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An analytical app

Center for General Education, Tainan Unive

E-mail: t00252@mail.tut.edu.tw

Cite this: RSC Adv., 2014, 4, 32117

Received 15th April 2014Accepted 11th June 2014

DOI: 10.1039/c4ra03430a

www.rsc.org/advances

This journal is © The Royal Society of C

roach for the energy spectrum andoptical properties of gated bilayer graphene

Cheng-Peng Chang*

First, we present an analytical approach to access the exact energy spectrum and wave functions of gated

Bernal bilayer graphene (BBLG), with all the tight-binding parameters included. To tackle the broken mirror

symmetry caused by a gated voltage (Vg) and interlayer interactions, we create a unitary transformation to

reduce the Hamiltonian matrix of BBLG to a simple form, which can offer the analytical energy spectrum.

The formulae generate the gated tunable energy bands and reveal that Vg changes the subband spacing,

produces the oscillating bands, and increases the band-edge states. Then, we employ the analytical

model to revisit the optical dipole matrix element and optical absorption spectra. In the absence of Vg,

the anisotropic dipole matrix element exhibits a maximum around the point M and zero value along the

high symmetry line GK in the first Brillouin zone. Vg effectively induces the nonzero dipole matrix

element along the high symmetry line GK, which makes a significant contribution in the absorption

spectra. Moreover, the application of Vg opens an optical gap and gives rise to a profound low-energy

peak in the absorption spectra. The dependence upon the gated bias Vg for the location and height of

this peak clearly emerges through the analytical model. Our exact analytical model can be further used

to study the many-body effect and exciton effect on the electronic and optical properties of BBLG.

1 Introduction

Graphene, a hit topic since 2004,1 having attracted a surge ofinterest for fundamental and experimental studies,2–9 is a oneatom-thick layer made up of carbon atoms arranged in ahoneycomb lattice. Stemming from the specular geometricalstructure, graphene is an extraordinary two dimensional mate-rial with many unique properties, e.g., linear energy dispersionscrossing at the Dirac point, electron–hole symmetry, Kleintunneling, high mobility at room temperature, and the novelquantum Hall effect.7–13 Its unusual electric properties makegraphene a possible candidate for fabricating future electronicdevices. Graphene is a zero band gap semiconductor due tolinear energy dispersions crossing at the Dirac point. Thepresence of valence and conduction bands in its energy spec-trum limits the on-off current ratios achievable and prevents theuse of graphene in making eld transistors. It is signicant forthe opening of a band gap in graphene to remove the limitationin using graphene material for semiconductor applications.Many breakthrough methods are proposed to generate a bandgap in monolayer graphene, e.g., using molecular doping,14,15

the application of strain,16,17 and patterning a graphene sheetinto a nanoribbon.18 On the other hand, a tunable band gap isinduced in Bernal bilayer graphene (BBLG) through the appli-cation of a perpendicular electric eld. BBLG is made up of two

rsity of Technology, 710 Tainan, Taiwan.

hemistry 2014

monolayer graphenes, held by weak Van der Waals forces in theAB-stacking. Experimental works demonstrate that a tunableband gap of up to 0.25 eV is achieved for electrically gatedbilayer graphene by a variable external electric eld.19–22

The energy dispersions of the gated BBLG are usually exploredwithin the tight-binding model.23–31 The minimal model,considering only the intralayer and main interlayer interactions,is widely employed to study the energy dispersions of BBLG.30,31

Around the Dirac points, the analytical low-energy dispersions ofgated BBLG are obtained through the continuummodel. A 4 � 4Hamiltonian matrix is solvable and gives out the analytical low-energy dispersions. A full tight-binding Hamiltonianmodel takesinto account all hopping integrals, including the additional skewinteraction (g3) and electron–hole asymmetry (g4) in themodel.23,27 g3 is the trigonal warping and g4 is the origin ofelectron–hole asymmetry. The trigonal warping can signicantlymodify the physical properties of BBLG.32–34Cserti has shown thattrigonal warping has a great effect on theminimal conductivity ofBBLG.32 Moreover, g3 gives rise to the electron–hole asymmetryand distorts the circular equi-energy contour to a trigonalsymmetry.33,34 A full tight-binding Hamiltonian matrix is a 4 � 4matrix and its four energy bands are usually obtained bynumerical diagonalization. Many important physical propertiesdo not directly emerge through the numerical calculationbecause of a lack of the exact energy spectrum. To fully andexactly describe the electric and optical properties in the fullenergy region, an analytical description of the energy dispersionsof the four-band model is expected.

RSC Adv., 2014, 4, 32117–32126 | 32117

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Here, we present a model to access the analytical form of theenergy spectrum and eigenstates of gated BBLG based on thetight-binding framework and all the tight-binding parameters,g0, g1, g3 and g4, are considered too (see the inset of Fig. 1(a)).Unlike AA-stacked grapheme,35,36 the skew interlayer interac-tions (g3) and gated bias (Vg) destroy the inversion symmetry ofgated BBLG. The broken symmetry increases the difficulty insolving the eigenvalue problem. Through a rotation operator,we generate a new set of tight-binding basis functions to dealwith the broken symmetry resulting from the g3 and Vg. Therenormalized Vg-dependent intralayer and interlayer interac-tions are derived based on the new set of basis functions. Mostimportantly, the new basis functions exactly reduce theHamiltonian matrix (H) into a band-storage matrix. As a result,the energy dispersions and wave functions are analyticallysolvable. Furthermore, the present work aims to provide full-frequency energy bands, which are beyond the effective massapproximation. The analytical form of the energy spectrum andeigenstates are further applied to study the optical absorptionspectra and explore the origin of optical transition channels.

The rest of the paper is organized as follows: we develop amodel to derive the analytical form of the energy spectrum ofgated BBLG in Section 2. Subsequently, the electronic propertiesare revisited in Section 3 through the analytical energy spec-trum. In Section 4, the energy spectrum and eigenstates areused to explore the dipole matrix element and optical absorp-tion spectra of gated BBLG. Finally, the conclusions are drawnin Section 5.

Fig. 1 (a) The energy spectra, obtained using eqn (11), are illustrated bythe red dashed curves. The black curves are the energy spectrum bynumerical diagonalization. The geometric structure of the AB-stackedbilayer graphene and the intralayer and interlayer interactions areshown in the inset. (b) The same plots as in (a) but with the gatedvoltage Vg ¼ 200 meV. The inset exhibits the first Brillouin zone. M isthe saddle point and K and K0 are the Dirac points.

32118 | RSC Adv., 2014, 4, 32117–32126

2 Theory and model

BBLG is a stack of two identical graphene layers with a AB-stacking, as shown in the inset of Fig. 1(a). On each grapheneplane, two basic atoms, A and B, are arranged in a honeycomblattice and the bond length between atoms A and B is b¼ 1.42 A.The distance between two graphene layers is assumed to be 3.35A.37 The lower or upper graphene layers are denoted by the indexl ¼ 1 or 2. Atoms A2 and A1 have the same (x, y) projection andthey are denoted as the dimer sites. The projections of the otherhalf, the B atoms, lie in the center of the hexagons in theadjacent sheets. BBLG is a periodic system in the xy plane andhas four atoms, A1, B1, A2 and B2, in its primitive cell. The rstBrillouin zone of BBLG, as illustrated in the inset of Fig. 1(b), isthe same as that of graphene, which is a hexagon with points G

located at (0, 0), K at�

4p

3ffiffiffi3

pb; 0�, and M at

�pffiffiffi3

pb;p

3b

�.

The electronic properties of BBLG subjected to a perpen-dicular electric eld are studied within the tight-bindingmethod framework. The Hamiltonian equation of BBLG isHF ¼ EF, where the Bloch function is F ¼ c1|A1i + c2|A2i +c3|B1i + c4|B2i, which is the linear combination of the periodicwave functions |A1i, |A2i, |B1i, and |B2i. HF ¼ EF is thentransformed into the matrix equation H |ci ¼ E |ci, here |ci ¼|c1, c2, c3, c4iT. The representation of the Hamiltonian matrixreads

H ¼�HAA HAB

HBA HBB

�; (1)

where H is a 4 � 4 matrix. The elements HAA, HAB, HBA, and HBB

are 2 � 2 matrices. The hopping integrals (gi) usually used todescribe Bernal bilayer graphene are illustrated in the inset ofFig. 1(a). g0 is the intralayer interaction. The hopping integral,g1, is the interaction between the dimer sites A1 and A2. Thetrigonal warping integral, g3, is the skew interlayer hoppingbetween atoms B1 and B2. g4 represents gA1,B2

and gB2,A1. The

matrices HAA, HAB and HBA are further expressed as

HAA ¼�Vg g1

g1 �Vg

�; (2a)

HAB ¼ H*BA ¼ g0 fk g4 f

*k

g4 fk g0 f*k

!; (2b)

HBB ¼ Vg g3 fk

g3 f*k �Vg

!; (2c)

where the on-site energy (Vg) is the electric potential caused by

the external perpendicular electric eld. fk ¼X3l¼1

expðik$blÞ is

the structure factor, where k ¼ (kx, ky) is the wavevector and blrepresents the nearest neighbor on the same graphene plane.The three neighboring atoms are located at

b1 ¼��

ffiffiffi3

pb

2;� b

2

�, b2 ¼ (0, b) and b3 ¼

� ffiffiffi3

pb

2;� b

2

�, and thus

fk ¼ 2 cos� ffiffiffi

3p

b2

kx

�exp�� iky

b2

�þ expðikybÞ.

This journal is © The Royal Society of Chemistry 2014

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To nd the analytical energy dispersions of the 4 � 4 matrixabove, we now construct the symmetrized basis functions (|j1i,|j2i, |j3i, |j4i)T, which are linear combinations of the periodicwave functions |A1i, |A2i, |B1i, and |B2i. The symmetrized basisfunctions are

jj1ijj2ijj3ijj4i

0BBB@

1CCCA ¼

0BBBBBBBBB@

sþ1 sþ2 0 0

0 0 sþ1f *k| fk|

sþ2fk

| fk|

0 0 s�1f *k| fk|

s�2fk

| fk|

s�1 s�2 0 0

1CCCCCCCCCA

jA1ijA2ijB1ijB2i

0BBB@

1CCCA; (3)

where the coefficients s�1 and s�2 are the components of theeigenvector of the matrix HAA (eqn (2a)). The eigen-equation ofHAA is

Vg g1

g1 �Vg

� �s�1s�2

� �¼ �3AA

s�1s�2

� �; (4)

where the eigenenergy is 3AA ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig1

2 þ Vg2

p. The related eigen-

vectors (s�1 , s�2 )

T are

�sþ1sþ2

�¼

0BBBBB@

g1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig1

2 þ �3AA � Vg

�2q3AA � Vgffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

g12 þ �3AA � Vg

�2q

1CCCCCA;

�s�1s�2

�¼

0BBBBBB@

g1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig1

2 þ �3AA þ Vg

�2q��3AA þ Vg

�ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig1

2 þ �3AA þ Vg

�2q

1CCCCCCA: (5)

Similarly, the eigenenergy related to HBB is

3BB ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig3

2| fk|2 þ Vg2

q.

The Bloch function, spanned by the new basis functions(|j1i, |j2i, |j3i, |j4i)T, is F ¼ d1|j1i + d2|j2i + d3|j3i + d4|j4i.According to the symmetrized basis functions |jii, the matrixequation has the form

0BB@

H 11 H 12 H 13 0

H 21 H 22 H 23 H 24

H 31 H 32 H 33 H 34

0 H 42 H 43 H 44

1CCA0BB@

d1d2d3d4

1CCA ¼ E

0BB@

d1d2d3d4

1CCA; (6)

where the matrix elements are H ij ¼ hji|H|jji.The diagonalmatrix elements are

8>>>>>>><>>>>>>>:

H 11 ¼ 3AA;

H 22 ¼ Vg2

3AAþ g1

3AA

g3

2| fk|2

�fk

3 þ f *3k

�;

H 33 ¼ �H 22;

H 44 ¼ �3AA:

(7a)

The off-diagonal terms are

This journal is © The Royal Society of Chemistry 2014

8>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>:

H 12 ¼ H 21 ¼�g0 þ

g1

3AAg4

�j fkj;

H 13 ¼ H 31 ¼ � Vg

3AAg4j fkj;

H 14 ¼ H 41 ¼ 0;

H 23 ¼ H *32 ¼

g1Vg

3AAþ g3

2j fkj2�3AA � Vg

�3AA

f *3k ��3AA þ Vg

�3AA

fk3

;

H 24 ¼ H 42 ¼ � Vg

3AAg4j fkj;

H 34 ¼ H 43 ¼�g0 �

g1

3AAg4

�j fkj:

(7b)

Notably, in the absence of the gated bias, Vg ¼ 0, H 13, H 31, H 24

and H 42 are all equal to zero.38 Moreover, a straightforwardcalculation shows that H 22

2 + H 23H 32 ¼ 3BB2 ¼ Vg

2 + g32|f|2

because the sub-matrix �H 22 H 23

H 32 H 33

�

is the transformation of HBB.

2.1 In the limit of g4 ¼ 0

The exclusion of electron–hole asymmetry, g4 ¼ 0, allows us toobtain a simple and exact analytical solution to the energyspectrum. The interactions H 13, H 31, H 24 and H 42 in Hred areall equal to zero as g4 ¼ 0. The reduced Hamiltonian matrix(Hred) is a band-storage matrix and is expressed as follows:

Hred ¼

0BB@

H 11 H 12 0 0

H 12 H 22 H 23 0

0 H 32 �H 22 H 12

0 0 H 12 �H 11

1CCA; (8)

where H 12 ¼ H 34 ¼ g0|f(k)|. The analytical energy dispersions,associated with Hred, read

EðkÞ�� ¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB2 � C

pq; (9)

where fB ¼ H 112�2þ H 12

2 þ �H 222 þ H 23H 32

��2

¼ g12

2þ g0

2j fkj2 þ g32jfkj22

þ Vg2;

C ¼ H 124 � 2H 11H 12

2H 22 þ H 112�H 22

2 þ H 23H 32

�:

¼ g04jfkj4 � 2g1g0

2g3jfkj3 cosð3fÞ þ g12g3

2jfkj2

þVg2�g1

2 þ Vg2 þ g3

2jfkj2 � 2g02jfkj2

:

where H 222 + H 23H 32 ¼ g3

2|f|2 + Vg2 is used. fk ¼ �� fk��eif and the

angle f is f ¼ tan�1 Im½ f ðkÞ�Re½ f ðkÞ�. There are four branches of energy

bands, which are in the sequence E(k)++ > E(k)+� > E(k)�� >E(k)�+. The former tow, E(k)++ and E(k)+�, are the energydispersions associated with the conduction bands. The lattertwo, E(k)�� and E(k)�+ are the energy spectra related to thevalence bands.

RSC Adv., 2014, 4, 32117–32126 | 32119

Fig. 2 The dashed curves are the low energy spectra at different Vgsacquired by eqn (11). For comparison, the energy spectra by numericaldiagonalization are drawn in the black curves.

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At the point K, B ¼ g12/2 + Vg

2 and C ¼ Vg2(g1

2 + Vg2). The

state energies are exactly E(K) ¼ �Vg and � ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig1

2 þ Vg2

p. Around

the point K and in the absence of the electric eld,B [ C andC z (g0

2|fk|2 � g1g3|fk|)

2. The two branches of the low energy

dispersions of BBLG are EðkÞ�� z �ffiffiffiffiffiffiffiC2B

r¼ � 1

g1ðg0

2��fk��2�

g1g3

��fk��Þ. They are the eigenenergies of the effective Hamilto-nian matrix

Heff ¼ 1

g1

0 g02jfkj2 � g1g3jfkj

g02j fkj2 � g1g3j fkj 0

!:

2.2 The effect of g4 on the energy dispersions

A plainly analytical energy dispersion of the Hamiltonianmatrix(eqn (6)), involving all hopping integrals, is in fact inaccessible.The energy dispersions are generally obtained by the numericaldiagonalization method. To obtain a simply analytical solutionto the energy spectrum, we treat g4 as a perturbation andneglect the matrix elements H 13, H 31, H 24 and H 42 becausethey make nonsignicant contributions to the energy spectrum.H 13, H 31, H 24 and H 42 are right dependent on the magnitudeof the gated voltage (Vg) and interlayer interaction (g4). Themagnitude of g4 is smaller than that of g0 and g3, so that H 13,H 31, H 24 and H 42 are negligible. g4 affects the matrix elements

H 12 and H 34 in such a manner as H 12 ¼ g0 þg1

3g4

� �� fk�� andH 34 ¼

�g0 �

g1

3g4

��fk��. The Hamiltonian equation with the

band-storage Hamiltonian matrix is expressed as follows:0BB@

H 11 H 12 0 0H 21 H 22 H 23 0

0 H 32 �H 22 H 34

0 0 H 43 �H 11

1CCA0BB@

d1d2d3d4

1CCA ¼ E�;�

0BB@

d1d2d3d4

1CCA: (10)

The eigenenergies, determined by the criterion Det|H | ¼ 0, arethe roots of the secular equation E4 � BE2 � B0E + C ¼ 0, whereB ¼ H 11

2 + H 222 + H 23H 32 + H 12

2 + H 342, B0 ¼ (H 11 +

H 22)(H 122 � H 34

2), and C ¼ H 122H 34

2 � H 11H 22(H 122 + H 34

2)+ H 11

2(H 222 + H 23H 32). The analytical form of the eigenener-

gies looks so complicated and tedious just because the coeffi-cient B0 s 0. To acquire the simple analytical result, we furtherapproximate B0 z 0 because H 12 z H 34. As a result, the eigen-energies, 3(k), are

3ðkÞ�� ¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB=2�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðB=2Þ2 � C

qr: (11)

To check out the accuracy of the afore-presented model, werst calculate energy dispersions 3(k)�� at Vg ¼ 0 in the highsymmetry line K–G–M–K, shown by the dashed red curves inFig. 1(a). There are four energy subbands 3(k)��. For compar-ison, energy dispersions, E��, obtained through the numericaldiagonalization of the Hamiltonian matrix (eqn (1)), arealso drawn by the black curves in Fig. 1(a). The tightbinding parameters (gis) used to calculate the energy bands are:g0 ¼ �3.12 eV, g1 ¼ 0.38 eV, g3 ¼ 0.280 eV, and g4 ¼ 0.12 eV.39

32120 | RSC Adv., 2014, 4, 32117–32126

Obviously, the red curves are not identical to those in black inthe higher energy region 3 > 3.0 eV. The higher the energy (3) is,the more discrepancy there is between the two curves. Plus, theenergy spectrum 3 in the presence of the gated voltage Vg ¼ 200meV are presented in Fig. 1(b). The applied gated voltage doesnot signicantly affect the prole of energy bands in the highsymmetry line K–G–M–K. The disagreement between the redand black curves reveals that the derived analytical formula cannot exactly describe the energy spectra in the energy regionE(eV) > 3.0 eV. Energy dispersions around the point K atdifferent Vgs are shown in Fig. 2 for comparison. The calculationresult illustrates that the presented analytical model can notreplicate the exact energy dispersions around the point K.Notably, in the absence of the gate voltage, the presentedanalytical model 3�� can not reproduce the analytical formulaof BBLG at Vg ¼ 0.38

We here modify and improve the aforementioned analyticalmodel to approach the exact energy dispersions at any Vg. In theabsence of the gated voltage Vg ¼ 0, the matrix elements H 23

and H 32 in eqn (10) play a minor role and can then be neglec-ted.38 E++ and E�� (E+� and E�+) belong to the symmetricalstates (the anti-symmetrical states). E++ and E�� depend only

upon H 12 ¼ H 21 ¼�g0 þ

g1

3g4

�� fk�� while E+� and E�+ depend

on H 34 ¼ H 43 ¼�g0 �

g1

3g4

�� fk��.38 We thus replace both H 34

and H 43 with H 12 and H 21 in eqn (10) during the calculation ofE++ and E�� at any Vg. Then, we obtain the energy dispersions

EðkÞþþ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB2 � C

pq; (12a)

EðkÞ�� ¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB2 � C

pq; (12b)

where B ¼ H 122 + (H 11

2 + H 222 + H 23H 32)/2 and C ¼ H 12

4 �2H 12

2H 11H 22 + H 112(H 22

2 + H 23H 32). The other twoenergy bands E+� and E�+ are acquired by utilizing

H 34 ¼�g0 �

g1

3g4

�� fk�� to calculateB0 ¼ H 34

2 + (H 112 + H 22

2 +

This journal is © The Royal Society of Chemistry 2014

Fig. 3 (a) The energy dispersions at Vg¼ 0 illustrated in the red dashedcurves are obtained from the exact analytical model. The black curvesare the energy spectra from numerical diagonalization. (b) Theconduction bands around the point M. (c) The same plot as in (a) butfor Vg ¼ 200 meV. (d) The conduction bands around the point M.

Fig. 4 The dashed and solid curves are energy dispersions obtainedfrom the exact analytical formulae and numerical diagonalization. Forcomparison, the exact energy dispersions are illustrated by the coloredcurves.

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H 23H 32)/2 and C0 ¼ H 34

4 � 2H 342H 11H 22 + H 11(H 22

2 +H 23H 32). E+� and E�+ are, respectively, expressed as

EðkÞþ� ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB 0 �

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB 02 � C 0

pq; (12c)

EðkÞ�þ ¼ �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB 0 þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiB 02 � C 0

pq: (12d)

The analytical formulae (eqn (12a–d)) are now employed tocalculate the energy dispersions of gated BBLG. We rst presentthe analytical energy dispersions E(k)�� at Vg ¼ 0 in Fig. 3(a).E(k)�� in the high-symmetry line K–G–M–K are drawn in red tobe distinguished from those obtained through the numericaldiagonalization of the Hamiltonian matrix in black. The redcurves are exactly identical to the black ones except for theconduction bands in the high-symmetry line KM, where 3.0 eV <E < 3.5 eV (Fig. 3(b)). The energy spectrum in the presence of thegated voltage Vg ¼ 200 meV is also exhibited in Fig. 3(c). Theidentity between the red and black curves reveals that thederived analytical formula can well describe the energy spec-trum of the gated BBLG, except for the conduction bands in theenergy region 2.6 eV < E < 3.4 eV (Fig. 3(d)). The results of thecalculation show that the analytical model is able to provide uswith the exact energy dispersions of the gated BBLG as E < 2.6 eVor E > 3.5 eV. The analytical energy dispersions and conjugatedeigenstates are useful for exploring the electronic properties,such as energy dispersions, density of states (DOS), andabsorption spectra.

3 Discussion: electronic properties

The characteristics of the low energy spectrum are explored. BBLGowns two pairs of parabolic conduction and valence bands, as

This journal is © The Royal Society of Chemistry 2014

shown in Fig. 4 in the dashed and solid curves, which areobtained, respectively, by the analytical formulae and the numer-ical diagonalization method. The dashed curves are in goodagreement with the solid curves. With the comparison shownabove, the exact solution to the low energy spectrum of BBLGemerges through our analytical model. While the high energybands arouse some focus of our discussion, we are more inter-ested in the lowest-energy electronic properties. At Vg ¼ 0, E(k)+�and E(k)�+, the rst pair located near the chemical potential m¼ 0,exhibit a tiny overlap. E(k)+� and E(k)�� are asymmetrical about m¼ 0, i.e., electron–hole asymmetry, which is caused by the inter-layer interaction g4. The minimum of E(k)+� and the maximum ofE(k)�+ are located at almost the same wavevector (�the K point).E(k)+� and E(k)�� along the high symmetry lines G–K andM–K arestrongly anisotropic, which originates in the vanishing of H 23 andH 32 in eqn (12a–d) along the line G–K.

The gated voltage (Vg) has a great inuence on the low energyspectrum. First, the low energy subbands are signicantlymodied by Vg and change from monotonically parabolicdispersions into oscillating ones, referred to as a Mexican hat.Then, the asymmetry between the conduction E(k)+� and thevalence E(k)�+ bands along the G–K direction about the chem-ical potential is also enhanced. In addition to the K point, twoextra band-edge states emerge in all directions. There exists aband-edge state with the minimum (maximum) energy alongthe G–K direction. The energy difference between the lowestband-edge state of E+�(k) and the highest one of E��(k) decidesthe band gap, Eg. The locations of the band-edge state are

determined byvEþ�vkx

|kc¼ 0 and

vE��vkx

|kv¼ 0. By taking the main

interactions, g0 and g1, into consideration, E+� (E��) has aminimum (maximum) at

kc ¼ Vg

ħv

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2g1

2 þ 4Vg2

g12 þ 4Vg

2

s;

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where the Fermi velocity (v) is ħv¼ 3bg0/2. Therefore, the energygap is

Eg ¼ 2g1Vgffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffig1

2 þ 4Vg2

q :

The size of the band gap, Eg ¼ E�+ (kc) � E�� (kv), at rst growsrapidly but then gradually declines with the increase in Vg.Another band-edge state along the KM direction belongs to thesaddle point. Such states are the critical points in the energy-wavevector space and thus have a high DOS.

The DOS is useful for understanding the essential physicalproperties of BBLG. DOS is dened as

DðuÞ ¼Xh1 ;h2

DðuÞh1 ;h2 ; (13a)

Dh1 ;h2ðuÞ ¼2

p

ð1st BZ

d2k

ð2pÞ2G�

Eh1 ;h2ðkÞ � u�2 � G2

; (13b)

where h1, h2 ¼ �, � represents the subband index. The broad-ening energy width (G) is set as 3.0 meV. The full density ofstate, D(u), is the summation of Dh1,h2(u), resulting from eachsubband, Eh1,h2. The D(u) and Dh1,h2(u) of BBLG at Vg ¼ 0 areillustrated in Fig. 5(a). The DOS exhibits logarithmic peaks,originating in the saddle point M around u ¼ �3 eV, whichdirectly reects the main features of the energy dispersions. Thedetailed structure of the DOS at low energy is presented inFig. 5(b). D+�(u) (D�+(u)) makes contributions to D(u) as 0 < u <g1 (�g1 < u < 0). BBLG at Vg ¼ 0 is a semi-metal due to the non-vanished DOS at u ¼ 0. The gated voltage (Vg) can open a bandgap and induce two van Hove singularities in the energy region�g1 < u < g1, as shown in Fig. 5(c–e), where the band gap is

Fig. 5 (a) DOS Dh1h2, associated with each subband Eh1h2, of BBLG atVg ¼ 0. (b)-(e) are the low-energy DOS at Vg ¼ 0, 100, 200, and 300meV. The vertical red bars indicate the energy band gap. The bluearrows label the van Hove singularities in the DOS.

32122 | RSC Adv., 2014, 4, 32117–32126

labeled by vertical red bars and van Hove singularities areindicated by the blue arrows. One canmodulate the band gap inBBLG using an external electric eld. The increase in Vg not onlyenlarges the size of the band gap, but also enhances the heightof van Hove singularities. The predicted size of the band gapand locations of van Hove singularities can be veried byexperimental measurements.

4 Absorption spectra

Utilizing the analytical model, we revisit the low-energyabsorption spectra of gated BBLG, which has been studiedthrough numerical methods.40–42 The absorption function ofbilayer graphene at zero temperature (T ¼ 0) is given by38

AðuÞfXi;f

ð1st BZ

dk

ð2pÞ2 Im

Ff � Fi

Eex � u� iG

� ��MfiðkÞ

��2; (14)

where G is the broadening parameter owing to various non-radiative processes, f (i) denotes the nal (initial) state and Ff(Fi) is the Fermi-Dirac distribution function. Eex¼ Ef(k)� Ei(k) is

the excitation energy. MfiðkÞ ¼*FfðkÞ

����� E$~Pme

�����FiðkÞ+, the dipole

matrix element, is the velocity operator between the initial andnal wave functions Fi(k) and Ff(k). The absorption spectrumAx(u) originates in electronic transitions that correspond toexcitations from the occupied valence bands to the unoccupiedconduction bands, excited by the electromagnetic eld. In thiswork, the direction of the electromagnetic eld is assumed to beparallel to the x-axis. Within the gradient approximation,43,44 the

dipole matrix element MfiðkÞ ¼*FfðkÞ

����vHvkx����FiðkÞ

+, where

E$~Pme

is derived from the gradient of the Hamiltonian operator,vHvkx

.

Because of the zero momentum of photons, the optical selec-tion rule is Dk ¼ 0. The optical transition channels are deter-mined by the thermal factor FF � Fi and the dipole matrixelement.

The dipole matrix element, by inserting the tight-bindingwave functions F ¼ c1|A1i + c2|A2i + c3|B1i + c4|B2i, is expressedas

Mfi ¼Xl;m

cfl

�vH

vkx

�lm

cim; (15a)

where�vHvkx

�lm

¼ vHlm

vkxand Hlm is the element of the Hamilto-

nian matrix (eqn (1)). Aer the action of the rotation operator(eqn (3)), the dipole matrix element is then changed into

Mfi ¼Xl;m

dflV lmdim; (15b)

where the analytical form of the matrix element V lm is givenexplicitly in the Appendix. |di,f1 , d

i,f2 , d

i,f3 , d

i,f4 i are the initial (nal)

eigenstates associated with the Bloch function, eqn (6). Theeigenstate corresponding to each energy dispersion (E��) canbe exactly specied as

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8>>>>>>>>>>>>><>>>>>>>>>>>>>:

d1 ¼ 1ffiffiffiffiffiffiNc

p ;

d2 ¼ 1ffiffiffiffiffiffiNc

pE�� � H 21

H 12

;

d3 ¼ 1ffiffiffiffiffiffiNc

pðE�� � H 22Þd2 � H 11

H 23

;

d4 ¼ 1ffiffiffiffiffiffiNc

pðE�� � H 33Þd3 � H 32d2

H 34

:

(16)

The analytical expression for the dipole matrix element can beused to efficiently evaluate the magnitude of M and the opticalabsorption spectra A(u). M depends on energy spectrum,velocity operator, and eigenstates.

The square of the absolute value of dipole matrix element|M(k)|

2 in the high-symmetry line G–K–M for the transition E��/ E+� at different Vgs is drawn in Fig. 6, where the solid(dashed) curves are calculated with the energy spectrum and theeigenstates are obtained through the numerical method(analytical model). The dashed curves are exactly in agreementwith the solid ones along the high-symmetry line GK, while theyexhibit some discrepancy around the pointM. In the absence ofthe gated voltage (Vg ¼ 0), |M(k)|

2 exhibits a highly anisotropiccharacteristic with a maximal value around the point M in thehigh-symmetry line G–K–M. |M| is equal to zero in the line G–K,along which the structure factor f(k) is a real number and thematrix elements H 23 and H 32 in eqn (6) are equal to zero. The 4� 4 Hamiltonian matrix is decomposed into two 2 � 2 blocks.The eigen-vector corresponding to the nal (initial) state is |dfi¼ |d1, d2, 0, 0i (|dii ¼ |0, 0, d3, d4i). The six matrix elements V 13,V 23, V 24, V 31, V 32, and V 42 are all equal to zero as Im[f(k)] ¼0 and Vg ¼ 0. The straightforward calculation exhibits that themagnitude of the dipole matrix element vanishes, i.e., |M| ¼ 0.

In the presence of the gated voltage, |M|2 shows different

aspects, as illustrated in Fig. 6(b–d). Vg alters the Hamiltonianmatrix elements, energy spectrum and eigenstates |dii and |dfi.

Fig. 6 (a)-(d) The solid curves are the square of the absolute values ofthe optical matrix |Mfi|, obtained by the numerical diagonalizationmethod. The same plots obtained using analytical formulae are givenin the dashed curves.

This journal is © The Royal Society of Chemistry 2014

Moreover, the Vg-dependent V lms are also modied by the gatedvoltage. As a result, at Vg ¼ 100 meV, |M|

2 (blue solid curve inFig. 6(b)) around the point M exhibits a different aspect andmagnitude. The maximum is located in the vicinity of the pointM. The increase in Vg does not alter the prole of |M|

2 aroundthe point M. Most importantly, the application of Vg enhances|M|

2 in the high-symmetry line KG. Vg introduces the non-zeromatrix elements H 23 and H 32 in eqn (6), which mixes thesymmetrical and anti-symmetrical wave functions, i.e., thealternation of the eigenstates. Moreover, Vg also produces thesix matrix elements V 13, V 23, V 24, V 31,V 32, and V 42, beingunequal to zero. As a result, Vg intensies |M|

2 in the symmetryline KG. Such a characteristic strongly modies the low energyabsorption spectra in the presence of Vg.

The low-energy absorption spectra A(u) of AB BLG reect thecharacteristics of the energy dispersions. There are six possibleinterband excitations resulting from four branches of energydispersion. The optical excitation from the subband E�+ to E��and transition between E+� and E++ are forbidden by thethermal factor Ff � Fi. Four allowed excitations, A1(u), A2(u),A3(u) and A4(u), are shown by the vertical lines in the inset onthe le panel of Fig. 7(a). A1(u) stems from the transitionbetween the highest valence subband (E��) and the topconduction subband (E++), and A2(u) originates in the excitationE�+ / E+�. Moreover, the transition from the subband E�� toE+� generates the spectra A3(u) and the excitation between thesubbands E�+ and E++ leads to A4(u). A1(u) [A2(u)] shows adiscontinuous structure at u z 380 meV, as illustrated inFig. 7(a). Such an optical gap originates in the interband tran-sition between the E�� and E++ subbands [E�+ and E�+ sub-bands]. A4(u), resulting from the transition from the E�+ to E++

Fig. 7 (a) A1(u), A2(u), A3(u), and A4(u), the sub-optical spectra, areshown by the blue, green, red and brown curves. Four allowed opticaltransitions are indicated by the vertical lines in the inset in the leftpanel. (b) The low-energy absorption spectra (Ai(u)) at Vg ¼ 0, 100,200, and 300meV are drawn in red, green, brown and blue. (c) The lowenergy absorption spectra with different tight-binding parameters.

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subband, exhibits an absorption edge at uz 800 meV. There isno optical gap found in the spectra A3(u) (red curve) because ofthe overlap between subbands E�� and E+�. As a result of thesuperposition of the sub-spectra, the total spectra exhibit twodiscontinuities at u z 380 and u z 800 meV, and no opticalgap is found in A(u).

The application of Vg has a great effect on the low-energyabsorption spectra A3(u), as depicted in Fig. 7(b). Vg not onlyopens a band gap, but also gives rise to a sharp peak in A3(u). Inthe presence of Vg, the steep rise of the spectra, due to thetransition between the band-edge states of E�� and that of theE+� subband in the GK direction, can be used to determine thesize of the energy gap, Eg. The energy gaps are equal to u¼ 0.17,0.25, and 0.28 eV at Vg ¼ 100, 200, and 300 meV. The rstabsorption peak is a compound one, consisting of a shoulder onthe low energy side and a main peak. The former (shoulders)result from the transition between the E�+ and E+� subbands,denoted by the vertical red arrow in Fig. 4. The excitationindicated by the vertical blue arrow in Fig. 4 brings about themain peak. At Vg ¼ 100, 200, and 300 meV, the main peaks arelocated at u ¼ 0.18, 0.30, and 0.35 eV. The main peak is in thelogarithmic form, stemming from the saddle point in energydispersions in the high-symmetry lineMK. The frequency of themain peak depends on themagnitude of Vg. These peaksmake ablue-shi with the increase in Vg. The height of the rst peakalso enhances with the increase in Vg because the DOS aroundthe band-edge state of E�+ (E+�) and |M|

2 in the high-symmetryline GK are enhanced by Vg, as shown in Fig. 5 and 6.

How the tight-binding parameters (gis) inuence the lowenergy absorption spectra (A(u)) is deliberated. The dashedbrown curve in Fig. 7(c) represents A(u) at Vg ¼ 200 meV,through the analytical model, including all the tight-bindingparameters, g0, g1, g3, and g4. The close of the interlayerinteraction g4 has no effect on A(u), as shown by the cyan curve

in Fig. 7(c). According to eqn (7b), the magnitude ofg1

3AAg4

�� fk��gradually declines with an increase in Vg. As a result, theabsorption spectra at Vg ¼ 200 meV are not affected by the

Fig. 8 The cross symbols are band gaps acquired through the ab initiocalculations (data are taken from APL 98, 2011, 263107). Eg is obtainedfrom the analytical model for g3 ¼ g4 ¼ 0 (black curve) and g3 s 0 andg4 s 0 (red curve).

32124 | RSC Adv., 2014, 4, 32117–32126

interlayer interaction g4. A(u), illustrated by the red curve,exhibits a larger band gap (Eg ¼ 270 meV) and a sharp peaklocated at u ¼ Eg as the main intralayer interaction (g0) andinterlayer interaction (g1) are taken into consideration. More-over, A(u) presents a simple feature and weaker intensity ofabsorption relative to those in the brown curve. The chief causeis that the close of the interlayer interaction g3 not only changesthe energy dispersions and wave functions, but also turns offsome optical transition channels.

A comparison of the study results with those obtainedthrough ab initio calculations is made. The analytical formulaecan exactly describe four energy dispersions and generate theenergy bands with “Mexican hat” structures. Our researchdemonstrates that the gated voltage, Vg, alters the subbandspacing, changes the energy band gap, produces the oscillatingbands, and induces more band-edge states. The aforemen-tioned electronic properties, e.g., band feature, tunable bandgap, and electric-eld-modied oscillating bands, are alsoproduced through the ab initio calculations.41 That is, the elec-tronic properties derived from the analytical model are inqualitative agreement with those given by the ab initio calcula-tions. To be more specic, we quantitatively compared the sizesof the gated tunable band gaps acquired by the two methods.The calculated tunable band gaps, shown in Fig. 8 in the crosssymbols, are taken from the ab initio calculations.45 The bandgap rst increases monotonically as the strength of the electriceld increases. Then, the energy gaps saturate to a value Eg z270 meV when the eld strength |F| is greater than 0.2 V A�1.We now evaluate Eg by employing the analytical model for g3 ¼g4 ¼ 0 (black curve) and g3 s 0 and g4 s 0 (red curve). The redcurve exhibits that Eg rst increases sublinearly when thestrength of the electric eld (|F| ¼ 2Vg/c and c is the layerdistance) is less than 0.05 V A�1. Meanwhile, the energy gapsapproach a value of Eg z 300 meV when |F| > 0.2 V A�1. Thebehavior of Eg predicted by the analytical method (red curve) issimilar to that of the ab initiomethod. The comparison betweenthe black and red curves illustrates that when the eld strengthis weaker than 0.05 V A�1, the tunable band gaps are indepen-dent of the hopping integral g3. Aer that, the hopping integralg3 reduces the size of Eg.

5 Conclusions

We develop a method to access the analytical form of the energyspectrum and eigenstates of gated BBLG within the tight-binding model with all the tight-binding parameters included.The trigonal warping (g3) and applied gated bias (Vg) destroy themirror symmetry of BBLG. To deal with the broken symmetrycaused by g3 and Vg, a new set of tight-binding basis functions isconstructed through a rotation operator. The renormalizedintralayer and interlayer interactions exhibit a gated biasdependence. Most importantly, the Hamiltonian matrix (H) istransformed into a band-storage matrix in the subspace span-ned by the new basis functions. The formulae of the energydispersions and wave functions are then obtained analytically.g4, the electron–hole asymmetry, is treated as a perturbationand included in the analytical formula. The electronic

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properties, density of states, optical dipole matrix element andoptical absorption spectra are explored by employing thisanalytical formula. The study shows that Vg can open an opticalgap and produce a profound low-energy peak in the absorptionspectra. The dipole matrix element (M) and absorption spectraare strongly modied by Vg. Finally, the characteristics of theabsorption spectra, e.g., the location and height of the low-energy peak, are signicantly presented and explored throughthe analytical model.

Appendix

The element V lm is the operator�vHvkx

�sandwiched by

the symmetrized basis functions hfl| and |fmi, i.e.

V lm ¼�fl

����vHvkx����fm

�, where H is the Hamiltonian representation

(eqn (1)). The analytical form of each V lm is listed as follows.

V 11 ¼ 0,

V 12 ¼ g0

2| f |

0BB@g1

2f*vf

vkxþ �3� Vg

�2fvf*

vkx3�3� Vg

�1CCA

þ g1

3g4

�f*

2| f |

vf

vkxþ f

2| f |

vf*

vkx

�;

V 13 ¼�g0

g1

3� g4

� f*

2| f |

vf

vkx� f

2| f |

vf*

vkx

�

��g4

Vg

3

��f

2| f |

vf*

vkxþ f*

2| f |

vf

vkx

�;

V 14 ¼ 0,

V 21 ¼ V *12,

V 22 ¼ g1

3

g3

2

�f*

| f |

vf*

vkx

f*

| f |þ f

| f |

vf

vkx

f

| f |

�;

V 23 ¼ g3

2

�f*

| f |

vf*

vkx

f*

| f |� f

| f |

vf

vkx

f

| f |

�

� Vg

3

g3

2

�f*

| f |

vf*

vkx

f*

| f |þ f

| f |

vf

vkx

f

| f |

�;

V 24 ¼�� g0

g1

3þ g4

� f*

2| f |

vf

vkx� f

2| f |

vf*

vkx

�

��g4

Vg

3

��f

2| f |

vf*

vkxþ f*

2| f |

vf

vkx

�;

This journal is © The Royal Society of Chemistry 2014

V 31 ¼ V *13,

V 32 ¼ V *23,

V 33 ¼ �V 22,

V 34 ¼ g0

2| f |

0BB@g1

2fvf*

vkxþ �3þ Vg

�2f*

vf

vkx3�3þ Vg

�1CCA

þ �g1

3g4

�f

2| f |

vf*

vkxþ f*

2| f |

vf

vkx

�;

V 42 ¼ V *24,

V 34 ¼ V *43,

V 44 ¼ 0.

Acknowledgements

The authors gratefully acknowledge the support of the TaiwanNational Science Council under the Contract no. NSC 101-2112-M-165-001-MY3.

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