Role of Phonon Scattering in Carbon Nanotube Field-effect Transistors

The role of phonon scattering in carbon nanotube field-effect transistors ͑CNTFETs͒ is explored by solving the Boltzmann transport equation using the Monte Carlo method. The results show that elastic scattering in a short-channel CNTFET has a small effect on the source-drain current due to the long elastic mean-free path ͑mfp͒ ͑ϳ1 ␮m͒. If elastic scattering with a short mfp were to exist in a CNTFET, the on current would be severely degraded due to the one-dimensional channel geometry. At high drain bias, optical phonon scattering, which has a much shorter mfp ͑ϳ10 nm͒, is expected to dominate, even in a short-channel CNTFET. We find, however, that inelastic optical scattering has a small effect in CNTFETs under modest gate bias. Carbon nanotube field-effect transistors ͑CNTFETs͒ are now being explored for high-performance electronics. 1,2 A recently reported CNTFET with a channel length of ϳ50 nm appears to deliver a near-ballistic current. 3 This is a surprising result, because under high drain bias the channel is expected to be several mean-free-paths ͑mfps͒ long. Previous studies have shown that the dominant scattering mechanism in a high-quality carbon nanotube ͑CNT͒ is phonon scattering. 4–7 Under low bias, the mfp is observed to be very long in CNTs ͑ϳ1 ␮m͒, and is thought to be nearly elastic and limited by acoustic phonon ͑AP͒ scattering. 5 Under high bias, optical phonon ͑OP͒ emission dominates, and short ͑ϳ10 nm͒ mfps result. In metallic CNTs, OP emission leads to a high-bias saturation current of ϳ25 ␮A for a long tube and a significant decrease of channel conductance for a short tube. 6,7 Studies of phonon scattering in CNTs have focused on metallic tubes or on long semiconducting tubes. 8 Phonon scattering in short-channel CNTFETs, which is important for nanoelectronic applications, remains unexplored. In this letter, we show that near-ballistic dc currents can be obtained for a short-channel CNTFET even under high source-drain bias in the presence of significant inelastic scattering. The results indicate that elastic scattering has a small effect on the source-drain current for a short-channel CNT-FET when the elastic mfp is long. If, however, a short elastic mfp were to exist in a CNTFET, elastic scattering would degrade the on current of CNTFETs much more severely than it does for a typical metal–oxide–semiconductor field-effect transistor ͑MOSFET͒. This difference results from the difference between one-dimensional ͑1D͒ transport in a CNT and two-dimensional ͑2D͒ transport in a MOSFET channel, …

The role of phonon scattering in carbon nanotube field-effect transistors ͑CNTFETs͒ is explored by solving the Boltzmann transport equation using the Monte Carlo method.The results show that elastic scattering in a short-channel CNTFET has a small effect on the source-drain current due to the long elastic mean-free path ͑mfp͒ ͑ϳ1 m͒.If elastic scattering with a short mfp were to exist in a CNTFET, the on current would be severely degraded due to the one-dimensional channel geometry.At high drain bias, optical phonon scattering, which has a much shorter mfp ͑ϳ10 nm͒, is expected to dominate, even in a short-channel CNTFET.We find, however, that inelastic optical scattering has a small effect in CNTFETs under modest gate bias.© 2005 American Institute of Physics.͓DOI: 10.1063/1.1923183͔Carbon nanotube field-effect transistors ͑CNTFETs͒ are now being explored for high-performance electronics. 1,2A recently reported CNTFET with a channel length of ϳ50 nm appears to deliver a near-ballistic current. 3This is a surprising result, because under high drain bias the channel is expected to be several mean-free-paths ͑mfps͒ long.5][6][7] Under low bias, the mfp is observed to be very long in CNTs ͑ϳ1 m͒, and is thought to be nearly elastic and limited by acoustic phonon ͑AP͒ scattering. 5Under high bias, optical phonon ͑OP͒ emission dominates, and short ͑ϳ10 nm͒ mfps result.In metallic CNTs, OP emission leads to a high-bias saturation current of ϳ25 A for a long tube and a significant decrease of channel conductance for a short tube. 6,7Studies of phonon scattering in CNTs have focused on metallic tubes or on long semiconducting tubes. 8Phonon scattering in short-channel CNTFETs, which is important for nanoelectronic applications, remains unexplored.
In this letter, we show that near-ballistic dc currents can be obtained for a short-channel CNTFET even under high source-drain bias in the presence of significant inelastic scattering.The results indicate that elastic scattering has a small effect on the source-drain current for a short-channel CNT-FET when the elastic mfp is long.If, however, a short elastic mfp were to exist in a CNTFET, elastic scattering would degrade the on current of CNTFETs much more severely than it does for a typical metal-oxide-semiconductor fieldeffect transistor ͑MOSFET͒.This difference results from the difference between one-dimensional ͑1D͒ transport in a CNT and two-dimensional ͑2D͒ transport in a MOSFET channel, and generally applies to all nanotube/nanowire transistors with 1D channel geometry. 9o treat phonon scattering in CNTFETs, we simulate semiclassical transport by the Monte Carlo ͑MC͒ method self-consistently coupled to the Poisson equation.The validity of a semiclassical approach for treating a short-channel CNTFET ͑L ch ϳ 20 nm͒ was first confirmed by a full quantum simulation under ballistic conditions. 10The MC simulation simulates stochastic carrier trajectories and has been ex-tensively applied to study the dissipative carrier transport in Si MOSFETs. 11It has also been applied to study carrier transport in CNTs, 8 and calibrated to experiments for a metallic tube with length down to ϳ50 nm. 6Two scattering mechanisms have been identified to be important in metallic CNTs, 6 and were included in this study, acoustic phonon scattering and OP scattering ͑including phonons with both small wave vector k ϳ 0 and large k near the Brillouin zone edge͒. 7e describe the first conduction subband E-k using a simple analytical expression derived from a p z orbital tightbinding model, 12 where E is the kinetic energy, ប is the Planck constant, F Ϸ 8.0ϫ 10 7 cm/ s is the Fermi velocity in a metallic tube, and k 0 =2/͑3d͒, where d is the diameter of the tube.The corresponding density of states ͑DOS͒ is where D 0 =4/͑ប F ͒ is the constant DOS of a metallic tube, and ⌬ is one-half of the band gap, and ⌰͑x͒ equals 1 for x Ͼ 0 and 0 for x ഛ 0. Only the lowest subband is treated; the effect of higher subbands will be discussed later.
In a metallic tube, the scattering rates and mfps are energy independent due to the constant DOS near the Fermi level, but in a semiconducting tube, the scattering rates and mfps are energy dependent.Note that the band structure of a a͒ Electronic mail: guoj@ece.ufl.eduFIG. 1. Scattering rate vs carrier kinetic energy in the lowest subband.
semiconducting tube at high energies approaches that of a metallic tube, so the mfps of a semiconducting tube at high energies, elastic high and OP high also approach those of a metallic tube.The scattering rate in a semiconducting tube, as shown in Fig. 1, is computed by where elastic high = 500 nm is a typical high-energy mfp for AP scattering, OP high = 15 nm is a typical high-energy mfp for OP scattering, and ប OP = 0.16 eV is a typical OP energy. 6Only OP emission was treated because at thermal equilibrium, ប OP ӷ k B T, so the phonon population is small.͑The effect of hot phonons will be discussed later.͒Pauli blocking is an important factor that is treated using a rejection technique as described by Lugli. 13The carrier distribution function is updated after each time step, so that when a scattering event occurs, the probability that the final state is available can be evaluated.A random number between 0 and 1 then determines whether the scattering is permitted.To treat transistor electrostatics, Poisson's equation is solved self-consistently with the transport simulation.Two types of contacts are treated.For a CNT MOSFET with doped tubes as source/drain shown in Fig. 2͑a͒, 14 the contacts are assumed to be ideal ͑without reflection͒.For CNT Schottky barrier ͑SB͒ FETs as shown in the inset of Fig. 4͑a͒, 15 we treat the tunneling of carriers through metal-CNT junctions as follows.For a carrier injected to a SB, the transmission probability through the SB is evaluated using the Wentzel-Kramers-Brillouin approximation.A random number between 0 and 1 is then generated to determine whether the carrier tunnels through the SB or gets reflected.Such an approach has been previously developed to treat Si SBFETs, and validated by experiments for a channel length of Ͻ30 nm. 16e first simulated a CNT MOSFET with doped tubes as source/drain extensions, as shown in Fig. 2͑a͒.To explore the role of elastic scattering in CNTFETs, we first treated only elastic scattering in the CNT channel and omitted OP scattering ͑ op high → + ϱ͒.For a long elastic mfp of elastic high = 500 nm ͓which is typical for a high-quality CNT ͑Ref.6͒, and results in a thermal average mfp ͗ elastic ͘ϳ90 nm near the top of the barrier͔, the transistor delivers ϳ80% of the ballistic on current, as shown by the dashed line in Fig. 2͑b͒.
Elastic scattering has a rather small effect because of the long elastic mfp.In contrast, for a short elastic mfp of elastic high =15 nm ͑which corresponds to a thermal average mfp of ͗ elastic ͘ϳ3 nm near the top of the barrier͒, the transistor only delivers ϳ10% of the ballistic current, as shown by the dashed-dotted line.
Compared to a Si MOSFET, elastic scattering has a much stronger effect for a CNTFET.It has been reported that Si MOSFETs with a channel length several times longer than the elastic scattering mfp ͑due to, for example, surface roughness scattering͒ can still operate at nearly 50% of the ballistic limit. 17,18This difference results from the difference between 1D carrier transport in a CNT channel and 2D in a MOSFET channel.For a MOSFET, the final k states for an elastic scattering event distribute in ͑k x , k y ͒ plane.For most final states, a carrier does not possess enough backward velocity along the channel direction after an elastic scattering event to overcome the barrier, and return to the source.For this reason, scattering near the drain has much less of an effect on I D than scattering near the source end of the channel. 19͑Of course, scattering near the drain causes the space-charge density to build up, which has an indirect, though potentially strong, effect on the current of a shortchannel MOSFET.͒ 20,21In contrast, for a carrier with a wave vector +k in a CNT, the only available final state after a scattering event is −k, because of the one-dimensional channel geometry.The magnitude of carrier velocity along the channel direction remains unchanged and the carrier can overcome the top of the barrier and return back to the source.Elastic scattering anywhere in the channel affects the sourcedrain current equally.The effect of a short-mfp-elastic scattering in a nanotube/nanowire transistor is much more severe that in a typical MOSFET.Because the elastic mfps are so long, however, we do not expect elastic scattering to have a strong effect on short-channel CNTFETs.
We next explore the role of inelastic optical phonon scattering, which has a much shorter mfp, and scatters carriers even in a short CNT at high biases. 6,7The circles in ers the carrier energy by ប OP , which results in a final state that is already occupied.Such a scattering event is prohibited by Pauli exclusion.After the carrier travels over the top of the barrier, OP emission can occur.Such OP emission, however, does not affect the dc source-drain current.At modest gate bias, the top of the barrier, E top , is only modestly below the source Fermi level ͑ S − E top Ͻប OP ͒.After OP emission, a carrier loses energy of ប OP ͑ϳ0.16 eV͒, and it does not have enough energy to overcome the barrier and return to the source.The large OP energy ͑ប OP ϳ 0.16 eV͒ in CNTs helps the transistor to deliver a near-ballistic dc current at modest gate biases, although a significant amount of scattering exists near the drain end of the channel.We also simulated the I D -V D characteristics in the presence of only OP scattering with OP high = 15 nm and AP high → ϱ, as shown by the crosses in Fig. 2͑b͒.The transistor essentially delivers a ballistic source-drain current, which further confirms that the effect of short-mfp OP scattering is small.
We also explored the role of phonon scattering under high gate voltages, which pushes the top of the barrier, E top , well below the source Fermi level, S , as shown in Fig. 3͑b͒.The results indicate that once S − E top Ͼប OP , OP scattering begins to significantly affect the source-drain current.For a carrier injected with energy E Ͼ E top + ប OP , after an OP emission event, the carrier is still energetic enough to overcome the top of the barrier and return back to the source.The result is that the current contributed by carriers with E Ͼ E top + ប OP is significantly reduced by OP scattering.
Because most CNTFETs, to date, operate like SB transistors, 15 it is important to examine whether our understanding of CNT MOSFETs is relevant to CNT SBFETs.As shown in the inset of Fig. 4͑a͒, the simulated CNTFET is similar to a recently reported CNTFET with Pd contacts, 3 except that coaxial-gate geometry is used.The main panel of Fig. 4͑a͒ plots the I D versus V D at the ballistic limit ͑the solid line͒, and with scattering ͑the dashed line͒.Again, we find that the CNT SBFET also delivers a near-ballistic sourcedrain current for the following reason.The elastic scattering has a small effect because of its long mfp.For OP scattering in the channel, a carrier loses energy of ប OP after OP emission.Because of the large OP energy ͑ϳ160 meV͒, the backscattered carrier faces a much thicker and higher tunneling barrier, as shown in Fig. 4͑b͒.Because the tunneling probability exponentially decreases with the SB thickness and height, the chance for the scattered carrier to return back to the source significantly decreases.OP scattering, therefore, has little effect on the source-drain current at modest gate biases in a SB CNTFET.
As mentioned earlier, only the first subband is treated in this study, but an energetic carrier near the drain end of the channel can scatter to higher subbands.After a carrier is scattered to a higher subband, however, the potential barrier between the carrier and the source increases, and it becomes more difficult for the carrier to return back to the source.Intersubband scattering, therefore, will not change the conclusion that inelastic scattering has a small effect on the dc current.As also mentioned earlier, only OP emission was treated, because essentially no optical phonons are present at thermal equilibrium.Optical phonons, however, can build up and be reabsorbed by source-injected electrons.Detailed treatment of hot phonon effects requires solving electronphonon-coupled Boltzmann transport equation, which is beyond the scope of this study.Simple estimation shows that most OPs are emitted when +k going electrons are backscattered, and therefore, they possess a wave vector along +k direction.When such OPs are reabsorbed, electrons are scattered toward the drain rather than back to the source, which does not lower the source-drain dc current.Our neglect of OP phonon emission and intersubband scattering is justified because this study is concerned with the effect of scattering on the dc current under high-bias conditions.

FIG. 3 .
FIG. 2. Scattering in CNT MOSFETs: ͑a͒ A coaxially gated CNTFET with doped source/drain.The tube diameter is d ϳ 1.4 nm with a band gap E g ϳ 0.6 eV.The total tube length is 50 nm, with an intrinsic channel of L ch = 20 nm, and a doped source/drain of L S = L D = 15 nm.The gate oxide thickness is t ox = 3 nm, and the dielectric constant is =16.͑b͒ The simulated I D vs V D at V G = 0.4 V for five cases: ͑i͒ solid: Ballistic transport, ͑ii͒ crosses: OP scattering only, with OP high = 15 nm and elastic → ϱ, ͑iii͒ dashed: Elastic scattering only, with elastic high = 500 nm and OP → ϱ, ͑iv͒ Circles: elastic and OP scattering with elastic high = 500 nm and OP high = 15 nm, and ͑v͒ dash-dot: Strong elastic scattering with elastic high = 15 nm and OP → ϱ.

FIG. 4 .
FIG.4.Scattering in CNT SBFETs: ͑a͒ I D vs V D at V G = 0.4 V for ballistic transport ͑solid line͒, and in the presence of both elastic and OP scattering with elastic high = 500 nm and OP high =15 nm ͑dashed line with circles͒.The inset shows the simulated CNTFET with metal source/drain.The SB height for electrons is Bn = 0, the intrinsic channel length is 50 nm, and the tube diameter is d ϳ 1.4 nm with E g ϳ 0.6 eV.The gate oxide thickness is 8 nm with a dielectric constant of =16.͑b͒ A snapshot of the electron distribution at the steady state for V D = V G = 0.4 V with OP and elastic scattering.