2.6 Summary and conclusions
3.1.1 Effective low-energy Hamiltonian of gapped single layer graphene 73
The low-energy Hamiltonian of graphene honeycomb lattice is derived using the tight binding model [92]. This honeycomb lattice is made from two sublattices, A and B, connected by time-reversal symmetry (SU(2) symmetry). Therefore, the Hamiltonian of gapless graphene is symmetric under the transformation of A←→B, and the massless Dirac fermions show gapless linear energy–momentum dispersion at the Dirac points. The zero gap dispersion implies that the conduction of electrons cannot be simply toggled by the external gate voltage, and this limits the use of graphene in electronic applications. This gapless energy dispersion of graphene has been derived by assuming that the on-site energy of electrons in the sublattices A and B are equal, HAA =HBB. Whenever HAA 6=HBB, a mass term exists which is responsible for opening of a gap in the energy spectrum. This mass term breaks the symmetry between the two sublattices A and B, and graphene becomes asymmetric under the transformation of A ←→ B. Now the graphene Hamiltonian will no longer be symmetric under this transformation. There are various ways to induce this mass term such as interaction of graphene sheet with the substrate upon which graphene sheet is deposited [188, 196], sandwiching a monolayer graphene between a pair of hexagonal Boron nitride layers [190] or applying an external electric field perpendicular to plane of the graphene sheet.
There are experimental and theoretical [189,193] research articles available on the topic of generation of a gap in the energy spectrum of graphene.
The effective low energy Hamiltonian [92] of gapped monolayer graphene can then be written as a pure graphene Hamiltonian and a term that describes the creation
of a mass term
H =H0+Hmt (3.1)
where, H0 = vF(~σ·~p) is the intrinsic graphene Hamiltonian and Hmt = vF2mtσz is the Hamiltonian of the mass term that may arise due to sublattice space asym- metry between A and B sublattices. The energy eigenvalues of Hamiltonian Eq.
(3.1) are derived in Refs. [189, 194] as given below:
E±(~k) = q
(vF2mt)2+γ2|f(~k)|2 (3.2) where f(~k) = eikxa√3 + 2e−ikxa√3 Cos(ky2a) and γ is the transfer integral. Equation (3.2) gives the parabolic dispersion for the charge carriers (Fig. 3.1).
+Δ
−Δ 𝟐 𝟐
-1.0 -0.5 0.0 0.5 1.0 -1.0
-0.5 0.0 0.5 1.0
k E
Figure 3.1: Left: Schematic of the graphene sheet deposited upon a sub- strate. ∆ is the on-site energy of the atoms on sublattice sites A and B that defines the intra-layer asymmetry between two sublattices and gives rise a mass term in the monolayer graphene Hamiltonian Eq. (3.1). Right: Energy band spectrum of single layer graphene with (solid green) and without asymmetry (dashed black). The energy spectrum of asymmetric single layer graphene has a parabolic dispersion and shows a band gap between the conduction and valance
band.
To describe the phenomenon of Rabi oscillations a semiclassical approximation is used, radiation is treated classically and matter fields are quantum. An in plane electric field is applied to the graphene sheet through a vector potential of the form A(t) =~ A(0)e~ −iωt, where A(0) =~ ec(Ax(0) +iAy(0)) is a complex amplitude of an external driving field. In second quantization, using two-component basis eigenstates of Hamiltonian Eq. (3.1) in momentum space of the form ψT(~k, t) =
(cA(~k, t) cB(~k, t)), the Hamiltonian Eq. (3.1) reads as Hˆ =X
~k
h
vF (k+−A∗(t))c†A(~k, t)cB(~k, t) +h.c.i +vF2mtX
~k
h
c†A(~k, t)cA(~k, t)−c†B(~k, t)cB(~k, t)i (3.3)
where k± = kx±iky, c(c†) are the annihilation(creation) operators on sublattice sites A and B and vice versa. If mass term is zero, the Hamiltonian Eq. (3.1) commutes with the helicity operator ˆh (projection of pseudo-spin in the direction of momentum) that preserves the SU(2) symmetry. In presence of a mass term, the Hamiltonian Eq. (3.1) does not commute with helicity operator ˆh. This breaks SU(2) symmetry of Hamiltonian Eq. (3.1). It means the time reversal symmetry has been broken or we can say that there exists a sublattice space asymmetry between two sublattices A and B. This also signifies that we cannot produce the crystal of graphene by simply repetition of a unit cell in two dimensions.
3.1.2 Bloch equations of gapped graphene
In this section, we derive the Bloch equations of population and polarization excess on sublattice sites A and B in gapped graphene. In second quantization, these quantities are expressed as follows,
ndif f(~k, t) = hc†A(~k, t)cA(~k, t)i − hc†B(~k, t)cB(~k, t)i
p(~k, t) = hc†A(~k, t)cB(~k, t)i (3.4) With the help of the equations of motion for operators i∂tρˆ= [ ˆρ, H], and simple anti-commutator algebra for fermions, the Bloch equations ofndif f(~k, t) andp(~k, t) reads as (setting ~= 1),
i∂tndif f(~k, t) = 2vF
h
(k+−A∗(t))p(~k, t)−h.c.
i
(3.5) i∂tp(~k, t) = vF (k−−A(t))ndif f(~k, t)−2vF2mtp(~k, t) (3.6) These are the rate equations of the population excess ndif f(~k, t) and polariza- tion p(~k, t) on sublattice site A and B. Where k± = kx±iky, A(t) =~ A(0)e~ −iωt, and A(0) =~ ec(Ax(0) +iAy(0)). The asymmetry affects only the rate equation of polarization p(~k, t) while the population equation remains unchanged. It means
asymmetry affects only the induced polarization in presence of an electric field while population is not affected by asymmetry. This is the main difference be- tween the Bloch equations of gapped and without gapped single layer graphene.
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 compar- ison to the inter-band transition frequency of the system. The Bloch equations in single layer graphene have already been solved in absence of gap [184]. Here, we solve these equations in presence of gap parameter (in presence of asymmetry parameter). We followed a process similar to the one described in Ref.[184].