Cavity-enhanced optical Hall effect in epitaxial graphene detected at terahertz frequencies

Accepted Manuscript Title: Cavity-enhanced optical Hall effect in epitaxial graphene detected at terahertz frequencies Author: Nerijus Armakavicius Chamseddine Bouhafs Vallery Stanishev Philipp Kuhne ¨ Rositsa Yakimova Sean Knight Tino Hofmann Mathias Schubert Vanya Darakchieva PII: DOI: Reference:

S0169-4332(16)32118-3 http://dx.doi.org/doi:10.1016/j.apsusc.2016.10.023 APSUSC 34117

To appear in:

APSUSC

Received date: Revised date: Accepted date:

7-8-2016 4-10-2016 4-10-2016

Please cite this article as: Nerijus Armakavicius, Chamseddine Bouhafs, Vallery Stanishev, Philipp Kddotuhne, Rositsa Yakimova, Sean Knight, Tino Hofmann, Mathias Schubert, Vanya Darakchieva, Cavity-enhanced optical Hall effect in epitaxial graphene detected at terahertz frequencies, (2016), http://dx.doi.org/10.1016/j.apsusc.2016.10.023 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

Cavity-enhanced optical Hall effect in epitaxial graphene detected at terahertz frequencies

ip t

Nerijus Armakaviciusa , Chamseddine Bouhafsa , Vallery Stanisheva , Philipp K¨ uhnea , b c a,c,d a,c Rositsa Yakimova , Sean Knight , Tino Hofmann , Mathias Schubert , Vanya Darakchievaa,∗ a

an

us

cr

Terahertz Materials Analysis Center, Department of Physics, Chemistry and Biology IFM, Link¨ oping University, Sweden b Semiconductor Materials, Department of Physics, Chemistry and Biology IFM, Link¨ oping University, Sweden c Department of Electrical and Computer Engineering, University of Nebraska-Lincoln, U.S.A. d Department of Physics and Optical Science, University of North Carolina at Charlotte, U.S.A.

Abstract

Ac ce pt e

d

M

Cavity-enhanced optical Hall effect at terahertz (THz) frequencies is employed to determine the free charge carrier properties in epitaxial graphene (EG) with different number of layers grown by high-temperature sublimation on 4H-SiC(0001). We find that one monolayer (ML) EG possesses p-type conductivity with a free hole concentration in the low 1012 cm−2 range and a free hole mobility parameter as high as 1550 cm2 /Vs. We also find that 6 ML EG shows n-type doping behavior with a much lower free electron mobility parameter of 470 cm2 /Vs and an order of magnitude higher free electron density in the low 1013 cm−2 range. The observed differences are discussed. The cavity-enhanced THz optical Hall effect is demonstrated to be an excellent tool for contactless access to the type of free charge carriers and their properties in two-dimensional materials such as EG. Keywords: THz optical Hall effect, epitaxial graphene, free charge carrier properties

1. Introduction

Graphene has attracted significant scientific interest due to its outstanding electronic properties, which arise from the linear electronic band structure resulting in massless Diractype fermion behavior [1, 2]. Epitaxial graphene (EG) grown on silicon carbide (SiC) by sublimation allows wafer-scale production of large-area homogeneous graphene on semiinsulating substrates that could be easily integrated in the current device fabrication technologies [3, 4, 5]. However, EG is significantly affected by the substrate properties and shows lower free charge carrier mobility compared to exfoliated graphene transferred on ∗

I am corresponding author Email address: [email protected] (Vanya Darakchieva)

Preprint submitted to Applied Surface Science

October 4, 2016

Page 1 of 12

ed pt ce Ac Page 2 of 12

Highlights Cavity-enhanced THz optical Hall effect reveals transport properties of graphene

•

Type of charge carriers, their mobility, density and effective mass were determined

•

Monolayer graphene has p-type doping, while multilayer graphene - n-type doping

•

Increased hydrophobicity of multilayer graphene explains its n-type conductivity

Ac ce p

te

d

M

an

us

cr

ip t

•

Page 3 of 12

Filename: Directory:

Highlights_re_VD.doc

Ac ce p

te

d

M

an

us

cr

ip t

C:\FMS\MNT_ELSEVIER_JOURNAL_APSUSC_34117_1\NEW_ORI GINALS Template: C:\TeesEls\Normal.dot Title: Subject: Author: kraftas Keywords: Comments: Creation Date: 9/30/2016 5:18 PM Change Number: 5 Last Saved On: 10/6/2016 1:43 PM Last Saved By: fs Total Editing Time: 1 Minute Last Printed On: 10/6/2016 1:43 PM As of Last Complete Printing Number of Pages: 1 Number of Words: 189 (approx.) Number of Characters: 1,080 (approx.)

Page 4 of 12

Ac ce pt e

d

M

an

us

cr

ip t

SiO2 [1] or BN [6]. It has been demonstrated that suitable SiC substrate preparation and hydrogen intercalation significantly improves free charge carrier mobility parameters [7, 8]. Electronic devices and wafer-scale integrated circuits based on EG have already been demonstrated [9, 10, 11]. In order to push this technology forward, an improved understanding of how the substrate and growth process conditions affect the free charge carrier properties of EG is very important. Existing contact-based techniques to assess transport characteristics of EG typically require complex processing that may alter its properties and provide only local information. There is a strong need for a reliable and contactless technique for determination of free charge carrier properties of graphene and related two-dimensional materials. Spectroscopic ellipsometry (SE) is a nondestructive and contactless technique, which provides access to the dielectric function of individual layers in complex layered structures [12]. SE in the ultraviolet, visible and near-infrared spectral regions have been used to study the critical points in the band structure and the electronic properties of EG [13, 14]. SE was also used for in-situ monitoring of graphene growth on metals [15]. Recently, a relatively new technique - the optical Hall effect at long wavelengths, where terahertz (THz) and mid-infrared generalized ellipsometry is employed in combination with strong magnetic fields, has been demonstrated to determine the free charge carrier properties of thin conductive layers [16, 17, 18, 19, 20]. The optical Hall effect (OHE) is the occurrence of transverse and longitudinal birefringence caused by the non-time-reciprocal response of electric charge carriers subjected to an external magnetic field [21, 20]. Hofmann et al. have shown that THz OHE measurements provide the free charge carrier type, concentration, mobility and effective mass parameters of EG layers [16, 17]. To study EG the authors employed a custom-built THz ellipsometer and an electromagnet reaching a magnetic field of 1.7 T [16, 17, 22]. Very recently we used an externally coupled Fabry-P´erot cavity to resonantly enhance the OHE signatures at THz frequencies. We demonstrated an enhancement of OHE on traditional Drude-like two-dimensional electron gas (2DEG) in AlGaN/GaN and AlInN/GaN high electron mobility transistor structures [23, 24]. The external cavity can simply be formed by placing the sample near the THz reflective metal surface of a permanent magnet. The physical cause of the OHE cavity enhancement is the retroreflected THz radiation from the magnet surface adding further cross-polarized electromagnetic field components to the reflected THz beam by passing the magneto-optic birefringent 2DEG transistor channel. The enhancement thus permits determination of, for example, 2DEG free charge carrier parameters at relatively small magnetic fields dispensing with the need for expensive superconducting, liquid He-cooled high-field electromagnets [23, 24]. 2. Experimental and modeling

In this work we employ cavity-enhanced (CE) THz OHE (CE-THz OHE) measurements to study the free charge carrier properties of EG samples with different number of graphene layers. CE-THz OHE measurements were performed employing a newly built THz ellipsometer at Link¨oping University. The experimental setup is schematically depicted in Fig. 1. The frequency-domain rotating analyzer ellipsometer measures the upper left 3×3 block of the Mueller matrix (Mij , i, j=1,2,3). The Mueller matrix describes the transformation of the 2

Page 5 of 12

d

M

an

us

cr

ip t

Stokes vector of the incident beam into the Stokes vector of the reflected or transmitted beams [25]. The ellipsometer can be operated in transmission and reflection mode at angles of incidence from 28◦ to 90◦ . The ellipsometer uses a frequency-tunable backward-wave oscillator (BWO) THz source operating in the frequency range from 100 GHz to 180 GHz providing output power up to 20 mW. The source is augmented with Schottky diode ×2 and ×3 frequency multipliers, which expand the spectral range up to 1 THz. Our ellipsometer design is a similar but improved version of the THz ellipsometer recently reported by K¨ uhne et al. [22]. In particular, our ellipsometer employs free-standing wire grid polarizers and was designed to avoid appearance of surfaces, parallel and perpendicular to the beam projection direction in order to suppress standing wave formation and thereby improve the signal-to-noise ratio over the previous design in Ref. [22].

Ac ce pt e

Figure 1: Schematic drawing of the THz ellipsometer at the THz Materials Analysis Center at Link¨oping University. The THz radiation is emitted from the backward-wave oscillator (BWO) and collimated by a parabolic mirror (PM1). After passing the polarization state rotator (PR) and a pair of plane mirrors (M1,M2) the radiation is focused onto the sample (S), which is mounted on a goniometer (G), through a polarization state generator (P). The reflected beam is passed through a polarization state analyzer, collimated by a parabolic mirror (PM3), reflected off a plane mirror (M3) and finally, focused by a parabolic mirror (PM4) into the golay cell detector (GC) input port. For the cavity-enhanced THz OHE measurements a permanent magnet (M) is placed behind the sample and serves as a sample stage. The magnetic field direction is counted positive along the z direction.

EG samples were grown on the Si-face of semi-insulating 4H-SiC (0001) substrates using high-temperature sublimation in argon (Ar) atmosphere. The growth temperature was 2000◦ C. A homogeneous, one monolayer (ML) graphene sample was grown using Ar pressure of 900 mbar while a thicker, 6 ML graphene sample was grown at Ar pressure of 55 mbar. The number of layers was determined using reflectance mapping technique [26]. The samples are mounted onto the north and south pole-faces of a neodymium magnet with B = 0.55 T (Bx,y = 0), using two thin strips of double-sided tape (15×2 mm2 ) approximately 100 µm thick thereby forming the external cavity between the back side of the sample and the magnet surface. A foam mask, which absorbs and scatters any unwanted THz radiation, is used to cover the edges of the sample and the otherwise exposed magnet surface. Data measured at 45◦ angle of incidence at positive, δM+ , and negative magnetic 3

Page 6 of 12

cr

ip t

field directions, δM− , as well as the differences between these two datasets, δM+ − δM− , are simultaneously analyzed using a stratified optical model consisting of air/EG/SiC/airgap/magnet, in order to determine the free charge carrier properties of EG. A parametrized dielectric function tensor is assigned to each layer in the optical model and the sample’s optical response in terms of Mueller matrix elements is calculated using 4 × 4 matrix algorithm. The dielectric function of the semi-insulating SiC substrate is described by a phonon contribution, while the neodymium magnet is treated as a metal with a Drude dielectric function [20]. The phonon parameters for 4H-SiC used in the data analysis were reported elsewhere [18]. The dielectric tensor of EG is expressed by the classical quasi-free electron gas (plasma) Drude model with the magnetic field induced contribution [20]:  −1 0 −Bz By 0 −Bx  ωc  εFC−MO (ω) = εDC I + ωp2 ×  − ω 2 I − iωγI +iω  Bz −By Bx 0 

an

us



(1)

Here εDC , ω, I, Bj (j = x, y, z) are the static relative permittivity, the angular frequency of the incident light, a 3×3 identity matrix, and the qmagnetic field components, respectively. q|B| m∗

nq2 , m ∗ ε0

γ=

q m∗ µ

is the plasma broadening

M

The plasma frequency parameter is given by ωp =

Ac ce pt e

d

is the cyclotron frequency parameter, where n is the volume free parameter and ωc = charge carrier concentration parameter, m∗ is the isotropic effective mass parameter, q is the elementary charge, ε0 is the vacuum permittivity, and µ is the carrier mobility parameter. For very thin films, the THz OHE is only sensitive to the sheet carrier density, Ns = nd, where d is the thickness of EG. Therefore, the parameters varied during data analysis of the EG samples are the sheet carrier density parameter, Ns , the mobility parameter, µ, the effective mass parameter, m∗ , the air-gap thickness parameter, dg , and the SiC thickness parameter, dsub . Due to limited sensitivity, in the case of the 1 ML graphene sample, the dependence of the free charge carrier effective mass on the density, Ns , is implemented in q the analysis using the equation m∗ = (h2 Ns )/(4πvf2 ) (h is the Planck constant and vf is the Fermi velocity) according to Ref. [2]. 3. Results and discussion

Figure 2 shows the experimental block-off-diagonal elements of CE-THz-OHE difference data1 δM+ − δM− (dots) and the corresponding best-match model data (solid lines) at B = 0.55 T for the two EG samples. In this representation of the data, the on-diagonalblock elements (not depicted) vanish for both EG samples, as is expected for in-plane (structurally) isotropic materials. The magneto-optic birefringence, causing THz polarization mode conversion, due to magneto-optical response of the free charge carriers within the EG layer is reflected in the off-diagonal-block elements δMij+ − δMij− , with ij = 13, 31, 23 1

For the definition of optical Hall effect difference data see Ref. [22]

4

Page 7 of 12

Ac ce pt e

d

M

an

us

cr

ip t

and 32. Furthermore, the off-diagonal-block elements are symmetric in this representation δMij+ − δMij− = δMji+ − δMji− in the absence of structural in-plane anisotropy and there+ − + − fore δM23 − δM23 and δM13 − δM13 are omitted in Fig. 2 for brevity. Good agreement between experimental and calculated data sets is found for all Mueller matrix elements. Cavity enhancement features around 830 GHz caused by interference of sample and cavity Fabry-P´erot modes are detected in the OHE spectra (Fig. 2). The spectral location of the Fabry-P´erot interference features in Fig. 2 is determined by the layer thicknesses of all sample constituents and the cavity gap. The shape and magnitude of these resonances are governed by the properties and distribution of the free charge carriers in the EG layer. The free charge carrier parameters, the cavity gap and SiC substrate thickness values of the two EG samples extracted from the best-fit to all experimental CE-THz OHE data are listed in Table 1. Fig. 2 displays quite different behaviors of the CE-THz OHE spectra for the two EG + − samples. The δM31 − δM31 spectrum for the 1 ML EG sample is practically zero, while the thicker 6 ML EG exhibits a single strong maximum in the same element. These differences for the two samples are related to the specific free charge carrier parameters and the respective values of the air gaps. We have shown previously that depending on the mobility, effective + − mass and density parameters, the minimum (maximum) of δM13,31 − δM13,31 may vanish for + − certain gap value in the case of a 2DEG [23]. Concerning δM32 − δM32 , the 1 ML graphene shows a strong minimum while the 6 ML graphene shows a strong maximum. The sign of the CE-THz OHE difference signal in this case is related to the carrier type, which is different in the two samples: p-type and n-type for the 1 ML and 6 ML graphene samples, respectively. The p-type doping of our 1 ML graphene can be related to doping from the ambient after growth. It is well known that directly after growth EG is intrinsically n-doped due to charge transfer from the substrate. However, after exposing EG to air its electron concentration is reduced or inverted to p-type conductivity as a result of compensation from p-type dopants such as oxygen, hydrocarbons, or water molecules [27]. The free hole mobility parameter in the 1 ML graphene sample is determined as µp = 1548±100 cm2 /Vs. Note that this parameter is obtained from a THz optical experiment, which averages the free charge carrier parameters over the entire sample surface area with size of 15×10 mm2 . In contrast to the 1 ML sample, the 6 ML graphene shows n-type conductivity with significantly higher free electron density of 2.0±0.2×1013 cm−2 (Table 1). Increased hydrophobicity of the thicker EG compared to the 1 ML sample may provide a possible explanation for the electron doping in this case. Although there is no experimental data for 6 ML, it has been shown that hydrophobicity of EG increases with increasing number of graphene layers [28]. Assuming a similar trend for 6 ML EG, one may expect that the adsorption of water and associated p-type dopants diminishes significantly compared to 1 ML EG. Consequently, free electron concentrations of the order of the intrinsic n-type doping of 1.0×1013 cm−2 can be expected [27]. Indeed the free electron density determined here by CE-THz OHE (Table 1) is very close to this value. The electron mobility parameter in 6 ML EG is determined to be µe = 474±40 cm2 /Vs, which is considerably lower compared to the hole mobility parameter in 1 ML graphene (Table 1). This reduction in mobility can be understood in terms of the much higher free charge carrier density in the thicker EG, 5

Page 8 of 12

ip t cr us an

M

Figure 2: Representative experimental (dots) and best-match model (solid lines) calculated block-off− + (i = 3; j = 1, 2) at B = 0.55 T for the EG − δMij diagonal optical Hall effect difference data δMij samples: 1 ML graphene - (a) and (b); and 6 ML graphene - (c) and (d).

Ac ce pt e

d

as well as the increased scattering between the graphene sheets in the stack. In addition, the band structure of the coupled graphene stack is parabolic resulting in massive Dirac fermions with increased carrier effective mass and reduced mobility. Indeed, a much higher effective mass of 0.14±0.03 m0 is determined for the 6 ML EG sample as compared to the here assumed, but well known value of 0.022 m0 for 1 ML EG (see Table 1). Table 1: Best-match model parameters for the two EG samples: carrier type, sheet density Ns , mobility µ, effective mass m∗ in EG, air-gap thickness dg , and 4H-SiC substrate thickness dsub . The errors correspond to the 90% confidence intervals. The effective mass for 1 ML EG was not varied but coupled to the sheet carrier density according to Ref. [2]

Sample

type

Ns [cm−2 ]

µ [cm2 /Vs]

m∗ [m0 ]

dg [µm]

dsub [µm]

1 ML 6 ML

holes electrons

1.0±0.1×1012 2.0±0.2×1013

1548±100 474±40

0.022 0.14±0.03

98.6±0.8 98.4±0.8

356.76±0.07 356.49±0.06

4. Summary In summary, cavity-enhanced optical Hall effect at THz frequencies was employed to study the free charge carrier properties in epitaxial graphene with different number of layers on 4H-SiC(0001). It was shown that 1 ML EG possess p-type conductivity with a free hole density of 1012 cm−2 and a mobility parameter as high as 1550 cm2 /Vs. On the other 6

Page 9 of 12

cr

ip t

hand, 6 ML EG shows n-type doping with much lower free electron mobility parameters of 470 cm2 /Vs and an order of magnitude higher free electron density in the low 1013 cm−2 range. The observed differences in the carrier type of the two samples are suggested to be due to increased hydrophobicity of EG with increasing number of graphene layers, which significantly diminishes adsorption of p-type dopants from the environment. The reduced mobility parameter in the thick EG sample was attributed to the higher free charge carrier density, higher effective mass and enhanced scattering between the graphene sheets in the 6 ML stack. The cavity-enhanced THz optical Hall effect was demonstrated to be an excellent tool for contactless access to the type of free charge carriers and their properties in twodimensional materials such as epitaxial graphene.

us

Acknowledgments

Ac ce pt e

d

M

an

The authors would like to acknowledge financial support from the Swedish Research Council (VR Contract 2013-5580), the Swedish Governmental Agency for Innovation Systems (VINNOVA) under the VINNMER international qualification program Grant No. 201103486 and 2014-04712, the Swedish foundation for strategic research (SSF) under Grants No. FFL12-0181 and No. RIF14-055, the Swedish Government Strategic Research Area in Materials Science on Functional Materials at Link¨oping University (Faculty Grant SFO Mat LiU No. 2009 00971), the FP7 EU project Nano-Rf under the Grant Agreement No. FP7-ICT-2011-8, the European Union Seventh Framework Programme under Grant Agreement No. 604391 Graphene Flagship, the National Science Foundation (NSF) through the Center for Nanohybrid Functional Materials (EPS-1004094), the Nebraska Materials Research Science and Engineering Center (DMR-1420645), and awards CMMI 1337856 and EAR 1521428. [1] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, A. A. Firsov, Electric Field Effect in Atomically Thin Carbon Films, Science 306 (5696) (2004) 666. doi:10.1126/science.1102896. [2] K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, S. V. Dubonos, I. V. Grigorieva, A. A. Firsov, Two-dimensional gas of massless dirac fermions in graphene, Nature 438 (2005) 197. doi.org/10.1038/nature04233. [3] C. Berger, Z. M. Song, X. B. Li, X. S. Wu, N. Brown, C. Naud, D. Mayou, T. B. Li, J. Hass, A. N. Marchenkov, E. H. Conrad, P. N. First, W. A. de Heer, Electronic Confinement and Coherence in Patterned Epitaxial Graphene, Science 312 (2006) 1191. doi:10.1126/science.1125925. [4] A. Tzalenchuk, S. Lara-Avila, A. Kalaboukhov, S. Paolillo, M. Syv¨ ajarvi, R. Yakimova, O. Kazakova, T. J. B. M. Janssen, V. Fal’ko, S. Kubatkin, Towards a quantum resistance standard based on epitaxial graphene, Nat. Nanotechnol. 5 (3) (2010) 186. doi:10.1038/NNANO.2009.474. [5] T. Yager, A. Lartsev, R. Yakimova, S. Lara-Avila, S. Kubatkin, Wafer-scale homogeneity of transport properties in epitaxial graphene on SiC, Carbon 87 (2015) 409. doi.org/10.1016/j.carbon.2015.02.058. [6] C. R. Dean, A. F. Young, I. Meric, C. Lee, L. Wang, S. Sorgenfrei, K. Watanabe, T. Taniguchi, P. Kim, K. L. Shepard, J. Hone, Boron nitride substrates for high-quality graphene electronics, Nat. Nanotechnol. 5 (2010) 722. doi.org/10.1038/nnano.2010.172. [7] F. Speck, J. Jobst, F. Fromm, M. Ostler, D. Waldmann, M. Hundhausen, H. B. Weber, T. Seyller, The quasi-free-standing nature of graphene on H-saturated SiC(0001), Appl. Phys. Lett. 99 (12) (2011) 122106. doi.org/10.1063/1.3643034.

7

Page 10 of 12

Ac ce pt e

d

M

an

us

cr

ip t

[8] E. Pallecchi, F. Lafont, V. Cavaliere, F. Schopfer, D. Mailly, W. Poirier, A. Ouerghi, High Electron Mobility in Epitaxial Graphene on 4H-SiC(0001) via post-growth annealing under hydrogen, Sci. Rep. 4 (2014) 4558. arXiv:1403.5059, doi:10.1038/srep04558. [9] L. Yu-Ming, A. Valdes-Garcia, S.-J. Han, D. B. Farmer, I. Meric, Y. Sun, Y. Wu, C. Dimitrakopoulos, A. Grill, A. Phaedon, K. A. Jenkins, Wafer-Scale Graphene Integrated Circuit, Science 332 (6035) (2011) 1294. doi:10.1126/science.1204428. [10] J. Hicks, A. Tejeda, A. Taleb-Ibrahimi, M. S. Nevius, F. Wang, K. Shepperd, J. Palmer, F. Bertran, P. Le F`evre, J. Kunc, W. A. de Heer, C. Berger, E. H. Conrad, A wide-bandgap metal-semiconductormetal nanostructure made entirely from graphene, Nat. Phys. 9 (1) (2013) 49. arXiv:1210.3532, doi:10.1038/nphys2487. [11] J. Baringhaus, M. Ruan, F. Edler, A. Tejeda, M. Sicot, A. Taleb-Ibrahimi, A. P. Li, Z. G. Jiang, E. H. Conrad, C. Berger, C. Tegenkamp, W. A. de Heer, Exceptional ballistic transport in epitaxial graphene nanoribbons, Nature 506 (7488) (2014) 349. arXiv:1301.5354, doi.org/10.1038/nature12952. [12] M. Schubert, Another century of ellipsometry, Ann. Phys. (Berlin) 15 (7-8) (2006) 480. arXiv:1301.5354, doi.org/10.1002/andp.200510204. [13] A. Boosalis, T. Hofmann, V. Darakchieva, R. Yakimova, M. Schubert, Visible to vacuum ultraviolet dielectric functions of epitaxial graphene on 3C and 4H-SiC polytypes determined by spectroscopic ellipsometry, Appl. Phys. Lett. 101 (1) (2012) 011912. doi.org/10.1063/1.4732159. [14] V. Darakchieva, A. Boosalis, A. A. Zakharov, T. Hofmann, M. Schubert, T. E. Tiwald, T. Iakimov, R. Vasiliauskas, R. Yakimova, Large-area microfocal spectroscopic ellipsometry mapping of thickness and electronic properties of epitaxial graphene on Si- and C-face of 3C-SiC(111), Appl. Phys. Lett. 102 (21) (2013) 213116. doi.org/10.1063/1.4808379. [15] M. Losurdo, M. M. Giangregorio, G. Bruno, Ellipsometry as a Real-Time Optical Tool for Monitoring and Understanding Graphene Growth on Metals, J. Phys. Chem. C 115 (44) (2011) 21804. doi.org/10.1021/jp2068914. [16] T. Hofmann, A. Boosalis, P. K¨ uhne, C. M. Herzinger, J. A. Woollam, D. K. Gaskill, J. L. Tedesco, M. Schubert, Hole-channel conductivity in epitaxial graphene determined by terahertz optical-Hall effect and midinfrared ellipsometry, Appl. Phys. Lett. 98 (4) (2011) 041906. doi.org/10.1063/1.3548543. [17] T. Hofmann, C. M. Herzinger, J. L. Tedesco, D. K. Gaskill, J. A. Woollam, M. Schubert, Terahertz ellipsometry and terahertz optical-Hall effect, Thin Solid Films 519 (9) (2011) 2593. doi:10.1016/j.tsf.2010.11.069. [18] T. Hofmann, P. K¨ uhne, S. Sch¨oche, J.-T. Chen, U. Forsberg, E. Janz´en, N. B. Sedrine, C. M. Herzinger, J. A. Woollam, M. Schubert, V. Darakchieva, Temperature dependent effective mass in AlGaN/GaN high electron mobility transistor structures, Appl. Phys. Lett. 101 (19) (2012) 192102. doi:10.1063/1.4765351. [19] P. K¨ uhne, V. Darakchieva, R. Yakimova, J. D. Tedesco, R. L. Myers-Ward, C. R. Eddy, D. K. Gaskill, C. M. Herzinger, J. A. Woollam, M. Schubert, T. Hofmann, Polarization selection rules for interLandau-level transitions in epitaxial graphene revealed by the infrared optical Hall effect, Phys. Rev. Lett. 111 (7) (2013) 077402. doi:10.1103/PhysRevLett.111.077402. [20] M. Schubert, P. K¨ uhne, V. Darakchieva, T. Hofmann, Optical Hall effect - model description: tutorial, J. Opt. Soc. Am. A 33 (8) (2016) 1553. doi.org/10.1364/JOSAA.33.001553. [21] M. Schubert, T. Hofmann, C. M. Herzinger, Generalized far-infrared magneto-optic ellipsometry for semiconductor layer structures: determination of free-carrier effective-mass, mobility, and concentration parameters in n-type GaAs, J. Opt. Soc. Am. A 20 (2) (2003) 347. doi.org/10.1364/JOSAA.20.000347. [22] P. K¨ uhne, C. M. Herzinger, M. Schubert, J. A. Woollam, T. Hofmann, Invited article: An integrated mid-infrared, far-infrared, and terahertz optical Hall effect instrument, Rev. Sci. Instrum. 85 (7) (2014) 071301. doi.org/10.1063/1.4889920. [23] S. Knight, S. Sch¨oche, V. Darakchieva, P. K¨ uhne, J.F. Carlin, N. Grandjean, C. M. Herzinger, M. Schubert, T. Hofmann, Cavity-enhanced optical hall effect in two-dimensional free charge carrier gases detected at terahertz frequencies, Opt. Lett. 40 (12) (2015) 2688. doi:10.1364/OL.40.002688.

8

Page 11 of 12

Ac ce pt e

d

M

an

us

cr

ip t

[24] N. Armakavicius, J.T. Chen, T. Hofmann, S. Knight, P. K¨ uhne, D. Nilsson, U. Forsberg, E. Janz´en, V. Darakchieva, Properties of two-dimensional electron gas in AlGaN/GaN HEMT structures determined by cavity-enhanced THz optical Hall effect, Phys. Stat. Sol C 13 (5-6) (2016) 369. doi.org/10.1002/pssc.201510214. [25] H. Fujiwara, Polarization of Light, John Wiley & Sons, Ltd, 2007 49-79. doi:10.1002/9780470060193.ch3. [26] I. G. Ivanov, J. U. Hassan, T. Iakimov, A. A. Zakharov, R. Yakimova, E. Janz´en, Layernumber determination in graphene on SiC by reflectance mapping, Carbon 77 (2014) 492. doi:10.1016/j.carbon.2014.05.054. [27] C. Riedl, C. Coletti, U. Starke, Structural and electronic properties of epitaxial graphene on SiC(0001): a review of growth, characterization, transfer doping and hydrogen intercalation, J. Phys. D: Appl. Phys. 43 (37) (2010) 374009. doi.org/10.1088/0022-3727/43/37/374009. [28] C. Giusca, V. Panchal, M. Munz, V. Wheeler, L. O. Nyakiti, R. L. Meyers-Ward, D. K. Gaskill, O. Kazakova, Water Affinity to Epitaxial Graphene: The Impact of Layer Thickness, Adv. Mater. Inter. 2 (16) (2015) 2196. doi.org/10.1002/admi.201500252.

9

Page 12 of 12