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Terahertz Kerr and Reflectivity Measurements on the Topological Insulator Bi2Se3

We report the first terahertz Kerr measurements on bulk crystals of the topological insulator Bi2Se3. At T=10K and fields up to 8T, the real and imaginary Kerr angle and reflectance measurements utilizing both linearly and circularly polarized incident radiation were measured at a frequency of 5.24meV. A single fluid free carrier bulk response can not describe the line-shape. Surface states with a small mass and surprisingly large associated spectral weight quantitatively fit all data. However, carrier concentration inhomogeneity has not been ruled out. A method employing a gate is shown to be promising for separating surface from bulk effects.


I. INTRODUCTION
The discovery 1-8 and realization of 3D strong topological insulators (TIs) [9][10][11][12] have generated much recent interest. A strong spin-orbit coupling can produce a topologically protected metallic surface state which has a Diraclike dispersion with a non-degenerate chiral spin state that can not be destroyed or gapped by (non-magnetic) impurity scattering. Since the spin direction of the quasiparticle is locked to the momentum direction, 10 suppression of Anderson localization results from the destructive interference associated with the quaisparticle's Berry phase upon backscattering. 13 With the action of local ordered magnetic moments, the surface state becomes gapped. 7 In zero magnetic field with the Fermi energy within the gap, a quantized Kerr angle associated with the half quantized Hall conductance is a predicted signature of the topological origin of the surface state. 7 These and other predicted properties have stimulated a recent flurry of activity to probe the exotic surface state. Although the surface states have been observed by angular resolved photo-emission spectroscopy (ARPES), [9][10][11][12] unambiguous separation of bulk from surface state conduction has plagued transport characterization of the surface state due to the rather large intrinsic charge density in the bulk conduction band in as-grown Bi 2 Te 3 and Bi 2 Se 3 . Various schemes to circumvent this difficulty in bulk crystals have either implemented counter doping during growth, 14 minimized Se vacancies, 15 or fine tuned annealing processes. 14 One intermediate goal has been to sufficiently deplete the bulk to allow gating or chemical doping of the surface in suitably thin crystals to move the chemical potential of the bulk into the gap. 16 There are several advantages of terahertz Kerr measurements over other transport measurements which allow separation of the surface from the bulk response. The surface states are expected to have a a different mass than the bulk, depending upon the location of the chemical potential, giving rise to two different resonant cyclotron frequencies. This effect can be further exaggerated by application of an electric field either from gating or chemically doping the surface causing 'band bending' effects, pushing the Dirac point closer to the chemical potential.
Secondly, the line-shape as a function of magnetic field associated with a surface state is qualitatively different than the bulk response. Thirdly, no electrical contacts are required. Lastly, Kerr measurements on thick crystals allow probing of a single surface state.
In this article, Kerr measurements are reported on two bulk crystals of Bi 2 Se 3 at an optical frequency of 5.24 meV in fields up to 8 T. Line-fits of the data show evidence of surface states. For a measured surface state cyclotron mass of m cs = 0.033 m e , the associated spectral weight is about a factor of 20 times larger than expected based upon the corresponding number density estimated from ARPES data. Measurements of other samples from the same as-grown crystal show a higher carrier concentration. Alternatively, a simple model of the optical response which assumes the presence of two crystallites with different carrier concentrations can qualitatively describe all of our data. Future work to optically characterize carrier density homogeneity as well as an analysis of surface state signals expected by gating are discussed.

A. Kerr System Description and Calibration
For calibration purposes, the Faraday rotation and ellipticity induced by a sample in magnetic field was measured utilizing a polarization modulation technique with a heterodyne detection scheme, a technique detailed elsewhere. 17 The complex Faraday angle and relative transmission were concurrently measured.
The instrument was modified allowing simultaneous measurement of the complex Kerr angle and relative reflection by the insertion of a beam splitter, polarizer, and collection optics which include an off-axis parabolic aluminum mirror and a 2 K silicon composite bolometer as depicted in Fig. 1 Kerr and Faraday angle are defined as follows: where r (t) is the reflection (transmission) complex Fresnel amplitude of the sample expressed in a circular polarization basis. The + and -represent right and left circular polarization states, respectively. Absolute measurement of the real part of the Faraday and Kerr angles is verified by inserting a linear polarizer into the system immediately downstream from the rotator. The measured rotation angle equals the polarizer angle for small angles which is a very good approximation for the signals measured in Bi 2 Se 3 . Large angle calibration may be accomplished in a similar manner, but corrections from higher order terms involving the Fresnel amplitudes must be explicitly taken into account in the modulated signal. 17 Calibration of the imaginary part of the Faraday and Kerr angles is accomplished by measuring the known response of a GaAs 2-DEG heterostructure. A NiCr antireflection (AR) coating was deposited on the backside to minimize multiple reflections within the GaAs substrate. There is a well defined relationship between the real and imaginary parts of the Faraday angle as well as the Kerr angle. Therefore, the imaginary part may be calibrated against the real part. In this way, calibration of the system is performed as a function of magnetic field in the same manner as samples of interest without the addition of other optical components. Utilizing the formalism developed in Sec. A 2, the real and imaginary part of the Faraday and Kerr angle data shown in Fig. 2(a-d) are simultaneously fit where the mobility µ, cyclotron mass m, 2D number density n, and sheet resistance of the AR coating R are free parameters. The B-field value of the resonance determines the mass m, the width of the resonance determines the scattering rate characterized by the mobility µ, the overall strength of the response determines the number density n, and deviations of the AR coating sheet resistance R away from the ideal value tends to mix the real and imaginary parts of the angles giving rise to anisotropies in the resonant peaks.
The fit parameters are found to be the following: n=2.0 × 10 11 cm −2 , µ = −1.19 × 10 5 cm 2 /V sec where the negative sign denotes the sign of the carriers, m=0.0695 m e , and R=1.01× R AR where m e is the mass of an electron and R AR is the sheet resistance for the perfect AR coating condition of GaAs given by Z 0 /(n GaAs −1) = 145 Ω/ where Z 0 = 377 Ω/ is the impedance of free space and n GaAs = 3.6 is the index of refraction for GaAs. The mass is consistent with the known mass of GaAs, 18 and the antireflection coating is very close to ideal. The fits are overlayed on top of the reflection and transmission data shown in Fig. 2(e,f). Essentially, all 6 curves are very well reproduced with only two parameters, n and µ. Together with the polarizer calibration, the 2-DEG GaAs calibration indicates that both the Kerr and Faraday experiments perform exceptionally well.
This particular data set also illustrates the robustness of the Kerr system. Instability associated with the two-laser system occurred between +8 and ∼+5.5 T appearing as a loss of power and commensurate increase of relative detector noise in the Kerr and reflection data sets. Although the reflection data was visibly affected, the Kerr signal was essentially unaffected. The reflection measurement is derived by the ratio of the signals from two detectors ('Kerr detector' and 'reference detector' in Fig. 1) and therefore susceptible to large spacial mode fluctuations, but the Kerr signal is derived from the modulated signal measured by a single detector.

B. Bi2Se3 sample description
Bulk crystals of Bi 2 Se 3 were grown by varying the heating conditions of the melt and stoichiometric ratio of bismuth and selenium. 15 Surfaces were cleaved perpendicular to he c-axis, and the backsides were cut and ground yielding a total bulk thickness of about 0.5 cm. Samples were mounted such that the applied magnetic field was parallel to the c-axis and the radiation propagation direction. The effective area measured was defined by the optical spot size ∼ 1.0 mm. During a time span of about 10 seconds, the surfaces of the crystals were cleaved using scotch tape in air and immediately inserted into a vacuum of ∼ 10 −7 Torr.

III. KERR AND REFLECTION MEASUREMENTS ON Bi2Se3 AND MODEL DESCRIPTION
The Kerr angle and concurrent reflection measurements of linearly polarized light are shown in Fig. 3(ac) and (e-g) for two samples both measured at 10 K in fields up to 8 T. A separate measurement was performed on Sample 1 where the waveplate in the rotator depicted in Fig. 1 was statically set producing elliptically (nearly circularly) polarized light. The reflection was then measured and normalized to the B=0 T value. The data is reported in Fig. 3(d). β characterizes the degree of mixing between left and right circular polarization states via (1 − β)R + + βR − where R ± is the reflection produced by purely circularly polarized light. In this measurement, β is found to be 0.13.
To analyze the results, a model is constructed where the bulk is considered to be a thick slab whose optical response is characterized by a dielectric function consisting of a Drude carrier response with an associated unscreened bulk plasma frequency ω pb , bulk cyclotron mass m cb , and bulk scattering rate γ b along with the addition of two TO phonon modes whose parameters have been previously determined. 19 The surface state resides at the interface whose admittance is modelled within a Drude model with an associated 2-D surface plasma frequency ω ps , surface cyclotron mass m cs , and surface scattering rate γ s . Details of the model are discussed in Sec. A 3.

IV. DISCUSSION
A. Analysis of the optical response Sample 1 data shown in Fig. 3(a-d) can not be quantitatively nor qualitatively reproduced by a homogenous bulk response with any arbitrary carrier concentration, cyclotron mass, or scattering rate. The blue dash-dot curves in Fig. 3(a-d) show a representative example of line-shapes expected from a simple bulk response, calculated with the specific parameters ω pb = 420 cm −1 (corresponding to a number density of n b ∼ 3.2 × 10 17 cm −3 ), γ b = 12 cm −1 , and m cb = .135 m e . Although the high field Kerr data in Fig. 3(a,b) is well described by this particular set of parameters, strong qualitative deviations exist at fields below |B| 4 T suggesting that another carrier response with a much smaller cyclotron mass may be superimposed. The reflection data in Fig. 3(c,d) also show very different behavior at low fields |B| 4 T compared to a bulk response. The small humps which appear at fields around ∼ ±1 T in Fig. 3(c) and ∼ −1 T in Fit with 2xBulk params.

Data
Fit with 1xBulk params.
Fit with 2xBulk params. (e-h) Sample 2 data (black dots) show a very different response but was only 2 mm away from Sample 1 in the as-grown crystal. A homogeneous bulk response (blue dash-dot) does not quite describe the data. Addition of a surface state response does not improve the fits. A weighted response of two crystallites with different carrier density (green dash-dot) more closely resembles the data. No reflectivity data using circular polarized light was taken for Sample 2. β is assumed zero in calculations in Fig. 3(h).

Fig. 3(d)
can not be qualitatively reproduced with any value of bulk carrier concentration, cyclotron mass, or scattering rate. The addition of a surface state with parameters ω ps = 41 cm −1 , γ s = 28 cm −1 , and m cs = 0.033 m e results in the red solid curves in Fig. 3(a-d) which fit all the data remarkably well. The surface state cyclotron mass corresponds to a resonance at B=1.5 T (for a frequency of 5.24 meV). The scattering rate associated with the surface is substantially larger than that of the bulk, a result which might be expected based upon Shubnikov de Haas measurements on similarly grown samples 15 and linewidths estimated from ARPES data. 11 Reasonable linefits may be achieved with a surface state plasma frequency as low as ω ps = 20 cm −1 although deviations from the data begin to become more pronounced.
From the ARPES measured dispersion of the surface states (see Eq. A4), 11 the cyclotorn mass given by k f /v f and number density given by πk 2 f /(2π) 2 may be parameterized in terms of the Fermi energy. A mass of 0.033 m e corresponds to a Fermi energy of ∼20 meV above the Dirac point, a number density of 10 11 cm −2 , and corresponding surface plasma frequency of 1.3 cm −1 . The bulk plasma frequency corresponds to a Fermi energy of approximately 20 meV above the bulk conduction band edge which is itself located ∼ 200 meV above the Dirac point.
ARPES observations correspond to a large occupation number of the surface states, n s ∼ 10 12 cm −2 . 11 This amount of charge acting on the bulk carriers would deplete the free carriers to a depth d = n s /n b ∼ 100 nm (where n b is the bulk number density). Since ARPES concurrently observed the bulk bands but the measurement technique is sensitive to depths of only ∼1 nm, it must be assumed that there is a fixed neutralizing positive surface charge coming from excess Se + or adsorbed ions, or possibly an intrinsic effect. Therefore, band bending effects are apparently small in vacuum cleaved crystals. However, for samples exposed to ambient conditions, band bending may be more severe 20 possibly due to oxidation of the surface. This chemical doping moves the Dirac point toward the chemical potential and leads to a depletion region in the bulk, presumably with d < n s /n b . Since the optical penetration depth is much larger than 100 nm at terahertz frequencies, 19 an insulating depletion layer has no significant optical response and the experi-ment sees the surface carriers only as a modification to the bulk.
More significantly, the surface plasma frequency obtained from the fits to the optical response are about a factor of 15 to 30 larger than predicted from conventional transport theory based on the ARPES measured surface state dispersion relation. A possible explanation is that the crystal has multiple surface layers within the optical penetration depth, perhaps separated by interstitial selenium, dividing the crystal into ∼10 thin flakes which are optically very close but electronically decoupled. The cleave process may favor the region with layers of flakes where these interstitial selenium may be more prevalent offering a weaker link between planes.
Given the unconventional spin-chiral nature of the eigenstates, which imply spin currents perpendicular to the charge currents, an unconventional matrix element for Landau level transitions may also lead to enhancement of the spectral weight, a possibility that needs further investigation.
As a possible alternative explanation, the structure observed at low field in the optical response of Sample 1 may arise from spatial variation of the bulk charge carrier concentration. Consider the possibility that the optical spot covers a region where two crystallites of different carrier concentrations exist. Modelling this scenario, a simple weighted average of two individual bulk responses without surface states is performed with the following parameters: one bulk response weighted by 0.84 with parameters ω pb1 = 334 cm −1 , γ pb1 = 6.5 cm −1 , and m cb1 = 0.126 with a second bulk response weighted by (1-0.84) with parameters ω pb2 = 548 cm −1 , γ b2 = 8.2 cm −1 , m pb2 = 0.148 m e . The green dash-dot curves in Fig. 3(ad) show all of the salient features of the data which was attributed to the surface states. The difference between the two bulk masses is unexpected for a simple parabolic band near a gapped band edge. The coarse qualitative features of the data may be reproduced by choosing the bulk masses to be the same with the other parameters chosen similarly although the fits are substantially worse. Therefore, even though the fit is not very good quantitatively, it is possible that a distribution of carrier concentrations may be the culprit giving rise to the observed line-shapes.
Perhaps lending more credence to the spatial variation of carrier concentration argument is that a completely different response was measured by a separate sample which was only 2 mm away from Sample 1 in the ∼ 1.5 cm as-grown crystal. The measurement results of Sample 2 are shown in Fig. 3(e-g). A single bulk response qualitatively describes all the features of the data, as depicted by the blue dash-dot curves, whose parameters are ω pb = 583 cm −1 , γ b = 7 cm −1 , and m cb = 0.145 m e . β is assumed zero in calculations in Fig. 3(h). This is a much higher bulk plasma frequency than found for Sample 1. Addition of a surface state response does not improve the fits. However, modelling the bulk carrier concentration variations as was done for Sample 1 where one bulk response is weighted by 0.70 with parameters ω pb1 = 583 cm −1 , γ b1 = 5.3 cm −1 , and m cb1 = 0.145 m e and a second bulk response weighted by (1-0.70) with parameters ω pb2 = 250 cm −1 , γ b2 = 15 cm −1 , m cb2 = 0.145 m e yields a good quantitative fit to all three data sets as shown by the green dash-dot curves in Fig. 3(e-g). A third sample measured 2 mm away from Sample 2 (and 4 mm away from Sample 1) yielded the same response as sample 2.
Controlling carrier concentrations to well within a factor of ∼ 10 over cm-sized crystals is very challenging. Using the Kerr and reflectivity measurements, it is not possible to determine whether such a response originates from a distribution of carrier concentration in a single crystallite, or the presence of two (or more) crystallites with different concentrations.

B. Other methods to characterize surface states
Using the Kerr measurement technique as a routine diagnostic of sample carrier concentration variation is not feasible since it is resource intensive. Since inhomogeneity is an important general issue, a routine diagnostic tool is required. One method presently being explored utilizes FTIR spectroscopic reflection measurements. As representatively shown in the reflectivity spectral measurements in figure 4 of Ref. 19, a large dip occurs at the plasma edge. The frequency location of the dip gives information regarding the carrier concentration. Measurements on different parts of a sample utilizing a small spot size ∼ 0.5 mm in the infrared spectral region is currently being investigated as a diagnostic tool to quantify the degree of carrier concentration variation.
Another method that may be favorable for observing surface states presently being developed utilizes a gate. If the surface states have a cyclotron mass and optical spectral weight commensurate with the ARPES measured dispersion, the line-fitting procedure as used for the above Kerr and reflection data for discerning the surface state will most likely be extremely difficult with the present signal-to-noise of the system, even for fairly small surface scattering rates (see Sec. B 1 for details).
By gating, the surface state optical response can be modulated with negligible effects produced by the bulk. Any modulated signal coming from the bulk would necessarily be due to the screening layer after the depletion layer which is of thickness ∼ 1/k s , the Thomas-Fermi screening length. k s ∼ 15 nm at n b = 10 17 cm −3 , a typical carrier concentration. The charge in this layer n d ∼ n b /k s ∼ 10 11 cm −2 which is much less than the surface state carrier density. More important, however, is that k s does not depend on gate voltage and therefore is not modulated. Secondly, by modulating a gate, signal-to-noise is expected to increase by over an order of magnitude. Averaging between gate-on and gate-off configurations in fixed field on relatively short time scales eliminates magnetic field drift issues and minimizes time drift issues. Thirdly, due to this large gain in sensitivity, even very high surface scattering rates should be detectable. Expected signals produced by gating, modelled in Sec. B 2, are expected to be well within experimental capability. Fourthly, the expected Kerr response from the surface states for bulk plasma frequencies ranging from 200 to 600 cm −1 have qualitatively the same line-shapes. Even with possible carrier inhomogeneities, the detection of surface states is possible by modulating a gate with modest shifts of the surface state Fermi energy. However, characterization of the surface states will require reasonably homogeneous carrier concentrations. Lastly, developing the technique is a necessary stepping stone for future experiments: measurement of the half-quantized Hall conductivity in topological insulators inside the gap created by a magnetic layer. 7

V. CONCLUSION
Kerr measurements using heterodyne detection in the terahertz regime are reported. Measurements of a 2-DEG GaAs heterostructure prove the system works exceptionally well. Results on Bi 2 Se 3 crystals show that the technique is sensitive to bulk carrier concentration. For one particular sample, the Kerr and reflectance data are very well described by including surface states. The measured surface spectral weight ω ps (with an associated surface cyclotron mass m cs = 0.033 m e ) is ∼20 times larger than expected for a single surface state based upon ARPES measurements. Spatial variation of carrier concentration can also qualitatively account for the data. Separation of bulk and surface states may be accomplished in the Kerr geometry with greater sensitivity by incorporating a gate, a proposed next step to clearly separate the bulk and surface state effects.

VI. ACKNOWLEDGEMENT
This work was supported by the CNAM, NSF (DMR-0030112), and MRSEC (DE-AC02-98CH10886). N. P. B. was supported by CNAM. We thank Michael Fuhrer and Tudor Stanescu for insightful discussions. The formalism for deriving the Fresnel reflection and transmission amplitudes for a system consisting of multiple layers of 'dielectric' slabs with thin films at the interfaces are derived for a general number of layers. The model will be applied to two specific cases: a GaAs 2-DEG heterostructure with a metallic antireflection coating and a bulk Bi 2 Se 3 crystal with a surface state.
Using Maxwell's equations for radiation at normal incidence, boundary conditions on the electric and magnetic field may be applied across any boundary between two dielectric slabs i and i + 1 of thicknesses db i and db i+1 with bulk complex indices n i and n i+1 and a metallic thin film at the interface with a sheet conductivity of σ i df i where df i is the thickness of the film. The vector notation (E r ,E l ) represents right and left travelling waves. The electric field may be expressed in either a linear polarization basis notated as (E r x ,E l x ) and (E r y ,E l y ) or in a circular polarization basis as (E r + ,E l + ) and (E r − ,E l − ). To derive the equations generally, we use the notation (E r α ,E l α ) to represent one of the polarization basis states y, x, +, or -.
The matrix M i,α which transfers an arbitrary E-field (E r α ,E l α ) across an interface consistent with Maxwell's equations is given by the following: where the admittance of the thin film is y i,α = Z 0 σ i,α df i and Z 0 is the impedance of free space.
The matrix K i,α transfers an arbitrary oscillating Efield (E r α ,E l α ) at frequency ω across a dielectric of thickness db i and complex index n i,α is given by the following: In general, an incident right travelling wave will be assumed incident on the first interface with a resulting net reflected travelling wave to the left from the first interface and a net right travelling transmitted wave from the last interface. The Fresnel transmission and reflection amplitudes for a stack of dielectrics of m interfaces with thin metallic films are then found by the relation:

GaAs 2-DEG
The above formalism can be applied to calculate the Kerr and Faraday angle as well as the reflection and transmission for the calibration sample. A GaAs 2-DEG is modelled as a thin metallic sheet whose sheet conductance is described within a simple Drude model located at the interface between vacuum and a GaAs dielectric slab. The backside of the sample consists of an interface between a GaAs dielectric slab and vacuum with an antireflection coating, a thin metallic NiCr film, modelled within a Drude model. The antireflection coating is fully characterized by the dc sheet resistance, R = 1/(σ df ), since the scattering rate of NiCr is very large compared Re to the optical frequency. The circular polarization basis is easiest to work with in applied magnetic field.
The Fresnel amplitudes r ± and t ± as a function of n, µ, R , m, and B are then found by substituting the above parameters into the following equations: The reflection R and transmission T for unpolarized light, reflection R ± and transmission T ± for circularly polarized light, and the Faraday θ F and Kerr θ K angles are calculated from the Fresnel amplitudes as follows: where '*' notates the complex conjugate.

Bi2Se3
Bi 2 Se 3 is modelled as a surface state Drude response on top of a very thick bulk slab whose response consists of a free carrier Drude term with two phonon resonances. The bulk phonon parameters as well as the approximate cyclotron mass and scattering rate have previously been measured. 19 The bulk dielectric function is given by the following: where j enumerates the phonon oscillators, ω j is the TO phonon frequency, S j is the phonon spectral weight, a subscript b refers to bulk electrons, ǫ ∞ is the dielectric constant at higher infrared frequencies, ω pb is the unscreened plasma frequency for free electrons: ω 2 pb = 4πN b e 2 /m * b , where N b , e and m * b are concentration, charge, and an effective mass of electrons; γ b and γ j are the damping rates; ω cb = eB/(m cb c) is the bulk cyclotron frequency, where B is magnetic field, m cb is bulk cyclotron mass, and c is the speed of light.
The reported phonon parameters are ω 1 = 67 cm −1 , S 1 = 3.94 × 10 5 cm −2 , γ 1 = 5 cm −1 , ω 2 = 134 cm −1 , S 2 = 7056 cm −2 , γ 2 = 2 cm −1 , and ǫ ∞ = 25.6. 19 The reported bulk cyclotron mass is approximately m cb ≈ 0.16 m e where m e is the bare electron mass and the approximate bulk scattering rate γ b ≈ 8 cm −1 , although both are left as parameters in the model. The relation m * b = m cb is assumed true, so that N b (or equivalently the plasma frequency, ω pb ) is the third adjustable parameter associated with the bulk response. With these values, ǫ ± is a function of radiation frequency ω and applied magnetic field B, and the bulk plasma frequency ω pb , mass m cb , and scattering rate γ b .
Since the surface state penetration depth is ∼nm which is much less than the optical wavelength for terahertz radiation, the surface state may be modelled as a two dimensional sheet of charge. The model of a 1 cm thick Bi 2 Se 3 slab in vacuum with a Drude surface state response at the first interface are as follows: n 1,± = 1, n 2,± = √ ǫ ± , n 3,± = 1, (Z 0 σdf ) 1,± = ω 2 ps /(γ s − ı(ω ± ω cs )), db 2 = 1 cm, (Z 0 σ df ) 2,± = 0 where ω cs = eB/(m s c) and ω ps = (4π/c)n s e 2 /m s . Even for very low densities of bulk carriers, the properties of the last interface do not effect the reflected optical response due to the large attenuation within the thick bulk crystal. Inserting the above values into Eqs. A1 give the reflection and transmission Fresnel amplitudes as a function of ω, B, ω pb , γ b , m cb , ω ps , γ s , and m cs . The Fresnel amplitudes are also given in Ref. 19, Eqs. 4 to 6.
Digitizing and fitting the reported ARPES measured dispersion relation for the surface state from Ref. 11 gives the following functional form: where the Fermi energy E F is measured in eV above the Dirac point and k f is measured inÅ −1 . n s (or ω ps ), m cs , v f , and k f are fully determined by the dispersion relation for any given Fermi energy.
Using the dispersion relation, R, T, R ± , T ± , θ F , and θ K are calculated from Eq. A2 as a function of ω, B, ω pb , γ b , m cb , E F , and γ s .  Figure 4 shows calculated results for the expected Kerr and reflection signals, both from linearly and circularly polarized light, as a function of magnetic field for various Fermi energies ranging from 0 to 400 meV above the Dirac point. All graphs in Fig. 4 are calculated assuming a bulk cyclotron mass value of m cb = 0.15 m e and bulk scattering rate γ b = 10 cm −1 . The surface state plasma frequency ω ps and cyclotron mass m cs are determined by the ARPES measured dispersion (Eq. A4) for a specific surface Fermi energy. Each row of graphs are calculated with a different bulk plasma frequency ω pb and surface scattering rate γ s .
The optical response for a pure surface state with γ s = 10 cm −1 is shown in Fig. 4(a-d) (the response of the bulk is negligible for ω pb = 10 cm −1 ). All signals are well within the capability of the Kerr measurements. Note that Re(θ K ) and Im(θ K ) show antiresonant-and resonant-like behavior. The cyclotron mass and strength of the response increase with Fermi energy. The reflection in purely circularly polarized light depicted in Fig. 4(d) exhibits a hump in the cyclotron resonant active mode. The reflection in linear polarization depicted in Fig. 4(c) (composed of equal amounts of left and right circularly polarized light) shows two humps since there is an active cyclotron mode in positive and negative fields. The qualitative structure of the Kerr and reflection signals is instructive since they appear superimposed on larger bulk responses at larger values of ω pb .
Row 2 of figures, namely Fig. 4(e-h), show the optical response expected with a bulk plasma frequency of ω pb = 400 cm −1 and surface scattering rate γ s = 3 cm −1 . The bulk is depicted by the black curves since no carriers are in the surface state at the Dirac point, E f = 0. The curves are offset for clarity: in zero magnetic field, the Kerr angle is always zero and the normalized reflection is always equal to unity. With such a low surface scattering rate, the sharpness of the surface state response is seen clearly superimposed upon the bulk response.
The next row of figures, Fig. 4(i-l), show the effect of increasing the surface scattering rate to γ s = 10 cm −1 . Again, the curves are offset for clarity. Although the effect of surface states is discernable in many of the curves, fitting the lineshapes with the current signal-to-noise of the measurement system becomes problematic.
The last row of figures, Fig. 4(m-p), show the effect of increasing γ s to 25 cm −1 . Fitting lineshapes on this level to discern the effects produced by surface states is not conducive to the current measurement scheme.

Expected Kerr signals with application of a gate
A gate which concurrently allows ample throughput optical power while inducing negligible Kerr or Faraday rotation has been accomplished. High resistivity silicon substrates with a 300 nm film of SiO 2 were ion implanted with peak concentrations > 10 19 /cm 3 resulting in a density profile thickness of ∼ 0.3 µm. Characterization of these devices show very small Faraday rotation up to 8 T, excellent optical throughput power, no carrier freeze out, and large break down voltages. The devices are currently being used to gate large CVD grown graphene in similar magneto-optical experiments.
Estimates of the shift in Fermi energy of the surface state associated with the application of electric fields commensurate with the typical breakdown voltages of SiO 2 depends upon the Fermi energy. For currently available samples the surface state charge density is sufficiently large that it dominates the charge transfer to the sample due to the gate. As a result the band offset is reasonably given by the change in the Fermi energy due to the charge transfer to the surface states. The depletion depth in the bulk is then given to a good approximation by d 2 = 2U 0 ǫ 0 /e 2 n b (where U 0 is the energy above the Dirac point and ǫ 0 is the bulk static dielectric constant). Therefore, the net charge which one can add or subtract from the surface state without breakdown of the insulator is very nearly equal to the maximum allowable charge transfer with the application of a gate. Using a maximum applicable electric field value of 3.3 × 10 8 V/m for SiO 2 is equivalent to a net change in surface charge density of 7 × 10 12 cm −2 . Based upon the ARPES measured dispersion, this gives a shift of Fermi energy ∆E f ≈90 meV at E f =0, ∆E f ≈30 meV at E f =200 meV, and ∆E f ≈15 meV at E f =350 meV. Since the gate voltage can be reverse biased as well, roughly twice these changes can be realized. The resulting depletion depth decouples the surface states, but is small enough that the insulating depletion layer is optically invisible at THz frequencies.
Based upon these estimates, the application of a gate is chosen to produce a change in Fermi energy of ∆E f = 30 meV. Fig. 5 shows the expected Kerr signals predicted by the ARPES measured surface state properties with bulk parameters of ω pb = 400 cm −1 , γ b = 10 cm −1 , and m cb = 0.15 m e . Each graph shows Fermi level shifts of 0-30 meV, 30-60 meV, 70-100 meV, 120-150 meV, 170-200 meV, 270-300 meV, and 370-400 meV. Graphs (a-d) show the effects of increasing the surface scattering rate from 10 cm −1 to 100 cm −1 . Reflectance measurements are not very sensitive, showing that R/R[0 T] 0.5% for γ s = 25 cm −1 and 0.1% for γ s = 50 cm −1 .
Changes in Kerr angle at the level of 0.1 mrad is very reasonably obtainable in this configuration. It should be noted that the signals are maintained at these general levels even for higher bulk plasma frequencies up to 600 cm −1 .