diff --git a/2D-Conductivity/conductivity_phase_calculations.m b/2D-Conductivity/conductivity_phase_calculations.m index ee132b1..8aa55b9 100644 --- a/2D-Conductivity/conductivity_phase_calculations.m +++ b/2D-Conductivity/conductivity_phase_calculations.m @@ -33,25 +33,25 @@ for x=1:length(x_vals) cond(x, :) = sheet_conductivity(x_vals(x),... % omega (rads-1) fermi_from_carrier_density(1.3e17, ev_to_j(3)),... % fermi_level (J) 300,... % temp (K) - 5e-12); % scatter_lifetime (s) + 1e-12); % scatter_lifetime (s) end if MULTIPLE_SERIES cond2 = zeros(length(x_vals), 2); for x=1:length(x_vals) cond2(x, :) = sheet_conductivity(x_vals(x),... % omega (rads-1) - fermi_from_carrier_density(1.3e17, ev_to_j(3)),... % fermi_level (J) + fermi_from_carrier_density(2.2e17, ev_to_j(3)),... % fermi_level (J) 300,... % temp (K) 1e-12); % scatter_lifetime (s) end - cond3 = zeros(length(x_vals), 2); - for x=1:length(x_vals) - cond3(x, :) = sheet_conductivity(x_vals(x),... % omega (rads-1) - fermi_from_carrier_density(1.3e17, ev_to_j(3)),... % fermi_level (J) - 300,... % temp (K) - 1e-13); % scatter_lifetime (s) - end +% cond3 = zeros(length(x_vals), 2); +% for x=1:length(x_vals) +% cond3(x, :) = sheet_conductivity(x_vals(x),... % omega (rads-1) +% fermi_from_carrier_density(1.3e17, ev_to_j(3)),... % fermi_level (J) +% 300,... % temp (K) +% 1e-13); % scatter_lifetime (s) +% end end if DISPLAY_HZ % divide radians back to hertz @@ -62,15 +62,9 @@ end %% RENDER %%%%%%%%%%%%%%%%%%%%%%%%%%% -RE_COLOUR = 'r-'; -IM_COLOUR = 'r--'; -MAG_COLOUR = 'r:'; -RE_COLOUR2 = 'g-'; -IM_COLOUR2 = 'g--'; -MAG_COLOUR2 = 'g:'; +RE_COLOUR = 'r'; +RE_COLOUR2 = 'g'; RE_COLOUR3 = 'b'; -IM_COLOUR3 = 'b--'; -MAG_COLOUR3 = 'b:'; LW = 2; figure(1); @@ -82,7 +76,7 @@ if strcmp(EXCITATION_TYPE, 'intra') if MULTIPLE_SERIES plot(x_vals, angle(cond2(:, 1)) .* (180/pi), RE_COLOUR2, 'LineWidth', LW); - plot(x_vals, angle(cond3(:, 1)) .* (180/pi), RE_COLOUR3, 'LineWidth', LW); +% plot(x_vals, angle(cond3(:, 1)) .* (180/pi), RE_COLOUR3, 'LineWidth', LW); end title('2D Intraband Sheet Conductivity Phase'); @@ -93,7 +87,7 @@ elseif strcmp(EXCITATION_TYPE, 'inter') if MULTIPLE_SERIES plot(x_vals, angle(cond2(:, 2)) .* (180/pi), RE_COLOUR2, 'LineWidth', LW); - plot(x_vals, angle(cond3(:, 2)) .* (180/pi), RE_COLOUR3, 'LineWidth', LW); +% plot(x_vals, angle(cond3(:, 2)) .* (180/pi), RE_COLOUR3, 'LineWidth', LW); end title('2D Interband Sheet Conductivity Phase'); @@ -104,7 +98,7 @@ else if MULTIPLE_SERIES plot(x_vals, angle(sum(cond2, 2)) .* (180/pi), RE_COLOUR2, 'LineWidth', LW); - plot(x_vals, angle(sum(cond3, 2)) .* (180/pi), RE_COLOUR3, 'LineWidth', LW); +% plot(x_vals, angle(sum(cond3, 2)) .* (180/pi), RE_COLOUR3, 'LineWidth', LW); end title('2D Sheet Conductivity Phase'); end @@ -112,12 +106,13 @@ end set(gca,'Xscale','log') % set(gca,'Yscale','log') axis tight +ylim([-90 90]) if MULTIPLE_SERIES -% legend('\phi(TTF)', '\phi(CoCp_2)'); -% legend('\phi(1x10^8m^{-2})', '\phi(1x10^{15}m^{-2})', '\phi(1.3x10^{17}m^{-2})'); -% legend('\phi(10K)', '\phi(300K)', '\phi(2230K)'); - legend('\phi(5x10^{-12} s)', '\phi(1x10^{-12} s)', '\phi(1x10^{-13} s)'); + legend('TTF \phi(\sigma)', 'CoCp_2 \phi(\sigma)'); +% legend('1x10^8m^{-2} \phi(\sigma)', '1x10^{15}m^{-2} \phi(\sigma)', '1.3x10^{17}m^{-2} \phi(\sigma)'); +% legend('10K \phi(\sigma)', '300K \phi(\sigma)', '2230K \phi(\sigma)'); +% legend('5x10^{-12} s \phi(\sigma)', '1x10^{-12} s \phi(\sigma)', '1x10^{-13} s \phi(\sigma)'); else legend('\phi'); end diff --git a/Report/antennae-power.jpg b/Report/antennae-power.jpg new file mode 100644 index 0000000..a8608e4 Binary files /dev/null and b/Report/antennae-power.jpg differ diff --git a/Report/references.bib b/Report/references.bib index 83f49d2..c3b1182 100644 --- a/Report/references.bib +++ b/Report/references.bib @@ -157,3 +157,81 @@ year = {2021} } +@inproceedings{gnrfet-low-power, + author = {Chen, Ying-Yu and Sangai, Amit and Gholipour, Morteza and Chen, Deming}, + booktitle = {International Symposium on Low Power Electronics and Design (ISLPED)}, + doi = {10.1109/ISLPED.2013.6629286}, + pages = {151--156}, + title = {Graphene nano-ribbon field-effect transistors as future low-power devices}, + url = {https://ieeexplore.ieee.org/document/6629286}, + urldate = {2021-04-26}, + year = {2013} +} + +@article{flexible-antennae-review, + abstract = {The field of flexible antennas is witnessing an exponential growth due to the demand for wearable devices, Internet of Things (IoT) framework, point of care devices, personalized medicine platform, 5G technology, wireless sensor networks, and communication devices with a smaller form factor to name a few. The choice of non-rigid antennas is application specific and depends on the type of substrate, materials used, processing techniques, antenna performance, and the surrounding environment. There are numerous design innovations, new materials and material properties, intriguing fabrication methods, and niche applications. This review article focuses on the need for flexible antennas, materials, and processes used for fabricating the antennas, various material properties influencing antenna performance, and specific biomedical applications accompanied by the design considerations. After a comprehensive treatment of the above-mentioned topics, the article will focus on inherent challenges and future prospects of flexible antennas. Finally, an insight into the application of flexible antenna on future wireless solutions is discussed.}, + article-number = {847}, + author = {Kirtania, Sharadindu Gopal and Elger, Alan Wesley and Hasan, Md. Rabiul and Wisniewska, Anna and Sekhar, Karthik and Karacolak, Tutku and Sekhar, Praveen Kumar}, + doi = {10.3390/mi11090847}, + issn = {2072-666X}, + journal = {Micromachines}, + number = {9}, + pubmedid = {32933077}, + title = {Flexible Antennas: A Review}, + url = {https://www.mdpi.com/2072-666X/11/9/847}, + urldate = {2021-04-26}, + volume = {11}, + year = {2020} +} + +@article{water-transfer-graphene-antennae, + author = {Wang, Weijia and Ma, Chao and Zhang, Xingtang and Shen, Jiajia and Hanagata, Nobutaka and Huangfu, Jiangtao and Xu, Mingsheng}, + doi = {10.1080/14686996.2019.1653741}, + eprint = { https://doi.org/10.1080/14686996.2019.1653741 }, + journal = {Science and Technology of Advanced Materials}, + note = {PMID: 31489056}, + number = {1}, + pages = {870--875}, + publisher = {Taylor \& Francis}, + title = {High-performance printable 2.4 GHz graphene-based antenna using water-transferring technology}, + url = {https://doi.org/10.1080/14686996.2019.1653741}, + urldate = {2021-04-26}, + volume = {20}, + year = {2019} +} + +@article{graphene-microwave, + author = {Bozzi, Maurizio and Pierantoni, Luca and Bellucci, Stefano}, + doi = {10.13164/re.2015.0661}, + journal = {Radioengineering}, + month = {09}, + pages = {661--669}, + title = {Applications of Graphene at Microwave Frequencies}, + url = {https://www.researchgate.net/publication/283181514_Applications_of_Graphene_at_Microwave_Frequencies}, + urldate = {2021-04-26}, + volume = {24}, + year = {2015} +} + +@article{graphene-modal-prop-drude, + author = {Araneo, R. and Burghignoli, Paolo and Lovat, Giampiero and Hanson, George}, + doi = {10.1109/TEMC.2015.2406072}, + journal = {IEEE Transactions on Electromagnetic Compatibility}, + month = {08}, + pages = {1--8}, + title = {Modal Propagation and Crosstalk Analysis in Coupled Graphene Nanoribbons}, + url = {https://www.researchgate.net/publication/276930151_Modal_Propagation_and_Crosstalk_Analysis_in_Coupled_Graphene_Nanoribbons}, + urldate = {2021-04-26}, + volume = {57}, + year = {2015} +} + +@misc{short-channel, + author = {{{Semiconductor Engineering}}}, + month = jul, + title = {Knowledge Center Short Channel Effects}, + url = {https://semiengineering.com/knowledge_centers/manufacturing/process/issues/short-channel-effects/}, + urldate = {2021-04-26}, + year = {2018} +} + diff --git a/Report/report.lyx b/Report/report.lyx index 9a25a33..2d7041f 100644 --- a/Report/report.lyx +++ b/Report/report.lyx @@ -182,7 +182,19 @@ University of Surrey \end_layout \begin_layout Abstract -abstract +Graphene has many desirable mechanical and electrical characteristics that + make it an exciting material for use in innovative devices. + In particular, its extremely high carrier mobility leads to high operating + frequencies that potentially make the material suitable for the gigahertz + and terahertz spectrum. + The use of graphene for high-frequency electronics applications is considered, + firstly as a channel material for digital logic transistor technology with + a focus on graphene nanoribbon FETs (GNRFET). + Graphene's applicability is then evaluated for flexible antennae with prospecti +ve use in wearable and IoT devices. + Part two models graphene's sheet conductivity, exploring how net carrier + concentration, temperature and scatter lifetime affect both intraband and + interband electrical characteristics. \end_layout \begin_layout Standard @@ -275,7 +287,7 @@ Introduction \begin_layout Standard Graphene is a 2D allotrope of carbon with highly interesting mechanical and electrical properties that have made it a popular target of research - over the last two decades since it's experimental discovery in 2004 + over the last two decades since its experimental discovery in 2004 \begin_inset CommandInset citation LatexCommand cite key "geim-04" @@ -289,7 +301,7 @@ literal "false" \begin_inset Formula $p$ \end_inset - orbitals. +-orbitals. This leaves one remaining \begin_inset Formula $\pi$ \end_inset @@ -307,9 +319,10 @@ noprefix "false" . As a result, graphene has a 0 energy bandgap, the conduction and valence - band meet at a single point known as the Dirac point. + band meet at a single point known as the Dirac point at the K points in + momentum space. Both bands linearly extend away from the Dirac point, a highly unusual - result that allows electrons to act as massless Fermions and move with + result that allows electrons to act as massless Fermions which move with extremely high mobility, as high as 200,000 cm \begin_inset script superscript @@ -337,7 +350,7 @@ s \end_inset - at room temperature + \begin_inset CommandInset citation LatexCommand cite key "graphene-review-2010" @@ -410,7 +423,7 @@ With these highly sought after properties, it is unsurprising that graphene \end_layout \begin_layout Standard -This work explores the suitability of graphene for high frequency applications. +This work explores the suitability of graphene for high-frequency applications. Section \begin_inset CommandInset ref LatexCommand ref @@ -421,7 +434,7 @@ noprefix "false" \end_inset - presents two applications of graphene that take advantage of it's electrical + presents two applications of graphene that take advantage of its electrical and mechanical behaviour at high frequencies. Section \begin_inset CommandInset ref @@ -434,6 +447,8 @@ noprefix "false" \end_inset presents an investigation into the 2D sheet conductivity of the material. + The net carrier concentration, temperature and scatter lifetime are varied + in order to visualise how this affects the materials electrical properties. \end_layout \begin_layout Section @@ -448,25 +463,20 @@ name "sec:Applications" \end_layout \begin_layout Standard -This section explores two uses of graphene for high frequency applications. - First, the applicability of graphene for field effect transistors will +This section explores two uses of graphene for high-frequency applications. + First, the applicability of graphene for field-effect transistors will be considered as a channel material. Throughout, a particular focus will be paid to use in digital logic and thus as a possible replacement for the current Silicon CMOS/MOSFET paradigm. - \end_layout \begin_layout Standard -\begin_inset Flex TODO Note (inline) -status open - -\begin_layout Plain Layout -second application -\end_layout - -\end_inset - - +Following its prospective use in digital logic, graphene's applicability + as a flexible antennae material will be considered. + As the space of wearable and bio-electronics develop, the need for strong, + flexible and efficient antennae at gigahertz frequencies is clear. + Existing flexible antennae technology will be considered before presenting + how graphene could be included. \end_layout \begin_layout Subsection @@ -553,8 +563,8 @@ name "fig:cpu-transistor-number" However, as transistors are made smaller, theoretical limits for many engineerin g challenges are approached. In 2015, the ITRS predicted that by 2021 the current push for smaller transisto -rs would no longer be economically viable, instead requiring innovative - 3D device structures +rs would no longer be economically viable, instead, innovative 3D device + structures would be required \begin_inset CommandInset citation LatexCommand cite key "transistors-21" @@ -566,13 +576,11 @@ literal "false" Some of the most important limiting factors in the current Silicon landscape are short-channel effects, a group of undesirable electrical properties that can occur when the channel length of a MOSFET device is of the same - order of magnitude as the depletion layer -\begin_inset Flex TODO Note (Margin) -status open - -\begin_layout Plain Layout -cite, reword? -\end_layout + order of magnitude as the depletion layer +\begin_inset CommandInset citation +LatexCommand cite +key "short-channel" +literal "false" \end_inset @@ -587,8 +595,8 @@ large-area graphene has no bandgap. As such it is unable to turn off, making it unsuitable as a channel material for a digital MOSFET. - Quantitatively, this can be measured with by the ratio of on current to - off current or the + Quantitatively, this can be measured with the ratio of on current to off + current or the \emph on on/off ratio \emph default @@ -615,8 +623,8 @@ literal "false" \end_layout \begin_layout Standard -Therefore, it is clear that in order to use graphene as a channel material - a bandgap must be formed. +Therefore, it is clear that to use graphene as a channel material a bandgap + must be formed. There are a number of ways to do so, a selection of band diagrams for such structures can be seen in figure \begin_inset CommandInset ref @@ -630,7 +638,7 @@ noprefix "false" . For digital applications, one of the most promising methods for creating - a band gap is by confining graphene in one dimension to create a + a bandgap is by confining graphene in one dimension to create a \emph on graphene nanoribbon \emph default @@ -657,7 +665,6 @@ noprefix "false" \end_inset . - \end_layout \begin_layout Standard @@ -764,11 +771,327 @@ name "fig:gnrget-structure" \end_layout \begin_layout Standard -\begin_inset Flex TODO Note (inline) +Table +\begin_inset CommandInset ref +LatexCommand ref +reference "tab:gnrfet-low-power-results" +plural "false" +caps "false" +noprefix "false" + +\end_inset + + compares transistor performance characteristics for both silicon and GNR-based + FET technology +\begin_inset CommandInset citation +LatexCommand cite +key "gnrfet-low-power" +literal "false" + +\end_inset + +. + High-quality GNRFETs can be seen to demonstrate the lowest sub-threshold + swing and a reasonably high value for +\begin_inset Formula $\nicefrac{I_{on}}{I_{off}}$ +\end_inset + +. + Further analyses from +\begin_inset CommandInset citation +LatexCommand cite +key "gnrfet-low-power" +literal "false" + +\end_inset + + suggest that GNRFET demonstrates better performance than silicon for low-power + applications with lower power consumption. +\end_layout + +\begin_layout Standard +\begin_inset Float table +wide false +sideways false status open \begin_layout Plain Layout -Advantages over silicon +\noindent +\align center +\begin_inset Tabular + + + + + + + + +\begin_inset Text + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +S (mV/dec) +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +I +\begin_inset script subscript + +\begin_layout Plain Layout +on +\end_layout + +\end_inset + +/I +\begin_inset script subscript + +\begin_layout Plain Layout +off +\end_layout + +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +V +\begin_inset script subscript + +\begin_layout Plain Layout +DD +\end_layout + +\end_inset + +(V) +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Plain Layout +Silicon CMOS (High Performance) +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +93.46 +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $3.49\times10^{3}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +0.7 +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Plain Layout +Silicon CMOS (Low Power) +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +86.96 +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $5.12\times10^{6}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +0.9 +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Plain Layout +GNRFET (Pristine) +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +66.67 +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $1.81\times10^{5}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +0.5 +\end_layout + +\end_inset + + + + +\begin_inset Text + +\begin_layout Plain Layout +GNRFET (Rough Edge) +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +140.85 +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +\begin_inset Formula $9.85\times10^{3}$ +\end_inset + + +\end_layout + +\end_inset + + +\begin_inset Text + +\begin_layout Plain Layout +0.5 +\end_layout + +\end_inset + + + + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Caption Standard + +\begin_layout Plain Layout +FET operating characteristics for two silicon-based (16 nm) and two GNR-based + transistor technologies. + Subthreshold swing, on/off ratio and threshold voltage reported +\begin_inset CommandInset citation +LatexCommand cite +key "gnrfet-low-power" +literal "false" + +\end_inset + + +\begin_inset CommandInset label +LatexCommand label +name "tab:gnrfet-low-power-results" + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Plain Layout + \end_layout \end_inset @@ -779,10 +1102,8 @@ Advantages over silicon \begin_layout Standard Some of the limitations of GNR-based FETs include the difficulty in fabricating high-quality graphene and specifically graphene nanoribbons. - On top of the typical issues in graphene fabrication including the requirements - for tight environmental control. - Edge roughness can significantly affect the channels properties, introducing - new scattering centres + Edge roughness can significantly affect the properties of the channel, + introducing new scattering centres \begin_inset CommandInset citation LatexCommand cite key "gnrfet-structure-image" @@ -790,8 +1111,28 @@ literal "false" \end_inset -. - However, the major constraint of GNRFETs is the limited mobility. +, this can be seen in table +\begin_inset CommandInset ref +LatexCommand ref +reference "tab:gnrfet-low-power-results" +plural "false" +caps "false" +noprefix "false" + +\end_inset + + with rough-edge graphene demonstrating a higher subthreshold swing than + both pristine graphene and the existing silicon technology. + A rough edge also increases delay and leakage power +\begin_inset CommandInset citation +LatexCommand cite +key "gnrfet-low-power" +literal "false" + +\end_inset + +, further emphasising the requirement for high-quality fabrication methods. + However, one of the major constraints of GNRFET technology is limited mobility. Although graphene has incredibly high carrier mobility, by opening the bandgap as seen in figure \begin_inset CommandInset ref @@ -809,17 +1150,162 @@ noprefix "false" \begin_layout Standard Despite these drawbacks, GNRFETs look to be one of the most promising avenues - for the post-Silicon, high-performance CMOS/MOSFET electronics. + for post-silicon, high-performance and/or low-power CMOS/MOSFET electronics. \end_layout \begin_layout Subsection -Terahertz Radiation +Flexible Antennae +\end_layout + +\begin_layout Standard +The miniaturisation and increased efficiency of electronics have pushed + for smaller-scale devices including IoT and wearable devices. + One critical component for these devices is efficient wireless technologies + and antennae in particular. +\end_layout + +\begin_layout Standard +Graphene has a number of properties that make it a suitable prospective + material for such purposes. + Being flexible, bio-compatible and highly stiff +\begin_inset CommandInset citation +LatexCommand cite +key "warda-gfet-review" +literal "false" + +\end_inset + +, graphene should be able to withstand much of the repeated physical deformation +s that could be expected in the wearable electronics domain. + The previously mentioned electrical properties are obviously also applicable + to the domain - high mobility, conductivity and operating frequencies suggest + that the material could be applicable to gigahertz applications and thus + Wi-Fi and Bluetooth usage. +\end_layout + +\begin_layout Standard +Existing technologies for flexible antennae include printed metal-ink components + including copper and silver +\begin_inset CommandInset citation +LatexCommand cite +key "flexible-antennae-review" +literal "false" + +\end_inset + +. + An advantage of metal-based inks are the inherently high conductivities + and some helpful mechanical properties. + However there are disadvantages, silver in particular is a valuable metal + that is too expensive to be used extensively in antennae +\begin_inset CommandInset citation +LatexCommand cite +key "flexible-antennae-review" +literal "false" + +\end_inset + +. + Copper also has disadvantages in its tendency to oxidise when exposed to + the environment as could be expected in wearable or IoT applications. + +\end_layout + +\begin_layout Standard +Printed graphene antennae using water-transfer technology have been demonstrated + that function at the 2.4 GHz WiFi/Bluetooth spectrum +\begin_inset CommandInset citation +LatexCommand cite +key "water-transfer-graphene-antennae" +literal "false" + +\end_inset + +, the power spectrum compared to a copper antenna can be seen in figure + +\begin_inset CommandInset ref +LatexCommand ref +reference "fig:antenna-power" +plural "false" +caps "false" +noprefix "false" + +\end_inset + +. + This room-temperature method demonstrates that as the performance is comparable + with metal-based inks, graphene's mechanical properties make it a highly + applicable material for wearable antennae. +\end_layout + +\begin_layout Standard +\begin_inset Float figure +wide false +sideways false +status open + +\begin_layout Plain Layout +\noindent +\align center +\begin_inset Graphics + filename antennae-power.jpg + width 60col% + +\end_inset + + +\end_layout + +\begin_layout Plain Layout +\begin_inset Caption Standard + +\begin_layout Plain Layout +Normalised radiation power for graphene and copper-based dipole antenna, + a) E-plane, b) H-plane +\begin_inset CommandInset citation +LatexCommand cite +key "water-transfer-graphene-antennae" +literal "false" + +\end_inset + + +\begin_inset CommandInset label +LatexCommand label +name "fig:antenna-power" + +\end_inset + + +\end_layout + +\end_inset + + +\end_layout + +\begin_layout Plain Layout + +\end_layout + +\end_inset + + \end_layout \begin_layout Subsection Summary \end_layout +\begin_layout Standard +From these applications, it is clear that graphene has the potential to + revolutionise cutting-edge electronics domains including high-performance, + high-efficiency transistor technologies and the wearable/IoT devices. + Although the material does not yet have significant penetration in consumer + electronics, it is likely that this is not far away. + +\end_layout + \begin_layout Section Sheet Conductivity Modelling \begin_inset CommandInset label @@ -832,8 +1318,8 @@ name "sec:Sheet-Conductivity-Modelling" \end_layout \begin_layout Standard -This section presents a model for graphene's high frequency conductivity - using the equation below below +This section presents a model for graphene's high-frequency conductivity + using the equation below \begin_inset CommandInset citation LatexCommand cite key "yao" @@ -866,16 +1352,6 @@ literal "false" \end_inset - -\begin_inset Flex TODO Note (Margin) -status open - -\begin_layout Plain Layout -cite -\end_layout - -\end_inset - . These two contributions are separated for reference below, \end_layout @@ -944,9 +1420,8 @@ noprefix "false" . These were varied within reasonable ranges in order to investigate how such variations affect the conductivity, both as a whole and individually. - Prior to these wider investigations, however, experimental data was recreated - in order both to validate the model and to allow better identification - of later differences from the planned variations. + Prior to these wider investigations, however, experimental data was simulated + in order both to validate the model. \end_layout \begin_layout Subsection @@ -1012,7 +1487,7 @@ noprefix "false" \end_inset respectively with both having a cutoff frequency of around 100 GHz. - Beyond the cutoff frequency the value is around 1 mS by 10 THz. + Beyond the cutoff frequency, the value is around 1 mS by 10 THz. The imaginary component peaks over the same frequency band that the real component declines and the two intersect at around 150 GHz at a conductivity of 31 mS with CoCp @@ -1044,7 +1519,7 @@ status open \end_layout \begin_layout Standard -Looking to the phase information (figure +Looking at the phase information (figure \begin_inset CommandInset ref LatexCommand ref reference "fig:david-phase" @@ -1230,7 +1705,7 @@ With Fermi velocity energy scale, \begin_inset Caption Standard \begin_layout Plain Layout -Carrier concentration values for dopants from +Carrier concentration values for n-type dopants from \begin_inset CommandInset citation LatexCommand citet key "david-paper" @@ -1418,7 +1893,7 @@ noprefix "false" Comparing the two, it can be seen that the interactions happen over largely separate frequency ranges. In general, the intraband conductivity can be seen to exist up to the 10 - THz portion of the spectrum while the interband has the majority of it's + THz portion of the spectrum while the interband has the majority of its contributions above the 10 THz range. The intraband can be seen to dominate the total conductivity, it is largely invisible in the previous combined view of figure @@ -1432,7 +1907,7 @@ noprefix "false" \end_inset . - The interband interactions begin after the 10 THz range, as the real component + The interband interactions begin after the 10 THz range - as the real component steps from 1 \begin_inset Formula $\mu S$ \end_inset @@ -1445,19 +1920,6 @@ noprefix "false" same spectral band. \end_layout -\begin_layout Standard -\begin_inset Flex TODO Note (inline) -status open - -\begin_layout Plain Layout -Static value continues past THz -\end_layout - -\end_inset - - -\end_layout - \begin_layout Standard \begin_inset Float figure wide false @@ -1650,7 +2112,7 @@ status open \begin_inset Caption Standard \begin_layout Plain Layout -Fermi level associated with different carrier concentrations +Fermi level associated with different net carrier concentrations \begin_inset CommandInset label LatexCommand label name "fig:fermi-concentration-func" @@ -2429,7 +2891,7 @@ m the higher gigahertz value smearing the lighter blue across a higher frequency band. This moves the cutoff from 100 GHz to around 1 THz. - For the imaginary component, at low carrier concentrations the peak value + For the imaginary component, at low carrier concentrations, the peak value decreases to around 1 \family default \series default @@ -2570,7 +3032,7 @@ m . Finally, as the carrier concentration further increases and the 100 GHz - intraband value increases in magnitude, the frequency for this high frequency + intraband value increases in magnitude, the frequency for this high-frequency imaginary conductivity drop begins to increase again. \end_layout @@ -2704,7 +3166,7 @@ noprefix "false" \end_inset - presents the conductivity for three graphene species of differing carrier + presents the conductivity for three graphene species of different carrier concentrations decomposed into the intraband and interband components. The blue series, \family roman @@ -2798,7 +3260,7 @@ noprefix "false" \end_layout \begin_layout Standard -Looking to the intraband interactions, both components can be seen to have +Looking at the intraband interactions, both components can be seen to have the same profiles as seen previously, the differences lie in magnitude. Higher net carrier concentrations can be seen to increase the magnitude exponentially, this behaviour was previously identified in figure @@ -2914,7 +3376,7 @@ noprefix "false" \end_inset . - From these it is clear that the dopant concentration has a significant + From these, it is clear that the dopant concentration has a significant effect on the conductivity's phase in the terahertz spectrum. As the carrier concentration is increased, the phase can be seen to increase to 90 degrees for longer throughout the 10 GHz decade and begin dropping @@ -3065,7 +3527,7 @@ noprefix "false" For intraband conductivity, temperature can be seen to have little effect throughout the inspected thermal range. The low and room temperature series' effectively overlap, while moving - to the upper temperature limit increases the conductivity by only 5 mS + to the upper-temperature limit increases the conductivity by only 5 mS or 10%. \end_layout @@ -3176,8 +3638,8 @@ name "fig:temp-inter" \begin_inset Caption Standard \begin_layout Plain Layout -Intraband (a) and interband (b) conductivity for low, room and high temperature - graphene using TTF doping and a a scatter lifetime of 1 ps +Intraband (a) and interband (b) conductivity for low, room and high-temperature + graphene using TTF doping and a scatter lifetime of 1 ps \begin_inset CommandInset label LatexCommand label name "fig:inter-intra-temperature" @@ -3300,7 +3762,7 @@ noprefix "false" \end_layout \begin_layout Standard -Looking to the real component, the scatter lifetime can be seen to affect +Looking at the real component, the scatter lifetime can be seen to affect both the cutoff frequency and the magnitude of the pre-cutoff value. As the lifetime increases, the cutoff frequency occurs at a lower value, from around 500 GHz to 50 GHz. @@ -3309,8 +3771,7 @@ Looking to the real component, the scatter lifetime can be seen to affect \end_layout \begin_layout Standard -Considering the imaginary component, a somewhat similar behaviour can be - seen. +Considering the imaginary component, somewhat similar behaviour can be seen. The same exponential growth in magnitude can be seen in the 100 GHz peak. With regards to the spectral behaviour, increasing scatter lifetime reduces the frequency of the leading edge of the peak, broadening the bandwidth. @@ -3453,7 +3914,7 @@ noprefix "false" seen in the intraband conductivity. The bandwidth of the imaginary component for the higher lifetime species is broadened while increasing the magnitude. - Looking to the interband contributions, the three series show no variation, + Looking at the interband contributions, the three series show no variation, the scatter lifetime has no effect. This can be seen in the surfaces of figure \begin_inset CommandInset ref @@ -3640,11 +4101,11 @@ noprefix "false" presents the complex phase of the conductivity for the three previous lifetimes. Following the observation that the interband conductivity is unaffected - by scatter lifetime, it follows that the phase is also unchanged across - the selected lifetimes. + by scatter lifetime, it follows that the high-frequency phase is also unchanged + across the selected lifetimes. the phase is affected in the lower gigahertz ranges, however, with a longer lifetime being associated with an earlier rising edge in the spectrum and - a longer bandwidth of 90 degree phase. + a longer bandwidth of 90-degree phase. \end_layout \begin_layout Subsection @@ -3652,16 +4113,36 @@ Discussion \end_layout \begin_layout Standard -\begin_inset Flex TODO Note (inline) -status open - -\begin_layout Plain Layout -Intraband and drude stuff, graphs look the same -\end_layout +The intraband transitions are derived from the Drude model of carrier transport + +\begin_inset CommandInset citation +LatexCommand cite +key "graphene-microwave,graphene-modal-prop-drude" +literal "false" \end_inset +. + This results in the characteristic real cutoff/imaginary peak behaviour + seen in figure +\begin_inset CommandInset ref +LatexCommand ref +reference "fig:david-intraband" +plural "false" +caps "false" +noprefix "false" +\end_inset + +. + From the simulated data from the literature, the gigahertz cutoff for intraband + conductivity can be seen to have an associated phase increase. + This suggests an increase in inductive reactance throughout this spectral + range which would have implications for applications employing graphene + at these frequencies and could suggest an increase in delay. + N-type doped material can be seen to show a negative phase peak at terahertz + frequencies indicating capacitive behaviour over this spectrum. + \end_layout \begin_layout Standard @@ -3730,8 +4211,9 @@ When considering n-type doping, the Fermi level is increased from the Dirac As interband conductivity involve transitions from the valence band or lower Dirac cone to the conduction band or upper cone (when considering n-type doping), in order for an electron to make a direct transition without - momentum change it must absorb at least two times the Fermi level energy - in order for the destination to be an empty state. + momentum change (straight up on a energy-momentum diagram) it must absorb + at least two times the Fermi level energy in order for the destination + to be an empty state. This restriction, more formally that incident photons of angular frequency, \begin_inset Formula $\omega$ @@ -3760,17 +4242,6 @@ literal "false" question. \end_layout -\begin_layout Standard -From the simulated data from literature, the gigahertz cutoff for intraband - conductivity can be seen to have an associated phase increase. - This suggests an increase in dynamic or reactive electrical behaviour throughou -t this spectral range which would have implications for applications employing - graphene at these frequencies. - This phase is reversed at higher frequencies before beginning to relax - as the imaginary component approaches zero which will have similar implications - for THz applications. -\end_layout - \begin_layout Paragraph Net Carrier Concentration \end_layout @@ -3831,19 +4302,6 @@ noprefix "false" carriers. -\end_layout - -\begin_layout Standard -\begin_inset Flex TODO Note (inline) -status open - -\begin_layout Plain Layout -Why? -\end_layout - -\end_inset - - \end_layout \begin_layout Standard @@ -3859,7 +4317,7 @@ n. The phase can also be kept positive at terahertz frequencies with a lower carrier concentration, although this severely limits the conductivity magnitude. This flexibility in reactive electrical characteristics could prove applicable - to high frequency applications. + to high-frequency applications. \end_layout \begin_layout Paragraph @@ -3873,26 +4331,13 @@ The conductivity spectrum as a function of temperature shows promising results The intraband conductivity, specifically, showed little variation in behaviour between 10 K and the highest stable temperatures with the previously reported 10% increase. - Looking to the interband interactions, these results showed the opposite + Looking at the interband interactions, these results showed the opposite trend with the magnitude being decreased as the temperature increased. - This would suggest that graphene could prove useful in high temperature + This would suggest that graphene could prove useful in high-temperature devices with special consideration being needed for terahertz applications where the phase can be more variable around the critical frequency. \end_layout -\begin_layout Standard -\begin_inset Flex TODO Note (inline) -status open - -\begin_layout Plain Layout -what does this mean -\end_layout - -\end_inset - - -\end_layout - \begin_layout Standard \begin_inset Float figure wide false @@ -4003,7 +4448,7 @@ noprefix "false" This higher conductivity is associated with the slightly lower energy required to make an interband transition as a result of the electron's higher energies. Additionally, as the critical temperature is approached, the higher temperature - series begins smoothly rising earlier than lower temperature series'. + series begins smoothly rising earlier than the lower temperature series'. These behaviours are a result of the statistical distribution of both the temperature of the electrons and the energy of the incident photons. \end_layout @@ -4018,39 +4463,70 @@ From the presented trends for how conductivity is affected by a varied carrier the magnitude and spectral behaviour of intraband conductivity. A longer scatter lifetime was shown to increase the magnitude of gigahertz conductivity, this can be justified by considering the meaning of the scatter - lifetime. + lifetime using the Drude model of charge transport. + The scatter lifetime is the average lifetime between collisions, more collision +s reduce the overall mobility of the material. + A longer lifetime suggests that electrons are able to travel for longer + distances at higher speed without being slowed by scatter events, this + explains the higher magnitude throughout the gigahertz spectrum. \end_layout \begin_layout Standard -\begin_inset Flex TODO Note (inline) -status open - -\begin_layout Plain Layout -Equation analysis -\end_layout +The interband interactions were seen to be unaffected by the scatter lifetime. + This can be seen from the model where, unlike equation +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:intra-conductivity" +plural "false" +caps "false" +noprefix "false" \end_inset - -\end_layout - -\begin_layout Standard -\begin_inset Flex TODO Note (inline) -status open - -\begin_layout Plain Layout -Why? -\end_layout +, equation +\begin_inset CommandInset ref +LatexCommand ref +reference "eq:inter-conductivity" +plural "false" +caps "false" +noprefix "false" \end_inset - + has no reference to a lifetime term. + As a higher energy interaction, electrons are less affected by the scattering + of atoms and thus scatter lifetime is not employed for these high-frequency + transitions. \end_layout \begin_layout Section Conclusion \end_layout +\begin_layout Standard +Two applications of graphene at high frequencies have been presented. + The use of graphene nanoribbons as a channel material for FET technology + was shown to be a suitable prospective material for challenging the current + silicon CMOS/MOSFET paradigm as short-channel effects become harder to + engineer around. + Following this, graphene's applicability to flexible antennae was evaluated. + With a less established consumer domain, graphene's bio-compatibility, + flexibility, stiffness and electrical properties could overtake the more + limited metal-based ink approaches. +\end_layout + +\begin_layout Standard +The sheet conductivity of graphene was modelled and the effect of varying + carrier concentration, temperature and scatter lifetime was evaluated. + Carrier concentration was seen to have a significant effect on conductivity + as a result of its relation to the materials Fermi level. + Temperature, however, was shown to have little effect, a promising result + for the stability of graphene over a wide thermal operating range. + Scatter lifetime was also shown to have a significant effect on intraband + conductivity as a result of its importance in the Drude modelling of carrier + transport. +\end_layout + \begin_layout Standard \begin_inset Newpage newpage \end_inset