Coherent non-linear optical response in SU(2) symmetry broken singleand bilayer graphene

Vipin Kumar n, Enamullah, Upendra Kumar, Girish S. SetlurDepartment of Physics, Indian Institute of Technology Guwahati, Guwahati, Assam 781039, India

a r t i c l e i n f o

Article history:Received 13 September 2013Received in revised form2 December 2013Accepted 2 December 2013Available online 8 December 2013

Keywords:GrapheneBilayer grapheneSU(2) symmetryNonlinear phenomenonRabi oscillation

a b s t r a c t

Anomalous Rabi oscillations in single and bilayer graphene, in the absence of time-reversal symmetry,are described. The main findings of this work are that intra-layer sublattice space asymmetry has aremarkable effect on anomalous Rabi frequency in single and bilayer graphene, namely it is offset by theasymmetry parameter. However, the conventional Rabi frequency is nearly independent of theasymmetry parameter. Inter-layer asymmetry in bilayer graphene has an even more significant effecton anomalous Rabi frequency. When inter-layer asymmetry is taken into account, the anomalous Rabifrequency versus the external field goes through a minimum. The induced current in the frequencydomain in these systems shows a finite threshold behavior even for vanishingly small applied fields.These offset oscillations are attributable to the asymmetry parameter in these systems, and areobservable only for weak applied fields. For stronger applied fields these phenomena tend towardsthose without asymmetry.

& 2013 Elsevier B.V. All rights reserved.

1. Introduction

Individual graphene flakes of graphite were first isolated adecade ago by the experimental group of Novoselov et al. [1]. Thereis widespread interest in graphene since it impacts many fieldsincluding material science, condensed matter physics, optical phy-sics, high-field and high-energy physics. Graphene is unique in thatat one end it has potential applications in carbon based electronicsand at the other it provides a tabletop laboratory of quantumrelativistic phenomena [2,3]. Graphene is an array of carbon atomspacked in a dense two-dimensional (2D) honeycomb lattice. Thishoneycomb lattice may be considered as composed of two trian-gular sublattices and the wave function amplitudes on thesesublattices may be described in terms of pseudo-spin, whichis analogous to a two-component real spin-1/2 elementary particle.The charge carriers (electron and hole) in graphene are masslessquasi-particles that follow the relativistic massless Dirac equation(Dirac–Weyl equation). These quasi-particles have specific (linear)energy momentum dispersion at each of the six corners, the so-called Dirac points, of the hexagonal-shaped Brillouin zone, wherethe valance and conduction band touch one another. This lineardispersion is governed by the Dirac Hamiltonian [2,3] HD ¼ ℏvFsαkα ,where s are the pauli spin matrices and α¼x, y. Only two of the sixDirac points are physically inequivalent. They are connected by

time-reversal symmetry at these points. Graphene CH [4], fluoro-graphene CF [5], BN [6] and MoS2 [7] also exhibit unusual chemicaland physical properties, and these nanomaterials may be useful forfuture nanotechnology applications. Similar to single layer graphenethese nanomaterials also have a 2D honeycomb lattice structure [8].

Subsequently, bilayer graphene (BLG), graphite with two gra-phene flakes, became a subject in its own right due to itsintriguing electronic properties. From a technological point ofview, it is a semiconductor with a gate-tuneable band gap betweenthe conduction and valance band [9–12]. It is an excellent nano-material for nonlinear optics and photonic applications becausethe field strength required to elicit a nonlinear response is verylow [13,14]. Bilayer graphene is composed of two honeycomblattices consisting of inequivalent lattice sites A1, B1 and A2, B2 inthe bottom and in the top layer, respectively. These two layers arearranged in a Bernal (A1–B2) stacking [10], and coupled by theweak Van der Waals forces. The charge carriers in bilayer grapheneare massive chiral Dirac fermions (quasi-particles) that obey thepseudo-relativistic Dirac equation. These quasi-particles also pos-sess the pseudo-spin degree of freedom found in the single layer.It is associated with the wave-function amplitude on two layersand turns twice as quickly as the direction of momentum [9]. Thequasi-particles in bilayer graphene show a parabolic dispersion atthe Dirac point of the hexagonal-shaped Brillouin zone.

Rabi oscillation is a nonlinear optical phenomenon well-knownin quantum optics [15,16] and conventional semiconductors [17].Graphene is analogous, in some sense, to conventional semicon-ductors. It is expected that it will also exhibit nonlinear optical

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Physica B

0921-4526/$ - see front matter & 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.physb.2013.12.002

n Corresponding author. Tel./fax: þ91 361 258 2715.E-mail address: k.vipin@iitg.ernet.in (V. Kumar).

Physica B 436 (2014) 140–148

phenomena similar to conventional semiconductors and two-levelatomic systems. The nonlinear optics of graphene is a nascent fieldin material science and condensed matter physics. However it isgaining importance due to its wide range of potential applications.The nonlinear optical response of graphene systems is notadequately described by either pure intra-band or pure inter-band electron dynamics but their intermediate role [18]. Ang et al.[13] studied the nonlinear optics of bilayer graphene in thefrequency ranging from terahertz to far-infrared regime andsuggested that it is a more suitable lattice structure for developinggraphene based photonics and optoelectronic devices [14] such asphotodiodes including solar-cell, photo-detector, and light-emitting devices. The nonlinear optics of monolayer and bilayergraphene is also described in the literature available on this topic[19–24].

Graphene is a zero gap semiconductor, so it cannot be used as aviable semiconductor in electronic devices since such devicesmade from zero gap semiconductors cannot be turned off. There-fore, to make use of graphene in electronic applications, it is veryimportant to open a tunable and sizable band gap in the electronicspectrum of graphene. This gap may be opened usually bysymmetry breaking through extrinsic effects [25] to generate adynamical mass. These relativistic massive carriers in graphenehave different electronic properties than massless carriers, withparabolic dispersion between energy and momentum [26]. Usingdensity functional theory, Quhe and collaborators [27] show thatwhen single layer graphene is properly sandwiched between apair of hexagonal boron nitride single layers, a band gap of 0.16 eVand 0.34 eV can be opened without and with electric field,respectively. Hexagonal boron nitride consists of single layers ofBN of a honeycomb structure [6,28,29] almost commensurate tosingle layer graphene [2]. Ongun et al. [28] studied the epitaxialgrowth mechanism of graphene with and without a substrateusing density functional theory and showed that the growthprocess is affected by the presence of substrate in a crucial way.

Mak et al. [30] studied the gap opening in the electronicspectrum of bilayer graphene by applying a perpendicular electricfield to the plane of bilayer graphene. They are able to induce a gapof 0.2 eV in the presence of a strong applied electric field, about1 V/nm. The inter-layer asymmetry has a dramatic effect on theenergy band structure of bilayer graphene. An inter-layer asym-metry gives rise to a ‘Mexican hat’ like structure in the bandstructure of the bilayer graphene [9,31–34]. The properties ofgapped monolayer and bilayer graphene are also discussed in theextensive literature available in this subject [35–39].

Pioneering work on the nanoscale super capacitor has beenproposed by Ongun et al. [40]. Recently, Lee et al. [41] fabricateda FET, based on MoS2 using h�BN as dielectric and graphene asa gate electrode. They found that MoS2 shows a very high mobilityat low operating gate voltage, and the heterostructure devicesbased on a stacking of MoS2/h-BN/graphene are highly transparentand flexible. These two dimensional heterostructures are alsoflexible and transparent and may be used in logic circuits. Apartfrom this, when multi-layer graphene sheets are sandwichedbetween Ni(1 1 1) surfaces, another interesting phenomenon hasbeen seen – when these layers undergo a sliding motion under aconstant pressure, they behave as a lubricant [42–44]. Therefore,this proposed model may be used to make a single layer nano-structured lubricant.

In the present work, we discuss the nonlinear optical response(mainly, the phenomenon of Rabi oscillation) in gapped monolayerand bilayer graphene in the presence of an intense opticalpump field. The phenomenon of Rabi oscillation is analysed intwo different regions of interest namely – the resonance and theoff resonance case. In the case of symmetric graphene, Rabioscillations close to resonance have already been discussed in

the literature [18,20,21] using rotating wave approximation (RWA)[16,17]. To analyze the phenomenon of Rabi oscillations in the offresonance case, an approximation is used called the asymptoticrotating wave approximation (ARWA) [45]. Rabi oscillations showanomalous behaviour off resonance. Off resonance, the gappedgraphene systems (graphene systems mean – monolayer andbilayer graphene) show offset oscillations even for vanishinglysmall electric fields – a feature absent in the symmetric graphenesystems. These offset oscillations are characterized by the asym-metry parameter in these systems. In bilayer graphene, inter-layerasymmetry has a dramatic effect on anomalous Rabi oscillations.Conventional Rabi oscillations however are not significantlyaffected by the asymmetry parameter in these graphene systems.

2. The low energy Hamiltonian of asymmetric monolayergraphene

The low energy Hamiltonian of the graphene honeycomblattice may be derived using the tight binding model [2]. Thishoneycomb lattice is made from two sublattices named A and B,and connected by the time-reversal symmetry (SU(2) symmetry).Therefore, the Hamiltonian is symmetric under the transformationof A2B, and the massless Dirac fermions show gapless linearenergy–momentum dispersion at the Dirac points. The zero gapdispersion implies that the conduction of electrons cannot besimply toggled by the external gate voltage, and this limits theuse of graphene in electronic applications. This gapless energydispersion of graphene has been derived by assuming that the on-site energy of electrons in the sublattices A and B is equal,HAA ¼HBB. Whenever HAAaHBB, a mass term exists which isresponsible for opening of a gap in the energy spectrum. Thismass term breaks the symmetry between the two sublatticesA and B, and graphene becomes asymmetric under the transfor-mation of A2B. Now the graphene Hamiltonian will no longer besymmetric under this transformation. There are various ways toinduce this mass term such as interaction of graphene sheet withthe substrate upon which graphene sheet is deposited [25,36],sandwiching a monolayer graphene between a pair of hexagonalBoron nitride layers [27] or applying an external electric fieldperpendicular to the plane of the graphene sheet. There areexperimental and theoretical [26,37] research articles availableon the topic of generation of a gap in the energy spectrum ofgraphene.

The effective low energy Hamiltonian [2] of monolayer gra-phene can then be written as a pure graphene Hamiltonian and aterm that describes the creation of a mass term,

H1 ¼H0þHmt ð1Þwhere H0 ¼ vF ðsxkxþsykyÞ is the intrinsic graphene Hamiltonianand Hmt ¼ vFmtsz is the Hamiltonian of the mass term that mayarise due to sublattice space asymmetry between A and Bsublattices. The energy eigenvalues of Hamiltonian equation (1)with the ‘mass’ term are derived in Refs. [26,38] as given below:

E¼ 7

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðvFmtÞ2þγ2jf ð k!Þj2

qð2Þ

where f ð k!Þ¼ eikxa=ffiffi3

pþ2e� ikxa=

ffiffi3

pcos ðkya=2Þ and γ is the transfer

integral. Eq. (2) gives parabolic dispersion for the charge carriers(Fig. 1, solid green).

To describe the phenomenon of Rabi oscillations a semiclassicalapproximation is used, radiation is treated as classical andmatter fields are quantum. An in plane electric field is applied tothe graphene sheet through a vector potential of the formA!ðtÞ ¼ A

!ð0Þe� iωt , where A!ð0Þ ¼ ðe=cÞðAxð0Þþ iAyð0ÞÞ is a complex

amplitude of an external driving field. In the second quantization,using two-component basis eigenstates of Hamiltonian equation (1)

V. Kumar et al. / Physica B 436 (2014) 140–148 141

in momentum space of the form ψ T ¼ ðcAð k!

; tÞ cBð k!

; tÞÞ, theHamiltonian equation (1) reads as

H1 ¼∑k-

vF ðkþ �AnðtÞÞc†Að k!

; tÞcBð k!

; tÞþh:c:� �

þvFmt∑k-

c†Að k!

; tÞcAð k!

; tÞ�c†Bð k!

; tÞcBð k!

; t� �

ð3Þ

where k7 ¼ kx7 iky, c and c† are the annihilation and creationoperators on sublattice sites A and B and vice versa. If mt ¼ 0, theHamiltonian equation (1) commutes with the helicity operator h(projection of pseudo-spin in the direction of momentum) thatpreserves the SU(2) symmetry. In the presence of the mass term,the Hamiltonian equation (1) does not commute with pseudo-spinprojection operator along the direction of momentum, and breaks SU(2) symmetry of Hamiltonian equation (1), which means that time-reversal symmetry has been broken or we can say that there exists asublattice space asymmetry between two sublattices A and B.

2.1. Derivation and solution of Bloch equations of asymmetricmonolayer graphene

In this section, we derive the Bloch equations of asymmetricmonolayer graphene and discuss their solution in the differentregions of interest namely – resonance and far from resonance. Toderive the Bloch equations of asymmetric monolayer graphene, wefollow a similar process discussed in the case of symmetricgraphene. The Bloch equations for asymmetric monolayer gra-phene may be written as follows:

i∂tndiff ð k!

; tÞ ¼ 2vF ½ðkþ �AnðtÞÞpð k!; tÞ�h:c:� ð4Þ

i∂tpð k!

; tÞ ¼ vF ðk� �AðtÞÞndiff ð k!

; tÞ�2vFmtpð k!

; tÞ ð5ÞThese are the rate equations of the population ndiff ð k

!; tÞ and

polarization pð k!; tÞ excess on sublattice sites A and B. ndiff ð k!

; tÞ¼ ⟨c†Að k

!; tÞcAð k

!; tÞ⟩� ⟨c†Bð k

!; tÞcBð k

!; tÞ⟩ and pð k!; tÞ ¼ ⟨c†A ð k!; tÞcB

ð k!; tÞ⟩. The asymmetry affects only the rate equation of polariza-

tion pð k!; tÞ. Now, we want to solve these equations near resonance,when the external driving frequency is nearly equal to the inter-band transition frequency of the system, and far from resonance,when the external driving frequency is very large in comparison tothe inter-band transition frequency of the system.

Solution of Bloch equations in the off resonance case. The Blochequations in this regime, identified by the limit ωb2Ek where Ek isthe energy eigenvalue of the system, are solved using an approx-imation which is alternative to the rotating wave approximation(RWA) called asymptotic rotating wave approximation (ARWA). For

this, we have broken the population ndiff ð k!

; tÞ and polarizationpð k!; tÞ into slow and fast varying terms.

F ¼ Fsð k!

; tÞþF þ ð k!

; tÞe� iωtþFn

� ð k!

; tÞeiωt ð6Þ

where F can be ndiff or p, s in the subscript denotes the slow varyingcoefficients. The solution of the fast coefficients of the populationand polarization will be of the form

nf ð k!

; tÞ � 2vFAnð0Þ

ωpn

s ð k!

; tÞ

pþ ð k!

; tÞ � � vFAnð0Þ

ð2vFmtþωÞnsð k!

; tÞ; p� ð k!

; tÞ � 2v2Fk�Að0Þωð2vFmt�ωÞpsð k

!; tÞ

The solution of the slow varying part of the polarization equationwill be

psð k!

; tÞ ¼ nsð k!

;0Þ ωRm

4vFkþ� cos ð2ΩtÞ� i

2ΩωRm

sin ð2ΩtÞ� �

ΩARWA ¼ 2 ðvF jkjÞ2þω2

Rm

4

� �1=2; ωRm ¼ 2vFmtþ

2v2F jAð0Þj2ω

ð7Þ

Eq. (7) gives the anomalous Rabi frequency in the off resonancecase. Exactly at the Dirac point of the Brillouin zone, the aboveformula reduces to ΩARF ¼ 2vFmtþ2ω2

R=ω where ωR ¼ vF jAð0Þj isthe zero detuned conventional Rabi frequency. In the absence of amass term, it gives the anomalous Rabi frequency of pure graphene.In the presence of the mass term, the anomalous Rabi frequency isoffset by the asymmetry parameter (mass term) even for vanish-ingly small applied field – a feature absent in symmetric graphene,and the value of the offset frequency may be identified with theasymmetry parameter. Typically, this asymmetry parameter is dueto sublattice space asymmetry Δ between A and B sublattices. Fig. 2depicts the offset frequency in asymmetric monolayer graphene.The actual value of offset frequency ΩARF is 0.21ω for a value of thefrequency ω¼6π�1014 rad/s of the applied field. In addition wechoose ωR¼2.02�1013 rad/s. This offset frequency is better obser-vable for weak electric fields. We can see that the mass term has aremarkable effect on the anomalous Rabi frequency only for weakfields, as we can see from Eq. (7). However, the asymmetryparameter has no significant effect on the conventional Rabifrequency as we shall see from the subsequent discussion.

Solution of Bloch equations near resonance. Here, the Bochequations are solved near resonance, when the frequency of theapplied field is nearly equal to the particle–hole excitation energyðω� 2EkÞ. To solve the Bloch equations near resonance, we adoptthe well-known rotating wave approximation also described in anearlier work. In the case of asymmetric monolayer graphene, the

E

k

Fig. 1. Left: Schematic of the graphene sheet deposited upon a substrate. Δ is the on-site energy of the atoms on sublattice sites A and B that defines the intra-layerasymmetry between two sublattices and gives rise a mass term in the monolayer graphene Hamiltonian equation (1). Right: Energy spectrum of monolayer graphene with(solid green) and without asymmetry (dashed black). The energy spectrum of asymmetric monolayer graphene has a parabolic dispersion and shows a band gap between theconduction and valance band. (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this paper.)

V. Kumar et al. / Physica B 436 (2014) 140–148142

conventional Rabi frequency near resonance is given by

ΩRWA ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiδ2þω2

Rþω2

R

E2kΔðΔ�2EkÞ

sð8Þ

where δ¼ω�2Ek and Ek ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiv2Fk

2þΔ2q

. Near resonanceω� 2Ek andalso Δ5ω. With these approximations, Eq. (8) reduces to ΩRWA ¼ωRð1�2Δ=ωÞ �ωR because of, Δ5ω. We can see that the conven-tional Rabi frequency near resonance is nearly independent of theasymmetry parameter Δ, as we argued earlier. Therefore, we mayconclude that only the anomalous Rabi frequency is significantlyaffected by asymmetry, and this is better observable for weak appliedelectric fields.

2.2. Current density in asymmetric monolayer graphene

Current density is an experimentally accessible quantity. The-oretically, this may be derived with the help of continuity equation

and Heisenberg equation of motion for the charge density.

J!ðtÞ ¼ sτevF∑

k-

pð k!; tÞ s!ABþh:c:

where vF is the Fermi velocity of Dirac fermions, s! is the Paulispin matrices, s and τ are the spin and valley degeneracy,respectively. Inserting the value of polarization pð k!; tÞ in theabove equation and performing a straightforward calculation, wemay easily write the slow part of current density in the frequencydomain, in the off-resonance case,

js!ðω′Þ � �2sτeAAnð0Þ s!AB

2ωRmvFωω′4�ω′2 2Δþ2ω2

R

ω

� �2" #1=2ð9Þ

Eq. (9) has the following explanation: the value of thethreshold frequency of the slow part of current density, inthe case of asymmetric monolayer graphene, is increased bythe asymmetry parameter (as we have seen earlier as well),and the threshold frequency ω′ of induced current is equal toð2Δþ2ω2

R=ωÞ which is exactly the anomalous Rabi frequency atthe Dirac point, as is clear from Eq. (7). Moreover, the currentdensity in asymmetric graphene exhibits a threshold behavioureven for vanishingly small applied fields. The asymmetryparameter does not affect the exponent at threshold but ithas a remarkable effect on the value of the threshold fre-quency. In Fig. 3, we have shown the variation of the inducedcurrent with a dimensionless quantity ηð ¼ω′=ωÞ. For this, wemake the quantity within the square bracket dimensionless.Realistic values of ω and ωR are 6π�1012 rad/s and 2.02�1013 rad/s, respectively. This explanation becomes more clearin Fig. 3.

In the above discussion, we have observed the effect of asym-metry on Rabi oscillations as well as on the induced current inasymmetric monolayer graphene, mainly in the off resonance case.The anomalous Rabi frequency is offset by the asymmetry para-meter in the off resonance case whereas asymmetry has nosignificant effect on the conventional Rabi frequency. The inducedcurrent in asymmetric monolayer graphene shows threshold beha-viour even in the absence of external applied electric field.

The subsequent sections describe the same phenomenon inasymmetric bilayer graphene. In bilayer graphene, there are twoasymmetries involved namely – inter-layer and intra-layer asym-metry. We will investigate the effect of these asymmetries

0.0 0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

1.5

2.0

2.5

3.0

x

Fig. 2. Schematic of the variation of the dimensionless anomalous Rabi frequency(ARF) ΩARF=ω versus a dimensionless quantity x ð ¼

ffiffiffi2

pωR=ω;ωR ¼ vF jAð0ÞjÞ in the

case of symmetric graphene (dashed black) and asymmetric monolayer graphene(solid green), at the Dirac point of the Brillouin zone. ARF in symmetric grapheneshows a zero trivial minimum in the absence of externally applied field whereas forasymmetric graphene it has a finite value. This shows that the ARF exhibits offsetoscillations that may be identified with the twice the asymmetry parameter Δ. Forclarity, we have chosen a large value of the offset frequency. A realistic value of theoffset frequency is related to asymmetry parameter through the relation,ΩARF=ω¼ 2Δ=ℏω¼ 0:21. (For interpretation of the references to colour in this figurecaption, the reader is referred to the web version of this paper.)

0 0.02 0.040

0.0010.002

Symmetric SLG

Asymmetric SLGR 0

0.0 0.1 0.2 0.3 0.4 0.50.00

0.05

0.10

0.15

0.20

0.25Symmetric SLG

Asymmetric SLGR 0

0 0.02 0.040

0.0010.002

0.0 0.1 0.2 0.3 0.4 0.50.00

0.05

0.10

0.15

0.20

0.25

η η

ω ω

Fig. 3. Schematic of the variation of the slow part of: (left) induced current versus a dimensionless quantity ηð ¼ω′=ωÞ, in the presence of an external applied field, and (right)in the absence of an externally applied field. This figure is plotted by assuming that ωR ¼ 1 and ω¼10, in arbitrary units. It is obvious from the left part of this figure that thethreshold frequency of the induced current is increased in the presence of asymmetry (solid green) in comparison to that of without asymmetry (dashed black) shown ininset for clarity. The threshold frequency ω′=ω is ðð2Δ=ℏωÞþð2ω2

R=ω2ÞÞ which is the anomalous Rabi frequency at the Dirac point, as may be seen from Eq. (7). The right part is

plotted in the absence of external field. It is clear from the right part of this figure that the induced current in symmetric graphene [45] (dashed black) loses its thresholdbehaviour when the external field is absent, whereas asymmetric monolayer graphene (solid green) still exhibits the threshold behaviour. The frequency of offset oscillationsof the current is identified by the asymmetry parameter Δ, and has the relation, 2Δ¼ 0:21ℏω. (For interpretation of the references to colour in this figure caption, the reader isreferred to the web version of this paper.)

V. Kumar et al. / Physica B 436 (2014) 140–148 143

on anomalous Rabi oscillations. The intra-layer asymmetry has asimilar effect on anomalous Rabi oscillations as in asymmetricmonolayer graphene, whereas the effect of inter-layer asymmetryon anomalous Rabi oscillations is dramatic.

3. Effective low-energy Hamiltonian of asymmetric bilayergraphene

In this section, we investigate the effect of asymmetry onanomalous Rabi oscillations in bilayer graphene. In bilayer gra-phene, there are two kinds of asymmetry – intra- and inter-layerasymmetry. The inter-layer asymmetry may be due to the dopingeffect or the external gate-voltage, whereas the intra-layer asym-metry arises due to the substrate and graphene sheet interactionupon which the graphene sheet is deposited. It could also arisedue to an applied electric field perpendicular to the plane of thegraphene sheet [30,33,35]. Bilayer graphene is composed of twosingle graphene sheets in a Bernal stacking [9]. The effective lowenergy Hamiltonian of bilayer graphene including the inter-layerand intra-layer asymmetry may be written as follows:

H2 ¼ � 12m

0 k2�k2þ 0

0@

1Aþ Δ 0

0 �Δ

!þU

12�v2kþ k�

γ21

!1 00 �1

� �

ð10Þ

The first term in Eq. (10) is the effective low energy two-component Hamiltonian of pure bilayer graphene that describes theindirect inter-layer hopping from A1 to B2 atomic site which includesa direct inter-layer hopping γ1 forming dimer-site between the atomicsites B1 and A2, and an intra-layer hopping velocity v. This two-component Hamiltonian was first derived by McCann et al. [9] usingGreen's function method for matrices. The second and third terms inEq. (10) are due to the intra- and inter-layer asymmetry that breaksthe sublattice space symmetry and the symmetry between two layers,respectively. The intra-layer asymmetry opens up a gap in the lowenergy spectrum of bilayer graphene. The effect of inter-layer asym-metry is dramatic, it leads to a ‘Mexican-hat’ structure in the lowenergy spectrum of bilayer graphene, Fig. 4. Our purpose is to describethe effect of asymmetry on Rabi oscillations in the presence of anintense applied electromagnetic field. We first discuss the effect ofintra-layer asymmetry on Rabi oscillations and the effect of inter-layerasymmetry is discussed afterwards.

The low-energy electronic band spectrum in Fig. 4 is derived inRefs. [9,10,32] by solving a four-component Hamiltonian that givesthe following energy bands when intra- and inter-layer asymmetry

have been taken into account, respectively,

ε2α ¼Δ2

4þγ21

4

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ4γ20jf ð k

!Þj2γ21

vuut þð�1Þα264

3752

ð11Þ

ε2α ¼γ212þU2

4þγ20jf ð k

!Þj2þð�1Þαffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiγ414þγ20jf ð k

!Þj2ðγ21þU2Þs

ð12Þ

where f ð k!Þ¼ eikya=ffiffi3

pþ2e� ikya=2

ffiffi3

pcos ðkxa=2Þ, a is the lattice con-

stant. α¼1 and 2 that describe low and high energy bands,respectively. Eqs. (11) and (12) show that intra-layer asymmetryonly gives a shift in the energy whereas inter-layer asymmetry givesrise to a ‘Mexican hat’ in the energy spectrum.

3.1. Effect of intra-layer asymmetry on Rabi oscillations

The model presented here, uses a semiclassical approximation,similar to the model presented in the case of asymmetric monolayergraphene. In the low-energy continuum limit, the Hamiltonian ofbilayer graphene including intra-layer asymmetry may be written as

H¼ � 12m

∑k-

ðk� �AnðtÞÞ2c†A1ð k!

; tÞ cB2ð k!

; tÞþh:c:� �

þΔ∑k-

c†A1ð k!

; tÞ cA1ð k!

; tÞ�c†B2ð k!

; tÞ cB2ð k!

; t� �

ð13Þ

where k7 ¼ kx7 iky, AðtÞ ¼ Að0Þe� iωt is the vector potential whichcouples to bilayer graphene in the Coulomb gauge and Að0Þ ¼ ðe=cÞðAxð0Þþ iAyð0ÞÞ, c and c† are annihilation and creation operators onsublattice sites A1 and B2. The second term in Eq. (13) is purely dueto intra-layer asymmetry. Just as we did for asymmetric monolayergraphene, here too we derive the Bloch equations of bilayergraphene in the presence of intra-layer asymmetry and solve theseequations in two different regions of interest – resonance and far fromresonance. Following a process similar to Section 2.1, we may writethe Bloch equations of population ndiff ð k

!; tÞ and polarization pð k!; tÞ

excess on sublattice site A1 in the bottom layer and B2 in the toplayer when intra-layer asymmetry is taken into account:

i∂tndiff ð k!

; tÞ ¼ � 1m½ðk� �AnðtÞÞ2pð k!; tÞ�h:c:� ð14Þ

i∂tpð k!

; tÞ ¼ � 12m

ðkþ �AðtÞÞ2ndiff ð k!

; tÞ�2Δpð k!; tÞ ð15Þ

As we can see from Eqs. (14) and (15) that asymmetry affects only thepolarization whereas population is not affected by asymmetry. Thepopulation excess ndiff ð k

!; tÞ and polarization pð k!; tÞ are defined

E

k

Fig. 4. Left: Schematic of bilayer graphene deposited upon a substrate. Δ is the on site energy of the atoms on sublattice sites A1 and B1 that defines the intra-layerasymmetry between two sublattices and opens up a gap at the Dirac point. U is the asymmetry between two layers that gives rise to a ‘Mexican’ hat structure in the energyspectrum. Right: Low energy spectrum of bilayer graphene with intra-layer asymmetry (solid green), without asymmetry (dashed black) and with inter-layer asymmetry(solid blue). (For interpretation of the references to colour in this figure caption, the reader is referred to the web version of this paper.)

V. Kumar et al. / Physica B 436 (2014) 140–148144

below:

ndiff ð k!

; tÞ ¼ ⟨c†A1ð k!

; tÞ cA1ð k!

; tÞ⟩�⟨c†B2ð k!

; tÞ cB2ð k!

; tÞ⟩pð k!; tÞ ¼ ⟨c†A1ð k

!; tÞ cB2ð k

!; tÞ⟩

Now we wish to solve the rate equations of population and polariza-tion in two different regions of interest, resonance and far fromresonance, as we have already been mentioned in Section 2.1 in thecase of asymmetric monolayer graphene.

Solution of Bloch equations in the off resonance case. Offresonance, when the frequency of the external field is very largein comparison to the inter-band transition frequency of the systemðωbk2=mÞ and also from the zero detuned Rabi frequency ofcarriers ðωbωRÞ, the Bloch equations are solved using an approx-imation called asymptotic rotating wave approximation (ARWA),as discussed in Section 2.1. In the case of bilayer graphene, whenwe decomposed the population ndiff ð k

!; tÞ and polarization pð k!; tÞ

into slow and fast varying, we have to keep terms up to the secondharmonic in the external frequency unlike single layer graphene,where we kept only the terms that oscillate with the firstharmonic in the external driving frequency, as it is clear fromthe Hamiltonian equation (13). Now, inserting Eq. (6) into theBloch equations, and following a similar process of Section 2.1.We may write the solution of fast coefficients of populationand polarization,

nf1ð k!

; tÞ ¼ �2kþAð0Þmω

pn

s ð k!

; tÞ; nf2ð k!

; tÞ ¼ A2ð0Þ2mω

pn

s ð k!

; tÞ

p1� ð k!

; tÞ ¼ 1mð2Δ�ωÞ

jkj2kþAnð0Þ

mωpsð k

!; tÞ

�

þkþ jAð0Þj2Anð0Þ2mω

psð k!

; t�

p2� ð k!

; tÞ ¼ � k2þAn2ð0Þ

8m2ωðΔ�ωÞpsð k!

; tÞ;

p1þ ð k!

; tÞ ¼ kþAð0Þmð2ΔþωÞnsð k

!; tÞ

p2þ ð k!

; tÞ ¼ � A2ð0Þ4mðΔþωÞnsð k

!; tÞ

The slow varying population and polarization equations are

i∂tnsð k!

; tÞ ¼ � 1m½k2�psð k

!; tÞ�k2þp

n

s ð k!

; t�

i∂tpsð k!

; tÞ ¼ �k2þm

nsð k!

; tÞ� 2Δþω2R

ωþ4

jkj2ωR

mω

� �psð k

!; tÞ

These two equations may be solved mutually by assuming that thesolution of the slow part of the populations equation nsð k

!; tÞ ¼

nsð k!

;0Þ cos ð2ΩtÞ. Then the solution of the slow part of polariza-tion equation may be written as follows:

psð k!

; tÞ ¼ k2þ4mEk

cos ð2ΩtÞþ i2ΩωR;asy

sin ð2ΩtÞ� �

;

nsð k!

;0Þ ¼ jkj42m2EkωR;asy

ΩARWA ¼ 2jkj22m

� �2

þω2

R;asy

4

" #1=2; ωR;asy ¼ 2Δþω2

R

ωþ4

jkj2ωR

mω

� �

ð16Þwhere ωR ¼ jAð0Þj2=2m, Ek ¼ ðjkj4=4m2þΔ2Þ1=2 is the energy eigen-value of the system and Δ denotes the intra-layer asymmetry thatopens up a gap between the valance and conduction band. Eq. (16)gives the anomalous Rabi frequency near the Dirac point of theBrillouin zone. Exactly at the Dirac point, when k

!¼ 0, Eq. (16)

reduces to ΩARF ¼ 2Δþω2R=ω. In the absence of asymmetry, this

gives the anomalous Rabi frequency of pure bilayer graphene thatvaries linearly with the square of the intensity of the applied field,whereas , in single layer graphene it is linear with intensity [45].In the presence of asymmetry, the system exhibits Rabi-likeoscillations even for vanishingly small applied fields. It meansthat, here also, similar to asymmetric monolayer graphene, theanomalous Rabi frequency is offset by an asymmetry parameter.The value of the offset frequency is identified with the twice theasymmetry parameter. Realistic values of ω, ωR and Δ are listed inTable 1. Offset oscillations in bilayer graphene with intra-layerasymmetry are shown in Fig. 5.

Solution of Bloch equations near resonance. To solve the Blochequations near resonance, we follow a procedure described inSection 2.1. In bilayer graphene, there are two possible resonances– one is at the first harmonic and another is at the secondharmonic in the external frequency, unlike in single layer gra-phene [45] where there is only one possible resonance. Theconventional Rabi frequency near first and second harmonicresonances are given below, respectively,

Ω1 ¼ δ2þ2ωRjkj2mE2k

jkj44m2þ2ΔðΔ�EkÞ� �" #1=2

Ω2 ¼ δ2þω2R

E2k

jkj44m2þ2ΔðΔ�EkÞ� �" #1=2

ð17Þ

Table 1This table contains realistic values of various energy scales inmonolayer and bilayer graphene. The value of Δ and U are adoptedfrom Refs. [25,34,46].

Energy Unit (eV)

SLG BLG

ℏω2R=ω 1.4�10�4 1.78�10�7

ℏωR 0.013 4.69�10�4

Δ 0.26 0.15U – 0.075ℏω 1.24 1.24

0.0 0.2 0.4 0.6 0.8 1.00.0

0.5

1.0

1.5

2.0

2.5

3.0

x

Fig. 5. Schematic of the variation of the dimensionless anomalous Rabi frequency(ARF) ΩARF=ω versus a dimensionless quantity x (intensity of applied field)ð ¼ωR=ω;ωR ¼ jAð0Þj2=2mÞ for symmetric (dashed black) and asymmetric bilayergraphene (solid green), respectively, at the Dirac point of the Brillouin zone. Thedimensionless ARF for symmetric bilayer graphene shows a zero trivial minimumin the absence of the applied field whereas for asymmetric bilayer graphene it has afinite value. This reveals that the ARF shows offset oscillations in the absence ofapplied field, and these offset oscillations are observed only for weak fields. Forclarity, we have chosen a large value of offset frequency. A realistic value of offsetfrequency is related to the asymmetry parameter by the relation, 2Δ=ℏω¼ 0:24.(For interpretation of the references to colour in this figure caption, the reader isreferred to the web version of this paper.)

V. Kumar et al. / Physica B 436 (2014) 140–148 145

where δ¼ nω�2Ek, n defines the order of harmonic resonancesand Ek is the energy eigenvalue of the system. When Δ¼0, Eq. (17)gives the conventional Rabi frequency without asymmetry inbilayer graphene. Near resonance, nω� 2Ek, the conventional Rabifrequencies at first and second harmonic resonances reduce to

Ω1 ¼ 2ωRjkj2m

� �1=2

1�2Δω

� �; Ω2 ¼ωR 1�2Δ

ω

� �

Since Δ5ω, we can see that the conventional Rabi frequency isnearly independent of the intra-layer asymmetry. This shows thatthe conventional Rabi frequency in both resonance cases is nearlythe same with or without asymmetry. Thus we can say that theasymmetry has a significant effect only on the anomalous Rabifrequency and is better observable for weak fields.

3.2. Effect of intra-layer asymmetry on current density

The compact form of current density in bilayer graphene canbe derived by using a process described in Section 2.2. The

x-component of which can be written as

⟨JxðtÞ⟩¼ � 12πm

∑k-

ðk� �AnðtÞÞpð k!; tÞþh:c:

Inserting pð k!; tÞ in the above equation and performing theintegration over the momenta k

!, we can write the final form of

the slow part of current density in the frequency domain asfollows:

Jsxðω′Þ ¼ � Að0Þ16π2

� �Aω2

32ω3R

ω′�2Δ�ω2R

ω

� �θ ω′�2Δ�ω2

R

ω

� �ð18Þ

Eq. (18) gives the induced current in bilayer graphene whenintra-layer asymmetry has been taken into account. If Δ¼0,Eq. (18) gives the induced current in bilayer graphene withoutasymmetry. Eq. (18) is valid only near the threshold. The exponentof induced current at threshold is equal to unity, whereas inmonolayer graphene [45] this is computed to be equal to 1/2. Theinduced current in Eq. (18) shows Rabi-like oscillations even in theabsence of external applied electric field ðωR ¼ 0Þ. This shows thatthe induced current exhibits offset oscillations – a feature absentin symmetric bilayer graphene. These offset oscillations arecharacterized by the asymmetry parameter Δ. The asymmetrydoes not affect the exponent at the threshold but changes thevalue of the threshold frequency. This explanation will becomemore clear from Fig. 6.

3.3. Effect of inter-layer asymmetry on Rabi oscillations

The third term in Hamiltonian equation (10) is due to inter-layer asymmetry in bilayer graphene, where U denotes thestrength of inter-layer asymmetry. The inter-layer asymmetryhas a dramatic effect on the energy spectrum of bilayer graphene.A ‘Mexican hat’ like structure appears in the low energy spectrumof bilayer graphene that gives the true value of the energy gap[10], Fig. 4. In the continuum limit, the Hamiltonian of bilayergraphene with inter-layer asymmetry may be written as follows:

H¼ � 12m

∑k-

K2� c

†A1ð k

!Þ cB2ð k!Þþh:c:

� �

þ∑k-

U2�Uv2

γ21K þK �

" #½c†A1ð k

!Þ cA1ð k!Þ�c†B2ð k

!Þ cB2ð k!Þ� ð19Þ

where K � ¼ k� �AnðtÞ and K þ ¼ Kn

� .Following a procedure described in Section 3.1 we first derive

the Bloch equations and then solve these equations near resonanceand far from resonance. Upon solving the Bloch equations far fromresonance, we obtain the following expression for the slowlyvarying part of polarization,

psð k!

; tÞ ¼ k2þ4mEk

cos 2Ωtþ i2Ωω′

Rsin 2Ωt

� �

where Ek is the energy eigenvalue of bilayer graphene with inter-layer asymmetry. The general expression of anomalous Rabifrequency in this case comes out to be

ΩARWA ¼ 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffijkj22m

� �2

þω′2R

4

s

ω′R ¼

jAð0Þj44m2ω

þ2jkj2jAð0Þj2m2ω

þU 1�v2

γ21jkj2þjAð0Þj2� � !" #

ð20Þ

Exactly at the Dirac point, the above expression for anomalousRabi frequency reduces to the following:

ΩARF ¼ U 1�ωR

γ1

� �þω2

R

ω; ωR ¼

jAð0Þj22m

ð21Þ

Unlike the anomalous Rabi frequency in bilayer graphene with intra-layer asymmetry, here we find an additional term containing ωR thatis responsible for a non-trivial minimum in the anomalous Rabifrequency. This non-trivial minimum is observed only for weakapplied electromagnetic fields – a characteristic absent in symmetricand intra-layer asymmetric bilayer graphene. For strong fields, ittends towards the anomalous Rabi frequency of bilayer graphene

SymmetricBLG

AsymmetricBLGR 0

0 0.010

0.01

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.70.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7SymmetricBLGAsymmetricBLG

R 0

0.0 0.1 0.2 0.3 0.4 0.50.0

0.1

0.2

0.3

0.4

0.5ω

ω

η η

Fig. 6. Schematic of the variation of the slow part of: (left) induced current versus a dimensionless quantity ηð ¼ω′=ωÞ, in the presence of the external applied field. Thisfigure is plotted by assuming that ωR ¼ 1 and ω¼10, in arbitrary units. It is obvious from the left part of this figure that the threshold frequency of the induced current isincreased in the presence of asymmetry (solid green) in comparison to without asymmetry (dashed black). The threshold frequency ω′=ω is ð2Δ=ℏωþω2

R=ω2Þ, where

ωR ¼ jAð0Þj2=2m. The right part is plotted in the absence of external field which shows that the induced current in symmetric bilayer graphene (dashed black) loses itsthreshold behaviour, whereas the asymmetric bilayer graphene (solid green) retains a threshold behaviour. The frequency of offset oscillations of the current is identifiedwith the asymmetry parameter Δ. The offset oscillations in the induced current are observable only for weak applied fields. For strong fields these parameters reduce to thosewithout asymmetry. A realistic value of the offset frequency is 2Δ=ℏω¼ 0:24. (For interpretation of the references to colour in this figure caption, the reader is referred to theweb version of this paper.)

V. Kumar et al. / Physica B 436 (2014) 140–148146

with intra-layer asymmetry. The effect of inter-layer asymmetry onanomalous Rabi frequency in bilayer graphene at the Dirac point ofthe Brillouin zone is shown in Fig. 7.

The effect of inter-layer asymmetry on the slow part of inducedcurrent can be seen from the following expression:

JsxðtÞ ¼A

2πmγ21

8πUv2ωR

4m2ωAð0Þ

U�UωR

γ1þω2

R

ω

� � ω′� U�UωR

γ1þω2

R

ω

� �

4ωR

mω�Uv2

γ21

!2

ð22Þ

The induced current in bilayer graphene with inter-layer asym-metry is also offset by the asymmetry parameter – a feature absentin symmetric bilayer graphene, and the offset value is identified

with the asymmetry parameter which is different from that of theintra-layer asymmetry parameter. A shift appears in the inducedcurrent from lower to higher threshold frequency and is wellobserved only for weak fields. Similar to the effect of intra-layerasymmetry on the induced current, the inter-layer asymmetry alsodoes not change the exponent at threshold but changes thethreshold frequency with different asymmetry parameters, ascan be seen from Eq. (22). The effect of inter-layer asymmetryon the induced current in bilayer graphene can be easily under-stood from Fig. 8.

The numerical values of various quantities given in Table 1 arecalculated at a fixed value of the frequency ω¼ 6π � 1014 rad=sof the external applied field, Emax¼250 kV/cm. The value of thezero-detuned Rabi frequency and anomalous Rabi frequency at theDirac point in monolayer graphene is 102 and 103 times larger thanthat of bilayer graphene, respectively.

So far we have discussed the phenomenon of Rabi oscillationsin monolayer and bilayer graphene deposited on a substrate, forexample hexagonal boron nitride. The substrate–graphene inter-action is responsible for opening of a gap, around 53 meV[6,28,29,47], in the electronic spectrum of these graphene systems.There are known substrates such as Ni that open a negligible gap[48,49] or no gap [36] in the electronic spectrum. Experimenters[25,50–53] observed a gap of 260 meV in the band spectrum ofepitaxial graphene on the SiC substrate due to the graphene–substrate interaction. A density functional theory (DFT) calculationconfirmed substrate-induced symmetry breaking [54]. Theirresults showed a gap in the band spectra of graphene of about200 meV, which is in agreement with recent experimentalobservations. Varykhalov et al. [36] have studied the photoemis-sion spectrum of a single graphene sheet grown on Ni and Cosubstrates, and demonstrated the absence of a band gap at theDirac point. They have shown that the effects that usually open agap at the Dirac point may also have the adverse effect of closing itin such situations. The gap opening is also dependent on thedistance between substrate and graphene sheet. So according tothe aforementioned literature on opening of gap in the electronicspectrum of graphene systems, the value of the gap parameterdepends on the nature of substrate chosen. When there is no gap,the anomalous and conventional Rabi oscillations in thesegraphene systems correspond to that of gapless graphene systems,and the current density will lose its threshold behaviour forthe vanishingly small applied fields. Depending upon the type ofsubstrate, therefore, the gap parameter may increase, decreaseor vanish altogether. Therefore, our results still valid even in

0.00 0.02 0.04 0.06 0.08 0.100.00

0.02

0.04

0.06

0.08

0.00 0.02 0.04 0.06 0.08 0.100.00

0.02

0.04

0.06

0.08

0.10

η η

Fig. 8. Schematic of the variation of the slow part of: (left) induced current versus a dimensionless quantity η¼ω′=ω, in the presence and (right) in the absence of the appliedfield. This figure is plotted assuming that ωR ¼ 1 and ω¼10, in arbitrary units. It is obvious from the left part of this figure that the threshold frequency of the induced currentis increased in the presence of asymmetry (solid green) in comparison to that of without asymmetry, similar to the intra-layer asymmetric bilayer graphene. This shift isobserved only for weak fields, for strong fields, the behaviour is same as without asymmetry. The threshold frequency ω′=ω is ðU=ℏω�UωR=ωγ1þω2

R=ω2Þ. Right part of this

figure shows that the induced current in symmetric bilayer graphene (dashed black) loses its threshold behaviour, whereas the bilayer graphene with inter-layer asymmetry(solid green) exhibits threshold behaviour even for vanishingly small external fields, and this is prominent only for the weak applied fields. The frequency of offsetoscillations of the current is identified by the asymmetry parameter U, and has the relation, U ¼ 0:06ℏω. (For interpretation of the references to colour in this figure caption,the reader is referred to the web version of this paper.)

0.00 0.05 0.10 0.15 0.20 0.25 0.300.00

0.02

0.04

0.06

0.08

x

Fig. 7. Schematic of the variation of the dimensionless anomalous Rabi frequency(ARF) ΩARF=ω versus a dimensionless quantity x ð �ωR=ωÞ for symmetric (dashedblack) and inter-layer asymmetric bilayer graphene (solid green), at the Dirac pointof the Brillouin zone. The dimensionless ARF for symmetric bilayer graphene showsa zero trivial minimum in the presence of the applied field. For inter-layerasymmetric bilayer graphene, the ARF goes through a non-trivial minimum thatoccurs at a value of x¼U=2γ1, and is observed only for weak applied fields. Forstrong applied fields, the effect of inter-layer asymmetry becomes similar to theintra-layer asymmetry, and if we further increase the strength of the applied fieldthe ARF tends towards to that without asymmetry. The offset frequency is relatedto asymmetry parameter by the relation, ΩARF=ω¼U=ℏω¼ 0:06, for a realistic valueof the ratio U=γ1 � 0:21 and frequency of the applied field ω¼ 6π � 1014rad=s. (Forinterpretation of the references to colour in this figure caption, the reader isreferred to the web version of this paper.)

V. Kumar et al. / Physica B 436 (2014) 140–148 147

these unusual situations with an appropriate choice of the gapparameters.

4. Conclusions

We have studied the effect of intra- and inter-layer asymmetryon Rabi oscillations in monolayer and bilayer graphene. The intra-layer asymmetry in monolayer and bilayer graphene comes eitherfrom the graphene–substrate interaction or simply by applying anelectric field perpendicular to the plane of the graphene sheet.The intra-layer asymmetry has a dramatic effect on (anomalous)Rabi oscillations far from resonance in monolayer graphene,whereas (conventional) Rabi oscillations near resonance are nearlyindependent of asymmetry. In the presence of asymmetry, offsetoscillations are seen far from resonance even for vanishingly smallapplied electromagnetic fields – a feature absent in symmetricmonolayer graphene. The offset frequency is identified with thetwice of the asymmetry parameter, and is easily observable onlyfor weak fields. For strong applied fields, the anomalous Rabifrequency tends towards that without asymmetry. The inducedcurrent exhibits the threshold behaviour even for vanishinglysmall applied electromagnetic field. Asymmetry does not changethe value of exponent at threshold but changes the value of thethreshold frequency.

In bilayer graphene, there are two kinds of asymmetry – intra-and inter-layer asymmetry. The effect of intra-layer asymmetry onRabi oscillations in graphene bilayer has a similar effect as inmonolayer graphene, only the difference is in the strength of theasymmetry parameter. The inter-layer asymmetry has an evenmore dramatic effect on Rabi oscillations in bilayer graphene. Inthe presence of inter-layer asymmetry, the ARF goes through anon-trivial minimum – a feature absent in asymmetric monolayerand intra-layer asymmetric bilayer graphene. Rabi oscillations inmonolayer and bilayer graphene at extreme non-resonance issignificantly affected by asymmetry, and is easily observable forweak applied electric fields. Usually, different substrates open gapsin the band structure whose size depends on the nature of thesubstrate–graphene interaction. Our work is able to probe thisaspect also through the phenomenon of Rabi oscillations.

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