Accepted Manuscript

Exact Landau spectrum and wave functions of biased AA-stacked multilayer

graphene

Cheng-Peng Chang

PII: S0008-6223(13)00395-3

DOI: http://dx.doi.org/10.1016/j.carbon.2013.04.086

Reference: CARBON 8017

To appear in: Carbon

Received Date: 20 January 2013

Accepted Date: 29 April 2013

Please cite this article as: Chang, C-P., Exact Landau spectrum and wave functions of biased AA-stacked multilayer

graphene, Carbon (2013), doi: http://dx.doi.org/10.1016/j.carbon.2013.04.086

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Exact Landau spectrum and wave functions of biased

AA-stack multilayer graphene

Cheng-Peng Chang ∗

Center for General Education, Tainan University of Technology, Tainan 710, Taiwan

Abstract

A novel model is presented to obtain exact energy spectrum, Landau levels (LLs)

and wave functions in AA-stacking N -layer Graphene under the influence of perpen-

dicular electric and magnetic fields. A particular unitary transformation to deal with

the broken mirror symmetry caused by a perpendicular electric field can transform an

AA-stacking N -layer graphene into N independent graphene-like layers. The exact

energy spectrum of a graphene-like layer is equal to E = EMLG + ε, where EMLG

is the energy spectrum of a monolayer graphene and ε is controlled by the magni-

tude of electric field and layer number. The associated wave functions are clearly

specified. Such a method is applied to analytically study the combined effect of the

vertical electric field and magnetic fields on LLs and wave functions in AA-stacking

N -layer graphenes. An analytical formula of the gate-tuned LLs clearly exhibits the

dependence of LLs on the number of layers, strength of gated field and magnetic

fields.

Corresponding author. Tel/Fax: +886-62-545329

E-mail address: t00252@mail.tut.edu.tw (C. P. Chang)

1

ed

1 Introduction

Graphene can provide a good platform for the study of the electronic properties of a pure

two-dimensional (2D) system [1]. It exhibits a variety of exotic electronic properties, such

as low-lying linear energy bands, electron-hole symmetry, Klein tunneling, and anomalous

quantum Hall effect [2–9]. Layered graphite and multilayer graphenes are the carbon

allotropes of graphene. Layered graphite is the stacking sequence of infinite layers of

graphene bound together by the van der Waals interactions. Naturally-layered graphite

exists in two different forms. The natural Bernal graphite adopts the AB-stacking sequence

of graphene [10,11], and the rhombohedral graphite has ABC stacking of the layers [11,12].

The low-energy electronic structures are strongly related to the stacking types [10–12].

Multilayer graphene is a superposition of several graphene layers. The physical properties

of the AB and ABC stacking multilayer graphenes have been explored over the last few

years [13–24]. And the findings illustrate that the low-energy band dispersions are very

sensitive not only to the stacking configuration but to the number of layers. The Bernal

bilayer graphene shows two groups of the parabolic band around the Dirac points [13, 23].

Because of the tiny overlap between valence and conduction bands, Bernal bilayer graphene

is semimetal. The application of a vertical electric field can induce a band gap in Bernal

bilayer graphene. In contrast, the gated field is not able to open a band gap in Bernal

trilayer graphene [21]. Still, an electrically tunable band gap in ABC-stacking trilayer

graphene is reported [22].

On the other hand, the fabrication of AA-stacking graphite [25] stimulates many studies

to revisit the fundamental properties of AA-stacking bilayer graphene, e.g. infrared and

Raman spectra, Landau-level energies, transport and magneto absorption spectra [26–31].

Owing to the different stacking types, the AA-stacking bilayer graphene shows some dis-

2

tinguishable physical properties from those of the AB-stacking bilayer graphene. Recently,

the synthesis of the AA-stacking multilayer graphenes (AAMLG) [32] has inspired us to

analytically study the combined effect of electric field and magnetic field on the electronic

properties of AAMLG. The electronic properties of multilayer graphenes with AB or ABC

stacking, eg. the effective mass, electron velocity, and density-of-state, are strongly mod-

ified with the application of a vertical electric field [20–22, 33]. Notably, the electronic

structures of gate-tuned multilayer graphene with AB or ABC stacking are usually studied

through numerical diagonalization of the tight-binding Hamiltonian matrix.

In this work, we propose a model to analytically describe the electronic structures of

AAMLG subjected to both vertical electric and magnetic fields. By utilizing a unitary

transformation, we decompose the Hamiltonian matrix of AAMLG into N 2× 2 diagonal

block matrices. The energy spectrum, LLs and associated wave functions are precisely

expressed in a close form. The dependence of electronic properties on the number of

layers, strength of magnetic field and magnitude of electric field clearly emerge through

the exact formula. The application of the current analytical model to the studies, eg.

optical conductivity, magneto-absorption spectra, minimal conductivity, and dynamical

conductivity of AAMLG, is now in progress.

2 Analytical Expression For Gate-Tuned Energy Spec-

trum Of AAMLG

Graphene is made up of carbon atoms precisely packed in a planar hexagonal lattice with the

lattice vector a =√

3b, where b = 1.42A is the nearest carbon-carbon distance. The lattice

contains two sublattices, and a primitive cell contains two atoms, A and B. An AAMLG

is obtained by stacking the N layer graphenes directly on each other with an interlayer

3

distance between graphenes c = 3.35 A [34], as shown in Fig. 1(a). A primitive cell

consists of 2N atoms, denoted as Al and Bl, where l is the layer index and l = 1, 2, · · · , N .

Atoms A (B) in the stacking direction form a linear chain with N carbon atoms [Fig. 1(b)].

The first Brillouin zone is the same as that of a graphene. In the presence of a vertical

electric field, the Hamiltonian of an AAMLG is written as

H = hA + hB + hAB, (1a)

where hA and hB are

hA =∑l,i

Vl A†l,iAl,i + α1A†

l,iAl±1,i +H.c., (1b)

hB =∑l,i

Vl B†l,iBl,i + α1B†

l,iBl±1,i +H.c., (1c)

hA and hB are the Hamiltonian operators of the linear chain with N atoms subjected to a

parallel electric field. Vl is the on-site potential energy caused by the external electric field

F . As depicted in Fig. 1(b), Vl = [l − (N+1)2

]V , where V = |e|Fc is the electric potential

difference between the adjacent layers caused by the external electric field. Then, V is

denoted as the gate voltage. The on-site potential energy, Vl, depends on the gate voltage

V , layer number N , and layer index l. A†l,i (Bl,j) denotes the creation (annihilation) of

particles at sites A and (B) in the layer l. The interlayer hopping parameter, α1, couples

the two A (or B) atoms from the two adjacent layers [Fig. 1(a)]. α3, the interlayer

interaction between atoms A and B from the two adjacent layers, plays a role in giving

rise to a weak electron-hole asymmetry in the unbiased AAMLG [29]. The values of the

hopping integrals are α1 = 0.361 eV and α3 = −0.032 eV, which are obtained from Ref.

34. Since α3 is weaker than the main interlayer interaction α1, the interlayer interaction

α3 is not taken into consideration here. The operator hAB, resulting from the intralayer

4

interaction, reads

hAB = α0

∑l,〈i,j〉

(A†l,iBl,j + B†

l,iAl,j) +H.c., (1d)

∑〈i,j〉 denotes summing over the nearest neighbors. α0 is the intralayer nearest-neighbor

hopping between atoms A and B on the same graphene layer. In the presence of an electric

field, the Hamiltonian representation of AA-stacking N -layer graphene is a 2N×2N matrix.

For instance, the 6× 6 Hamiltonian matrix of AA-stacking trilayer graphene, spanned by

periodic Bloch functions |A1〉, |A2〉, |A3〉, |B1〉, |B2〉, |B3〉, reads

H =

V α1 0 αk 0 0

α1 0 α1 0 αk 0

0 α1 −V 0 0 αk

α∗k 0 0 V α1 0

0 α∗k 0 α1 0 α1

0 0 α∗k 0 α1 −V

, (2)

where αk = α0f(k) = α0

∑3j=1 exp(ik · bj). bj represents the three nearest neighbors on

the same graphene plane and k is the in-plane wave vector.

Now, by applying an appropriate unitary transformation, we build an analytical model

to reduce the 2N × 2N Hamiltonian matrix [Eq.(1a)] to N block diagonal matrices, which

reads H = H1⊕H2⊕ · · · ⊕HN . All the block diagonal matrices, H1,H2, · · · , and,HN , are

a 2 × 2 matrix. To acquire the reduced Hamiltonian matrices, we, first, generate the new

basis functions in such a way as follows

|ψj(Λ)〉 =N∑

l=1

sj,l|Λl〉 = sj,1|Λ1〉+ sj,2|Λ2〉+ sj,3|Λ3〉+ · · ·+ sj,N |ΛN〉, (3)

where Λ = A or B and |Λl〉 is the periodic Bloch function in the layer l. The coefficient, sj,l,

is the site amplitude of atom A or B located at the lth layer. sj,l is one of the components

5

of the bracket |Sj〉 = |sj,1, sj,2, sj,3, · · · sj,N〉, which is the eigenvector of the operators hA

and hB. That is, |Sj〉 is the eigenfunction of the eigenvalue equation

hΛ|Sj〉 = εj|Sj〉. (4)

For example, the eigenvalue equation associated with the linear chain with three atoms isV α1 0

α1 0 α1

0 α1 −V

sj,1

sj,2

sj,3

= εj

sj,1

sj,2

sj,3

. (5)

The analytical expression of eigenvalues εj and associated eigenvectors |Sj〉 are listed as

follows:

ε1 =√

2α21 + V 2 |S1〉 =

1√N1

|α21 + V (ε1 + V )

α21

,ε1 + V

α1

, 1〉,

ε2 = 0 |S2〉 =1√N2

|1, −Vα1

,−1〉,

ε3 = −√

2α21 + V 2 |S3〉 =

1√N3

|α21 + V (ε3 + V )

α21

,ε3 + V

α1

, 1〉,

(6)

where N1, N2, N3 are normalized constants. The coefficients sj,l are used to construct the

new basis functions ψ1(A), ψ1(B), ψ2(A), ψ2(B), ψ3(A), and ψ3(B) according to Eq. (3).

Then, the new Hamiltonian matrix is calculated by sandwiching Hamiltonian operator

between the new basis functions, which are arranged in the order of [ψ1(A), ψ1(B), ψ2(A),

ψ2(B), · · · , ψN(A), ψN(B)]. Finally, the reduced Hamiltonian matrix contains N 2 × 2

block diagonal matrices Hj. Each Hj reads

Hj =

εj αk

α∗k εj

. (7)

Notably, the on-site energy εj is the eigenvalue of Eq. (4). The exact solution to the

energy spectrum of each subsystem, Hj, is E = εj + EMLG, where EMLG = ±α0|f(k)| is

the energy dispersions of the monolayer graphene. The exact energy spectrum is a copy

6

of the monolayer graphene energy bands shifted up or down by εj, the gate-tuned on-site

energy. Around the Dirac point K, the diagonal block is for q = k - K

Hj =

εj −~vF (qx + iqy)

−~vF (qx − iqy) εj

, (8)

where ~vF = 32α0b is the Fermi velocity. The low-lying energy dispersions associated with

Hj are

E = εj ± ~vF |q|, (9)

where |q| =√q2x + q2

y . The energy dispersions are a pair of linear bands crossing at E = εj.

The on-site energy ε, the eigenvalue of Eq. (4), associated with AAMLG, is plotted in

Fig. 2 for the layer number N = 2, 3 and 4. The values of the intralayer interaction and

main interlayer interaction used in the calculation are α0 = 2.569 eV and α1 = 0.361 eV [34],

respectively. As N = 2, the analytical expression of eigenvalues is εj = ±√α2

1 + V 2/4 [the

blue curve in Fig. 2]. As ε1 is symmetrical to ε2 about E = 0, only ε1 ≥ 0 is shown in

Fig. 2. The eigenvalues of the N = 4 case are εj = ±√B±

√B2 − C, where B =

3α21

2+ 5V 2

4

and C = α41 + 3

4α2

1V2 + 9

16V 4 [the black dashed curves in Fig. 2]. As N = 3, an eigenvalue

ε2 = 0 is equal to zero and independent of the strength of gate potential V [the red curve in

Fig. 2]. Such a characteristic is found in AAMLG with the odd layer. Both the analytical

formula and numerical calculation exhibit that ε will rise (reduce) as V gradually grows

[Fig. 2]. That is, εj is controllable through variation of the magnitude of gate potential.

The wave functions of the N -layer AAMLG can be clarified. According to Eq. (7),

the wave function of the subsystem Hj with energy dispersions E = εj ± α0|f(k)| is

1√2

∑Nl=1 sj,l(|Al〉 + f∗(k)

|f(k)|Bl〉). The eigenvector |Sj〉 = |sj,1, sj,2, sj,3, · · · sj,N〉 dominates the

characteristics of the wave functions. Figure 3 exhibits the coefficients sj,1, sj,2, and sj,3

of wave functions of the AA-stacking trilayer graphene. |sj,l|2 related to εj reveals the

7

probability to find the charge carriers located at the sublattice Λ = A or B on the l

graphene layer. The calculation, resulting from the analytical expressions (Eq. (6)), shows

that the coefficients, sj,1, sj,2, and sj,3, depend on the εj and magnitude of gate potential

V . Obviously, s2,2 is equal to zero at V = 0; i.e., the charge carriers distribute on the first

and third graphene layers. The charge carriers move to the second graphene layer (layer

2), s2,2 6= 0, as V grows.

The low-lying energy dispersions of AAMLG with the layer number N show N pairs

of linear bands, and each pair of linear bands is specified by Eq. (9). As illustrated in

Fig. 4, there are three pairs of linear bands for the trilayer AAMLG. At the Dirac point

K, where q = 0, each pair of linear bands intersects at εj [the dashed curves in Fig. 4].

The vertical electric field does not destroy the feature of linear bands; it only shifts the

linear bands up or down [the solid curves in Fig. 4]. The pair linear bands crossing at

E = 0 are not affected by the vertical electric field [the black dashed curves and red solid

curves]. Moreover, Eq. (9) can be used to determinate the Fermi momentum kf , on which

the energy dispersions is equal to zero, E = 0. The Fermi momentum is kf = K + qf ,

where the magnitude of qf is the root of the equation εj ± ~vF |qf | = 0.

3 Gate-Tunable Landau Level Energies Of AAMLG

Finally, we apply the developed method to analytically study the combined effect of vertical

electric field F and perpendicular magnetic field B on the LLs of AAMLG. The AA-stacking

trilayer graphene is selected for the demo study. In the absence of magnetic field (B = 0),

8

the Hamiltonian matrix, in the vicinity of the Dirac point K, is

H =

V α1 0 −~vFq 0 0

α1 0 α1 0 −~vFq 0

0 α1 −V 0 0 −~vFq

−~vFq∗ 0 0 V α1 0

0 −~vFq∗ 0 α1 0 α1

0 0 −~vFq∗ 0 α1 −V

, (10)

where q = qx + iqy. The effect of uniform perpendicular magnetic field is coupled to the

Hamiltonian through usual Peierls substitution,

P = (p− eA(r)), (11)

where P and p are canonical and kinematic momenta, and A(r) is the vector potential.

Within the Landau gauge, the vector potential is A = (−By, 0, 0) and the envelope function

is in the form of exp(iqxx)[φA1(y),φA2(y),φA3(y),φB1(y),φB2(y), φB3(y)]T . The Hamiltonian

representation of the AA-stacking trilayer graphene subjected to both vertical electric and

magnetic fields is

H =

V α1 0 −~vF Π+ 0 0

α1 0 α1 0 −~vF Π+ 0

0 α1 −V 0 0 −~vF Π+

−~vF Π− 0 0 V α1 0

0 −~vF Π− 0 α1 0 α1

0 0 −~vF Π− 0 α1 −V

, (12)

where Π+ and Π− represent Π+ = ∂y+qx+eBy/~ and Π− = −∂y+qx+eBy/~, respectively.

By defining lB =√

~eB

, η = ylB−lBqx, a = −1√

2(∂η+η), and a† = −1√

2(−∂η+η), the Hamiltonian

9

representation after transformation is

H =

V α1 0 ~ωca 0 0

α1 0 α1 0 ~ωca 0

0 α1 −V 0 0 ~ωca

~ωca† 0 0 V α1 0

0 ~ωca† 0 α1 0 α1

0 0 ~ωca† 0 α1 −V

, (13)

where ~ωc is the cyclotron energy ~ωc =√

2~vF

lB=√

23b2α0

√eB~ . The solution, or enve-

lope function, has the structure [c1φn−1(A1), c2φn−1(A2), c3φn−1(A3), c4φn(B1), c5φn(B2),

c6φn(B3)]T , where φn(Λl) is the wave function of a harmonic oscillator located at the sub-

lattice Λ on the l layer, and n(= 0,±1,±2, · · · ) is the subband index. The Hamiltonian

equation is transformed into an eigen-equation and given by

V α1 0√n~ωc 0 0

α1 0 α1 0√n~ωc 0

0 α1 −V 0 0√n~ωc

√n~ωc 0 0 V α1 0

0√n~ωc 0 α1 0 α1

0 0√n~ωc 0 α1 −V

c1

c2

c3

c4

c5

c6

= E

c1

c2

c3

c4

c5

c6

, (14)

To reduce the 6×6 matrix to N block diagonal matrices, we generate the basis functions

Ψj,n−1(A) =∑N

l=1 sj,lφn−1(Al) and Ψj,n(B) =∑N

l=1 sj,lφn(Bl). The new envelope functions

of the AA-stacking trilayer graphene are [d1Ψ1,n−1(A), d2Ψ1,n(B), d3Ψ2,n−1(A), d4Ψ2,n(B),

10

d5Ψ3,n−1(A), d6Ψ3,n(B)]T . After some algebra is done, the reduced Hamiltonian matrix is

ε1

√n~ωc 0 0 0 0

√n~ωc ε1 0 0 0 0

0 0 ε2

√n~ωc 0 0

0 0√n~ωc ε2 0 0

0 0 0 0 ε3

√n~ωc

0 0 0 0√n~ωc ε3

d1

d2

d3

d4

d5

d6

= E

d1

d2

d3

d4

d5

d6

. (15)

The Hamiltonian matrix of the AA-stacking N -layer graphene is reduced to H = H1 ⊕

H2 ⊕H3 ⊕ · · ·+HN , and each 2× 2 Hj matrix satisfies the eigenvalue equation εj

√n~ωc

√n~ωc εj

dj,1

dj,2

= Ej,n

dj,1

dj,2

. (16)

Accordingly, the Landau-level energy Ej,n and the associated eigenvector are

Ej,n = εj + sig(n)√|n|~ωc, (17a)

(dj,1, dj,2) = (1√2,

1√2). (17b)

The envelope function related to each Hj or εj is exactly specified as 1√2

∑Nl=1 sj,lφn−1(Al)

1√2

∑Nl=1 sj,lφn(Bl)

. (18)

AAMLG with N layers exhibits N groups of LLs, which depend on the layer number

N , subband index n, and the strength of electric and magnetic fields. Figure 5(a) shows

the gate-potential-dependent LLs Ej,n of the trilayer AAMLG under B = 20 T. There are

three distinct groups of the LLs labeled by group I, II, and III and, respectively, illustrated

in the blue, red, and green curves. LLs with the level-index |n| ≤ 10 are shown. In

11

the absence of the electric field (V = 0), three groups of LLs are E1,n =√

2α1 ±√|n|~ωc,

E2,n = ±√|n|~ωc, and E3,n = −

√2α1±

√|n|~ωc. They are the shifts of LLs of a monolayer

graphene, En = ±√|n|~ωc, with energies ε =

√2α1, 0, and−

√2α1, respectively [29]. In

the presence of a vertical electric field, exact Landau spectra are specified and they are

E1,n =√

2α21 + V 2 ±

√|n|~ωc, E2,n = ±

√|n|~ωc, and E3,n = −

√2α2

1 + V 2 ±√|n|~ωc.

The two LLs E1,n=0 =√

2α21 + V 2 and E3,n=0 = −

√2α2

1 + V 2, the n = 0 LLs of the

group I and III, reflect how the electric field affects the energies of LLs. The group I (III)

LLs will rise (reduce) with the increase of gate potential V , while LLs of the group II are

independent of the magnitude of V .

The electric field also causes the crossing between different LLs; that is, the two different

LLs have the same energy. The crossing is determined by the criterion

εj + sig(nj)√|nj|~ωc = εi + sig(ni)

√|ni|~ωc,

where i, j = 1, 2, 3 denotes the group index and i 6= j. For instance, the crossings between

the n1 = −9 level of group I and the n2 levels of group II, as shown in Fig. 5(a), are

determined by the following relation:√2α2

1 + V 2 − 3~ωc =√n2~ωc,

where the cyclotron energy is ~ωc =√

23α0b2lB

≈ 0.03√B[T] = 0.134 eV. The calculated

result shows that the n1 = −9 level of group I intersects the n2 = 1 (n2 = 4) level of group

II at V = 0.163 eV (V = 0.434 eV). The crossing between LL of the groups I and that of

III abides by the relation:

sig(n3)√|n3| − sig(n1)

√|n1| =

2√

2α21 + V 2

~ωc

.

The gate-tuned Landau-level energies are also dependent on the magnitude of the mag-

netic field, as depicted in Fig. 5(b), for Landau-level energies exhibit√B-dependence.

12

Some Landau-level energies go to zero as they satisfy the condition

εj + sig(nj)√|nj|~ωc = 0.

The zero-energy LLs are induced by changing V or B. The cyclotron energy is ~ωc ≈ 0.190

eV at B = 40 T. The LL of group III (I) with the index n3 = 8 (n1 = −9), for example,

turns into zero-energy LL at V = 0.168 eV (V = 0.253 eV).

The gate-potential-dependent LLs of the quad-layer AAMLG are shown in Fig. 5(c).

There are four groups of LLs (group I, II, III, and IV) marked in the blue, red, green,

and black curves. The group-I (group-II) LLs are symmetrical to group-IV (group-III) LLs

about E = 0. LLs of the groups II [red curves] are almost not affected by the gate potential

in the region 0 < V < 0.2 eV. They will rise as the gate potential V > 0.2 eV. On the

contrary, the gate potential V has a noteworthy influence on the group-I LLs [blue curves].

The gate potential V significantly lifts the group-I LLs in the calculated region 0 < V < 0.6

eV. The crossing of LLs and the occurrence of zero-energy LLs are also found in Fig. 5(c).

4 Conclusions

In this work, an analytical model is presented to study the combined effect of vertical electric

and magnetic fields on the energy spectrum, Landau-level energies and wave functions of

the AAMLG. The vertical electric field breaks the mirror symmetry of AAMLG and the

magnetic field modifies the in-plane tight-binding hopping. First, we adopt an appropriate

unitary transformation to deal with the broken-mirror-symmetry caused by the gated field.

Such a unitary transformation can totally decompose the AA-stacking N -layer graphene

into N independent equivalent graphenes with the gate-potential-modified on-site-energy.

Each equivalent graphene exactly described by a 2 × 2 matrix provides us with the close

form of the state energies and wave functions of each subsystem. Next, the combined effect

13

of electric and magnetic fields on the Landau spectrum and wave functions in AAMLG

are explored. The analytical expression of gate-tuned Landau spectrum is derived. The

exact Landau spectrum is the copy of the monolayer graphene Landau-level energies shifted

up or down by a gate-tuned on-site energy. Above all, the predicted energy spectrum and

associated electronic properties could be eventually verified by experimental measurements.

Acknowledgements

The author gratefully acknowledges the support of the Taiwan National Science Council

under the Contract Nos. NSC 99-2112-M-165-001-MY3.

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Figure Captions

FIG. 1. (a) The geometric structure of the AA-stacking multilayer graphene and the intralayer

and interlayer interactions. (b) TheN -site linear atomic chain subjected to an electric

field in the stacking direction.

FIG. 2. (a) The eigen-energies ε versus the gate voltage V for a linear chain with N atoms,

where N = 2, 3, and 4. One of eigen-energies ε vs. V for N = 2 , 3, and 4 are shown

in (b), (c), and (d).

FIG. 3. The gate-tunable eigenvectors |sj,1, sj,2, sj,3〉 for a linear tri-atom chain are presented

in (a), (b), and (c).

FIG. 4. Dashed curves are the low energy dispersions of the trilayer AAMLG around the

point K. The gate-tuned energy bands are shown in solid curves.

FIG. 5. As illustrated in (a) are the gate-tuned Landau spectrum of the AA-stacking trilayer

graphene at B= 20 T . (b) Same plot as (a) but for B= 40 T . (c) Four groups of LLs

of the AA-stacking quad-layer graphene at B= 40 T .

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