Carbon Nanotube and Two-Dimensional Material FETs

1. Theory

• Definition: These are field-effect transistors whose channel is a semiconducting carbon nanotube (CNTFET) or an atomically thin 2D semiconductor such as monolayer $\mathrm{Mo}{\mathrm{S}}_2$, $\mathrm{W}{\mathrm{S}}_2$ or $\mathrm{WS}{\mathrm{e}}_2$.
• Scope: Graphene, which has no bandgap, has its own page.
• Nanometre-thin channel: Both channels are only about 1 nm across, so the gate has very strong electrostatic control and a very short scale length.
• Nanotube bandgap: A semiconducting nanotube’s bandgap varies inversely with its diameter (roughly $0.8/d$ eV with $d$ in nm).
• Quantum capacitance: The channel has few electronic states, so its quantum capacitance adds in series with the oxide and caps the total gate capacitance.
• Contact-dominated: Schottky barriers and Fermi-level pinning at the metal contacts often set most of the device resistance.
• 2D TMD properties: Monolayer $\mathrm{Mo}{\mathrm{S}}_2$ has a wide bandgap ($\approx 1.8$ eV) and large effective mass, which suppress tunnelling leakage at short channel lengths.

2. Working Principle

• CNTFET conduction: Carriers move nearly ballistically along the tube; even a perfect tube has a minimum resistance of $h/4q^2\approx 6.45\mathrm{k\Omega }$ from its four conducting channels.
• Schottky-barrier operation: The gate thins the contact barriers, so many CNT and 2D FETs operate as Schottky-barrier FETs; Pd contacts give barrier-free p-type CNTFETs.
• Quantum-capacitance limit: With thin high-k wrap-around gates, the total gate capacitance saturates at the quantum value.
• 2D short-channel control: The sub-nanometre channel keeps the scale length small, giving good on/off ratios (up to about $10^8$) at short gate lengths.
• 2D limitations: Contact resistance from pinning and the van der Waals gap, and modest mobility (tens of $\mathrm{c}{\mathrm{m}}^2/V\mathrm{s}$) from charged impurities and phonons.
• Uniformity challenge: CNT chirality sorting and alignment, and large-area 2D growth, are the main manufacturing obstacles.

3. Applications

• Current status: Mainly research devices, not yet in mainstream production.
• Beyond-silicon logic channels (research): 2D TMDs are candidates for future stacked nanosheet channels; CNT FETs have been used to build demonstration microprocessors.
• Flexible and printed electronics: Random-network CNT thin-film transistors on plastic substrates.
• Chemical and biological sensors: The all-surface channel makes CNT and 2D FETs highly sensitive to adsorbed molecules.

Carbon Nanotube FET (CNTFET)

Name	Formula	Related Formulae	Variables/Constants
Nanotube diameter and bandgap	$d=\frac{a}{\pi }\sqrt{n^2+nm+m^2}$ $E_g=\frac{2a_{cc}{\gamma }_0}{d}\approx \frac{0.8\mathrm{eV}\mathrm{nm}}{d}$	$\begin{array}{r}\text{metallic}\text{if}(n-m)\text{is}\text{a}\text{multiple}\text{of}\text{3,}\\ \text{otherwise}\text{semiconducting}\end{array}$	$(n,m):chiralindicesofthenanotube$ $a:graphenelatticeconstant,about0.246nm$ $a_{cc}:carbon-carbonbondlength,about0.142nm$ $\begin{array}{r}{\gamma }_0:nearest-neighbourtight-binding\\ hoppingenergy,about2.7to3eV\end{array}$ $d:nanotubediameter$
Ballistic drain current	$I_D=\frac{4q{k}_{B}T}{h}{\sum }_i\left[\ln \left(1+e^{(E_{F,S}-E_i)/k_{B}T}\right)-\ln \left(1+e^{(E_{F,D}-E_i)/k_{B}T}\right)\right]$	$G_0=\frac{4q^2}{h}\approx 155\mu \mathrm{S},R_0=\frac{h}{4q^2}\approx 6.45\mathrm{k\Omega }$	$\begin{array}{r}{E}_i:bottomofthei-thone-dimensional\\ subbandatthetopofthebarrier\end{array}$ $\begin{array}{r}{E}_{F,S},E_{F,D}:Fermilevelsofthe\\ sourceanddraincontacts\end{array}$ $\begin{array}{r}4:degeneracyfromspinandthetwo\\ valleys(Kand{K}^{\prime })\end{array}$ $\begin{array}{r}{G}_0,R_0:conductanceandresistance\\ quantumofasinglenanotubechannel\end{array}$ $k_B:Boltzmannconstant$
Landauer resistance with scattering	$R=\frac{h}{4q^2}\left(1+\frac{L}{{\lambda }_{eff}}\right)$	$\frac{1}{{\lambda }_{eff}}=\frac{1}{{\lambda }_{ac}}+\frac{1}{{\lambda }_{op,abs}}+\frac{1}{{\lambda }_{op,em}},\mathcal{T}=\frac{{\lambda }_{eff}}{{\lambda }_{eff}+L}$	$\begin{array}{r}R:two-terminalresistanceofone\\ nanotubeincludingthecontactlimit\\ h/4q^2\end{array}$ $L:channellength$ $\begin{array}{r}{\lambda }_{eff}:effectivemeanfreepath\\ fromacousticandopticalphonon\\ scattering\end{array}$ $\mathcal{T}:channeltransmissionprobability$
Gate capacitance (geometry)	$C_{ox}^{\prime }=\frac{2\pi {\epsilon }_{ox}}{\ln \left(\frac{r+t_{ox}}{r}\right)}\text{(wrap-around gate)}$ $C_{ox}^{\prime }=\frac{2\pi {\epsilon }_{ox}}{\ln \left(\frac{2t_{ox}}{r}\right)}\text{(nanotube on a planar gate, }{t}_{ox}\gg r\text{)}$		$\begin{array}{r}{C}_{ox}^{\prime }:oxidecapacitanceperunit\\ lengthofnanotube\end{array}$ $r:nanotuberadius$ $t_{ox},{\epsilon }_{ox}:oxidethicknessandpermittivity$ $\begin{array}{r}\text{(the}\text{planar}\text{form}\text{assumes}\text{a}\text{uniform}\\ \text{dielectric}\text{around}\text{the}\text{tube;}\text{for}\text{a}\text{tube}\\ \text{lying}\text{on}\text{the}\text{oxide}\text{with}\text{air}\text{above,}\text{use}\\ \text{an}\text{effective}\text{permittivity}\\ {\epsilon }_{eff}\approx ({\epsilon }_{ox}+{\epsilon }_0)/2\text{)}\end{array}$
Quantum capacitance and total gate capacitance	$C_Q^{\prime }=\frac{8q^2}{h{v}_F}\approx 4\times 10^{-10}F/m$ $\frac{1}{C_G^{\prime }}=\frac{1}{C_{ox}^{\prime }}+\frac{1}{C_Q^{\prime }}$		$\begin{array}{r}{C}_Q^{\prime }:quantumcapacitanceperunitlength\\ ofametallicnanotube,equaltothat\\ oftheconductingsubband\end{array}$ $v_F:Fermivelocity,about8\times 10^{5}m/s$ $C_G^{\prime }:totalgatecapacitanceperunitlength$
Approximate threshold voltage	$V_{TH}\approx \frac{E_g}{2q}\approx \frac{0.4\mathrm{V}\mathrm{nm}}{d}$		$\begin{array}{r}{V}_{TH}:gatevoltageatwhichtheFermi\\ levelinthechannelreachesthefirst\\ subbandforamidgapSchottkycontactor\\ anohmiccontactwithasymmetricgate\end{array}$ $d:nanotubediameterinnanometres$

Two-Dimensional Semiconductor FETs (TMD)

Name	Formula	Related Formulae	Variables/Constants
Quantum capacitance and sheet density	$C_Q=\frac{q^{2}g{m}^{\ast }}{2\pi {\hslash }^2}$ $n_s=\frac{C_{ox}\left(V_{GS}-V_{TH}\right)}{q}$		$\begin{array}{r}{C}_Q:quantumcapacitanceperunitarea\\ ofaparabolictwo-dimensionalband\end{array}$ $g:valleyandspindegeneracyoftheband$ $m^{\ast }:density-of-stateseffectivemass$ $n_s:sheetelectrondensitywhen{C}_Q\gg C_{ox}$
Electrostatic scale length	$\ell =\sqrt{\frac{{\epsilon }_{ch}{t}_{ch}{t}_{ox}}{N_g{\epsilon }_{ox}}}$		$\ell :naturallengthoftheatomicallythinchannel$ $\begin{array}{r}{\epsilon }_{ch},t_{ch}:permittivityand\\ thicknessofthetwo-dimensionalchannel\end{array}$ $\begin{array}{r}{N}_g:numberofgates(1forasingle\\ gate,2foradoublegate)\end{array}$ $\begin{array}{r}{t}_{ox},{\epsilon }_{ox}:gateinsulator\\ thicknessandpermittivity\end{array}$
Subthreshold swing of a thin channel	$S=\frac{k_{B}T}{q}\ln 10\left(1+\frac{C_{dep}+C_{it}}{C_{ox}}\right),C_{it}=q^2{D}_{it}$	$S\approx \frac{k_{B}T}{q}\ln 10\left(1+\frac{C_{it}}{C_{ox}}\right)\text{(}{C}_{dep}\approx 0\text{ for a monolayer)}$	$\begin{array}{r}{C}_{dep}:depletioncapacitance,nearly\\ zeroforanatomicallythinbody\end{array}$ $C_{it}:capacitanceofinterfaceandbordertraps$ $\begin{array}{r}{D}_{it}:interfacetrapdensityperunit\\ areaandenergy\end{array}$ $k_B:Boltzmannconstant$
Source-drain direct tunneling	$T_{SD}\approx \exp \left(-\frac{2L_b\sqrt{2m^{\ast }{\Phi }_b}}{\hslash }\right)$		$\begin{array}{r}{T}_{SD}:probabilitythatcarrierstunnel\\ directlythroughthegate-controlled\\ barrierintheoffstate\end{array}$ $\begin{array}{r}{L}_b:barrierlength(shorterthanthe\\ channellengthatthetopofthe\\ barrier)\end{array}$ ${\Phi }_b:barrierheight$ $\begin{array}{r}{m}^{\ast }:effectivemassalongtransport(a\\ largermasslowerstheleakage)\end{array}$
Metal contact barrier with Fermi-level pinning	${\Phi }_{Bn}=S_{\Phi }\left({\Phi }_M-\chi \right)+\left(1-S_{\Phi }\right)\left(E_g-{\varphi }_0\right)$	$S_{\Phi }=\frac{d{\Phi }_{Bn}}{d{\Phi }_M}$	$\begin{array}{r}{\Phi }_{Bn}:Schottkybarrierheightfor\\ electronsatthemetal-semiconductor\\ interface,involts\end{array}$ $\begin{array}{r}{S}_{\Phi }:pinningfactor,1foran\\ unpinnedcontactandnear0fora\\ stronglypinnedone\end{array}$ $\begin{array}{r}{\Phi }_M,\chi :metalworkfunctionand\\ semiconductorelectronaffinity\end{array}$ $\begin{array}{r}{\varphi }_0:chargeneutralitylevelmeasured\\ fromthevalencebandedge\end{array}$ $\text{(see Metal Semiconductor Contacts page)}$
Contact transfer length	$R_c=\frac{\sqrt{{\rho }_c{R}_{sh}}}{Z}\coth \left(\frac{L_c}{L_T}\right),L_T=\sqrt{\frac{{\rho }_c}{R_{sh}}}$		$R_c:resistanceofonecontact$ ${\rho }_c:specificcontactresistivity$ $\begin{array}{r}{R}_{sh}:sheetresistanceofthe\\ two-dimensionalchannelunderthecontact\end{array}$ $L_c,L_T:contactlengthandtransferlength$ $Z:contactwidth$

Important Questions

Ques: Why does the quantum capacitance matter for a carbon nanotube gate stack?

A nanotube has so few electron states that charge injected by the gate raises the channel Fermi level appreciably, so the channel adds its own capacitance $C_Q^{\prime }$ in series with the oxide, and $1/C_G^{\prime }=1/C_{ox}^{\prime }+1/C_Q^{\prime }$. With $C_Q^{\prime }\approx 4\times 10^{-10}F/m$, the total gate capacitance saturates at the quantum value however thin the oxide is, so $C_G^{\prime }$ cannot exceed $C_Q^{\prime }$. For a thin high-$\kappa$ oxide on a wrap-around gate, $C_{ox}^{\prime }$ becomes comparable to or larger than $C_Q^{\prime }$, and the device is said to operate near the quantum capacitance limit. Reference: H.-S.P. Wong and D. Akinwande, Carbon Nanotube and Graphene Device Physics, Cambridge University Press, 2011.

Ques: What sets the minimum resistance of a nanotube transistor, and how do the contacts matter?

Even a ballistic nanotube with perfect contacts has a resistance of $h/(4q^2)\approx 6.45\mathrm{k\Omega }$ because only four conducting channels (two spin, two valley) feed the tube from the wide contacts. Scattering adds $(h/4q^2)L/{\lambda }_{eff}$. In practice Schottky barriers at the metal contacts add more, and are strongly dependent on the gate voltage; Pd contacts have a low barrier for hole injection and so give a p-type, nearly barrier-free device. Reference: P. Avouris, Z. Chen, and V. Perebeinos, “Carbon-based electronics,” Nature Nanotechnology 2, 605 (2007); H.-S.P. Wong and D. Akinwande, Carbon Nanotube and Graphene Device Physics, Cambridge University Press, 2011.

Ques: Why do monolayer MoS${}_2$ transistors give a good on/off ratio at short channel length?

A monolayer channel is only about 0.65 nm thick, so $t_{ch}$ is very small and the natural length $\ell$ is small, which gives good gate control without a deep depletion region. The wide bandgap (about 1.8 eV) and large effective mass restrict band-to-band and source-drain tunneling, keeping the off-state current low, and the on/off ratio reaches $10^8$ in a top-gated device. The limits are the contact resistance caused by Fermi-level pinning and the van der Waals gap, and the carrier mobility, which is typically a few tens of $\mathrm{c}{\mathrm{m}}^2/V\mathrm{s}$ in practice because of charged impurities and interface traps (the higher values reported for the first devices were later shown to be overestimated). Reference: B. Radisavljevic, A. Radenovic, J. Brivio, V. Giacometti, and A. Kis, “Single-layer MoS${}_2$ transistors,” Nature Nanotechnology 6, 147 (2011); M.S. Fuhrer and J. Hone, “Measurement of mobility in dual-gated MoS${}_2$ transistors,” Nature Nanotechnology 8, 146 (2013); S.M. Sze and Kwok K. Ng, Physics of Semiconductor Devices, 3rd ed., Wiley, 2007.

Related pages in this vault: Graphene Field Effect Transistor, Schottky Barrier and Dynamic Threshold_MOSFET, Metal Semiconductor Contacts, Junctionless Transistor, Steep Slope FETs (Tunnel FET, Impact Ionization MOS, Negative Capacitance FET), Carbon Nanotube Random Access Memory (NRAM), Thin-Film Transistors (a-Si, LTPS, IGZO, Organic), Two-Dimensional Material Memristor (Atomristor and Memtransistor).