Optical Absorption by Interlayer Density Excitations in Bilayer Graphene

The optical absorption spectrum for light with electric field perpendicular to a bilayer graphene is shown to be quite different from that for parallel polarization. This arises partly from the difference in the selection rule and from the important depolarization effect due to induced polarization. The depolarization effect is most prominent for the sharp transition between the lowest to the first excited conduction band, which are almost parallel to each other.


§1. Introduction
−6) The purpose of this paper is to study optical absorption for light with electric field perpendicular to the layer and to clarify the difference from that for light incident normal to the layer.
−14) Transport properties in such an exotic system are intriguing and the conductivity with/without magnetic field including the Hall effect, 15,16) the dynamical transport, 17) and quantum corrections to the conductivity 18) were theoretically investigated.The system was shown to exhibit various properties different from conventional two-dimensional systems. 19)−37) The paper is organized as follows: In §2, the dynamical conductivity describing optical transitions is introduced and some of its properties are discussed for a symmetric bilayer.Explicit numerical results in bilayers with asymmetric potential self-consistently determined are presented in §3.Results are discussed and a short summary is presented in §4.§2.Formulation

Effective-mass description
We consider a bilayer graphene which is arranged in the AB (Bernal) stacking.The upper layer is denoted as 1 and the lower layer denoted as 2. In a monolayer graphene the conduction and valence bands consisting of π orbitals cross at K and K' points of the Brillouin zone, where the Fermi level is located. 38,39)−14) For the inter-layer coupling, we include only the coupling between vertically neighboring atoms.Then, electronic states are described by the k•p equation: 20,25,37) with and k± = kx ± i ky , ( where γ is a band parameter, k = ( kx , ky ) = −i∇ is a wave-vector operator, Δ represents the inter-layer coupling, and eF d represents the potential difference between layers 1 and 2 (F is the effective electric field and d = 3.34 Å is the interlayer distance), as illustrated in Fig. 1 (a).Parameters γ and Δ are related to tight-binding parameters γ 0 and γ 1 through γ = ( √ 3/2)aγ 0 and Δ = γ 1 , where a is the lattice constant given by a = 2.46 Å, γ 0 ≈ 3 eV, 40) and γ 1 ≈ 0.4 eV. 41)We shall completely neglect other interlayer couplings because they do not play important roles as has been discussed previously. 25,42)he states are specified by the set of quantum numbers j and k, with k the wave vector and bands j = 1, 2, 3, 4 in the order of increasing energy (j = 1 and 2 for the valence bands and j = 3 and 4 for the conduction bands).The energy dispersion for varying values of the potential difference eF d is shown in Fig. 2.
We consider the situation that electron concentration n s is varied by the bottom gate with a fixed value of the top-gate voltage giving external field F ext .The asymmetry between two layers causes difference in the electron density and then leads to that in the electrostatic potential.Therefore, eF d should be self-consistently determined for each electron concentration. 35)Some examples of such calculations were carried out in ref. 37 for the static dielectric constant of the environment, κ = 2.We shall use this result for explicit calculations to be Submitted to Journal of Physical Society of Japan presented in the following.

Optical Absorption for Perpendicular Polarization
We apply external electric field E ext (ω)e −iωt + c.c. perpendicular to the layer, where 'c.c.' stands for complex conjugate.Let n 1 and n 2 be the electron density in layers 1 and 2, respectively, per unit area.The asymmetry in the density distribution Δn(ω) is defined as (2.4) This induces electric field in the region −d/2 < z < +d/2, where κ is the static dielectric constant of the environment.This is illustrated in Fig. 1 (b).
We shall define the polarizability α(ω) through where E tot (ω)e −iωt +c.c. is the total electric field.Then, we have giving With the use of the equation of continuity, we have induced current in the region −d/2 < z < +d/2, with two-dimensional conductivity Then, we have and

.14)
The power absorption per unit area is given by (2.15) The dynamical conductivity can easily be calculated in a linear response theory.Define Then, we have where α and β stand for a set of quantum numbers (j, k), g s = 2 and g v = 2 are the spin and valley degeneracy, δ is phenomenological broadening, and We have used the fact that both K and K' points give the same contribution.Explicitly, we have In the above, we have introduced cutoff function, with cutoff energy ε c ≈ 3γ 0 corresponding to the half of the π-band width and n c should be chosen in such a way that the integral should converge.The typical magnitude of the conductivity becomes where we have used a = 2.46 Å, d = 3.34 Å, Δ ≈ 0.4 eV, and γ 0 ≈ 3 eV in the last expression.

Symmetric Bilayer
In the symmetric case, in particular, we can make some analytic treatments.Let us define where ψ is 0 at k = 0 and approaches π/2 when γk Δ/2.Then, the eigenenergies are given by and the corresponding wave functions are given by where L 2 is the area of the system, and ) We have the correspondence shown in Table I between quantum number j and (s, n) with s = ± and n = 1, 2.
The matrix elements of τ are calculated as in Table II.
Consider the case that the Fermi level lies in the band j = 3 or (+, 1).For the transition (+, 1) → (+, 2), in particular, we have independent of k.Further, we have (+, 2|τ |+, 1) = sin ψ. (2.28) Define n 0 s be the electron concentration when the Fermi level reaches the bottom of band (+, 2) or ε F = Δ, i.e., Then, the transition strength averaged over states up to the Fermi energy becomes where • • • means the average over states with energy lying between ε = 0 and ε F .Then, we have with This shows that the resonance energy is shifted from Δ to √ 1+βΔ due to the depolarization effect.For n s /n 0 s = 1, in particular, we have sin 2 ψ = 0.725 • • • and β = 0.68 • • •, giving about 30 % shift of the resonance energy to the higher energy side.
For the transition (−, 1) → (+, 1), we have Then, we have 35) This shows that the absorption exhibits a step-function-like behavior with a spectral edge suppressed by the logarithmic divergence of ε(ω).
For the transition (−, 2) → (+, 2), we have Then, we have (2.37) showing that the absorption exhibits a step-function-like behavior with a suppressed edge.

Optical Absorption for Parallel Polarization
The dynamical conductivity describing optical absorption for electric field parallel to the layer was calculated previously, 30) and therefore we shall give only the final results.The dynamical conductivity becomes (2.40) This can be separated into the Drude (intra-band) part and the inter-band part

41)
The squared matrix element for a symmetric bilayer is listed in Table III.At the band edges k = 0 or ψ = 0, the selection rule for the parallel and perpendicular polarized light is completely different.The dynamical conductivity for ε F = 0 has essentially no prominent structure except a step-like increase corresponding to transitions from (−, 1) to (+, 2). 30)With increase in ε F , a delta-function peak appears at hω = Δ with intensity proportional to |ε F |, corresponding to allowed transitions (+, 1) → (+, 2).§3.Numerical Results In the present system we have electron-hole symmetry in the energy bands.As a result the dynamical conductivity describing optical properties has symmetry and therefore we shall exclusively consider the case n s ≥ 0. The results for n s < 0 can be obtained by using the above relation.Figure 3 shows some examples of calculated dynamical conductivity for (a) perpendicular and (b) parallel polarization in a symmetric case eF d = 0, with several values of the Fermi energy.The dashed lines in (a) represent σ zz (ω) without depolarization effect and the solid lines σzz (ω) with depolarization effect.We assume broadening parameter δ/Δ = 0.02.One most prominent feature is the appearance of a sharp peak corresponding to transitions from (+, 1) to (+, 2) with the increase of ε F .In the clean limit δ → 0, the peak becomes a δ function because the bands are parallel to each other.For perpendicular polarization, this sharp peak is shifted to the higher energy side due to the depolarization effect.When the Fermi level reaches the bottom of band (+, 2), this shift is as large as 30 % as has been estimated in the previous section.
Figure 4 shows the dynamical conductivity for varying electron concentration n s for F ext = 0, i.e., when the bilayer is symmetric for n s = 0 and the asymmetry potential eF d increases with the change of n s .The energies where interband transitions start to appear are shown by thin dotted lines.For transitions from (+, 1) to (+, 2), i.e., 3 → 4, the upper edge where the transition disappears is also included.
For the parallel polarization, the absorption corresponding to transitions (+, 1) → (+, 2) becomes broadened with increase of n s due to the gap opening.For the perpendicular polarization, on the other hand, this broadening is not significant and exhibits a similar amount of shift due to the depolarization effect.This is because the resonance, i.e., the zero point of ε(ω), occurs at an energy higher than the interband continuum due to the depolarization effect.A weak structure appears near the line denoted as 2 → 4 with the increase of n s .This corresponds to the transition from (−, 1) to (+, 2), which becomes allowed by the band-gap opening as mentioned in the end of the last section.
Similar results for eF ext d/Δ = 1 are shown in Fig. 5.For the parallel polarization, the absorption corresponding to transitions (+, 1) → (+, 2) is considerably broadened due to further increase of the gap with n s .The absorption for the perpendicular polarization tends to have a sharper peak near the higher-energy edge of the interband continuum and exhibits a sharp peak for sufficiently large n s where the zero of ε(ω) is well separated from the continuum.The transition from (−, 1) to (+, 2) denoted as 2 → 4 is now much more prominent.The peak appearing at hω/Δ ∼ 1 at n s = 0 for the parallel polarization is due to the opening of gap ∼ |eF d|.
Figure 6 shows results for eF ext d/Δ = −1 corresponding to asymmetry opposite to Fig. 5.In this case the asymmetry becomes weaker with n s and therefore the peak corresponding to transitions (+, 1) → (+, 2) becomes sharper with n s .Further, the transition from (−, 1) to (+, 2) denoted as 2 → 4 for the perpendicular polarization becomes weaker with the increase of n s .§4. Discussion and Summary We have calculated the optical absorption spectrum for light with electric field perpendicular to bilayer graphene and shown that the spectrum is quite different from that for the parallel polarization.This arises partly from the difference in the selection rule and from the important depolarization effect due to induced charge imbalance between two layers.The depolarization effect is most prominent for the sharp transition (+, 1) → (+, 2) almost parallel to each other.
The strength of the absorption for the perpendicular polarization is determined by conductivity σzz (ω) of the order of σ 0 , while that for the parallel polarization by σ xx (ω) of the order of (g v g s e 2 )/(4πh).Because σ 0 ∼ 0.022 × (g v g s e 2 )/(4πh), the absorption intensity for perpendicular polarization is much smaller.As a result, for light incident with certain angle from the vertical direction the absorption by the field component perpendicular to the layer is small and usually negligible.
This smallness of σzz (ω) mainly comes from the fact that inter-layer coupling γ 1 (∼ 0.4 eV) is much smaller than intra-layer coupling γ 0 (∼ 3 eV) in spite of the fact that inter-layer separation d (= 3.34 Å) is not so much different from the intralayer lattice constant a (= 2.46 Å).This is quite in contrast to conventional space-charge layers on semiconductor surfaces where σzz (ω) is not so much different from σ xx (ω). 19)Usually, σzz (ω) can be experimentally measured using the configuration of a transmission line. 43,44)The weak absorption intensity discussed above requires the use of bilayer graphene with large area.

Table I
The correspondence between level index j and (s, n).

Table II
2alculated matrix elements of operator τ .Table III Calculated current matrix elements |(jk|Σ x | j k)|2averaged over the direction of the wave vector.