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ORIGINAL PAPER

Refractive Index Dispersion of Hexagonal Boron Nitridein the Visible and Near-Infrared

Seong-Yeon Lee, Tae-Young Jeong, Suyong Jung, and Ki-Ju Yee*

Hexagonal boron nitride (h-BN) is being widely utilized as a platform for opticaland electrical devices exploiting two-dimensional layered materials. By analyzingtransmission spectra of h-BN flakes transferred onto quartz plates with thetransfer-matrix method, the refractive index of h-BN is determined in thewavelength range from 450 to 1200nm. The transmission spectra are reproducedby applying the single oscillator model of n(λ)2¼ 1þAλ2/(λ2�λ0

2) as thewavelength-dependent refractive index. Averaging over the parameters fromdifferent samples, the parameters of λ0¼ 164.4 nm and A¼ 3.263 are obtained.

1. Introduction

Hexagonal boron nitride (h-BN) is a layered material of the samecrystal structure as graphene with the lattice constant beinglarger only by �1.8%.[1] Recently, thin films of h-BN have beenwidely used as a base or capping layer for graphene[2,3] andtransition metal dichalcogenides (TMDC)[4–6] to enhance theiroptical and electrical properties.

Becausemonolayers of atomic thickness are flexible and adaptto the roughness of their substrates, the monolayer is likely to bedeformed according to the morphology of the substrate. Theresulting strain due to the distortion can induce position-dependent variation of the energy gap within the monolayer andinhomogeneous broadening of resonance profiles.[7,8] Inelectrical measurements, substrate roughness, via inhomoge-neous charge distributions, can accelerate electron scatteringthrough Coulomb interactions with local charges. Suchroughness-dependent inhomogeneous broadening and electronscattering could be drastically reduced by employing atomicallysmooth h-BN films as a substrate. For example, grapheneencapsulated in h-BN exhibited the highest room temperaturecarrier mobility of a selection of layered materials.[2] It wasreported that by encapsulatingmonolayerWSe2 orMoSe2 withinh-BN layers, their exciton resonance was sharpened to the limitof intrinsic homogeneous width.[5,6] Moreover, Scuri et al.[9] have

S.-Y. Lee, T.-Y. Jeong, Prof. K.-J. YeeDepartment of PhysicsChungnam National UniversityDaejeon 34134, Republic of KoreaE-mail: kyee@cnu.ac.kr

T. Y. Jeong, Dr. S. JungKorea Research Institute of Standards and ScienceDaejeon 34113, Republic of Korea

DOI: 10.1002/pssb.201800417

Phys. Status Solidi B 2018, 1800417 © 21800417 (1 of 6)

demonstrated an exceptionally large reflec-tance at the exciton resonance of a high-quality MoSe2 monolayer encapsulated byh-BN.

When incorporating h-BN layers withinoptoelectronic devices, precise clarificationof the basic and fundamental opticalconstant parameters is required, as theyhave critical effects on the performance ofthe devices. For example, in TMDC mono-layers where the electronic wavefunction isvery diffuse and overlaps with the sur-rounding material, the exciton bindingenergies of the monolayers strongly de-pend on the dielectric constants of their

environments.[10,11] Utilization of photonic or plasmonic modesin h-BN-based optical resonators demands precise knowledge oftheir dielectric constants.[12–14] Although there are severalreports on the dielectric constant of h-BN,[15–17] only a fewrelate to its systematic dispersion in the visible and near-infraredrange, where exciton resonances are located for popular 2Dsemiconductors MoS2, MoSe2, and WSe2.

In this paper, we present the refractive index dispersionof h-BNin thevisibleandnear-infraredspectral rangeof450–1200nm.Thetransfer matrix method (TMM) was applied in order to extractthe dispersion relation of the refractive index from the transmit-tance spectrum. The refractive index dispersion was modeled bythe Sellmeier equation with a single oscillator,[18] whichsuccessfully reproduced the measured transmittance.

2. Experimental Section

h-BN flakes were mechanically exfoliated from a bulk crystal andtransferred onto quartz substrates for transmission measure-ments. Atomic force microscopy (AFM) was used to estimate thethickness of the h-BN flakes. For optical characterization, whitelight from a super continuum laser (Compact, NKT photonics)passed through a monochromator, and an objective lens withNA¼ 0.4 was used to focus the beam onto the sample. Thetransmitted intensity in the visible range was measured with asilicon photodiode, and that in the near-infrared range with a Gephotodetector. For the low temperature measurements, thesample was mounted inside an optical cryostat cooled by liquidnitrogen. In order to compensate for measurement uncertaintyand improve the data reliability, we measured the transmittedintensity at two points in close proximity, one on h-BN/quartz andthe other onbare quartz; thus,wewere able to analyze the values ofthe transmission contrast (TC), defined as the ratio (Tbnþq/Tq) ofthe transmittances of h-BN/quartz (Tbnþq) and quartz (Tq).

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3. Results and Discussion

Figure 1(a) shows optical images of the four different h-BNflakes on quartz that we investigated in this study. The topology,scanned via AFM, provided a thickness estimation for each flake,which varies from 0.87 to 1.17 μm. We note that the thicknesscan be nonuniform within each layer and depends to a certainextent on the measurement point. The Raman spectrum of flakeA in Figure 1(b) with strong in-plane E2g optical phononmode at1366 cm�1 confirms the hexagonal structure of the BN crystalsunder this study.[19] Fourier transform infrared (FTIR) spectros-copy was also applied to characterize the h-BN crystal. The FTIRtransmittance in Figure 1(c), obtained from a h-BN flake of about990 nm thickness transferred onto a silicon substrate, indicatesstrong absorption at the E1u(TO) resonance of the hexagonalcrystal lattice.[20]

Figure 1. a) Optical microscope images of the four h-BN flakes (A–D) oninvestigated in this study. AFM scanning profile results of the estimated filmoverlaid on the lower left of each image; the thickness steps displayed in eacedge indicated by the dashed line overlaid on the flake. b) Raman spectrum ofthe in-plane optical phonon mode with E2g symmetry of hexagonal sttransmittance spectrum of a h-BN flake with 990 nm thickness on silicon s

Phys. Status Solidi B 2018, 1800417 1800417 (

Figure 2(a) shows the TC spectrum in the wavelength range of450–1200 nm for the h-BN flake of�1.17 μm thickness (Flake A).The refractive index of h-BN is larger than that of quartz, andmultiple reflections occur at the air/h-BN and h-BN/quartzinterfaces. The wavelength-dependent modulation of TC inFigure 2(a), alternating peaks and troughs, originates from aninterference effect of the multiple reflections from the h-BNinterfaces. The points of maximum TC, corresponding todestructive interference between the reflections from the twointerfaces, are very close to the line of TC¼ 1, which indicates noabsorption loss of the h-BN layer in the visible and near-infraredrange, with h-BN being a wide gap material. The minimumpoints of TC, on the other hand, are located at the wavelengths atwhich constructive interference occurs. If the refractive index nof h-BN was higher, the trough of TCwould be located at a lowervalue, because the reflectivity would be increased at both

quartz that werethicknesses are

h case are for theflake A indicatingructure. c) FTIRubstrate.

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interfaces. By applying this relation, the nvalues can be determined at the wavelengthsof troughs of TC. Then, the variation of theminimum TC in Figure 2(a), which has asmaller value at shorter wavelengths, corre-sponds to the case of the positive materialdispersion of the refractive index. While themodulation depth of TC is determined solelyby the parameter n, the wavelength separationbetween adjacent peaks or adjacent troughsalso depends on the h-BN thickness d.

If the refractive index dispersion and thethickness of a h-BN flake are known, thetransmission spectrum can be calculated byapplying the TMM.With solving the boundarycondition problem for the wave propagationthrough the air/h-BN/quartz system in thisstudy, the electric filed amplitudes for theforward (Ai) and backward wave (Bi) at eachpoint designated in Figure 2(b) satisfy thefollowing equations

A1

B1

" #¼

nþ 12

n� 12

n� 12

nþ 12

2664

3775� A2

B2

" #;

A2

B2

" #¼

exp � inωdc

� �1

1 expinωdc

� �26664

37775

� A3

B3

" #;

A3

B3

" #¼

nq þ n2n

nq � n2n

nq � n2n

nq þ n2n

264

375

� A4

B4

" #ð1Þ

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A1

B1

" #¼

nþ 12

n� 12

n� 12

nþ 12

2664

3775�

exp � inωdc

� �1

1 expinωdc

� �26664

37775

�nq þ n2n

nq � n2n

nq � n2n

nq þ n2n

264

375� A4

B4

" #

¼ T11 T12

T21 T22

" #� A4

B4

" #ð2Þ

Then, the matrixT11 T12

T21 T22

" #directly relates the field

amplitudes at air,A1

B1

" #, to those at quartz,

A4

B4

" #, and is

obtained by multiplying the three sub-matrices that representthe electric field transfer at the air/h-BN interface, through theh-BN layer, and at the h-BN/quartz interface, respectively. If weassume that the light is incident from the left air side, thenB4¼ 0. In this case, the reflectance R is calculated as R¼ |B1/A1|

2¼ |T21/T11|2.

If the refractive index dispersion and the thickness of theh-BN flake are known, the transmission spectrum can beeasily calculated by applying the TMM. According to theTMM approach, the matrix elements ABCD for lightpropagation from air to quartz through h-BN are given asthe multiple of three sub matrices, as in Equation (3), whichcorrespond to the electric field transfer at the air/h-BNinterface, through the h-BN layer, and at the h-BN/quartzinterface, respectively.

Figure 2. a) TC spectrum obtained from the h-BN flake A. b) Schematicindicates the electric field amplitudes for the forward (Ai) and backwardinterface of air/h-BN and at that of h-BN/quartz.

Phys. Status Solidi B 2018, 1800417 1800417 (

T11 T12

T21 T22

" #¼

nþ 12

n� 12

n� 12

nþ 12

2664

3775

�exp � inωd

c

� �1

1 expinωdc

� �26664

37775

�nq þ n2n

nq � n2n

nq � n2n

nq þ n2n

264

375 ð3Þ

Here, ω is the angular frequency, c is the speed of light, and nand nq are the refractive indices of h-BN and quartz, respectively.With the ABCD elements determined, the reflectance R andtransmittance T are given as R¼ |C/A|2 and T¼ 1-R.

In the investigated region, the imaginary part of the dielectricconstant is negligibly small because the photon energies aremuch smaller than the energy gap of h-BN. But, the high-energyoptical transitions contribute to the real part through theKramers–Kronig relation. Contributions from the lattice can beneglected because the optical frequency under this study ismuchhigher. It was shown that the electronic response can bedescribed through the single-oscillator approximation when thephoton energy is much smaller than the lowest energy gap of thematerial.[21] Because the energy gap of h-BN is high enough, weassume the single oscillator model for the optical response in thevisible and near-infrared. According to the single oscillatormodel, the dispersive dielectric constant can be described asfollows

e1 λð Þ ¼ n λð Þ2 ¼ 1þ Aλ2

λ2 � λ20ð4Þ

where the parameters λ0 and A are related to the energy and

illustration whichwave (Bi) at the

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strength of the oscillator, respectively. Thus,the material dispersion, under the singleoscillator model, is determined from theparameters λ0 and A.

In order to extract the oscillator parameters,we have simulated the TC spectrum, varyingλ0, A, and the thickness d. The dispersionmodel that best matches the experimental datawas obtained after comparing the simulationwith the measurements. Figure 3 shows thefitting parameters and the simulation resultsfor the TC spectrum of the four h-BN flakes inFigure 1(a). The simulation reproduces theexperimental result well, confirming thevalidity of the single oscillator model.

Figure 4 presents the refractive indexdispersion plots of the four sets of λ0, Aparameters shown in Figure 3. The averageover the four samples yields λ0¼ 164.4 nm andA¼ 3.263; the refractive index dispersion forthis average is plotted as a black line in

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Figure 3. Simulated (red lines) and measured (black scatters) spectra of the transmission contrast for the four h-BN flakes A–D. Fit parameters areshown on each plot: d is the h-BN flake thickness, and the parameters λ0 and A define the material dispersion in the single oscillator model.

Figure 4. Refractive index dispersions for the best-fit simulations of thefour h-BN flakes (dashed lines). The solid line is the dispersion with theparameters λ0¼ 164.4 nm and A¼ 3.263, representing the average overthe samples measured. Several reported dispersions obtained throughthe reflectance measurement,[23] the ellipsometry technique,[24] and thetheoretical calculation[16] were added for comparison.

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Phys. Status Solidi B 2018, 1800417 1800417 (

Figure 4. The value λ0¼ 164.4 nm implies that the electronictransitions in h-BN is approximated by a single transition at7.54 eV, which is somewhat higher than the literature value forthe absorption edge.[22] According to the Kramers–Kronigtransformation, all the optical transitions contribute to the realpart of the dielectric constant though the relation,e1 ωð Þ ¼ 1þ 2

π PR10

ω0e2 ω0ð Þω02�ω2 dω

0, where e1(ω) and e2(ω) are, respec-tively, the real and imaginary part of the dielectric constant, Pdenotes the principal part. Because the optical transitions locatecontinuously above the absorption edge and each transitioncontributes to the refractive index, an effective single oscillatorthat accounts for the above gap transitions is located at an energysomewhat higher than the absorption edge of h-BN crystal.

In our results, sample-to-sample fluctuations in the refractiveindex are less than 0.5% over the entire wavelength region.Recently, Segura et al.[23] have reported on the refractive indexdispersion values of h-BN crystal from the constructiveinterference condition in the transmittance spectrum, withassuming a predefined crystal thickness. Their values of therefractive index are roughly 3.3% larger than those in this study.We note that the inaccuracy in their thickness estimation directlyleads to the uncertainty of the refractive index in their study,while the layer thickness was self-consistently determined in oursimulation. For comparison, we have added in Figure 4 severalreported refractive index dispersions of h-BN that were obtained

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Figure 5. Simulated (red line) and measured (black scatter) spectra ofthe transmission contrast for flake E with a thickness of less than 25 nm.The inset shows an optical microscope image and AFM scanning profileof the flake.

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through the reflectance measurement,[23] the ellipsometrytechnique,[24] and the theoretical calculation.[16]

Though the refractive index dispersion was obtained fromh-BN flakes with thicknesses around 1 μm, the results could beapplied to much thinner samples. Figure 5 shows theexperimental TC spectrum and the simulation for h-BN flakeEwith a thickness less than 25 nm.We find that the TC spectrumis successfully reproduced using the averaged λ0 and A valuesand a thickness of 22.2 nm, with confirming validity of theextracted dispersion to other thicknesses.

When the sample wasmeasured at low temperatures, down to80K, the refractive index did not deviate beyond the experimen-tal error (�0.004) with respect to the data at room temperature.We note that for the case of fused silica, for which the energy gapis smaller than h-BN, the refractive index in the visible range varyby less than 0.002 over the temperature range 30–300K.[25]

Because the energy gap of h-BN is larger than that of fused silica,the temperature-induced effect is expected to be comparable orsmaller. These facts indicate that the refractive index dispersionextracted in this study, with an uncertainty error of 0.004, can beutilized even at low temperatures.

4. Conclusion

We have investigated the refractive index dispersion of h-BN inthe visible and near-infrared range from the thin filminterference observed in the transmission spectrum. Bycomparing the experimental TC spectra with simulations basedon the TMM approach, we demonstrated that the Sellmeiersingle oscillator equation, n(λ)2¼ 1þAλ2/(λ2�λ02) with λ0¼164.4 nm and A¼ 3.263, reliably explains the refractive indexdispersion of h-BN in the wavelength range of 450–1200 nm. Weexpect that the refractive index established in this work will beapplied in the design and characterization of 2D optoelectronic

Phys. Status Solidi B 2018, 1800417 1800417 (

devices that incorporate h-BN layers in pursuit of high quality ornovel functionality.

AcknowledgmentsThis work was supported by the National Research Foundation of Korea(NRF-2016R1A2B4009816) and the research fund of 2017 ChungnamNational University.

Conflict of InterestThe authors declare no conflict of interest.

Keywordshexagonal boron nitride, refractive index dispersion, single oscillatormodel

Received: August 7, 2018Revised: October 10, 2018

Published online:

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