Quantum Modeling and Design of Carbon Nanotube (CNT) Embedded Nanoscale MOSFETs
Akin Akturk, Gary Pennington and Neil Goldsman
Department of Electrical and Computer Engineering
University of Maryland, College Park, MD20742, USA
We propose a novel MOSFET design that embodies single wall zig-zag semiconducting Carbon Nanotubes (CNT) in the channel, as shown in Fig. 1. Investigations show that CNTs have high low-field mobilities, which can be as great as 2x105 cm2/Vs. Thus we predict that MOSFET performance can be improved by embedding CNTs in the channel. To investigate the performance of the new CNT-MOSFET device, we develop a methodology that connects CNT modeling to MOSFET simulations. Our calculations indicate that by forming high mobility regions in the channel, MOSFET performance can be boosted.The gain in the drive current is as much as 30-40% over standard MOSFETs. Our predictions are based on calculations obtained from ourCNT Monte Carlo (MC) simulator -andquantum based device solver -.
Keywords:Single-Wall Zig-Zag Carbon Nanotube, CNT Mobility Model, CNT Embedded MOSFET Design, Simulation.
As we approach the end of the semiconductor roadmap, investigators are exploring device technologies to push electronics into the 21st century. Carbon nanotubes(CNTs) are being explored as a structure that may play a leading role in future electronic systems. CNTs are planar graphite sheets (graphene) that are seamlessly wrapped into tubes, as shown in Fig. 2.CNTs possess favorable electrical characteristics and can be fabricated in dimensions as small as 4-8Å in diameter. The electrical characteristics of a CNT vary with its diameter and the wrapping angle of the graphene . Both the diameter and the wrapping angle can be described by the tube’s fundamental indices (n,m).Theory indicates that CNTs can be metallic or semiconducting depending on the fundamental tube indices (n,m), and bandgap depends on the diameter. Analysis shows semiconducting CNTs have very high low-field mobilities, with peak drift electron velocities that can be as much as five times higher than that of silicon. It has also been shown that tubes can be doped by donors and acceptors [7,8]. Experiments and calculations also indicate that CNTs could facilitate devices with large transconductances and high drive currents [4,5], -. Experiments have demonstrated the viability of CNT-based FETs [12,13], and CNT-SOI type MOSFETs [14,15]. Preliminary research has been done to model, design CNT embedded bulk MOSFETs [4,5],
We propose a novel CNT-MOSFET device that is shown in Fig. 1. Ourcalculations indicate that the CNT-MOSFET can have improved device performance over the conventional MOSFET [4,5]. To investigate the potential attributes of the new design, we developed a methodology for modeling nanoscale CNT-MOSFETs. It includesdetermination of the electrical characteristics of single wall zig-zag CNTs, and the merging of the CNTresults into our quantum device solver. To electrically characterize the CNT, we developeda Monte Carlo (MC) simulator for CNTs. We first calculate electron and phonon dispersion relations for single wall zig-zag CNTs with different tube indices n. We then derive the selection rules and scattering matrix elements, using Fermi’s Golden Rule. Using MC results, we derive analytical models for CNT parameters such as mobility and density of states. Once we obtain CNT parameters, we import them to our quantum device solver.Our device solver is based on the semiconductor equations, modified to account for quantum effects, and the Schrödinger equation. We solve these coupled equations on a discretized mesh of our CNT-MOSFET device shown in Fig. 1. The solution gives results which include: CNT-MOSFET current voltage curves, electron concentration profile, in the bulk MOSFET, and in the CNT enhanced channel.
We summarize our algorithm in Fig. 3.
As mentioned above, we first employ a Monte Carlo (MC) simulator - to characterize fundamental transport properties of CNTs, and then we incorporate these properties into our device simulator. These properties include electron drift velocity versus electric field curves as well as CNT mobility forzig-zag single wall CNTs, which is the most studied topology. To obtain these properties, we begin with the physical CNT system, where electrons are confined around the circumference, and move relatively freely along the tube in the direction of the longitudinal axis. Therefore, one can write the appropriate plane wave solutions that satisfy periodicity for the given CNT circumference. However along the tube, electrons are not confined. Thus the wavevector can be written as follows:
, () (1)
Here is parallel to the tube axis, and is the unit vector along the circumference. Discrete values around the circumference, β, are bounded by the fundamental tube index n, to take advantage of the symmetry lines in the CNT Brillouin zone. Note thatn count as one unique zone boundary. A zig-zag single wall CNT and its corresponding Brillouin zone can be found in Figs. 2 and 4, respectively.We consider one-dimesional semiclassical charge transport along the tube.
As external field is applied, electrons are accelerated along the tube up until they scatter with phonons. Here we consider scattering by optical, and acoustic inter and intra subband, and intervalley. Both scattering mechanisms are treated within the deformation potential approximation using the Fermi’s Golden Rule. In this work, we use 3eV and 9eV for the nearest-neighbor π-hopping integral and the deformation potential, respectively. For more details of our MC simulations we refer the reader to [1,2].
A. CNT Bandstructure:
To account for the CNT related quantum effects in the CNT-MOSFET channel, we need to determine the band-structure of the CNTs. Due to confinement, the bandstructure splits into a system of subbands when graphene is wrapped into a CNT. Each of the subbands has a characteristic effective mass, mobility and band energy minima.
We determine the energy levels of CNTs by applying zone-folding methods to graphene. From the two-dimensional graphene band diagram, we cut one-dimesional slices, whose numbers and locations are set by the fundamental tube indices (n,m).
We use zone-folding of the graphene conduction band to calculate the electronic bandstructure of CNT. The following formula gives the energy dispersion for a zig-zag CNT:
Where a (=2.46Å) is the lattice constant of two dimensional graphite. It has been shown that this analytical formula agrees with experimental results .
To extract the pertinent information that can be easily integrated into our device simulator, we transform Eqn. (2) into a quadratic form which is consistent with traditional semiconductor analysis:
However we use a parabolic energy bandstructure, instead of the one in Eqn. 2. Because it is too complicated for use in an MC code and the parabolic dispersion curves well approximate the tight-binding bandstructure for energies of our interest, which is up to 2eV above the conduction band minimum. In Fig. 5, we show how well the parabolic bandstructure fits the results of the tight-binding model. The parabolic energy dispersion relation for different subbands can be written as follows:
Here is the conduction band minimum, is the parabolicity factor, and is the effective mass in subband β for a tube with index n. Conduction band minimum, effective mass and parabolicity factor can be calculated using Eqn. 2 for different subbands. For example, conduction band minimum for each subband can be found by setting kx to zero in Eqn. 2, and choosing the appropriate signs for the expression below such that E is minimum:
Likewise, effective mass is the curvature of the energy dispersion relation given in Eqn. 2 around kx=0. We obtain the following formula for the effective masses of each subband:
Our calculations indicate that only the lowest three subbands are effective in the electron transport calculations along the tube. Thus we only deal with those subbands, which correspond to βo, 2(n-βo), and 2(2βo-n), where βo is equal to 2n/3 rounded to the nearest integer. They are written in an order starting from the subband that has the lowest energy minimum and increasing. Note that these subbands are not around the gamma point, but rather correspond to higher frequencies. In Table 1, we give a list of conduction band minimums and effective masses for the first three subbands of different tubes.
B. CNT Mobility Model:
We derived a mobility model based on our MC simulation results of drift velocity versus electric field curves. They are plotted in Fig. 6 for CNTs with tube indices 10 to 58. They are equivalently ranging in diameter from 8Å to 46Å. Simulation indicates that electron drift velocity first increases linearly with the applied field, reaches a maximum, and then rolls off, showing a negative differential mobility (NDM). We find that peak electron velocities are as much as four times higher than what they are in silicon. Electrons reach velocities as high as 5x107cm/s in large diameter CNTs (n=58), while the maximum velocity, which is approximately 3x107cm/s for n=10, drops for smaller diameter tubes. However this is still larger than the corresponding values in other semiconductors. Calculated results also show that the critical field, where we have the peak drift velocity, increases from 1kV/cm to 10kV/cm as we enlarge the tube diameter from8Å to 46Å.
Figure 6 shows three main characteristics of the CNTs. First, CNTs attain larger drift velocities than other semiconductors. Investigations show that is due to decreased scattering rates, which is a result of the one-dimensional system we have. Second, electrons in large diameter tubes have higher velocities than the ones in small diameter tubes for a given applied field, unless the applied field is too large. This leads to higher low-field mobilities for larger diameter tubes. Analysis show that is due to increased density of states of the large diameter tubes. Third, all CNTs show NDM. It is similar to GaAs in that respect, where conduction band velocity of the first subband is larger than that of the second.
We developan analytical mobility model considering the two lowest subbands, which dominate the conduction. By that means, we embed the effects of NDM in our model. We then express the final mobility using Mathiessen’s rule, as follows:
Where and refer to the mobilities in the first and second subbands, respectively. They are functions of fundamental tube index, n, and the electric field, F. The mobility of the first subband is:
Here is the low-field mobility, and Fc(n) is the critical electric field.The critical electric field corresponds to the peak electron drift velocity. We have the following expressions for the low-field mobility and the critical field:
, (cm2/Vs) (8)
, (V/cm) (9)
Here ρ=1-gcd(n+1,3) (= 0, -2), where gcd(n+1,3) is the greatest common divisor of n+1 and 3.
We can write the mobility of the second subband as follows:
, (V/cm) (10)
Here Vmax is the maximum drift velocity of the electrons, shown in Fig. 6. It is formulated below as:
, (cm/s) (11)
In Fig. 7, we show our calculated mobility versus field curves, using Eqns. 6-11. CNTs can sustain large mobilities, larger than all known semiconductor devices. For n=58, low-field mobility is as high as 2x105cm2/Vs, while for n=10 it is approximately 4x103cm2/Vs. So they can be used to form high mobility regions.
C. CNT Intrinsic Carrier Concentration:
To embed a CNT in our device solver, we need a value for the intrinsic carrier concentration of the CNT. Thus we develop a methodology to obtain the intrinsic concentrations of different tubes. We start from the parabolic energy dispersion relation in Eqn. 3. However we ignore the parabolicity factor to simplify calculations. The density of states for each subband is:
We also assume that we have a Maxwellian type distribution. Thus we get the following expression for the intrinsic carrier concentration, where Fermi level is set to zero:
The integral in Eqn. 13b can be recognized as the gamma function with an argument equal to ½, and integral is equal to . The expression in Eqn. 13 gives one-dimensional carrier concentration, whereas we need the concentration per volume. Thus we further assume that the vertical cross section of the tube can be fit in a square, whose side is one diameter long. Finally, we arrive at the following formula for the intrinsic carrier concentration, which is only a function of fundamental tube index n:
III-Quantum Device Simulator:
We develop a quantum device solver - based on the Schrödinger equation and drift-diffusion equations. They are listed below in the order of Schrödinger, Poisson, electron current-continuity, hole current continuity and lattice heat flow equations.
Here the newly introduced variables, ψ, n-p, Jn-p, T, D, GRn-p, C and κ are electrostatic potential, wave function, electron-hole concentrations, electron-hole current densities, lattice temperature, net dopant concentration, electron-hole net generation-recombination rates, heat capacity and thermal diffusion constant, respectively. In Eqn. 15, x is parallel to the Si-SiO2interface in MOSFET, and y is normal to x.
Solution of the CNT-MOSFET system requires proper handling of two dynamics. First one is the quantum well formed at the Si-SiO2 interface that causes band splitting thus lowering of the carrier concentration. Second one is the heterojunction formed at the CNT-silicon interface that reflects the different bandstructures of the CNT and silicon.
We first solve our system classically. This includes the coupled solution of Eqns. 16-19. At this stage, we also resolve the effects of heterostructure, through the use of a revised Scharfetter-Gummel discretization scheme. The details can be found in . We keep the Scharfetter-Gummel discretization intact, but modify electron current density as follows:
Here no(x,y) is the intrinsic carrier concentration at a grid point on our device, and is the intrinsic carrier concentration of silicon.
Investigations show that carrier confinement at the Si-SiO2 interface can significantly reduce carrier concentration adjacent to the interface . Thus we also solve for the quantum effects using the classical solution as our initial guess. To inherent quantum effects in our device model, we solve the Schrödinger equation, Eqn. 15, in addition to Eqns. 16-19. We solve the one-dimensional Schrödinger equation along the normal direction of Si-SiO2 interface, starting from the interface and going down the substrate about 200Å. We apply the same procedure to different mesh points along x in the channel, as well. The physical system and the quantum well can be found in Fig. 8.
We treat the quantum induced effects like the formation of position dependent heterostructures in the quantum well. Thus we sum quantum effects in the carrier concentration term, where quantum confinement is reflected as bandgap broadening. This methodology introduces an extra term within the parentheses of the gradient in Eqn. 20, as follows:
Here subscripts refer to quantum (QM) and classical (CL) solutions. We calculate the quantum carrier concentration using the solution of the Schrödinger equation, which gives energy eigenvalues and eigenfunctions.We then combine the resulting subband structure with the occupation densities, which can be seen in Fig. 9. Thus we get the following formula for the quantum carrier concentration, where we have a sum over different subbands:
In summary, we first solve the coupled Poisson, electron current continuity, hole current continuity and heat flow equations, classically. During the simulation, we also include the effects of the heterostructure formed by the CNT and silicon. We achieve this by using an effective potential, argument of the gradient in Eqn. 20, instead of the classical electrostatic potential, , in the drift-diffusion equations. After we have the coupled solution of the device equations, we use it as an initial guess for the quantum system. We iteratively solve Schrödinger, Poisson, quantum electron and hole current continuity, and heat flow equations until coupled solution of the Schrödinger equation and quantum electron current continuity equation give the same electron concentration. Carrier concentration using the solution of the Schrödinger equation is calculated as shown in Eqn. 22. Whereas we get the carrier concentration within the modified drift-diffusion (DD) formalism as follows:
Here noQM(x,y) is the intrinsic carrier concentration at a grid point on our device, that includes the effects of the heterostructure, as well as the quantum.
Once we have the coupled solution of the system, our quantum device solver outputs current-voltage curves, and profiles of quantum electron concentration, electrostatic potential, hole concentration and such. In Fig. 10, we show our calculated electron concentration in the vertical direction of the MOSFET channel, starting from the Si-SiO2 interface. We applied 0.7V to the gate terminal, and grounded others. CNT-MOSFETs with large diameter tubes have higher concentrations in the channel. Thus they are likely to sustain large transconductances but may be at the expense of weaker isolation in the “off” state of the MOSFET. We also obtain the current-voltage characteristics of the 0.1μm CNT-MOSFETs in the linear and saturation regions.We show the drain current density versus applied drain voltage curves in Fig. 11. There are three configurations for comparison. First one is the conventional MOSFET without any CNTs in the channel. The other two are the CNT-MOSFETs with small (8Å) and large (46Å) diameter CNTs in the MOSFET channel.We find that CNT-MOSFETs utilizing smaller diameter tubes attain higher drive currents than the ones having larger diameter tubes, followed by the conventional MOSFET. The ordering within CNT-MOSFETs can be attributed to the height of the barrier formed at the CNT-silicon heterojunction. However they both have improved drive current characteristics than the conventional MOSFET due to larger channel electron concentrations shown in Fig. 10 and larger mobility values on CNTs. In Fig. 12, we plot the subthreshold characteristics of these three devices.Large diameter tubes have higher leakage currents than others, making them undesirable for low power applications. However small diameter tubes give rise to improved performance in subthreshold, linear and saturation regions, making them favorable for low power and high frequency applications.