PHYSICAL REVIEW B 94, 205408 (2016)

Influence of out-of-plane response on optical properties of two-dimensional materials:First principles approach

Lars Matthes,1 Olivia Pulci,2 and Friedhelm Bechstedt11Institut fur Festkorpertheorie und -optik, Friedrich-Schiller-Universitat, Max-Wien-Platz 1, 07743 Jena, Germany

2Dipartimento di Fisica, Universita di Roma Tor Vergata, and I.N.F.N., sezione di Roma Tor Vergata,Via della Ricerca Scientifica 1, I-00133 Rome, Italy

and CNR-ISM, Via Fosso del Cavaliere 100, I-00133 Rome, Italy(Received 21 August 2015; revised manuscript received 16 October 2016; published 8 November 2016)

The ab initio calculation of optical spectra of sheet crystals usually arranges them in a three-dimensionalsuperlattice with a sufficiently large interlayer distance. We show how the resulting frequency-dependent dielectrictensor is related to the anisotropic optical conductivity of an individual sheet or to the dielectric tensor of acorresponding film with thickness d . Their out-of-plane component is taken into account, in contrast to usualtreatments. We demonstrate that the generalized transfer-matrix method to model the optical properties of a layersystem containing a sheet crystal accounts for all tensor components. As long as d � λ (λ-wavelength of light)this generalized formulation of the optical properties for anisotropic two-dimensional (2D) systems of arbitrarythickness reproduces the limits found in literature that are based either on electromagnetic boundary conditionsfor a conducting surface or on an isotropic dielectric tensor. For s-polarized light, the results are independent ofthe sheet description. For oblique incidence of p-polarized light, the tensor nature of the optical conductivity (orthe dielectric function) of the sheet crystal strongly impacts on reflectance, transmittance, and absorbance dueto the out-of-plane optical conductivity. The limit d → 0 should be taken in the final expressions. Examplespectra are given for the group-IV honeycomb 2D crystals graphene and silicene.

DOI: 10.1103/PhysRevB.94.205408

I. INTRODUCTION

The remarkable properties of honeycomb crystals such asgraphene have renewed interest also in other inorganic, two-dimensional (2D) materials with unique electronic and opticalproperties. There is enormous progress in the production ofatomically thin crystals that can be viewed as individualplanes of atomic-scale thickness exfoliated from bulk crystalslike graphite, h-BN, several transition-metal dichalcogenides(TMDCs), or complex oxides [1–4].

In addition, elemental analogs of graphene, namely, sil-icene, germanene, stanene, silicongraphene, and phosphorene,have been predicted and tried to prepare [5–8]. By stackingvarious 2D crystals on top of each other it is possible tocreate multilayer heterostructures. They improve the promisefor novel materials with tailored properties, e.g., electronicand optical ones. Concepts for novel devices for application innanoelectronics, optoelectronics, and photovoltaics have beenproven [9,10].

Optical properties are very important for both spectroscopicstudies of the 2D crystals and optoelectronic applications.Optical studies have been performed for many 2D crystalssuch as graphene [11–13] and its alloys with hexagonalboron nitride [14,15]. TMDC layers have been studiedby reflectance measurements and spectroscopic ellipsometry[16–20] observing strong excitonic effects. Silicenelike over-layers on Ag(111) substrates [21], but, in particular, silicene-like stripes on Ag(110) surfaces, have been investigated bymeans of the differential reflectivity [22].

Whereas the measured spectra can be reasonably explainedin terms of atomic geometry, electronic structure, and electron-electron interaction, the propagation of light in and throughatomically thin layers and their optical response are in generalcontroversially discussed. The main problems are related to

the optical response perpendicular to the sheets and the for-mulation of the boundary conditions [23–27]. Typically, onlythe in-plane component of the optical conductivity is taken intoaccount [23]. Then, the 2D system is described by a conductingsurface of the substrate by means of electromagnetic bound-ary conditions. Other approaches start with a questionableisotropic dielectric function of the material with vanishingthickness, for which the transfer-matrix method can beused [24,28]. Ab initio calculations, however, indicate that evenin the limit of vanishing effective thickness d → 0, the tensorcharacter with two (or three, for anisotropic 2D planes) inde-pendent components of the optical constants is conserved [25].The resulting different approaches are discussed in this paper.

We demonstrate that superlattice calculations for isolatedsheet crystals always yield a tensor of the 2D optical conduc-tivity. Even for isotropic sheet crystals aside from the in-planeconductivity, also an out-of-plane conductivity appears. With afinite effective sheet thickness d � λ (λ-wavelength of light)the optical response of such a sheet can be also describedby a frequency-dependent dielectric tensor. The calculationof the reflectance (R), transmittance (T ), and absorbance (A)of an atomically thin layer can be therefore performed eitherby modeling the layer as a 2D conducting interface betweentwo media, and applying boundary conditions for the electricand magnetic fields, or by the transfer-matrix approach for alayer of thickness d embedded between two media, lettinga posteriori d → 0. We will show that the two differentdescriptions of the boundary conditions give the same resultsjust for s polarization, whereas for p polarization differencesare derived. We demonstrate that, in particular, the neglect ofthe perpendicular component of the optical conductivity leadsto deviations in the high-energy part of the calculated opticalspectra and suggest experiments to confirm our predictions.

2469-9950/2016/94(20)/205408(8) 205408-1 ©2016 American Physical Society

LARS MATTHES, OLIVIA PULCI, AND FRIEDHELM BECHSTEDT PHYSICAL REVIEW B 94, 205408 (2016)

II. CALCULATION OF RESPONSE FUNCTIONS:DIELECTRIC TENSOR AND OPTICAL

CONDUCTIVITY (d �= 0)

The linear optical properties are determined by thefrequency-dependent dielectric tensor εSL(ω) where SL standsfor superlattice, which is used to model isolated crystal sheets.In the ab initio calculations of the atomic geometry andthe electronic structure, the properties of an isolated sheetcrystal are simulated by an artificial superlattice arrangementof such sheets, with an interlayer distance L (see, e.g.,Ref. [29]). It allows the expansion of the electronic wavefunctions in sets of plane waves. The typical separation ofthe sheets by L = 20A vacuum avoids artificial interactionsbetween the periodic images of the sheet crystals. Based onthe resulting electronic structure, the dielectric tensor can becomputed in independent-particle, independent-quasiparticleapproximation or even including excitonic effects [30]. Forisotropic 2D crystals, e.g., group-IV honeycomb crystals, oneobtains a diagonal dielectric tensor with two independentcomponents ε⊥

SL(ω) and ε‖SL(ω) for light polarization parallel

or perpendicular to the superlattice axis [25,31,32]. Modeling2D sheet crystals with rectangular Bravais lattice such asphosphorene [33], the generalization to two in-plane opticalfunctions is straightforward and, therefore, should not bepresented here.

In independent-(quasi)particle approximation, the diagonalelements can be written within the Ehrenreich-Cohen formula(using SI units) [30]

εj

SL(ω) = 1 + 2e2�

2

ε0m2

1

LA

∑c,v

∑k

|〈ck|pj |vk〉|2[εc(k) − εv(k)]2

×∑

β=+,−

1

εc(k) − εv(k) − β(�ω + iη)(1)

with wave functions |νk〉 and eigenenergies εν(k) of com-pletely empty conduction (c) and filled valence (v) bands,A as the sheet unit-cell area, and pj the j th component ofthe momentum operator. In (1) we only study the interbandcontribution. Drude terms are here not considered. This isvalid for semiconducting and insulating 2D crystals but alsofor undoped multivalley zero-gap semiconductors such asgraphene and silicene. The spectra that we present are calcu-lated within the density functional theory [34]. Explicitly, theVienna ab initio simulation package (VASP) is applied [35]. Forthe structural optimizations we use an exchange-correlation(XC) functional according to Perdew, Burke, and Ernzerhof(PBE) [36]. The optical matrix elements are computed withall-electron wave functions following Gajdos et al. [37].Excitonic and quasiparticle effects are neglected. There is aclear tendency for a compensation of quasiparticle (on topof a local or semilocal XC approximation) and excitoniceffects, especially for the peak positions in graphene [38,39].Consequently, we present optical response functions derivedwithin the PBE treatment of the electronic structure.

For sufficiently large superlattice periods L only one layerof k points is taken in the Brillouin zone, i.e., the k summationin (1) characterizes the 2D nature of the contributing sheetsalready within the numerical description. Of course, the

quantum-mechanical wave functions describing the Blochstates |νk〉 and, hence, the space integral in the optical matrixelements are still three dimensional.

In order to model the optical properties of individual sheetcrystals, we apply the procedure of constructing the frequency-dependent dielectric tensor of the superlattice by those of thecorresponding layer materials forming the superlattice. Sinceone layer material is vacuum, the tensor εSL(ω) is directlyrelated to that of the sheet material ε2D(ω), if an effective thick-ness d of the sheet is assumed. Indeed, in the interpretationof the spectral ellipsometry of a graphene layer an effectivethickness d = 3.35A (corresponds to the sheet distance ingraphite) is taken into account [40]. Such a thickness can bealso introduced for stacks of atomically thin layers [41]. Inthe limit d � λ with λ = 2πc/ω the wavelength of the light,the tensor ε2D(ω) with two independent components can bedirectly related to that of the superlattice arrangement withlattice constant L within an effective medium theory [42]

ε‖2D(ω) = 1 + L

d

[ε

‖SL(ω) − 1

],

1

ε⊥2D(ω)

= 1 + L

d

[1

ε⊥SL(ω)

− 1

]. (2)

The description (2) of the frequency-dependent properties of asheet depends on the empirical parameter thickness d, but thequantity d[ε‖

2D(ω) − 1] is not.For individual sheets one is interested in thin layers and

large distance between them. It is then convenient to introducethe optical conductivity of a sheet, at least its in-planecomponent σ

‖2D(ω). According to Ohm’s law

j2D(ω) = σ‖2D(ω)E‖(ω) (3)

with the in-plane component E‖(ω) of the electric field E(ω) itis directly related to an in-plane current density j2D(ω) of the2D electron gas in the sheet. It fulfills the continuity equationin Fourier space as

ρ2D(ω) = q · j2D(ω)/ω (4)

with the wave vector q of the propagating light and the induced(2D) surface charge density ρ2D(ω).

The in-plane 2D conductivity in (3) is directly related tothe corresponding component σ

‖SL(ω) of the three-dimensional

(3D) conductivity of the superlattice by [25]

σ‖2D(ω) = Lσ

‖SL(ω). (5)

This relation accounts for the different volumes of the 2D and3D current densities.

According to the Maxwell equations it holds ε‖SL(ω) = 1 +

iε0ω

σ‖SL(ω). The combination of the two equations leads to the

relation

σ‖2D(ω) = −iε0ωL

[ε

‖SL(ω) − 1

]. (6)

Using an explicit expression for the dielectric functions of thesuperlattice, e.g., the Ehrenreich-Cohen formula (1) [30], oneimmediately sees that expression (6) is independent of L (andof d).

From many-body theory we know that local field effectsgive rise, for polarization perpendicular to the sheet layer, to a

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INFLUENCE OF OUT-OF-PLANE RESPONSE ON OPTICAL . . . PHYSICAL REVIEW B 94, 205408 (2016)

different formulation for σ⊥2D(ω) [43]:

σ⊥2D(ω) = −iε0ωL

[1 − 1

ε⊥SL(ω)

]. (7)

In Sec. IV, we will present results within the generalizedtransfer-matrix approach, for a 2D sheet of finite width d. Forp-polarized light, the evaluation of the transfer matrices in thelimit d � λ naturally leads to the quantity defined in (7), withthe dimension of a 2D conductivity, that can be identified asthe out-of-plane component σ⊥

2D(ω) of this optical conductivity.Despite the difference in the formal descriptions of the in-planeand out-of-plane conductivities (6) and (7), the expressionsbecome equivalent in the limit L → ∞ (but still L � λ).Indeed, for L → ∞ it holds L[1 − 1

ε⊥SL

] ≈ L[ε⊥SL − 1]. The

equivalence can be easily demonstrated using the explicitexpression (1) for the dielectric function of the superlattice.Even with the assumption of optical isotropy within the sheets,one has to deal, at least, with a diagonal tensor of rank 3, ε2D(ω)or σ2D(ω). The out-of-plane component cannot be, in general,omitted. Despite the wording “2D crystal,” microscopicallyone deals with the three-dimensional response of the sheetcrystal because of the extent of the quantum-mechanical wavefunctions in all space directions.

The two components of the 2D optical conductivity havedifferent physical meanings. The in-plane component σ

‖2D(ω)

gives rise to an in-plane current (3) if an electric field withan in-plane component is applied. For graphene, its real parthas been measured for normal incidence of light. It shows,in particular, the pronounced asymmetric resonance peaked at4.62 eV, which is interpreted as a 2D saddle-point exciton withthe electron-hole pair in π∗ and π bands [12,13]. The out-of-plane optical conductivity σ⊥

2D(ω) is associated with an electricdipole operator which describes electron excitation in normaldirection [44]. This fact explains the zero optical conductivityin a wide frequency range for not too large photon energies asclearly shown in Fig. 1 for graphene and freestanding silicene.

0 5 10 150

2

4

6

8

10

12

14

0 5 10 15

-8

-6

-4

-2

0

2

4

6

8

0 5 10 15 20 25 30

-8

-6

-4

-2

0

2

4

6

8

0 5 10 15 20 25 300

2

4

6

8

10

12

Opt

ical

cond

ucti

vity

σ(ω

) /σ

Photon energy ω (eV)

Opt

ical

cond

ucti

vity

σ(ω

) /σ

(b)

Photon energy ω (eV)

(d)

(a) (c)

FIG. 1. (a), (c) Real and (b), (d) imaginary part of the opticalconductivity of graphene and silicene, respectively, in a wide rangeof photon energies. In-plane component σ

‖2D(ω): black lines, out-of-

plane component σ⊥2D(ω): red lines. The spectra are normalized to

σ0 with σ0 = ε0cπα as dc conductivity and α as the Sommerfeldfine-structure constant.

In contrast to the in-plane component of the optical conduc-tivity σ

‖2D(ω), only high-energy van Hove singularities [around

�ω ≈ 10eV (graphene) and 5 eV (silicene) and beyond] mayinfluence the optical properties of the out-of-plane componentσ⊥

2D(ω). The drastic changes in the line shape of the parallel andperpendicular component of the optical conductivity, visiblein Fig. 1, are dominated by a seemingly opened gap in thereal parts at high energies in contrast to the imaginary parts,which approach the material-independent value σ0 = e2/4�

for ω → 0 [32]. They are consequences of the fact that theparallel (perpendicular) component is mainly characterizedby the corresponding (inverse) dielectric function of thesuperlattice arrangement reduced by the vacuum value 1. Thedrastic red-shift of the real parts of σ⊥

2D(ω) compared to σ‖2D(ω)

has been also observed in recent theoretical studies [44,45].However, an experimental confirmation is still missing.

III. OPTICAL PROPERTIES OF ATOMICALLY THINFILMS (d → 0): BOUNDARY CONDITIONS

First, we focus on the optical properties of a 2D sheetcrystal characterized by the optical conductivity tensor σ2D(ω)neglecting the extent of this layer. The sheet is modeled bya conducting surface of a substrate. That means the limitd → 0 is taken before calculating the optical properties ofa layered system including the sheet crystal. We study sucha sheet embedded by two isotropic, nonmagnetic dielectricsas illustrated in Fig. 2. They are characterized by complexdielectric functions εj (ω) or by the corresponding complexindex of refraction (j = 1,2)

nj = √εj (ω) = nj (ω) + iκj (ω) (8)

with its real part, the index of refraction nj (ω), and itsimaginary part, the extinction coefficient κj (ω) � 0.

At the conducting interface, describing the atomically thin2D sheet crystal between the two media j = 1,2, the tangentialcomponents of the electric field E and the normal componentof the magnetic induction B are continuous. In contrast, forthe tangential components of the magnetic field H and thenormal component of the displacement field D it holds near

dielectric 1( )

dielectric 2( )

2D 2D

z

FIG. 2. Light propagation in a system consisting of a 2D sheetcharacterized by the conductivity σ2D that is embedded between thedielectrics j = 1,2 with refraction indices nj (ω). The direction of thearrows illustrates incident, reflected, and transmitted light.

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LARS MATTHES, OLIVIA PULCI, AND FRIEDHELM BECHSTEDT PHYSICAL REVIEW B 94, 205408 (2016)

the sheet plane [46]

n × [H2 − H1] = n × [ j2D × n],

n · (D2 − D1) = ρ2D (9)

with n as the normal vector in the positive-z direction andthe sheet quantities j2D and ρ2D obeying the continuityequation (4). The presence of the latter ones in the boundaryconditions (9) describes the influence of the infinitely thin butconducting 2D crystal. With Ohm’s law, the discontinuitiesare determined by the tangential components of the magneticfield at the interface. Interestingly, in the approximation ofan infinitely thin layer d → 0, only the in-plane opticalconductivity σ

‖2D(ω) remains in (9).

In order to derive the expressions for the reflectance,transmittance, and, hence, the absorbance, we generalize themethod in Refs. [46,47] to conserve the correct analyticalproperties by a sign convention of the dielectric functions andthe optical conductivities. Since perpendicular to the normaldirection isotropy is assumed, we choose the yz plane spannedby the layer normal and the light propagation direction to studythe individual reflected and transmitted light beams. For thereflectance R(ω), transmittance T (ω), and absorbance A(ω),we find [25]

R =∣∣∣∣ η1 − η2 − σ

η1 + η2 + σ

∣∣∣∣2

,

T = Re η2

Re η1

∣∣∣∣ 2η1

η1 + η2 + σ

∣∣∣∣2

,

A = 1 − R − T (10)

with the normalized quantities

ηj (ω) ={qj,z (s polarized),

εj /qj,z (p polarized),

qj,z =√

εj − ε1 sin2 θ1 (branch: Im qj,z � 0), (11)

and

σ (ω) = σ‖2D(ω)/ε0c, (12)

where θ1 is the angle of incidence. Here, the validity of Snell’slaw

√ε1(ω) sin θ1 ≡ √

ε2(ω) sin θ2 has been assumed. Thein-plane conductivity in (12) is normalized with the impedanceε0c of the vacuum. With expression (11), formula (10) canbe applied for both s- and p-polarized light. In the specialcase of two dielectrics with real-valued dielectric constantsεj (ω) (j = 1,2) the results (10) agree with the predictionsin Refs. [23,24,27], where the out-of-plane component ofthe conductivity is not taken into consideration. Thereby, themisprint in expression (3) of Ref. [27] has to be removed.The dielectric function has to be replaced by its squareroot. In Refs. [23,24,27] the 2D crystal, there graphene, isonly characterized by the in-plane optical conductivity. Thecomparison with our results therefore shows that the use ofthe boundary conditions (9), where the 2D crystal is onlysimulated by an infinitely thin but conducting film, leads to theexpressions (10) independent of the out-of-plane conductivity.Just σ

‖2D(ω) appears (by construction).

0 5 10 15 20 25 30

-0.20

-0.10

0.00

0.10

0.20

0 5 10 15-0.30

-0.20

-0.10

0.00

0.10

0.20

0.30

×10 ×10

Opt

ical

spec

tra

)b()a(

Photon energy ω (eV) Photon energy ω (eV)

FIG. 3. Frequency dependence of the optical properties R (blacksolid line), T − 1 (green dashed line), and A (red solid line)for suspended (a) graphene and (b) silicene. Normal incidence isassumed.

For normal incidence θ1 = 0 the optical properties R, T ,and A are independent of the light polarization. For suspended2D crystal sheets with εj ≡ 1, it holds [25]

R =∣∣∣∣ σ /2

1 + σ /2

∣∣∣∣2

,

T = 1

|1 + σ /2|2 ,

A = Re σ

|1 + σ /2|2 . (13)

For graphene and silicene, the corresponding spectra aredisplayed in Fig. 3. Here, in contrast to Ref. [25] wherethe nonlocal hybride HSE06 approach [48] is applied toexchange and correlation, PBE results are presented. Theyindicate that already an atomically thin layer has a significantinfluence on the optical properties for photon energies �ω �20 (graphene) or 10 (silicene) eV. In the IR spectral range,the transmitted light beam is weakened by 2.3% in agreementwith measurements [11] and calculations [32]. More drasticeffects occur for photon energies close to the main resonancesalready appearing in the absorption spectrum Re σ

‖2D(ω) or

A(ω). The low-energy peaks near 5 eV (graphene) and 2 eV(silicene) are mainly due to π → π∗ interband transitions nearthe M points, while those near 16 eV (graphene) and 5 eV(silicene) are related to σ → σ ∗ transitions close to the � pointin the Brillouin zone. The underlying van Hove singularitiescorrespond to saddle points as well as band extrema in the jointdensity of states. In the graphene case, the two main resonancesagree well with energies of the π and π + σ plasmons found inexperimental and theoretical energy-loss spectra [49,50]. Thelow-energy peak is due to a π → π∗ saddle-point exciton [12].

IV. OPTICAL PROPERTIES BEYOND STRICTLY2D SHEET CRYSTALS: TRANSFER-MATRIX

DESCRIPTION (d �= 0)

When the condition of an infinitely thin conducting layer islifted, the optical properties of the 2D crystal with thicknessd can be modeled by that of a uniaxial crystal with a diagonaldielectric tensor with the two independent components (2).Then, the out-of-plane component of the optical conductivityof the sheet crystal should influence the optical properties

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INFLUENCE OF OUT-OF-PLANE RESPONSE ON OPTICAL . . . PHYSICAL REVIEW B 94, 205408 (2016)

of the layer system. Light propagating in such a crystalexperiences birefringence. Incoming light in parallel (i.e., p)polarization experiences a different effective index of refrac-tion than light in perpendicular (i.e., s) polarization, and isthus refracted at a different angle. The s- and p-polarizedelectromagnetic waves obey different dispersion relations

q2x + q2

y

ε‖2D(ω)

+ q2z

ε‖2D(ω)

= ω2

c2(s-polarized, ordinary wave),

q2x + q2

y

ε⊥2D(ω)

+ q2z

ε‖2D(ω)

= ω2

c2(p-polarized, extraordinary wave)

(14)

with (qx,qy,qz) as the light propagation vector in the 2Dmedium characterized by the dielectric tensor with the compo-nents ε

‖2D(ω) and ε⊥

2D(ω) [Eq. (2)]. Therefore, we introduce theindex o (e) for the ordinary (extraordinary) rays, correspondingto the s- (p-)polarized light.

In order to demonstrate the generalization of the re-sults (10), again we investigate the situation of two dielectricsseparated by a 2D crystal as shown in Fig. 2 but now with afinite thickness d and characterized by the dielectric tensor (2).Again, the propagation direction of the light is assumedin the yz plane (qx ≡ 0). A three-layer arrangement of thehalf-spaces 1 and 2 embedding the thin film of thicknessd is in the main focus. An efficient method to treat theoptical properties of the layer system is the transfer-matrixmethod [47] as demonstrated for systems containing graphenelayers [24,26,41]. The global transfer matrix is given as theproduct of the transfer matrices for each (now nonconducting)interface and the propagation matrix within each opticallayer [47]. The total transfer matrix through the 2D crystalis given by the product

Ttotal = D1,2DP2DD2D,2, (15)

where the dynamical matrices are

D1,2D = 1

2η1

(η1 + η2D η1 − η2D

η1 − η2D η1 + η2D

), (16)

D2D,2 = 1

2η2D

(η2D + η2 η2D − η2

η2D − η2 η2D + η2

), (17)

and, respectively, the propagation matrix

P2D =(

exp( − i ω

cq2D,zd

)0

0 exp(i ω

cq2D,zd

))

(18)

with the normal component of the propagation constant q2D,z.In the case of light propagating through an anisotropic, uniaxial2D crystal, Eqs. (11) need to be replaced by the generalizeddimensionless quantities

ηj (ω) ={

qoj,z (s polarized),

ε‖j /q

ej,z (p polarized)

(19)

with j = 1,2, 2D, and

qj,z =

⎧⎪⎨⎪⎩

qoj,z =

√ε

‖j − ε1 sin2 θ1 (s polarized),

qej,z =

√ε

‖j − ε

‖j

ε⊥j

ε1 sin2 θ1 (p polarized)(20)

for the two possible polarizations of the light. As in Eqs. (11),the branch Im q

o/e

j,z � 0 is chosen. If the media are isotropic it

holds ε‖j = ε⊥

j and expressions (11) for isotropic systems arerecovered.

Since the relation d � λ is fulfilled even for stacks withseveral atomic layers and photon energies in the far-ultravioletspectral region, the limit d → 0 can be finally applied formany 2D systems. The more explicit result can be derivedfrom expressions (16), (17), and (18) in the limit d � λ. For aneffective thickness of d = 3.34 A [40], this limit is fulfilled upto photon energies in the vacuum UV region. While evaluatingthe matrix products in (15) for s- and p-polarized light, thein- and out-of-plane conductivities as introduced in Eqs. (6)and (7) appear. For not too small wavelengths, now even inthe limit d → 0 the lower equation (20) lets over an influenceof the out-of-plane component of the dielectric function, andhence of the conductivity for p-polarized light, in contrast tothe application of the boundary conditions (9).

For oblique incidence and p polarization, even the out-of-plane oscillations influence the optical properties of anatomically thin layer, in contrast to the common belief. Wegive the 2×2 transfer matrices for s and p polarization:

T stotal = 1

2η1

(η1 + η2 + σ ‖ η1 − η2 + σ ‖

η1 − η2 − σ ‖ η1 + η2 − σ ‖

), (21)

Tp

total = 1

2η1

((η1 + η2)ch + (σ ‖ + ag)sh (η1 − η2)ch + (σ ‖ − ag)sh(η1 − η2)ch − (σ ‖ − ag)sh (η1 + η2)ch − (σ ‖ + ag)sh

)(22)

with the abbreviations

ch = cosh(√

ε1

√σ ‖σ⊥ sin θ1), (23)

sh = sinh(√

ε1

√σ ‖σ⊥ sin θ1)

√ε1

√σ ‖σ⊥ sin θ1

, (24)

ag = ε1σ⊥η1η2 sin2 θ1, (25)

σ ‖/⊥(ω) = σ‖/⊥2D (ω)/ε0c. (26)

These transfer matrices are directly related to the incident andreflected electric fields at both sides of the interface and, hence,can be connected to A, R, and T via the elements of the 2×2matrices T s

total [Eq. (21)] and Tp

total [Eq. (22)] [47].We have to point out that results of the general transfer-

matrix approach represent a generalization of those ob-tained using electromagnetic boundary conditions. Indeed,in expression (22) also the out-of-plane component σ⊥

2D(ω)appears. If the matrix elements tij of the transfer matrix T

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LARS MATTHES, OLIVIA PULCI, AND FRIEDHELM BECHSTEDT PHYSICAL REVIEW B 94, 205408 (2016)

of the system are known, the reflectance is given as R =|t21/t11|2, whereas the transmittance is obtained by means ofT = Re η2/Re η1×1/|t11|2. The absorbance A = 1 − R − T

is related to R and T as given in (10). Interestingly, in theunrealistic case σ⊥

2D(ω) = σ‖2D(ω) assumed in Ref. [26] for

graphene similar formulas for the transfer matrices have beenderived. We have to point out that in the limit σ⊥

2D(ω) = 0 thes- and p-transfer matrices (21) and (22) become equal but stillgive different results due to the different definitions of ηj fors and p. Nevertheless, in this limit the optical functions R, T ,and A coincide with the expressions (10) obtained by means ofthe boundary conditions (9). In principle, the above formalismcan be also applied to surfaces and interfaces, but only forsystems where bulk and surface/interface electronic propertiesare well separated.

In summary, in the more general case of anisotropictensors σ2D(ω) and ε2D(ω), even in the limit d � λ, the twocomponents of the optical conductivity appear for arbitrarypropagation directions. For normal incidence θ1 = 0, bothtransfer matrices (21) and (22) are equal. Only the in-planecomponent rules the optical properties. The results for theoptical properties R, T , and A are the same as given in (10).For oblique incidence, this also happens if σ⊥

2D(ω) ≡ 0 isassumed. For p polarization and oblique incidence, as wellas σ⊥

2D(ω) = 0 differences between the two approaches (9)and (14) appear despite the limit d → 0, which however hasto be done in the final expressions for R, T , and A.

The effect of the out-of-plane component is demonstratedfor the absorbance of the two sheet crystals graphene andsilicene in Fig. 4 for θ1 = 45◦. It is clearly visible that forgraphene (silicene) the approximation of an infinitely thinconducting layer is valid for energies below 10 eV (5 eV).According to Fig. 1 for graphene as well as silicene, this agreeswell with the absence of the out-of-plane conductivity σ⊥

2D(ω)in this frequency range as stated before.

The out-of-plane conductivity has a strong impact onthe optical properties of p-polarized light for higher photonenergies, for graphene (silicene), �ω � 10 eV (5 eV), andoblique incidence. In the case of s polarization we find nodifference between both approaches for the description of thesheet crystal since the interaction of s-polarized light withthe 2D sheet crystal is mediated solely by the in-plane 2Dconductivity σ

‖2D(ω). However, there is an angular dependence

which scales the spectra according to (10). The opticalproperties for s and p polarization are equivalent for normalincidence, but deviate for oblique incidence as indicatedby (10) and (11) without σ⊥

2D(ω). The differences are morepronounced using the transfer-matrix method. The transfermatrix (22) is really modified by the out-of-plane conductivityσ⊥

2D(ω). As a consequence, the absorbance spectra in Fig. 4show drastic modifications above the characteristic energies�ω ≈ 10 eV (graphene) and �ω ≈ 5 eV (silicene). Therefore,we suggest optical measurements of free-standing graphene foroblique incidence but especially in the high-energy range �ω >

10 eV, i.e., in the vacuum ultraviolet region of the σ → σ ∗transitions. Then, an increase of the absorption compared tothe prediction without the perpendicular component of theconductivity is expected.

0 5 10 15 20 25 300.00

0.05

0.10

0.15

0.20

0.25

0 5 10 15 20 25 300.00

0.05

0.10

0.15

0.20

0.25

Photon energy ω (eV)

Abs

orba

nce

Photon energy ω (eV)

Abs

orba

nce

(b)

(a)

FIG. 4. Frequency-dependent optical absorbance of (a) grapheneand (b) silicene for p-polarized light at an angle of incidence ofθ1 = 45◦ surrounded by air, i.e., εj ≡ 1 (j = 1,2). The red curveshows the result with the full frequency dependence of σ

‖/⊥2D (ω),

whereas the black curve results for the assumption σ⊥2D(ω) = 0.

V. SUMMARY AND CONCLUSIONS

We started from the results of typical ab initio calculations,in which the optical response functions of 2D systems aresimulated by their superlattice arrangements. The results areindependent of the superlattice thickness L (provided L islarge enough). The resulting frequency-dependent dielectrictensor εSL(ω) is used to describe the optical properties (i) of aninfinitely thin crystal by an optical conductivity tensor σ2D(ω)with in-plane and out-of-plane components or (ii) a thin butanisotropic (uniaxial) crystal with effective thickness d � λ

described by a dielectric tensor ε2D(ω).Correspondingly, we have applied two different methods to

describe the influence of a 2D crystal on the optical propertiesof a three-layer system. In the first method (assuming d → 0from the beginning), the infinitely thin crystal is describedby a conducting surface of a half-space by modifying theelectromagnetic boundary conditions by a correspondingelectric current and charge densities. In agreement withearlier findings, only the component σ

‖2D(ω) occurs in the

expressions for R, T , and A. In the second case, we apply the

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INFLUENCE OF OUT-OF-PLANE RESPONSE ON OPTICAL . . . PHYSICAL REVIEW B 94, 205408 (2016)

transfer-matrix method and describe the 2D crystal with finitethickness d by a dielectric tensor. The limit d → 0 is easilytaken in the final expressions. In the resulting transfer andpropagation matrices, a cancellation of the thickness d in thedefinition of the dielectric tensor and the thickness for lightpropagation in this film happens. For s-polarized light, onlyσ

‖2D(ω) appears. For s-polarized light, we have shown that both

descriptions lead to the same expressions for the reflectance,transmittance, and absorbance for two dielectrics separated bysuch a 2D crystal.

For p polarization, the transfer-matrix method gives rise toa generalization. The expressions for transfer and propagationmatrices are also in influenced by the out-of-plane component

σ⊥2D(ω). Consequently, the two different descriptions of the 2D

crystal give different results for the propagation of light withp polarization and oblique incidence due to the appearance ofthe out-of-plane optical conductivity. The effects are illustratedversus photon energy for prototypical 2D crystals, the group-IV honeycomb ones, graphene and silicene. We suggest opticalexperiments in the high-frequency region, where σ⊥

2D(ω) isrelevant, to prove the theoretical predictions.

ACKNOWLEDGMENT

O.P. acknowledges EU for funding within the MSC RISEproject CoExAN (Project No. GA 644076).

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