THE CARBON NANOTUBE FET, CNFET, OR CNTFET
We now come to the ultimate 1-dimensional FET, the FET that has a carbon nanotube (CNT) as a channel and is surrounded by an oxide from all sides. Before tackling the CNTFET we give a brief summary of the properties of CNTs.
a. Electronic properties of CNTs
A CNT is produced by rolling up a graphene sheet (i.e. a single layer of atoms of the graphite structure) along one of its 2-dimensional lattice vectors R = na +n2a2 as shown for example in figure 8.15 where a (6,4) CNT is depicted. This process will produce an infinite tube of nanometric diameter. We then expect, given the 1-dimensionality of the CNT, to obtain a 1-dimensional Brillouin zone. Flow does this come about? The argument is the following—the planar graphene sheet has a planar Brillouin zone, the hexagon depicted in figure 8.16a. The unit vectors bh b2 in к space are related to the unit lattice vectors a, by the relation bj a, = 27i8,y (see the corresponding equation in chapter 2, equation 2.18). When the rolling up of a CNT along a lattice vector R is performed and a CNT of radius r = |R|/2K is produced, then the atom at the origin and at the tip of R become one and the same atom. Their wavefunctions must be equal and this is guaranteed (using Bloch’s theorem) by the relation
where m is an integer. This will discretize the hexagonal Brillouin zone of graphene into a series of line segments as shown in figure 8.16b.
FIGURE 8.15 The 2-dimensional lattice of grapheme showing a typical lattice (n,m) vector along which if a graphene is rolled it will produce an (n,m) CNT.
FIGURE 8.16 (a) The hexagonal Brillouin zone of graphene and (b) its transformation into a series
of line segments when a CNT is produced by the rolling of a graphene sheet.
A simple mathematical proof is as follows. We can analyze к into components along the folding vector R and perpendicular to the folding vector R. Denote these by kx and ky, respectively. Then we have that Ry=0 and equation 8.35 takes the form
which means that ky may assume any value whereas kx is discretized in multiples of 2k/|jR|. Hence, the allowed к vectors are line segments perpendicular to the folding vector R. They are contained within the graphene hexagonal Brillouin zone, and are separated by a successive distance of 2ix/|l?|. As the magnitude |R| of JR tends to large values we recover the 2-dimensional character of the graphene but for |R| of the order of nanometers the number of line segments inside the hexagon is finite and each can be considered as a sub-band of the (nlyn2) nanotube. The ky values are also discrete but as the length of the CNT increases the ky tend to a continuum.
Carbon nanotubes can be either metallic or semiconducting. The analysis of the previous paragraph allows us to deduce a very simple rule as to whether a (nlyn2) CNT is metallic or semiconducting. The vertices of the hexagonal Brillouin zone (BZ) are the wavevectors at the Fermi energy, i.e. the kp. So if the line segments corresponding to the allowed к values cross (or contain) the vertices of the hexagon there are allowed states at the Fermi energy and the CNT is metallic. If not, they are semiconducting. Figure 8.17 is a magnification of the allowed states near the bottom right corner of the hexagonal BZ of figure 8.16b. The broken line shows the direction of the folding vector R, so the line segments containing the allowed states are perpendicular to it.
Consider an arbitrary state given by к and denote by A/q( and Ak± the components of |к-/ср| along the line segment and perpendicular to it. Clearly if Ak± is zero the
FIGURE 8.17 Magnification of figure 8.16b near one of the corners of the hexagonal BZ.
particular line segment contains the particular vertex and the CNT is metallic. The component A/cj. equals the projection of |fc —fcp| onto R. Then
The particular vertex we have encircled is given by
From 8.36 we then get
In the last, before the last line of 8.38 we have used the relation between the unit direct and reciprocal lattice vectors discussed above (equation 2.18 of chapter 2)
Hence we get
So if(«i — и2) is a multiple of 3, Ak± = 0 and the CNT is metallic, otherwise it is semiconducting with a bandgap Es that depends on the folding vector. It should be noted that the result is independent of the particular vertex chosen for the proof.
The band structure of the CNTs can be approximated by the band structure of graphene (i.e. curvature effects are omitted) by an application of the LCAO method that we have presented in detail in chapter 2. This is left as an exercise to the student, see problems at the end of the chapter. The result is
where E0 and t are the diagonal and off-diagonal elements respectively of the p orbitals of graphene and a is the interatomic distance (see the instructions for the corresponding problem). The density of states of the semiconducting (10,0) CNT is shown in figure 8.18a and that of the metallic (9,0) CNT in figure 8.18b. Equation 8.40 takes
FIGURE 8.18 (a) Density of states of a semiconducting (10,0) CNT and (b) of a metallic (9,0) CNT. Calculations by R. Saito et al, Appl. Phys. Lett. 60, 2204 (1992).
into account the discretization in one of the directions of the 2-dimensional к vector, but it does not take into account the curvature of the CNT. As a general trend, it can be stated that in some metallic (according to the above rule) CNTs curvature may open up a small gap if their diameter is small. We will not deal with this subject, instead assuming that all CNTs that form a FET channel are semiconducting.
b. The concept of a quantum capacitance
The concept of a quantum capacitance was introduced by S. Luryi in 1998  as a correction to the gate capacitance of a MOSFET. Flowever, this notion has become important in connection to the functioning and modeling of the CNTFET so we will give a brief introduction to the concept. We have seen in the previous chapter that as the gate voltage VG is increased above Vo, the conduction band of Si is populated and an almost surface charge Q„ is created at the Si-Si02 (Si-Flf02 nowadays) interface. This charge screens the electric field lines emanating from the gate and leads to the stabilization of the width of the space charge layer in Si. The situation is depicted in a more general form in the following figure 8.19a of a three-plate capacitor. If the middle plate is indeed metallic any increase of VG will increase the electric field only in the space covered with the dielectric £i.
However, if the charge of the middle plate is due to the quantum well of a semiconductor, the screening is not complete and the physics of the situation may be represented by the equivalent diagram of figure 8.19b where CQ is defined as
Then, assuming that the capacitance C2 can be neglected, the capacitance Cc seen by the gate is just Q and Cq in series, so
FIGURE 8.19 (a) Two capacitors in series in which the middle plate is not perfectly conducting and
(b) its equivalent circuit.
FIGURE 8.20 Comparison of the intrinsic carrier density n( of several semiconductors. Figure reproduced from D. Akiwande et al .
c. DC model of CNTFET
Carbon nanotubes may be doped producing N- and P-type semiconductors but it is not necessary to dope CNTs to produce a FET. The intrinsic carrier concentration of (я,0) CNTs is as high as that of semiconductors with common doping, see figure 8.20. Obviously the high values of и, (T) for CNTs displayed in figure 8.20 stem from their very low volume. Doped CNTs however are used as the source and drain contacts of a FET. Indeed, the two most common forms of CNTFETs are shown in figure 8.21. In figure 8.21a the source and drain contacts are formed by overdoped CNTs and an intrinsic CNT constitutes the channel. On the other hand, in 8.21b the source and drain are formed by metals. A brief discussion is necessary of these metal contacts before proceeding to a model of the CNTFET
We have more than once emphasized that a well-designed MOSFET needs ohmic contacts at the source and drain in order to function properly. This is usually achieved by overdoping the semiconductor which reduces the length of the Schottky barrier at
FIGURE 8.21 Two forms of CNT-FETs.
FIGURE 8.22 Reduction of the barrier length between a metal and a CNT by the application of a gate voltage in the initial stage of a CNT-FET research when ohmic contacts were not available. Figure reproduced from Avouris et al .
the metal-semiconductor junction. Unfortunately, in the initial stages of research into the CNTFET development, the Schottky barrier lengths were far from being permeable by tunnelling. However the Schottky barrier lengths were reduced by the application of the gate voltage VG, , see figure 8.22. This led to a device which was Schottky barrier controlled instead of a usual FET. Fortunately procedures and materials are available today which may reduce the Schottky barrier to almost zero. We now come to the evaluation of the current ID. We follow closely the analysis of Akinwande et al .
The band diagrams of the CNTFETs which are shown in figure 8.21 are given in figure 8.23. These band diagrams assume a Schottky barrier of almost zero. Note that because the Fermi level of the intrinsic CNT is almost at the middle of its band gap and that of the heavily doped CNT very near EG, the difference EG - £/-- = Eg/2 where Ef is the Fermi level of the channel. The dispersion relations of the subbands (those arising from the discretization of kx) of both the conduction and the valence band are shown in figure 8.24. Only the first two sub-bands are shown for each band. The effect of the application of a gate voltage Vc > 0, as shown in figure 8.24, is to shift the electron sub-bands down, This will populate the conduction band predominantly as can be seen from an inspection of figure 8.23. We may safely assume that the number of holes is negligible. The amount by which the conduction sub-bands move down is just the surface potential cps (see equations 7.1a and 7.1b of chapter 7 and figure 7.5b).
Then the current ID will be given by the Landauer formula just as we did for the MOSFET, see equations 7.43-7.47 of chapter 7. Furthermore, we take account of
FIGURE 8.23 Band diagrams of the CNTFETs shown in figure 8.21.
FIGURE 8.24 Dispersion relations (E-k) of the subbands of a CNT.
contributions from only the two lowest conduction subbands. Denoting the equilibrium Fermi-Dirac distribution by/(£/.-,£) and by Ea and Ea the subband minima, see figure 8.24, we have
Note that in equation 8.43 above it was no longer necessary to distinguish between the direction of transport x and other directions. There is only one direction, that along which transport takes place which is normal to the barrier. As with 7.43 g(E), v(£) stand for the 1-dimensional density of states and the velocity at energy E respectively. The factor M(E) is the subband degeneracy and is equal to M(£) = 2 for carbon nanotubes. Assuming ballistic transport and performing the simplifications we have used in going from 7.43 to 7.47 we get (choosing the Fermi level at the source E?s = £f)
where h in the denominator in 8.44 is Plank’s constant (not h bar).
Contrary to the situation encountered in 7.47 the resulting integrals involve only the Fermi-Dirac distribution / and can be performed analytically (see the formula for the supply function in chapter 5, equations 5.67-5.68). We get
It is worthwhile noting the following: 1) the above equation does not explicitly show the Vb dependence of Id, this is hidden in the surface potential (ps, 2) the current from drain to source is only important in the linear region, it can be dropped in the saturation region.
To complete the model the gate voltage dependence of cps must be evaluated. The latter can be obtained by either a self-consistent numerical calculation (Poisson- Schroedinger type) or can be obtained by a compact (empirical) model. It has been shown that
s can be given by the following analytic compact model ,
where Cq is the quantum conductance discussed in the previous subsection.
The above model reproduces the (/-V) curves of CNTFETs only moderately. Its primary deficiency is the omission of optical phonon scattering near the drain end of the channel . In fact this is a common characteristic of most nano-FET transistors, which brings us to our original question in chapter 5 of where the power is dissipated. It is dissipated near the drain end of the channel and in the thermodynamic reservoirs where electrons are thermalized.
8.3 Use the LCAO theory of chapter 2 to prove the dispersion relation (equation 8.39) of carbon nanotubes neglecting curvature effects (i.e. using grapheme as a model of a CNT). You may ignore the s orbitals as well as the px, py orbitals (the ones lying on the plane of the carbon atoms.
8.4 A model of CNT is shown in the following diagram (The model was produced by researchers of the Universities of Purdue, Cornell and Stanford). Prove that