaves like derivatives of σ klxy by μ.

In Fig. 3 we show numerical results of σ klxy for N = 4 as

functions of the Fermi energy μ, where the bias is applied tothe first layer, and B ′ ≡ 2v2eB/c. We have assumed B = 14 T,T = 0 K, = 0.01t⊥, t = 3.16 eV, t⊥ = 0.39 eV, and takenDC limit → 0. We can see the above three features for ABstacking in Fig. 3(c). Further, σ 14

xx is finite without a magneticfield, meaning that property (ii) is broken when a magneticfield is turned off, as shown in the inset of Fig. 3(c). Theconductivity vanishes when B ′ ∼ .

We mention that in principle one should include the Poissonequation and calculate self-consistently the distribution of thecharge over the different layers of graphene in the presence ofa finite chemical potential, but when we stay close to theDirac point, these effects are negligible. We also calculatelayer-resolved conductivities for other stacking structures. ForABC stacking, although the matrix decomposition techniqueis no longer available, we can diagonalize the Hamiltoniannumerically to obtain the layer resolved conductivities. In thiscase, as shown in Fig. 3(d), the electric currents are inducedin every layer. Moreover, the conductivities for ABC-stackingdo not exhibit negative values even in the vicinity of zeroenergy.

For AA stacking, the matrix decomposition techniquesimilar to AB stacking can also be used.20,23 We calculatethe conductivities by setting the value of interlayer hoppingas tAA

⊥ = tAB,ABC⊥ /2. In this system, negative conductivity also

appears as shown in Fig. 3(e). In this case, it is difficult to sum-marize the features of layer-resolved conductivities in terms ofsimple rules, because the effective Hamiltonian consists of N

monolayer systems with different Fermi energies, so that thenumber of Landau levels near zero energy depends strongly onthe strength of a magnetic field. In contrast to this, the Landaulevel structure is an intrinsic property for AB and ABC-stackedsystems.

Analytical results for AB stacking. In order to understandthe above results for AB stacking in more detail, we calculatethe first quantum Hall step of σ kl

xy analytically for = 0. Weconsider the following four cases: (a),(b) (k,l) = (odd, odd) forN = even/odd, respectively, (c) (k,l) = (odd, even), (d) (k,l) =(even, even). Since layer indices grow in the direction of thefield, such classification is unique. We obtain the followingresults, defining σ k,l

xy ≡ sgn(eB) h4e2 σ

k,lxy ,

σ k,l(a)xy =

[r2

4(δl′,k′±1 − δk′,1δl′,1) + 1 + r2

2δk′,l′

]

×∑m

U4k′−2,mU4l′−2,m

1 + (λmr)2(11a)

(a) (b)(c)AB (d)ABC (e)AA

0.02t⊥0.01t⊥

μ 0σ1,4

xx

B(T)

σx

y4e2

/h

σx

x4e2

/πh

μ/√

B

-0.5

0

0.5

1

layer1layer2layer3layer4

0

0.2

0.4

0.6

0.8

-1

0

1

2

3

4

0

0.4

0 7-2.5-2

-1.5-1

-0.5 0

0.5 1

1.5 2

0 0.2 0.4 0.6 0.8 1 1.2 1.4

0

0.5

1

1.5

0 0.2 0.4 0.6 0.8 1 1.2 1.4-1

0

1

2

3

4

0 0.2 0.4 0.6 0.8 1 1.2 1.4

k-2k-1k

k+1k+2

odd=k

B

k-2k-1k

k+1k+2

even=k

B

FIG. 3. (Color online) Schematic illustration of intensity of the layer-resolved conductivity for AB-stacked multilayer graphene applyingbias to the k-th layer where k is (a) odd and (b) even layers counting along the direction of a magnetic field. The layer-resolved Hall (σ kl

xy) andlongitudinal (σ kl

xx) conductivities of 4-layer graphene in a magnetic field (B = 14 T) for (c) AB, (d) ABC, and (e) AA stacking. The bias isapplied to the 1st layer. An inset in (c) shows σ 14

xx as functions of a magnetic field.

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BRIEF REPORTS PHYSICAL REVIEW B 85, 033403 (2012)

σ k,l(b)xy =

[r2

4(δl′,k′±1−δk′,1δl′,1−δk′, N+1

2δl′, N+1

2)+ 1 + r2

2δk′,l′

]

×[∑

m

U4k′−2,mU4l′−2,m

1 + (λmr)2+ 2(−1)k

′+l′

N + 1

], (11b)

σ k,l(c)xy = r2

4(δl′,k′ + δl′,k′−1)

∑m

U4k′−2,mU4l′,mλm

1 + (λmr)2, (11c)

σ k,l(d)xy = δk′,l′

∑m

U4k′,mU4l′,m2 + (λmr)2

4[1 + (λmr)2], (11d)

where k′ ≡ [ k+12 ]G and l′ ≡ [ l+1

2 ]G. From these results, it is ap-parent that interlayer conductivity is finite only within nearestor second nearest layers [property (ii)]. For the simplest exam-ple, the values of σ k,l

xy for three layer system obtained from theabove formalism are σ 1,1

xy = (2 + 3r2 + r4)/(4 + 8r2), σ 1,2xy =

r2/(4 + 8r2), σ 2,2xy = (2 + 2r2)/(4 + 8r2), σ 1,3

xy = −r4/(4 +8r2), with σ 2,3

xy = σ 1,2xy and σ 3,3

xy = σ 1,1xy . In this case, σ 1,3

xy

becomes negative. By summing up all contributions, we obtain∑k,l σ

k,lxy = 3/2 which consists of contributions from effective

monolayer (1/2) and bilayer (1). For other N cases, theseanalytical results coincide with the general results summarizedin Figs. 3(a) and 3(b).

Conclusion and discussion. To conclude, we have studiedthe interlayer electronic transport properties of multilayergraphenes in a magnetic field for variety of stacking orders.

The behavior of the layer-resolved conductivity dependsstrongly on the stacking structure. For AB stacking, variousinteresting properties, such as negative response and suppres-sion of the Hall conductivity, are identified. The breakdownof the quantization of the Hall conductivity indicates thatinterlayer conductivity has different features from those ofthe total response, and is not protected by topology.

Finally, we discuss the possibility of the experimentalobservation of the layer-resolved conductivities. Though itwould be rather challenging to connect leads to particularlayers, it is certainly easier to measure the layer-resolvedconductivity between the top and the bottom layers. The factthat the longitudinal conductivity greatly changes accordingto a magnetic field [for example σ 1,4

xx as shown in the insetof Fig. 3(c)] means that multilayer systems may be appliedas a switching device. We think that our work providesa comprehensive understanding of transport properties ofmultilayer graphene.

Acknowledgments. M.N. acknowledges support fromGlobal Center of Excellence Program “Nanoscience andQuantum Physics” of the Tokyo Institute of Technology andGrants-in-Aid No. 23540362 by MEXT. B.D. was supportedby the Hungarian Scientific Research Fund Nos. K72613,K73361, K101244, CNK80991, and ERC-259374-Sylo, NewSzechenyi Plan Nr. TAMOP-4.2.1/B-09/1/KMR-2010-0002,and by the Bolyai program of the Hungarian Academy ofSciences.

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