Terahertz wafer-scale mobility mapping of graphene on insulating substrates without a

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Introduction
Graphene synthesis technology is rapidly approaching a commercially viable level, where continuous graphene films of high structural integrity are routinely produced on wafer-scale [1][2][3][4][5][6] or even m 2 scale [7,8], with projected throughput of order 10 5 m 2 /year for CVD production tools [9]. Even so, no viable means for assessing consistency in electronic properties across these length scales or area production rates are currently offered by the prevailing electronic characterization methods, typically based on field-effect or Hall mobility measurements in lithographically defined devices. Over the past five years, several demonstrations of non-contact techniques for characterization of graphene electrical properties have been reported [10][11][12][13][14][15]. Most techniques, however, gauge the graphene conductance rather than the other primary parameters of interest, which are the carrier mobility, µ, and sheet carrier density, Ns. Recently we presented spatial mapping of fieldeffect mobility and carrier density of large-area chemical vapor deposition (CVD) graphene by non-contact, back-gated terahertz time-domain conductance spectroscopy (THz-TDS) [16]. While potentially very useful in the development and optimization phase of 2D material synthesis and transfer, the approach requires deposition of the graphene film on a gate-stack substrate, and is thus not directly applicable to the many applications and production scenarios involving graphene films on insulating and flexible substrates. These include some of the most lucrative technologies such as graphene photovoltaics, displays, and flexible electronics, where in-line characterization and quality-control require the THz-TDS analysis to be carried out on the native substrate. The present work is a step in that direction, presenting a novel non-contact approach for rapid spatial mapping of carrier drift mobility (µdrift) and carrier density of four inch graphene wafers. The technique, which is based on direct measurement of the terahertz carrier dynamics by time domain spectroscopy [13,17,18], does not require the use of carrier density control by a back gate, and is compatible with a broad range of common graphene substrates, including various polymers, glasses, and insulating metal oxides. In addition to providing a means of non-destructive and rapid measurement of graphene drift mobility and carrier density in scenarios without a gate or external magnetic field, the method is significantly faster than previously demonstrated methods [16], and provides additional information on transport dynamics such as carrier scattering rate and mean free path.

Terahertz time-domain spectroscopy
We employ a THz time-domain spectrometer based on photoconductive antennas [13], to map the electromagnetic response from 0.25 to 1.2 THz of charge carriers in an approximately 95 mm wide CVD graphene film residing on a 4-inch wafer substrate. As shown in Fig. 1(a), the substrate consists of 525 µm high resistivity silicon, 50 nm p + -doped polycrystalline silicon, and 165 nm low pressure chemical vapor deposited stoichiometric silicon nitride (Si3N4). The THz-transparent back gate, consisting of the p + doped polysilicon layer [16], was not needed in this study, however, and was left at floating potential. An approximately 10 mm wide area near one edge of the silicon wafer is kept free from graphene in the transfer process. As indicated in Fig. 1(a), the sample is raster scanned in the focal plane of the THz beam, which is focused to a diffraction limited spot with a frequency-dependent full-width-at-half-maximum (FWHM) ranging from approximately 1.0 mm at 0.25 THz to 0.3 mm above 1.0 THz. The entire 4-inch wafer is raster-mapped in steps of 400 µm, where the transmitted THz transient is recorded in each pixel of the map. Figure 1(b) shows an example of such a transmitted THz transient as well as a comparison to a THz transient transmitted through the non-graphene-covered area of the 4-inch sample. As described elsewhere [13,17], THz-TDS is particularly sensitive to free carrier absorption, and thus the fast Fourier transforms (FFT) of THz time-domain transients, , transmitted through silicon substrate and silicon substrate covered with graphene, respectively, can be related directly to the complex sheet conductance, ( ) s σ ν  , of the graphene film at THz frequencies as: Sheet conductance spectra, such as the ones shown in Fig. 1(c), are extracted in each pixel of the mapped area using Eq. (1).

Van der Pauw device fabrication & Hall measurements
As shown in Fig. 1(d), Ti/Au contacts were deposited onto the graphene film subsequent to THz-TDS analysis, by electron beam evaporation through an aluminum shadow mask. The graphene film was patterned by picosecond laser ablation into 5 mm wide square van der Pauw (vdP) devices, suitable for dual-configuration sheet conductance and Hall effect measurements to extract carrier mobility and carrier density [19]. Dual configuration measurements suppress geometrical errors in both sheet conductance [20] and Hall effect measurements [21] and allow the electrical continuity to be determined [13,18,22,23]. With a constant applied external magnetic field of B ext = 255 mT, the resistances R A , R B and R C were measured in the three electrode configurations, defined from Fig. 1 I  I  I  I  I  I where q is the electronic charge. Crucially, the chosen method allows for definition of vdP devices, without affecting the electronic properties of the active area of the graphene film by introduction of photolithographic or electron beam lithographic resist, solvents, or other chemicals.

Results and discussion
Consistent with previous observations in THz-TDS [18,24,25] and FTIR [10,15,[26][27][28] studies of graphene, we find the conductance dynamics to be well described by the Drude model, given by where σ DC is the sheet conductance at DC, τ sc is the carrier momentum relaxation time, and ω is the angular frequency. By fitting the Drude model to the real part of the conductance spectrum in each pixel of the recorded map, σ DC and τ sc can be mapped, as shown in Fig. 2. The imaginary part of the measured sheet conductance was not used in the fitting routine, because of large systematic errors due to variations across the sample in the substrate thickness. In principle, inclusion of the imaginary part of the extracted conductance could increase the reliability of the fit. However, even the slightest variations of substrate thickness across the wafer results in a significant systematic error in the extraction of the imaginary part of the conductance, which is not the case for the real part. While this systematic error can to some extent be quantified and corrected [16], in the present case, a fit to the real part of the conductance alone results in robust estimates of the DC conductance and scattering time. In the limit of B F k T E  , the dc conductance can be expressed as [10,24,[29][30][31][32] ( ) where v F is the Fermi velocity taken to be 10 6 m/s [29]. Thus, carrier drift mobility and carrier density can be evaluated across the 4-inch wafer, as shown in Fig. 3(a), from measurements of σ s,dc and τ sc , by using the relations It should be noted that this analysis does not provide any information on the dependence of τ sc and thus µ drift on N s , because the THz-TDS measurements are carried out at constant N s . For transport in graphene films limited by long-range carrier scattering on charged impurities, however, the scattering rate is expected to scale as sc s N τ ∝ , yielding a drift mobility that is independent of carrier density. Figure 3(b) shows histograms of drift mobilities and carrier densities normalized to their average values, constructed from measured values within the dashed rectangles in Fig. 2(a). The measured drift mobility follows a Gaussian distribution with an average value of 845 cm 2 /Vs, and a standard deviation of 244 cm 2 /Vs, corresponding to a relative standard deviation of 0.28. Meanwhile, the carrier density displays a distribution skewed towards higher concentration, which is more accurately described by an extreme value distribution. The graphene film exhibits a high unintentional doping level, with an average carrier density of 2.25 × 10 13 cm −2 . The standard deviation of measured carrier densities is 0.56 × 10 13 cm −2 , resulting in a relative standard deviation of 0.25. This is slightly lower, but of the order of the standard deviation found for µ drift .
While the determination from THz-TDS carrier dynamics provides the drift carrier mobility and carrier density, as defined in the Drude model, conventional device measurements of carrier mobility and density typically provide either the Hall or field-effect mobility and density. These may differ from the parameters as defined in Drude theory, due to several effects. Firstly, it is known from theory [33,34] and analysis [35] of 3D semiconductor materials that Hall mobility and drift mobility can in some cases differ fundamentally by a factor known as the Hall scattering factor. Secondly, geometrical effects, related to e.g. inhomogeneities of the graphene film can strongly affect the measured device transport properties measurements, so that the inferred material properties deviate significantly from the intrinsic properties [13,18]. To investigate how conventional device measurements of carrier mobility and carrier density relate to the THz-TDS measurements of µ drift and N s , van der Pauw structures (shown in Fig. 1(c)) suitable for dual-configuration Hall measurements were fabricated in the same graphene film shown in Figs. 2 and 3. Measurements of σ s,dc , µ Hall , and N s,Hall in 47 individual 5 x 5 mm 2 devices are compared to the average values of σ s,dc , µ drift , and N s , obtained by THz-TDS inside the active device regions, in Figs. 4(a), 4(b), and 4(c), respectively. As can be observed in Fig. 4, the spread in σ s,dc , µ Hall , and N s,Hall measured in vdP structures is quite large compared to the significantly smaller spread in σ s,dc , µ drift , and N s measured by THz-TDS. As we have shown in previous correlation studies [13,18], however, defects on the scale of the device dimensions can greatly affect the electronic parameters determined by vdP device measurements [22,23,36], while the THz-TDS outcome is much less affected, due to its much smaller characteristic carrier transport length (tens to hundreds of nm). The ratio of resistances R A /R C , measured in two different electrode configurations provides a unique signature of the presence of such extended defects in the devices, where a device with a continuous active region, free from such extended defects will fulfil the criterion of electrical continuity [18], which in this case is 1 A C