Graphene Field Effect Transistor

1. Theory

Definition: A graphene field-effect transistor (GFET) is a transistor whose channel is a single sheet of carbon atoms.
Zero bandgap: Graphene is a semimetal whose conduction and valence bands touch at the Dirac point, with a linear (Dirac) energy-momentum relation.
Ambipolar conduction: The gate shifts the Fermi level into either band, so the channel carries electrons or holes with a conductance minimum at the Dirac point.
Cannot turn off: A minimum conductivity of about $4 q^{2} / π h$ remains at the Dirac point, limiting the on/off ratio to about 10 for large-area sheets.
Quantum capacitance: The density of states vanishes at the Dirac point, so quantum capacitance in series with the oxide reduces gate control.
High mobility: Intrinsic phonon-limited mobility is around $1 0^{5} c m^{2} /V s$ at room temperature, but charged impurities limit graphene on $Si O_{2}$ to about $1 0^{4} c m^{2} /V s$ .
Bandgap engineering: Nanoribbons, bilayer graphene under a field, or substrate effects can open a small gap at the cost of mobility.

2. Working Principle

Gate tuning: The gate voltage sets the carrier density $n_{s}$ and type, with the Fermi level moving as $n_{s}$ .
V-shaped transfer curve: Drain current falls to a minimum at the Dirac point and rises on both sides.
Weak saturation: Current saturates only weakly with $V_{D S}$ , and the channel can change polarity near the drain, giving large output conductance.
High $f_{T}$ , low $f_{ma x}$ : High carrier velocity gives high current-gain cutoff frequencies, but the poor output conductance keeps $f_{ma x}$ and voltage gain low.
Sensing mode: Molecules or charges on the exposed surface shift the Dirac point, which is read as a change in current.

3. Applications

Current status: Mostly research; graphene cannot replace silicon in digital logic because it does not switch off.
Biosensors and chemical sensors: GFET sensors for ions, gases, DNA and proteins, some reaching early commercial use.
RF and THz components (research): High- $f_{T}$ amplifiers, frequency multipliers, mixers and THz detectors that exploit ambipolarity and high velocity.

Name	Formula	Related Formulae	Variables/Constants
Dirac dispersion and density of states	$E (k) = \pm ℏ v_{F} ∣ k ∣$ $D (E) = \frac{2 ∣ E ∣}{π ℏ ^{2} v _{F}^{2}}$	$v_{F} \approx 1 \times 1 0^{6} m/s$	$E (k) : e n er g y o f t h e l in e a r co n d u c t i o n (+) an d v a l e n ce (-) ban d s m e a s u r e d f r o m t h e D i r a c p o in t$ $v_{F} : F er mi v e l oc i t y$ $k : w a v e v ec t or m e a s u r e d f r o m t h e K p o in t$ $D (E) : d e n s i t y o f s t a t es p er u ni t a r e a an d e n er g y (s p in an d tw o v a l l ey s in c l u d e d)$
Carrier density and Fermi level	$n_{s} = \frac{k _{F}^{2}}{π} = \frac{E _{F}^{2}}{π ℏ ^{2} v _{F}^{2}}$ $E_{F} = ℏ v_{F} π n_{s}$		$n_{s} : s h ee t c a r r i er d e n s i t y o f e l ec t r o n s (or h o l es)$ $k_{F} : F er mi w a v e v ec t or$ $E_{F} : F er mi l e v e l m e a s u r e d f r o m t h e D i r a c p o in t$
Quantum capacitance	$C_{Q} = q^{2} D (E_{F}) = \frac{2 q ^{2} ∣ E _{F} ∣}{π ℏ ^{2} v _{F}^{2}}$	$C_{Q} = \frac{2 q ^{2}}{ℏ v _{F}} \frac{n _{s}}{π}$	$C_{Q} : q u an t u m c a p a c i t an ce p er u ni t a r e a a t z er o t e m p er a t u r e, w hi c h v ani s h es a t t h e D i r a c p o in t an d g r o w s l in e a r l y w i t h ∣ E_{F} ∣$ $D (E_{F}) : d e n s i t y o f s t a t es a t t h e F er mi l e v e l$
Gate voltage and carrier density	$V_{GS} - V_{D i r a c} = \frac{q n _{s}}{C _{o x}} + \frac{ℏ v _{F} π n _{s}}{q}$	$n_{s} = \frac{C _{o x} ( V _{GS} - V _{D i r a c} )}{q} (C_{Q} ≫ C_{o x})$	$V_{D i r a c} : g a t e v o l t a g e a t w hi c h t h e D i r a c p o in t l i es a t t h e F er mi l e v e l (c ha r g e n e u t r a l i t y p o in t)$ $C_{o x} : g a t e o x i d e c a p a c i t an ce p er u ni t a r e a$ $(second term: the Fermi level shift in volts, which matters for thin gate oxides)$
Resistance of a graphene FET	$R_{t o t} = R_{c} + \frac{L}{Z} \frac{1}{q μ n _{0}^{2} + n _{s}^{2}}$	$σ_{min} \approx \frac{4 q ^{2}}{π h} (ballistic theory; measured values are close to 4 q^{2} / h)$	$R_{c} : t o t a l co n t a c t r es i s t an ce$ $μ : c a r r i er m o bi l i t y$ $n_{0} : r es i d u a l c a r r i er d e n s i t y f r o m e l ec t r o n - h o l e p u dd l es a t t h e D i r a c p o in t$ $σ_{min} : minim u m co n d u c t i v i t y a t t h e D i r a c p o in t p r e d i c t e d f or ba l l i s t i c t r an s p or t in a s h or t w i d e s am pl e$ $Z, L : c hann e l w i d t h an d l e n g t h$
Drain current with velocity saturation	$I_{D} = Z q n_{s} v, v = \frac{μ E}{1 + μ E / v _{s a t}}$	$v_{s a t} \approx 1 to 5 \times 1 0^{5} m/s (decreasing with carrier density; depends on the substrate)$	$I_{D} : d r ain c u r r e n t a t a p o in t in t h e c hann e l, co n s t an t a l o n g t h e c hann e l$ $v : c a r r i er d r i f t v e l oc i t y$ $E : l oc a l l a t er a l f i e l d$ $v_{s a t} : s a t u r a t i o n v e l oc i t y l imi t e d b y o pt i c a l p h o n o n e mi ss i o n$
Ambipolar transfer characteristic	$I_{D} \propto ∣ V_{GS} - V_{D i r a c} ∣ (for ∣ V_{D S} ∣ small)$	$V_{D S} > V_{GS} - V_{D i r a c} : the channel is p-type near the drain and n-type near the source$	$(conduction is by electrons for V_{GS} > V_{D i r a c} and by holes for V_{GS} < V_{D i r a c}, with a minimum, not zero, current at the Dirac point)$
Cut-off and maximum oscillation frequency	$f_{T} = \frac{g _{m}}{2 π ( C _{g s} + C _{g d} )}$	$f_{ma x} = \frac{f _{T}}{2 g _{d s} ( R _{g} + R _{s} + R _{i} ) + 2 π f _{T} C _{g d} R _{g}}$	$f_{T} : c u r r e n t - g ain c u t - o f f f r e q u e n cy f or n e g l i g ib l e p a r a s i t i c c a p a c i t an ce$ $f_{ma x} : ma x im u m f r e q u e n cy o f osc i l l a t i o n (p o w er g ain o f u ni t y)$ $g_{m}, g_{d s} : t r an sco n d u c t an ce an d o u tp u t co n d u c t an ce$ $C_{g s}, C_{g d} : g a t e - so u r ce an d g a t e - d r ain c a p a c i t an ce$ $R_{g}, R_{s}, R_{i} : g a t e, so u r ce an d in t r in s i c c hann e l r es i s t an ce$
Charged-impurity limited mobility	$μ_{im p} \approx \frac{C}{n _{im p}}$	$C \approx 5 \times 1 0^{15} V^{- 1} s^{- 1}$	$μ_{im p} : m o bi l i t y l imi t e d b y sc a tt er in g f r o m c ha r g e d im p u r i t i es n e a r t h e g r a p h e n e$ $n_{im p} : d e n s i t y o f c ha r g e d im p u r i t i es p er u ni t a r e a (ab o u t 1 0^{12} c m^{- 2} f or g r a p h e n e o n S i O_{2})$ $C : e m p i r i c a l co n s t an t (t h e v a l u e i s a pp r o x ima t e an d d e p e n d s o n t h e d i e l ec t r i c e n v i r o nm e n t)$
Acoustic phonon limited resistivity	$ρ_{A} = \frac{π ^{2} D _{A}^{2} k _{B} T}{2 h e ^{2} ρ _{s} v _{s}^{2} v _{F}^{2}}$		$ρ_{A} : in t r in s i c r es i s t i v i t y f r o m a co u s t i c p h o n o n sc a tt er in g, l in e a r in T an d in d e p e n d e n t o f t h e c a r r i er d e n s i t y$ $D_{A} : a co u s t i c d e f or ma t i o n p o t e n t ia l$ $ρ_{s} : a r e a l ma ss d e n s i t y o f g r a p h e n e$ $v_{s} : so u n d v e l oc i t y o f t h e a co u s t i c m o d e$ $e : e l e m e n t a r y c ha r g e$ $k_{B} : B o l t z mann co n s t an t$
Bandgap opening	$E_{g} = \frac{α}{W - W ^{*}} (nanoribbon)$		$E_{g} : t r an s p or t g a p o f a l i t h o g r a p hi c nan or ibb o n$ $W : r ibb o n w i d t h$ $α, W^{*} : e m p i r i c a l co n s t an t s t ha t d e p e n d o n t h e e d g e r o ug hn ess an d t h e s u b s t r a t e$ $(a perpendicular electric field opens a gap in bilayer graphene of up to about 0.25 eV (optical gap); transport gaps are typically tens of meV)$

Important Questions

Ques: Why does a graphene FET not turn off, and why is its output conductance poor?

Graphene has no bandgap, so the Fermi level at the Dirac point sits where the conduction and valence bands touch and the channel always has carriers, with a minimum conductivity of about $4 q^{2} / (π h)$ . The drain current therefore has a minimum rather than zero current, and the on/off ratio stays at most about 10 for large-area sheets. Because the current saturates only weakly and because the channel changes polarity near the drain at high $V_{D S}$ (ambipolar behaviour), the output conductance $g_{d s}$ is large, so $f_{ma x}$ stays well below $f_{T}$ and the voltage gain is small. Reference: F. Schwierz, “Graphene transistors,” Nature Nanotechnology 5, 487 (2010); K.S. Novoselov et al., “Electric Field Effect in Atomically Thin Carbon Films,” Science 306, 666 (2004).

Ques: What does the quantum capacitance do to the gate control of graphene?

The density of states of graphene is zero at the Dirac point, so $C_{Q}$ is small near charge neutrality and $ℏ v_{F} π n_{s} / q$ adds to the gate voltage needed for a given $n_{s}$ . The gate capacitance is $C_{G} = C_{o x} C_{Q} / (C_{o x} + C_{Q})$ , which is below $C_{o x}$ and is limited by $C_{Q}$ for a thin high- $κ$ oxide. This lowers the transconductance for a given oxide thickness compared with a conventional channel with a large density of states. Reference: T. Fang, A. Konar, H. Xing, and D. Jena, “Carrier statistics and quantum capacitance of graphene sheets and ribbons,” Appl. Phys. Lett. 91, 092109 (2007); H.-S.P. Wong and D. Akinwande, Carbon Nanotube and Graphene Device Physics, Cambridge University Press, 2011.

Ques: What limits the mobility of graphene on a substrate?

Charged impurities in or on the substrate scatter carriers and give a mobility that varies as $1/ n_{im p}$ , and surface polar phonons of the substrate and ripples add further scattering, so graphene on SiO $_{2}$ is limited to about $1 0^{4} c m^{2} /V s$ . The intrinsic limit, from acoustic phonons, is a resistivity $ρ_{A}$ that increases linearly with temperature and gives a mobility of the order of $1 0^{5} c m^{2} /V s$ at room temperature, which is reached only in suspended graphene or on hexagonal boron nitride. Reference: J.-H. Chen, C. Jang, S. Xiao, M. Ishigami, and M.S. Fuhrer, “Intrinsic and extrinsic performance limits of graphene devices on SiO $_{2}$ ,” Nature Nanotechnology 3, 206 (2008); S. Adam, E.H. Hwang, V.M. Galitski, and S. Das Sarma, Proc. Natl. Acad. Sci. USA 104, 18392 (2007).

Quartz 5

Explorer