Non-contact method for measurement of the
microwave conductivity of graphene
L Hao, J Gallop, S Goniszewski, O Shaforost, N Klein and Rositsa Yakimova
Linköping University Post Print
N.B.: When citing this work, cite the original article.
Original Publication:
L Hao, J Gallop, S Goniszewski, O Shaforost, N Klein and Rositsa Yakimova, Non-contact
method for measurement of the microwave conductivity of graphene, 2013, Applied Physics
Letters, (103), 12.
http://dx.doi.org/10.1063/1.4821268
Copyright: American Institute of Physics (AIP)
http://www.aip.org/
Postprint available at: Linköping University Electronic Press
http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-100317
Non-contact method for measurement of the microwave conductivity
of graphene
L. Hao,1,2,a)J. Gallop,1S. Goniszewski,1,2O. Shaforost,2N. Klein,2and R. Yakimova3
1
National Physical Laboratory, Hampton Road, Teddington TW11 0LW, United Kingdom
2
Imperial College London, South Kensington Campus, London SW7 2AZ, United Kingdom
3
Department of Physics, Chemistry and Biology (IFM), Link€oping University, S-581 83 Link€oping, Sweden
(Received 14 May 2013; accepted 28 August 2013; published online 17 September 2013)
We report a non-contact method for conductivity and sheet resistance measurements of monolayer and few layers graphene samples using a high Q microwave dielectric resonator perturbation technique, with the aim of fast and accurate measurement. The dynamic range of the microwave conductivity measurements makes this technique sensitive to a range of imperfections and impurities and can provide rapid non-contacting characterisation. As a demonstration of the power of the technique, we present results for graphene samples grown by three different methods with widely differing sheet resistance values. [http://dx.doi.org/10.1063/1.4821268]
The remarkable properties of single- and few-layer gra-phene thin films have led to an explosion of activity.1–4 A number of different methods for preparing graphene thin films have appeared, and a wide range of experiments are being car-ried out on them.5–12There is a great deal of variability in the quality of films prepared, and even when identical methods are used, the film properties between successive batches may be quite different. The accepted method for characterising the electrical properties of graphene films is to measure the mobil-ity. However, this generally requires patterning of the films and making electrical contact. The provision of a gate voltage to tune the carrier density is also often necessary. The addi-tional processes mean that quality assessment of the films is time consuming and requires physical intervention on the gra-phene wafer which may affect its properties.
Here, we report development of a fast, non-invasive, and non-contacting method for measurement of the microwave surface impedance (and hence the conductivity) and sheet re-sistance of graphene thin films including monolayer and few layers graphene samples. Moreover, it is applicable to large area samples and provides an average conductivity and sheet resistance values for the entire sample. A similar technique has been used previously to examine a variety of other materi-als.13,14 However, the advantage of our method is that exact solution of the mode geometries is not required. Under the conditions which are specified below, it is not necessary to carry out a detailed mode matching or finite element electro-magnetic model to derive the electrical parameters of the thin film. The conductivity and sheet resistance are derived by ref-erence to measurements on a similar bare substrate to that on which the graphene sample is deposited.
Consider the arrangement shown in Figure1(a). A single crystal sapphire puck (diameter 12 mm, height 3 mm) acts as a microwave dielectric resonator, contained within a cylin-drical copper housing. Its crystallographic a-axis is aligned parallel with that of the copper housing from which it is sep-arated by a short quartz spacer tube. The sapphire has rela-tive permittivities of 11.6 (a-axis) and 9.4 (b-c plane) with a
loss tangent of <105. It is usually used as a TE011
micro-wave resonator with the electric field in the b-c plane. A plain low loss dielectric substrate, of thickness ts, can be
brought to a fixed position in relation to this sapphire puck resonator (see Figure1(b)), and the resulting shift in both the resonant frequency Dfs and the linewidth Dws can be
meas-ured, provided the quality factor of the sapphire resonator is high enough compared with the losses contributed by the substrate. Now take another, nominally identical substrate coated with a uniform layer of graphene of thicknesstg(see
Figure1(c)). Place it in the same position relative to the sap-phire puck and make further measurements of the resonant frequency shift Dfgand linewidth shift Dwg. Finally, measure
the unperturbed sapphire puck resonant frequency f0 and
linewidthw0with only puck and support quartz tubes in the
copper housing (see Figure1(d)).
Sincetg tsand alsotg dg, the electromagnetic skin
depth of graphene at microwave frequencies, we may assume, to a good approximation, that the field distributions in the bare substrate and graphene coated substrate situations are the same. Thus, we may apply perturbation theory to evaluate the surface impedance of the graphene, provided that the complex permittivity and thickness of the bare sub-strate are known
Dfs¼ f0 ðe0 s 1Þ Ð E2dV W ; (1)
where E is the field within the substrate and the integral is over the substrate volume.W is the total stored energy in the puck and substrate system. e0sis real part of the substrate
per-mittivity. As the assumptions above that the graphene layer is very thin, its presence in the second measurement will not significantly perturb the total field distribution in the system. However, the complex permittivity of the thin film may con-tribute significantly to the frequency shift and the linewidth shift Dfg¼ f0 ðe0 s 1Þ Ð E2dVþ ðe0 g 1Þ Ð E2dv W ; (2) a) E-mail: ling.hao@npl.co.uk 0003-6951/2013/103(12)/123103/4/$30.00 103, 123103-1
where now the second integral is over the volume of the gra-phene film only, and e0gis real part of the permittivity of the
graphene. Since the volume of the graphene is always far smaller than the substrate volume whereas the real permittiv-ities are likely to be similar in magnitude, the second term in the above integral can be ignored to first order.
Similar expressions can be written for the linewidth shifts in both cases
Dws¼ f0 e00 s Ð E2dV W ; (3) Dwg¼ f0 e00sÐE2dVþ e00 g Ð E2dv W ; (4)
where e00sand e00gare imaginary parts of the permittivity of
the substrate and the graphene, respectively. Note that in the case of the imaginary components, we cannot ignore the second term in the integral since graphene is a conductor with conductivity comparable to that of a metal in which case e00g e00s.
The readily measurable quantities are Dfs and (Dwg
-Dws) and the latter may be expressed as
Dwg Dws¼ f0 e00g Ð E2dv W ¼ e00 g Dfstg ðe0 s 1Þ ts : (5)
Here, we have assumed that both for graphene and bare substrate, the electric field is uniform throughout the thick-ness; a reasonable assumption provided the substrate is much thinner than the height of the puck. This is confirmed by both finite element calculations and reversing the graphene sample so that the graphene is on the substrate surface further from the puck. The deduced conductivity of the graphene is the same (within 610%) for both orientations. The aim of this measurement is to derive a surface resistance (or sheet resistance)Rsvalue for the graphene film since this
will also enable us to estimate the graphene conductivity (and the mobility if we are able to estimate the carrier density). We know that
Rs¼
1 r tg
: (6)
There is a simple relationship between conductivity r and imaginary component of the dielectric constant eg00
r¼ 2p f0e0e00g ¼
2pf0e0ðDwg DwsÞ ðe0s 1Þ ts
Dfstg
: (7)
So finally, our expression forRs becomes independent
of graphene thicknesstg
Rs¼
Dfs
2pf0e0ð Dwg DwsÞ ð e0s 1Þ ts
: (8)
The TE011 resonance in the sapphire dielectric puck
occurs at around 10.6 GHz. The presence of a 10 10 mm bare quartz substrate, placed directly on top of the sapphire resonator, shifts the frequency (downwards) in the range 50–200 MHz, depending on the spacing between them. This produces no significant reduction in the quality factor (Q) of the resonance, which at room temperature is around 1 104.
The presence of a single layer of graphene on such a quartz substrate produces a large reduction inQ value and a further small shift in the resonant frequency. In Figure 2, the upper trace shows an example of a resonance with plain quartz sub-strate and the lower trace shows a similar trace with a sample of CVD grown graphene12transferred onto a nominally iden-tical quartz substrate. Note the reduction Q by a factor of approximately 10 in the latter case.
The resonant frequency and linewidth are measured in transmission with a vector network analyser (VNA) being used to measure S12as a function of frequency. The internal
software of the VNA (HP 8720) is used to collect the centre frequency and 3 dB linewidth, but in addition, the full trace data (200 frequency points) are downloaded to a computer and a non-linear least squares fit routine is used to fit the data to a skewed Lorentzian lineshape. By this method, we have found that the uncertainty in centre frequency and in FIG. 1. (a) Photo of sapphire puck and quartz spacer tubes inside copper
housing (lid removed).(b)-(d) Schematic diagram of the high-Q sapphire dielectric resonator for measurement of the surface impedance of graphene samples. (b) A plain substrate, (c) an identical substrate with graphene film, and (d) with neither substrate, just the dielectric resonator and support structures.
FIG. 2. Measured S12transmission versus frequency for both a bare quartz
substrate (upper trace) and a quartz substrate with transferred CVD graphene (lower trace). The narrower resonance (higher Q) corresponds to the bare quartz sample and note that the linewidth is approximately 3 times greater for the CVD sample compared with bare quartz. Crosses are experimental points and solid lines are the Lorentzian fit to the data (NB: the two are almost indistinguishable to the eye).
linewidth is reduced by a factor of 10, compared with the results output directly from the VNA.
For graphene samples with rather high conductivity, it has sometimes been necessary to reduce the influence of the graphene layer on the quality factor of the system by insert-ing a small quartz tube spacer a few mm in height, between the sapphire puck and the graphene coated and bare sub-strates (as shown schematically in Figure1). In this way, the influence on the resonant properties is reduced so that theQ value can be accurately measured.
In order to demonstrate the utility of the above dielectric resonator technique, we have made measurements on a num-ber of samples of graphene grown by different techniques, possessing very different transport properties. In TableI, we compare liquid-phase grown graphene oxide (GO), subse-quently reduced (rGO),15 CVD graphene grown on a copper catalyst layer,12and then transferred to a clean quartz substrate (relative permittivity 4.4, 10 10 mm, and 0.5 mm thick). Finally, two separate samples of epitaxially grown graphene on SiC16 (relative permittivity 9.66, 5 5 mm, and 0.5 mm thick) are also shown. Note that the range of sheet resistance values measured spans almost four orders of magnitude, dem-onstrating the great sensitivity and dynamic range of the method. In our method, we can derive sheet resistanceRs
with-out the need to measure the thickness of the graphene since sheet resistance derived by Eq.(8)is independent of thickness. It is clear from the results in Table I that, as would be expected, the rGO sample has a conductivity considerably higher than the as-grown GO sample for same thickness. It is the reduction process which converts the sample into a semi-metallic state. Comparing the rGO and CVD sample conduc-tivities, it is clear that the latter is more metallic than the for-mer, again unsurprising given the nature of the wafer-scale growth process for CVD. For monolayer graphene on SiC, the conductivity is two orders of magnitude greater than the monolayer CVD graphene and the sheet resistance is nearly three orders of magnitude lower than the CVD sample.
The analysis presented above, and the results shown in TableI, confirms that the dielectric resonator technique pro-vides a quick and straightforward method for analysing the conducting properties of graphene samples. Note that no pat-terning or electrical contacts are required, which may dam-age or compromise the sample quality. The sample can be placed on top of the puck or quartz spacer without requiring adhesive, another potentially damaging addition. The method has great sensitivity and dynamic range. We may estimate the expected uncertainty in the measurements made with this method. First, it is clear that the dimensions of the plain substrate and the graphene coated substrate should be as similar as possible. Any difference in dimensions will
feed directly into the calculation of conductivity and sheet resistance through Eq. (8). Note that a 1% change in sub-strate thickness will produce a 1% change in calculated transport quantity. Equally, if graphene covers only X% of the substrate (X < 100%), the deduced graphene conductiv-ity will be only X% of the full coverage value. Obviously some method is required to determine the coverage if possi-ble but note that, if not, the measurement willpessimistically estimate the conductivity of the graphene, a satisfactory result from the point of view of quality control. A second in-dication of the expected uncertainty comes from a compari-son of samples taken from equivalent regions on a larger wafer. These show conductivity and sheet resistance values which are usually within 10% of each other, suggesting that this is on the order of the upper limit on uncertainty to be expected. In addition, the reproducibility of samples with respect to a sequence of positional adjustments shows that this uncertainty is at most a few percent. Importantly, com-plex electromagnetic mode modelling is not required for this method, thus it shows great promise for rapid quality control and characterisation. It may be extended in future to cryo-genic or elevated temperature measurement.17 We are also investigating the possibility that, based on using higher order puck resonator modes, it may prove to deduce the graphene mobility in a contact-free way, as well as sheet resistance and conductivity.
We thank Dr. C. Mattevi of Imperial College for provid-ing graphene oxide samples. This work was funded by the UK NMS Programme, the EU EMRP Project MetNEMS (NEW-08), and also EU FP7 Project Concept Graphene. The EMRP is jointly funded by the EMRP participating countries within EURAMET and the European Union.
1
K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, Y. Zhang, S. V. Dubonos, I. V. Grigorieva, and A. A. Firsov, Science 306, 666–669 (2004).
2
A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim,Rev. Mod. Phys.81, 109–162 (2009).
3K. S. Novoselov, A. K. Geim, S. V. Morozov, D. Jiang, M. I. Katsnelson,
I. V. Grigorieva, S. V. Dubonos, and A. A. Firsov,Nature (London)438, 197–200 (2005).
4
Y. Zhang, Y. W. Tan, H. L. Stormer, and P. Kim,Nature (London)438, 201–204 (2005).
5G. Eda, G. Fanchini, and M. Chhowalla,Nat. Nanotechnol.
3, 270–274 (2008).
6
Y. Hernandez, V. Nicolosi, M. Lotya, F. M. Blighe, Z. Sun, S. De, I. T. McGovern, B. Holland, M. Byrne, Y. K. Gun’Koet al.,Nat. Nanotechnol.
3, 563–568 (2008).
7
J. T. Robinson, M. Zalalutdinov, J. W. Baldwin, E. S. Snow, Z. Wei, P. Sheehan, and B. H. Houston,Nano Lett.8, 3441–3445 (2008).
8C. Berger, Z. M. Song, X. B. Li, X. S. Wu, N. Brown, C. Naud, D. Mayo,
T. B. Li, J. Hass, A. N. Marchenkov et al.,Science 312, 1191–1196 (2006).
TABLE I. Summary of properties of monolayer CVD graphene, reduced graphene oxide on quartz substrates, and monolayer graphene on SiC.
Sample Graphene thickness (nm) f0(GHz) D f (MHz) D wg- D ws(MHz) Conductivity r (S/m) Sheet resistance Rs(X/ⵧ)
1-layer reduced GO 0.4 10.5504 1.169 0.0243 4.82 104 48222
1-layer CVD 0.4 10.4596 140.7 10.91 1.92 105
13038
1-layer on SiC (sample 1) 0.4 10.5619 7.463 47.30 2.03 107 61.7
1-layer on SiC (sample 2) 0.4 10.5600 5.057 73.38 4.64 107
9A. Reina, X. Jia, J. Ho, D. Nezich, H. Son, V. Bulovic, M. S. Dresselhaus,
and J. Kong,Nano Lett.9, 30–35 (2009).
10
S. Park, J. An, J. W. Suk, and R. S. Ruoff,Small6, 210–212 (2010).
11A. Bagri, C. Mattevi, M. Acik, Y. J. Chabal, M. Chhowalla, and V. B.
Shenoy,Nat. Chem.2, 581–587 (2010).
12
H. Kim, C. Mattevi, M. R. Calvo, J. C. Oberg, L. Artiglia, S. Agnoli, C. F. Hirjibehedin, M. Chhowalla, and E. Saiz,ACS Nano6(4), 3614–3623 (2012).
13L. Hao, J. C. Gallop, A. J. Purnell, and L. F. Cohen,J. Supercond.14,
29–33 (2001).
14J. Krupka, W. Strupinski, and N. Kwietniewski,J. Nanosci. Nanotechnol.
11, 3358–3362 (2011).
15
C. Mattevi, G. Eda, S. Agnoli, S. Miller, K. A. Mkhoyan, O. Celik, D. Mastrogiovanni, G. Granozzi, E. Garfunkel, and M. Chhowalla, Adv. Funct. Mater.19, 2577–2583 (2009).
16
C. Virojanadara, M. Syv€ajarvi, R. Yakimova, L. I. Johansson, A. A. Zakharov, and T. Balasubramanian,Phys. Rev. B78, 245403 (2008).
17L. Hao, C. Mattevi, J. Gallop, S. Goniszewski, Y. Xiao, L. Cohen, and
N. Klein,Nanotechnology23, 28570 (2012).
