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Theoretical calculations of low-field electroreflectance of ultra-thin hexagonal BN films at the fundamental absorption edge

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Journal of Physics: Condensed Matter J. Phys.: Condens. Matter 26 (2014) 045301 (9pp)

doi:10.1088/0953-8984/26/4/045301

Theoretical calculations of low-field electroreflectance of ultra-thin hexagonal BN films at the fundamental absorption edge Vl A Margulis1 , E E Muryumin2 and E A Gaiduk2 1 2

Department of Physics, Mordovian Ogarev State University, Saransk 430005, Russia Department of Chemistry, Mordovian Ogarev State University, Saransk 430005, Russia

E-mail: [email protected] (Vl A Margulis) Received 28 August 2013, in final form 7 November 2013 Published 6 January 2014 Abstract

A theory is developed to describe the electric field effect on the reflectivity of ultra-thin films of hexagonal boron nitride (h-BN) at the fundamental absorption edge. The formulation of the theory is based on the original nonlinear optical approach to electroreflectance (ER) worked out by Aspnes and Rowe (1972 Phys. Rev. B 5 4022). Within the framework of the approach, the electric-field-induced change in the reflectivity in the low-field regime is expressed in terms of the third derivative of the linear optical dielectric function of the system. An explicit closed-form expression for this function is derived within the independent-layer approximation using the tight-binding representation for the π -electron energy bands of h-BN atomic layers. Incorporating this result with the general formalism of Aspnes and Rowe enables the electro-optical response function to be obtained in an explicit analytic form convenient to further numerical analysis for any particular set of input empirical parameters. The results of such an analysis suitable for the ER effect in few-layer h-BN films are presented and discussed in the context of important information they are able to provide for band structure parameters of this material. Our findings (e.g. distinct resonant field-invariant spectral features in the ER near the fundamental bandgap of the material under study) suggest that the ER technique can be used as a sensitive tool to help characterize the electronic structure of atomic hexagonal layers built from boron and nitrogen. Keywords: electroreflectance, hexagonal BN layers, optical properties

1. Introduction

or boronitrene, is a two-dimensional (2D) wide-bandgap semiconductor, which is expected to be a promising material for the designs of novel nano-optoelectronic devices operating at ultraviolet (UV) wavelengths. A clear understanding of the optical properties of atomically thin h-BN is, therefore, prerequisite for such designs to be actually realized. Experimentally, electromodulation of reflectance or, for short, electroreflectance (ER) has long affirmed itself as the most powerful modulation spectroscopy technique for studying the electronic structure of semiconductor materials and identifying the nature of optical transitions in the vicinity of critical points in the joint density

Atomically thin films of hexagonal boron nitride (h-BN) are currently generating considerable interest owing to their unique physical properties and potential nanotechnological applications. Several different techniques, including mechanical exfoliation and chemical vapor deposition, are available at present to produce such ultra-thin h-BN films with thickness down to a single atomic layer [1–16] (for a brief but very informative and readable review of the subject, see [17]). Due to its ionicity and the inequivalence of its lattice sites, a single atomic sheet of h-BN, often referred to as ‘white graphene’ 0953-8984/14/045301+09$33.00

1

c 2014 IOP Publishing Ltd Printed in the UK

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of electronic states [18, 19]. In the past few decades, ER measurements have been widely used to explore semiconductor systems of different dimensionalities from bulk (three-dimensional) crystals to quasi-zero-dimensional quantum-dot structures [20]. However, to our knowledge, no experimental observation of the ER effect has yet been reported for any of the nanometer-scale forms (nanotubes, single atomic layers, few-layer films, etc) of BN materials. Since such experiments can be expected to be performed in the near future, it may be worthwhile to present theoretical information on the ER effect in ultra-thin h-BN films, and this is the main objective of the present paper. The calculation we report here is based on the original nonlinear optical approach to ER, developed by Aspnes [21], together with Rowe [22], in the early 1970s (for a review, see [23]). The starting point of the treatment given in those pioneering papers is that the low-field ER effect may be thought of as a special case of the third-order optical mixing described by a nonlinear susceptibility χ (3) (−ω; 0, 0, ω), where ω is an optical frequency, and the zeros represent a zero-frequency electric field. Accepting this standpoint proved extremely useful, leading to a formal analytical expression for the ER of a semiconductor crystal in terms of the third derivative of the zero-field linear dielectric function of the system, an expression which can be applied to a variety of different semiconductor materials provided their unperturbed linear optical ‘constants’ have already been determined either experimentally or theoretically. It is for this reason that the low-field ER technique has been called thirdderivative modulation spectroscopy, unlike piezoreflectance, thermoreflectance, and wavelength-modulated reflectance, which are all first-derivative modulation techniques [23]. The nonlinear optical interpretation of low-field ER, suggested by Aspnes and Rowe [21–23], has important bearing for nonlinear optics itself, providing an alternative starting point for calculating the electric-field-induced refractive-index change (the so-called quadratic electro-optic Kerr effect), which is expressed in terms of the same nonlinear susceptibility χ (3) (−ω; 0, 0, ω). Although there are, of course, other suitable theoretical methods for the computation of the χ (3) , the Aspnes–Rowe approach has the advantage of being simpler and leading to a more tractable expression for the χ (3) . In particular, using this approach, Zarifi et al [24, 25] succeeded in deriving an analytical formula for χ (3) (−ω; 0, 0, ω) in the case of single-walled carbon nanotubes, a formula which predicts some new resonant features of the dc Kerr effect, as compared to those reported earlier [26]. In the present paper, we take advantage of the Aspnes–Rowe approach to study the ER effect in few-layer h-BN films within the independent-layer approximation. Our primary motivation here is to stimulate experimental interest in the problem at hand. The theoretical treatment presented in this paper is not the most sophisticated possible, but should serve as a useful guide. The calculations are straightforward and could be readily adapted to the case of gapped graphene, which can be described by the same π -electronic band structure model as for boronitrene [27, 28]. However, a more

fundamental need is for some experimental data in order to corroborate the theoretical analysis outlined below or to suggest what refinements, if any, are required to bring the theory to an agreement with experiment. It is our hope that future experiments will reveal most clearly the potential utility of low-field ER as a sensitive tool to probe the electronic structure of atomic hexagonal layers built from boron and nitrogen. The design of the paper is as follows. In section 2, we describe the model adopted in this study and develop the formalism. Section 3 comprises numerically calculated results for the ER effect and their discussion. The key points of the paper are summarized in section 4. 2. Model and formalism

Before we proceed to develop the theory describing the low-field ER effect in few-layered h-BN, we shall first specify the underlying model for the electronic structure of the system. In this study, we employ the model originally proposed by Semenoff in his seminal work [29], which has become very popular recently in the context of graphene research [30]. The model is based on the tight-binding approximation for the π-electron energy bands of a single BN sheet. In our previous paper [31], we have used this model to calculate the optical absorbance of few-layer h-BN films. The simplicity of the model enables a closed analytical expression for the absorbance to be derived within the independent-layer approximation, an expression which shows a pronounced single peak feature in the UV (at about 5.6 eV), in good agreement with recent optical absorption measurements for h-BN films consisting of two to five atomic layers [11]. This suggests that the same model may prove to be valid for calculating ER spectra of few-layered h-BN as well. In what follows, we specify the model used by writing down its main outcome—the π -electron-energy dispersion Esk for the valence (s = v) and conduction (s = c) bands of the planar honeycomb network formed by the hybridized spxy (σ ) bonds and the pz (π ) bonds between B and N atoms. Restricting our consideration to the low-energy electronic states near the two inequivalent corners K and K0 of the 2D hexagonal Brillouin zone of the h-BN sheet, we can approximate Esk at wavevector k by the expression q (1) Esk = ± 12 + (h¯ v0 k)2 with √ 3 hv t0 a, (2) ¯ 0= 2 where a is the lattice constant of 2D h-BN. The two energy bands of equation (1) are parametrized by the difference 21 of energies (Coulomb integrals) of π electrons localized on the B and N sites and by the transfer (resonance) integral t0 between π orbitals of nearest-neighboring B and N atoms. The upper (lower) sign on the right-hand side of equation (1), as well as of the subsequent one, refers to the c(v) band. 2

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is a normalization volume of the system, and, finally, γ is the damping factor of the excited electronic states, which is assumed to be independent of k. Also the v(c) band is assumed to be completely filled (empty). This is a legitimate assumption inasmuch as the bandgap energy h¯ ωG of 2D h-BN considerably exceeds the thermal energy kB T (kB is the Boltzmann constant) even at room temperature (kB Troom ≈ 26 meV). The ε(ω) of equation (8) represents the diagonal part of the in-plane optical dielectric tensor εαβ (ω), which is expressed in terms of the corresponding optical conductivity tensor σαβ (ω) as follows:

The eigenvectors corresponding to the energy eigenvalues Esk of equation (1) are given by the following two-component pseudospinor:   hk (1) ! √ ψsk 1   =√ (3)  pEck ∓ 1  , (2) 2E ck ψsk ± Eck ∓ 1 where
hk = h¯ v0 kx + iky exp(i2π/3)

(4)

near the K point and
hk = −h¯ v0 kx − iky exp(−i2π/3)

εαβ (ω) = δαβ + i4π σαβ (ω)/ω,

(5)

where δαβ is Kronecker’s delta. Due to the D3h symmetry of 2D h-BN, either of the tensors is symmetric and diagonal, with the diagonal elements being equal in magnitude, so that we have

near the K0 point, those points being located at the opposite corners of the 2D h-BN Brillouin zone √ and defined by the wavevectors K = (2π/a)(1/3, 1/ 3) and K0 = −K, respectively. To accomplish the specification of the model adopted in this study, we must record the periodic part Usk (r) of the Bloch wavefunction around the K and K0 points. On a low-energy scale, we may write Usk (r) in the form [32] X (j) (j) Uskτ (r) = ψsk Uτ (r), (6)

ε(ω) ≡ εxx (ω) = εyy (ω),

(j)

where the modulation amplitude Uτ (r) of the Bloch function at wavevector τ(=K, K0 ) is further expressed in terms of atomic 2pz orbitals φ(r) as follows:

(7)

Here, N is the total number of the lattice sites occupied by B and N atoms, Rn stands for the position vector of the nth unit cell of the 2D lattice under consideration, and dj denotes the position vector of the jth site, counted from the end of the vector Rn . In line with the Aspnes–Rowe perturbative approach to low-field ER [21–23], we first consider the optical dielectric function ε(ω) for our model in the zero-field limit. The ε(ω) which we shall use here is of the form ε(ω) ≡ εαα (ω+ )  α (k)X α (k) Xcv 4π e2 X vc = 1+ V Evk − Eck + h¯ ω+ k α (k)X α (k)  Xvc cv , + Evk − Eck − h¯ ω+

(8)

(12)

where Ek = Eck /1 is the normalized π-electron energy, ϕk = tan−1 (ky /kx ) defines the orientation of the wavevector k = (kx , ky ), and the upper (lower) sign on the right-hand side of the equation refers to the Xcv (k) near the K (K0 ) point.

where ω+ = ω + iγ , Z ∂ α ∗ Xcv (k) = Uck (r) Uvk (r) dr, ∂kα

εxy (ω) = εyx (ω) = 0.

It is to be noted that the above formula for ε(ω) (equation (8)) is obtained by use of equation (11) and the expression for σαβ (ω), which was derived by Genkin and Mednis [33] several decades ago in their theory of the optical conductivity of bulk semiconductors. The Genkin–Mednis treatment is based on the band-theory formalism in the crystalmomentum representation [34], which greatly facilitates the calculation of optical properties of crystalline solids3 . In this representation, the electron-position operator r may be conveniently divided into two parts: one which describes intraband motion of the electron and another which involves its interband dynamics. It is the second part that defines the α (k) of equation (10), which couples a pair matrix element Xcv of electronic states with the same value of k in the c and v bands involved through the k-space gradient operator ∂/∂kα . α (k), we need to know the atomic To calculate the Xcv wavefunction φ(r − Rn − dj ) associated with the lattice site at Rn + dj . Instead of using an explicit form for the φ, we shall assume that two localized atomic orbitals have zero overlap unless they are centered at the same lattice cite. This assumption is justified in calculating the optical transition matrix element when the photon energy h¯ ω approaches the bandgap energy h¯ ωG [37]—the case which will be of primary interest for us further. By virtue of equation (12), it is enough to calculate only α (k), which we denote here simply the xth component of Xcv Xcv (k). Using equations (3)–(7), we easily find   h¯ v0 1 cos ϕk ∗ ∓ i sin ϕk , (13) Xcv (k) = −Xvc (k) = 21 Ek Ek

j=1,2

N 1 X (j) φ(r − Rn − dj ) Uτ (r) = √ N n=1   × exp −iτ(r − Rn − dj ) .

(11)

(9) (10)

3 In a recent paper by Goupalov et al [35], the advantage of using that

α (as well as β in the next equation below) indices the αth Cartesian component of vectors (or tensors) in the x–y plane of 2D h-BN, e is the magnitude of the electron charge, V

representation to describe interband optical transitions has been demonstrated in the case of carbon nanotubes (cf another approach to the same problem in [36]). 3

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With the above expression for Xcv (k), it is a simple matter to obtain a closed-form expression for the complex dielectric function ε(ω) = ε1 (ω) + iε2 (ω) or, by using equation (11), for the complex optical conductivity σ (ω) = σ1 (ω) + iσ2 (ω). We find the last quantity to be even more convenient for the purpose we pursue in this paper, namely, to provide an analytical description for the ER effect in few-layer h-BN films. Thus, we may make use of the expression for the real part of σ (ω) derived in our previous work [31]. Below, we quote this result in the slightly rearranged form 2  σ1 (ω) 1 ω 61 (ω) ≡ = σ0 π ω2 + γ 2  2 2 2 ω − 3γ × ω2 + γ 2 + ωG Q1 (ω) ω2 + γ 2  2 2 γ 2 2 2 3ω − γ + ω + γ + ωG 2 Q2 (ω) 2ω ω + γ2   ωG + 4γ ωG 1 − (14) 2ωc

The imaginary part of the optical conductivity, σ2 (ω), also may be evaluated in closed form to give  2 1ω ω σ2 (ω) =− 62 (ω) ≡ σ0 π γ ω2 + γ 2  2 2 2 2 2 ω − 3γ Q1 (ω) × ω + γ − ωG 2 ω + γ2  2 2 γ 2 2 2 ω − 3γ − ω + γ + ωG 2 Q2 (ω) 2ω ω + γ2    ωG γ2 (19) − 2γ ωG 1 − 1− 2 2ωc ω with Q1 (ω) and Q2 (ω) defined by equations (15) and (16), respectively. Again, as ω ≈ ωG , the case appropriate to the study of the ER effect in few-layer h-BN films, we can simplify the above equation by retaining only the dominant first and second terms in the square brackets on its right-hand side. As a result, we find that if, in addition ω  γ , then ! 2 ωG 1ω 62 (ω) = − 1 − 2 Q1 (ω) πγ ω ! 2 ωG 1 + 1 + 2 Q2 (ω). (20) 2π ω

with π e2 , (15) 2 h     2ωc − ω 2ωc + ω Q1 (ω) = tan−1 + tan−1 γ γ     −1 ωG − ω −1 ωG + ω − tan − tan , (16) γ γ    (ω + 2ωc )2 + γ 2 (ω − ωG )2 + γ 2    , (17)  Q2 (ω) = ln (ω − 2ωc )2 + γ 2 (ω + ωG )2 + γ 2 σ0 =

With the above expressions for the real and imaginary parts of σ (ω), we may easily evaluate the reflectance of the system under study in the zero-field case. We shall assume that the normally incident electromagnetic radiation is linearly polarized in the plane of h-BN atomic layers supported on the substrate with refractive index of nS . Using the results obtained in [38–41], we can record the following expression for the normal incidence reflectance R0 (ω): 1 − N (ω) 2 (21) R0 (ω) = 1 + N (ω)

where h = 2π h¯ is Planck’s constant and h¯ ωc is the cutoff energy, equal to Eck of equation (1) taken at k = kc , with kc being an artificial cutoff wavevector introduced to restrict the k-space summation in equation (8) to the neighborhood of the K (K0 ) point, where our model is just valid. It is reasonable to suppose that the wavevector range over which the π -electron-energy dispersion is√ well described by equation (1) extends up to kc = δc (π 3/a), where δc should be considered to be about 0.1–0.2. In the above-cited paper [31], it has been shown that it is this form of kc which provides the best fit to the experimental data reported by Song et al [11]. Thus, we adopt the same form of kc in the present paper as well. Note that we can simplify the result in equation (14) by restricting our consideration to the near-bandgap photonenergy range (ω ≈ ωG ), where the calculated 61 (ω) spectrum exhibits the same single peak structure as observed experimentally [31]. We shall also assume that the inequality ω  γ is satisfied. We then find that both the second and the third terms in the square brackets of equation (14) are much less than the first term so that the 61 (ω) is well approximated by ! 2 ωG 1 61 (ω) = 1 + 2 Q1 (ω). (18) π ω

with N (ω) = nS + π αR

σ (ω) , σ0

(22)

where αR = e2 /h¯ c is the fine-structure constant. After a bit of manipulation, the R0 (ω) may be cast into the form  2 A− + 61 (ω) + 622 (ω) R0 (ω) =  , (23) 2 A+ + 61 (ω) + 622 (ω) where A± =

1 (nS ± 1). π αR

(24)

With the foregoing formulas, we are able to calculate the normalized differential change in the reflectivity, 1R(ω, F)/R0 (ω), produced by a dc electric field F applied to the system under consideration in the x–y plane (say, in parallel to the x axis). The 1R/R0 quantity is that measured directly in ER experiments so its magnitude and spectral behavior in the vicinity of the bandgap energy of the system are of primary interest for us here. In the analysis of ER 4

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by Aspnes and Rowe [21–23], the change 1R/R0 was expressed in terms of the field-induced change 1ε(ω+ , F) in the optical dielectric function ε(ω+ ). We rather choose to use here a slightly different but equivalent language of optical conductivity theory. In this language, we have 1R = β1 (ω)161 (ω, F) + β2 (ω)162 (ω, F), R0

ER, as presented in [22], is entirely within the scope of one-electron theory. The problem of extending their approach to include many-particle interaction effects is far from simple and, to our knowledge, has not yet been addressed4 . Still, we feel that the original approach of Aspnes and Rowe should provide a reasonable first-order description for the ER effect in ultra-thin h-BN films, since from ab initio calculations [43] it is known that quasi-particle self-energy corrections and excitonic effects nearly cancel each other out in a single sheet of h-BN. Thus, at least in this case, we are justified in treating the optical dielectric function ε(ω) within the independent-particle approximation, as has just been done in equation (8) above. Whether this is true in the case of few-layer h-BN films, which are in the intermediate 3D-to-2D range, is not certain yet and requires more theoretical investigation. In this context, it is worthwhile to note that the recent study of bilayer h-BN, carried out by Alem et al [16], showed that the edges of the h-BN bilayer undergo an unusual reconstruction arising from spontaneously formed covalent B–N bonds across the adjacent layers. As ˚ shown in [16], this sub-Angstrom edge reconstruction has a significant impact on the electronic properties of bilayer h-BN, restoring those expected for a single layer of h-BN with no edge effects. Going further into the calculation of 1ε(ω+ , F), we must first consider the electro-optic frequency 0 characteristic of our model. With the form of Esk in equation (1), we find

(25)

where β1 (ω) and β2 (ω) are the Seraphin-type coefficients (cf those in [19]) defined by β1,2 (ω) =

∂ ln R0 (ω), ∂61,2

(26)

and 161 (ω, F) and 162 (ω, F) represent, respectively, the real and the imaginary part of the field-induced change in the normalized optical conductivity 6(ω) ≡ σ (ω)/σ0 . Using equations (23) and (26), it is quite simple to show that β1,2 (ω) = 2(A+ − A− )B1,2 (ω)/C(ω),

(27)

where B1 (ω) = A+ A− + (A+ + A− )61 (ω) + 612 (ω) − 622 (ω), (28)   B2 (ω) = A+ + A− + 261 (ω) 62 (ω), (29) o n 2 C(ω) = A− + 61 (ω) + 622 (ω) o n 2 × A+ + 61 (ω) + 622 (ω) . (30)

(eFv0 )2 . 4h1 ¯ We next introduce the normalized complex frequency 30 =

Our next and final task is to compute 161,2 (ω, F). The nonlinear optical approach to low-field ER [21–23], which we rely on in this study, relates the complex dielectric function ε(ω+ ) to its complex field-induced change 1ε(ω+ , F) by the expression   1 ∂ 3 2 1ε(ω+ , F) = 0 [ω ε(ω+ )], (31) ∂ω 3ω2

ω γ +i (37) ωG ωG to recast the 1ε(ω+ , F) of equation (31) into dimensionless form. After a little manipulation, we have     " 3 t0 2 eFa 2 1 ∂ε(Z+ ) 1ε(ω+ , F) = 4 hω h¯ ωG Z 2 ∂Z+ ¯ G # 1 ∂ 2 ε(Z+ ) 1 ∂ 3 ε(Z+ ) + + . (38) 2 3 Z ∂Z+ 6 ∂Z+ Z+ = Z + i0 =

which has been derived by means of perturbation theory. In this expression, 0 ≡ (k)|k=0 ,

(32)

where (k) is the so-called electro-optical frequency, which depends on the electric field strength F and the interband optical transition energy Ecv (k) = Eck − Evk as follows: [(k)]3 =

e2 (F · ∇k )2 Ecv (k). (2h¯ )3

We then use the full expression in equation (8) to evaluate ε(Z+ ) directly (after converting, in the usual fashion, the sum over k to an integral). Upon inserting the result obtained into equation (38), carrying out the required triple Z+ differentiation, and using equation (11), we finally arrive at the following expressions for the real and imaginary parts of the field-induced change 16(ω, F) in the normalized optical conductivity:     3 t0 2 eFa 2 0 161 (ω, F) = F1 (Z, 0), (39) 2π h¯ ωG h¯ ωG     3 t0 2 eFa 2 F2 (Z, 0), (40) 162 (ω, F) = − 4π hω h¯ ωG ¯ G

(33)

The perturbative third-derivative result in equation (31) is valid in the weak-field regime determined by the inequality 0  γ .

(34)

In addition, the applicability of perturbation theory itself to the description of interband electron dynamics in a dc electric field requires that the following condition should be fulfilled: eFa  h¯ ωG .

(36)

(35) 4 An approximate perturbative treatment of excitonic effects on ER has

One more limitation of the result in equation (31) arises from the fact that the treatment by Aspnes and Rowe of

been developed by Rowe and Aspnes [42] within the framework of the Slater–Koster ‘contact’ interaction model. 5

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J4 (Z, 0) =

Z

3

h

f (Ek ) (Ek − Z)2 − 0 2

i

1

× [D(Ek − Z, 0)]2 dEk , (47) Z 3 h i J5 (Z, 0) = f (Ek )(Ek − Z) (Ek − Z)2 − 30 2 1

× [D(Ek − Z, 0)]3 dEk , Z 3 J6 (Z, 0) = f (Ek )

(48)

1

h i × (Ek − Z)4 − 60 2 (Ek − Z)2 + 0 4 × [D(Ek − Z, 0)]4 dEk ,

(49)

where 3 = 2ωc /ωG is the dimensionless cutoff parameter, and the auxiliary functions f (Ek ) and D(Ek − Z, 0) are given by ! 1 1 1+ 2 , (50) f (Ek ) = Ek Ek D(Ek − Z, 0) =

fundamental absorption edge. The real (61 ) and imaginary (62 ) parts of the conductivity are plotted versus the normalized radiation frequency ω/ωG in (a) and (b), respectively. The results shown in (a) and (b) are obtained using equations (14) and (19), respectively. The parameters used to generate the plots of this figure are as follows: t0 = 2.4 eV, 1 = 2.78 eV, hω ¯ G = 5.56 eV, and 0 = 10−3 .

where the functions F1 (Z, 0) and F2 (Z, 0) are given by (41)

F2 (Z, 0) = Z

(42)

−1

I4 (Z, 0) + 2I5 (Z, 0) + ZI6 (Z, 0)

with Il (Z, 0) = Jl (Z, 0) + (−1)l Jl (−Z, 0), l = 1, 2, . . . , 6.

(43)

Here Jl (Z, 0) with different ls is defined by Z 3 J1 (Z, 0) = f (Ek )(Ek − Z)

3. Numerical results and discussion

1

× [D(Ek − Z, 0)]2 dEk , Z 3 h i J2 (Z, 0) = f (Ek ) 3(Ek − Z)2 − 0 2

(44)

In all the calculations reported here, we take as the representative tight-binding model parameters t0 = 2.4 eV and 1 = 2.78 eV (the bandgap energy h¯ ωG is assumed to be 21 = 5.56 eV), and take the applied electric field strength |F| to be 3 kV cm−1 , nS equal to 1.46, which is characteristic ˚ As a dimensionless of a SiO2 substrate, and a as 2.504 A. measure of phenomenological lifetime broadening we use the parameter 0 = 10−3 . It follows from our previous study [31] that the above set of parameter values allows the calculated absorbance of few-layer h-BN films to be fitted closely to

1

× [D(Ek − Z, 0)]3 dEk , (45) Z 3 h i J3 (Z, 0) = f (Ek )(Ek − Z) (Ek − Z)2 − 0 2 1

× [D(Ek − Z, 0)]4 dEk ,

(51)

It does not seem possible to evaluate the integrals in equations (44)–(49) analytically so as to obtain the closed-form expressions for 161 and 162 in equations (39) and (40), respectively, in terms of elementary and/or special functions. However, this is not of great significance, since the above formulas are readily programmed to enable numerical results to be obtained, and these will be shown and discussed in section 3. At this point, we only note one technical detail, which follows directly from the numerical analysis reported below. Namely, the dominant contribution in equations (41) and (42) comes from the l = 3 and 6 terms, respectively, provided we are concerned with a limited frequency range around ωG . Before concluding this section, which proved rather heavily mathematical, it is worthwhile to catalog our principal results that form a basis for our numerical study of the ER effect in ultra-thin h-BN films, which we describe next. These are the formulas in equations (25), (27), (39) and (40), and (14), (19) and (23).

Figure 1. Optical conductivity of h-BN atomic layers at the

F1 (Z, 0) = Z −1 I1 (Z, 0) + I2 (Z, 0) + 2ZI3 (Z, 0),

1 . (Ek − Z)2 + 0 2

(46) 6

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the experimental data [11]. As to the electric field intensity adopted here, it is low enough for both the conditions expressed by equations (34) and (35) to be met. We start the presentation of our results with figure 1, where the spectra of the real and imaginary parts of the normalized optical conductivity 6(ω) are shown in the near-bandgap photon-energy range. While we have used the full expressions in equations (14) and (19) to obtain these results, as we have seen we could instead have used the simpler forms in equations (18) and (20), respectively, without sacrificing much accuracy. The 61 (ω) spectrum displayed in figure 1 is a replica of the spectrum obtained in our recent paper [31]. We reproduce it here not only for completeness of the picture and ease of the reader, but also to emphasize one important aspect of the problem at hand, which has already been mentioned in the preceding section. We have in mind the justification of treating the optical conductivity of few-layered h-BN at the simplest level of the independent-particle approximation. It would certainly not be reasonable to make any firm statements about the validity of the approach based only on the consistency of the results obtained thereof (in particular, for the position and the magnitude of the peak structure in the 61 (ω) spectrum) with the available experimental data on the absorbance of few-layer h-BN films [11], since the parameters of the model used are just fitted to those data. Our argument for the approach is somewhat different. Appealing to figure 1, we notice no marked asymmetry in the shape of the 61 (ω) spectrum, which matches the experiment [11] without a special fitting. Meanwhile, it is by now a common view that such asymmetry is indicative of an excitonic nature of the optical absorption in graphene in the UV region [44–46]. Since boronitrene resembles graphene in many aspects, a similar exciton effect on the shape of the optical absorption spectrum might be expected to be observed in few-layer h-BN films as well. However, no sign of such an effect has been found in experiment [11]. This suggests that the independent-particle approximation, on which we rely in this study, has a chance to be valid, at least as a first-order approximation, in calculating the ER effect in few-layered h-BN due to the effective cancellation of repulsive electron–electron and attractive electron–hole interactions in that system [43]5 . A plot of the 62 (ω) spectrum in figure 1 shows the behavior that contrasts with the behavior of 61 (ω) in the sense that the peak structure near ω = ωG becomes negative (anti-resonant) and clearly asymmetric. This is consistent with what would be expected from the Kramers–Kronig equation connecting the real part of σ (ω) with its imaginary part. It is worth noting that unlike the 61 (ω) spectrum, which was measured in experiment [11], no one seems to have reported experimental data on the magnitude and the frequency dispersion of 62 , which could also help discriminate between the two above-mentioned alternative treatments (with and without many-particle effects included)

of the optical response of few-layered h-BN. Besides this, the lack of such data does not allow optical constants of boronitrene to be reconstructed, as has been done by Kravets et al [45] in the case of graphene. Thus, there is imperative necessity to perform spectroscopic measurements of 62 (ω) to complement the data obtained by optical absorption spectroscopy technique [11]. Figure 2 displays the real and imaginary parts of the field-induced change 16(ω, F) in the normalized optical conductivity 6(ω). We see that the major effect of the electric field on 61 (ω) is that a dip structure surrounded by two subsidiary peaks is produced at ω = ωG instead of the single peak structure observed in the 61 (ω) spectrum. In contrast, the 162 (ω, F) passes through zero at ω = ωG , reaching its maximum positive (negative) value just below (above) that frequency. Such a sudden switching of the 162 (ω, F) in the neighborhood of the fundamental absorption edge is the most dramatic effect of applying the electric field to the system we are considering here. The origin behind the above-mentioned changes in the optical conductivity in the weak-field regime can be understood taking account of the proportionality of the calculated 16(ω, F) to the third derivative of the unperturbed optical conductivity 6(ω), resulting in generally sharper and more richly structured signals as compared to those observed in the spectra of the real and imaginary parts of 6(ω) themselves. In figure 3, we show the Seraphin-type coefficients β1 (ω) and β2 (ω), which represent the spectral weights of the contributions to 1R/R0 originating from the real and the imaginary part of 16(ω, F), respectively. The graphs plotted in this figure clearly demonstrate that neither of the contributions should be neglected in evaluating 1R/R0 . As a reference point for considering the ER effect, we have plotted the form of R0 (ω), i.e. the reflectance in the absence of a constant external electric field, in figure 4. The calculated R0 (ω) spectrum exhibits a distinctly asymmetric resonant structure just above the fundamental absorption edge. This is consistent with what one would expect from the graphs in figure 1, since R0 (ω) depends on both the real and imaginary parts of the optical conductivity 6(ω) (see equation (23)). As seen from figure 4, the maximum value of R0 , expressed as a percentage, is very close to 16. Finally, the main outcome of our calculation –the ER spectrum of h-BN atomic layers –is depicted in figure 5. The presence of two well defined extrema on the 1R/R0 curve –one is just beneath the fundamental absorption edge and the other is just above it –is the most prominent feature of this spectrum. Apart from the resonant enhancement of the ER in the near vicinity of the point ω = ωG , we see that increasing the photon energy from below to above the bandgap energy changes the sign of 1R/R0 from negative to positive. It should be stressed that both the above-mentioned qualitative features of the spectral behavior of the 1R/R0 are by no means unique properties of the ER effect in the system under consideration, but rather general features of ER spectra of all crystalline semiconductors in the neighborhood of critical points of their electronic structure [23]. How the specific shape of the 1R/R0 curve, as observed in figure 5,

5 Such a cancellation effect appears to account for some essential features of

optical absorption in carbon and boron nitride nanotubes as well [47–49]. 7

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Figure 4. Reflectance of h-BN atomic layers at the fundamental

absorption edge in the absence of an external dc electric field. The graph for R0 (ω) follows equation (23). The parameters used to generate the plot of this figure are the same as given in the caption of figure 1.

Figure 2. Electric-field-induced change in the optical conductivity

of h-BN atomic layers at the fundamental absorption edge. The real (161 ) and imaginary (162 ) parts of the change are plotted versus the normalized radiation frequency ω/ωG in (a) and (b), respectively. The results shown in (a) and (b) are obtained using equations (39) and (40), respectively. The electric field strength |F| is fixed at 3 kV cm−1 . The other parameters used to generate the plots of this figure are the same as given in the caption of figure 1. Figure 5. Electroreflectance of h-BN atomic layers at the

fundamental absorption edge. The graph for 1R/R0 follows equation (25). The parameters used to generate the plot of this figure are the same as given in the caption of figure 1.

spectroscopy may be important for further development of the field, since there is still no general consensus about their precise values (see [31] and references therein). In this context, the result shown in figure 5 is informative, clearly demonstrating the magnitude and characteristics of the ER effect which can be expected to be observed in few-layered h-BN. Figure 3. Frequency dependence of the Seraphin-type coefficients

β1 (ω) and β2 (ω) near the fundamental absorption edge. The graphs for β1 (ω) and β2 (ω) follow equation (27). The refractive index nS of a SiO2 substrate is fixed at 1.46. The other parameters used to generate the plots of this figure are the same as given in the caption of figure 1.

4. Summary

In summary, we have reported here a simple theory that perturbatively treats the effect of a dc electric field on the reflectivity of ultra-thin h-BN films. In our formulation of the theory, we have followed the approach developed by Aspnes and Rowe [22], benefiting from their seminal idea of considering ER as a special case of the general third-order optical nonlinearity. The band structure model necessary for evaluating the ER effect in the considered system has been incorporated into the theory at the simplest level of

can be used to determine the bandgap energy h¯ ωG and the lifetime broadening γ to high accuracy in experiment has been discussed in some detail by Aspnes [23] and need not be repeated here. We only note that the determination of those parameters for boronitrene by using low-field ER 8

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the nearest-neighboring tight-binding approximation for π electrons. With this background, we have derived an explicit closed-form expression for the normalized differential change 1R/R0 in the reflectivity measured in ER experiment. The implications of our findings in the case of h-BN atomic layers have been illustrated by the ER spectrum calculated numerically for the model parameter values appropriate to few-layer h-BN films deposited on a SiO2 substrate. When the photon energy of the incident light is close to the bandgap energy of the material here studied, the theory predicts a resonant enhancement of 1R/R0 , reaching the order of 5 × 10−4 , which is comfortably detected by means of standard experimental techniques [18, 19]. Moreover, the sign of 1R/R0 changes on sweeping the photon energy through the inter-π -band resonant condition ω = ωG . In line with original proposals by Aspnes [23], the experimental observation of the above-mentioned field-invariant spectral behavior features of the 1R/R0 would in principle allow a more precise measurement of the band structure parameters of h-BN atomic layers than optical absorption spectroscopy techniques are able to provide for these parameters. Such information would be of value since there is still no full agreement between theory [43] and experiment [11] in regard to the bandgap energy of this innovative material.

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Acknowledgments

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