Modeling of graphene electron devices

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Introduction

Graphene is an infinite single layer of Carbon atoms arranged in a honeycomb lattice. Although its electronic properties are known for sixty years, only in 2004 it has been possible to isolate it [1], despite it was believed impossible, since a pure bi-dimensional structure was judged thermodynamically unfavor-able [2]. Graphene adds to the already rich scenery of carbon allotropic forms that have attracted the attention of scientists in the last 20 years, such as the zero-dimensional form of carbon, the fullerene (1985) and the one-dimensional form of carbon, the carbon nanotubes [3] (1991). Graphene can be considered the bi-dimensional form of carbon allotropic forms and the building brick of the other n-dimensional carbon structures, as we can see in Fig. 1. Graphite is formed by stacking graphene layers, carbon nanotubes are rolled graphene sheets and fullerenes are cuts of graphene bulk in the form of a ball. Since its isolation, graphene has attracted the attention of the scientific community, due to its unique electronic [4] and physical properties [2], such as unconventional integer quantum Hall effect [5, 6], high carrier mobility at room temperature [1] and potential for a wide range of applications [7, 8, 9].

Among the many exceptional properties, some are very promising for nanoelec-tronic applications. First, it has a mobility of 200, 000 cm2_{/Vs, that exceeds} by more than 100 times that of silicon. This value of mobility has been pre-dicted [10] in the absence of charged impurities and ripples, and a mobility of 106_cm2_{/Vs was recently reported [11] for suspended graphene. Second, it has} a planar structure and therefore can be patterned with standard lithography techniques. Third, it has a low contact resistivity therefore can be used as material for interconnections.

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Figure 1: Graphene as the fundamental building brick of the other carbon allotropic forms: fullerene, carbon nanotubes and graphite. Figure adapted from [2].

However, as far as its possible use as a channel material in electron devices is concerned, graphene presents the problem of being a semi-metal, i.e., a zero-gap material. Transistors and diodes need to be made of a semiconductor with a gap sufficiently large to suppress interband tunneling, that can undermine the possibility of switching the device off. For this reason, several attempts have been made, in recent years, to induce an energy gap in graphene. One possibility is represented by carbon nanotubes [12], but this kind of structures have faced different problems. The main one is the inability to control the tube chirality and thus its electronic nature. Moreover, it is difficult to pat-tern it in a reproducible way and exhibits a large contact resistance. Another way to induce a gap in graphene is cutting it into stripes forming the so-called nanoribbons [13]. With this material as channel a very interesting field-effect transistor behavior has been obtained [14, 15] such us an Ion/Ioff ratio of up to

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Introduction

107_{. However, narrow nanoribbons are required [16] with single-atom precision} since a difference of only 1 dimer line may yield a quasi-zero gap nanoribbon as shown in Fig 2.

Figure 2: Ion/Ioff up to 107 for nanoribbons width lower than 10 nm. Plot of the energy gap in function of the nanoribbon width: the difference of only 1 dimer line may yield a quasi-zero gap nanoribbon. Figures extracted from [14, 15].

Recently, another way to induce a gap was attempted, that is using a bilayer graphene [17, 18, 19] in the presence of a perpendicular electric field. Unfortu-nately in the range of applicable bias we can obtain a gap of only 100 meV [20]. In 2007 Zhou et al. [21] have experimentally demonstrated, through angle-resolved photo-emission spectroscopy measurements, that a graphene layer epi-taxially grown on a SiC substrate exhibits a gap of about 0.26 eV. The fabrica-tion method is also very interesting from the point of view of manufacturability, as they indeed highlight in their work the easiness and reproducibility of the fabrication. To obtain a similar energy gap with the other graphene conductors we would need an armchair nanoribbon width smaller than 3 nm or nanotubes diameter smaller than 2 nm.

An other possible way to exploit the interesting electronic properties of graphene is to use it as a channel material in tunnel field-effect transistors (TFET’s) [22].

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Graphene is indeed characterized by small bandgap and small effective mass, with symmetric dispersion relations for electrons and holes. TFET is a gated p − i − n diode where the gate voltage modulates the position of energy bands in the intrinsic channel in order to control interband tunneling between the p-doped source and the n-doped drain. When the device is ON, current is dominated by interband tunneling, which instead is inhibited when the device is OFF. Small energy gaps and small effective masses can allow to achieve large ON current, comparable to that of mundane semiconductor field-effect tran-sistors. The subthreshold swing (SS) is the gate volatge variation required to increase by a factor 10 the drain current. In a standard FET, where current is basically limited by thermionic emission, the minimum achievable SS=kBT

q ln10, which is 60 mV/decade at room temperature. An impressive advantage of TFETs is a Subthreshold Swing (SS) much smaller than 60 mV/decade. The very small SS would also allows to implement adequate switches with very small supply voltages, and therefore to operate at extremely low power. Finally, carbon channels very often have the same dispersion relations and mobility characteristics for electrons and holes, which is desirable for the optimization of complementary logic gates in terms of switching speed and power consump-tion.

This thesis

This thesis is largely based on my papers on numerical and analytical mod-eling of graphene-based devices, to consider possible approaches to engineer a bandgap in graphene and to evaluate the perspectives of different technological options toward graphene nanoelectronics.

The thesis is organized as follows:

In Chapter 1 some basic concepts are presented. In the first section the basic operation of a field effect transistor (FET) and of a tunnel field effect transistor (TFET) are described. In the second section the graphene lattice geometry, the primitive lattice vectors of both the real and reciprocal lattice space are presented. The third section is dedicated to the description of the Tight Binding (TB) method applied to graphene.

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Introduction

In Chapter 2 contains an analytical model for a bilayer-graphene field-effect transistor. This kind of approach is suitable for exploring the design parameter space for a device structure with promising performance in terms of transistor operation. The model, based on the effective mass approximation and ballistic transport assumptions, takes into account bilayer-graphene tun-able gap and self-polarization, and includes all band-to-band tunneling current components, which are shown to represent the major limitation to transistor operation, because the achievable energy gap is not sufficient to obtain a large Ion/Ioff ratio.

Chapter 3is presented an analytical model of a nanoscale FET based on epitaxial graphene on SiC, and assess the achievable performance in the case of fully ballistic transport.

Chapter 4 presents an accurate model and a exploration of the design parameter space for a fully ballistic graphene-on-SiC Tunnel Field-Effect Tran-sistor (TFET). The DC and high frequency figures of merit are shown. The steep subthreshold behavior can enable Ion/Ioff ratios exceeding 104even with a low supply voltage of 0.15 V, for devices with gate length down to 30 nm. In-trinsic transistor delays smaller than 1 ps are obtained. These factors make the device an interesting candidate for low-power nanoelectronics beyond CMOS.

In Chapter 5 shows an accurate estimation of intrinsic mobility in the low-field limit, analyzing the effects of temperature and of charge concentration on phonon-limited mobility. Full-band calculations are performed, using the tight binding approximation, both for electrons and longitudinal and transverse phonons (acoustic and optical). For the electron-phonon scattering we use the Su-Schrieffer-Hegger (SSH) model and according to that approach, longitudinal and transverse modes have different angle dependencies for the electron-phonon coupling. This model allows us to consider the effect on mobility of all phonons modes. Finally, the low-field mobility has been calculated by the quantum mechanical Kubo-Greenwood formula adapted for bidimensional systems.

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Chapter 1

Basic concepts

1.1 Introduction

This chapter is a tutorial for those not familiar with electronic devices or graphene. Concepts are presented that are used in the next chapters. In the first section the basic operation of a field effect transistor (FET) and of a tunnel field effect transistor (TFET) are described. In the second section the graphene lattice geometry is presented, the primitive lattice vectors of both the real and reciprocal lattice space are reported. In the third section the Tight Binding method and its application in the case of graphene are derived.

1.2 FET and TFET

A solid-state device consists of a crystal with regions containing different type of impurities (dopants).

A Field Effect Transistor (FET) is a semiconductor electronic device that trans-forms a change in input voltage (Vg) into a change in output current. The gain of an FET is measured by its transconductance, defined as the ratio of change in output current to change in input voltage. The field-effect-transistor is so named because its input terminal (called gate) influences the flow of current through the transistor by projecting an electric field across an insulating layer.

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Figure 1.1: Schematic FET structure.

As shown in Fig. 1.1 a FET device consists of two electrodes, one of metal and one of extrinsic silicon, separated by a thin layer of a silicon dioxide. The metal electrode forms the gate, while the semiconductor slab forms the back-gate or body. The insulating oxide layer between the two is called the back-gate dielectric. By selectively doping the silicon on either side of the gate, two re-gions are formed. One of these rere-gions is called the source (S) and the other one is called the drain (D). We can define the threshold voltage Vththat is the gate-to-source bias required to just form a channel with the backgate of the transistor connected to the source. The threshold voltage depends on a number of factors, such as the backgate doping, dielectric thickness, gate material and excess charge in the dielectric.

Imagine that the source and the backgate are both grounded and that a positive voltage is applied to the drain. As long as the gate-to-backgate voltage remains less than the threshold voltage, no channel forms. The pn junction formed be-tween drain and backgate is reverse-biased, so very little current flows from

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Chapter 1. Basic concepts

drain to backgate. If the gate voltage exceeds the threshold voltage, a channel forms beneath the gate dielectric. This channel acts like a thin film of n-type silicon shorting the source to the drain. A current consisting of electrons flows from the source across the channel to the drain.

The performance of a transistor can be graphically illustrated by drawing a family of I-V curves. When I is the drain current (Id) and V is the drain-to-source voltage Vds, the curve is called the transistor characteristic (Fig. 1.2).

Figure 1.2: Transistor characteristic curve.

Each curve represents a specific gate-to-source voltage Vgs. At low drain-to-source voltages the FET channel behaves resistively, and the drain current increases linearly with voltage. This region of operation is called the linear region or triode region. The drain current levels off to an approximately con-stant value when the drain-to-source voltage exceeds the difference between the gate-to-source voltage and the threshold voltage. This region is called the

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Threshold Voltage

V

60 mV/dec

leakage current

I

TH OFF

Gate Voltage (V

_G

)

Drain Current (Id) LogScale

Figure 1.3: FET transfer characteristics.

saturation region.

In the linear region the channel acts as a film of doped silicon with a character-istic resistance that depend on the carrier concentration. The current increases linearly with voltage , exactly as one would expect of a resistor. Higher gate voltages produce larger carrier concentrations and therefore lessen the resis-tance of the channel.

In the saturation region, instead, while the Vds remains small, a depletion re-gion of uniform thickness surrounds the channel. As the drain becomes more positive with respect to the source, the depletion region begins to thicken at the drain end. This depletion region intrudes into the channel and narrows it. Eventually the channel depletes all the way through and it is said to have pinched off. Carriers move down the channel propelled by the relatively weak electric field along it. When they reach the edge of the pinched-region, they are pushed to the drain by the strong electric field. The voltage drop across the channel does not increase as the drain voltage is increased; instead the

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pinched-off region widens. Thus, the drain current reaches a limit and ceases to increase.

Another I-V curve usefull to understand the transistor performance is the transfer characteristic curve, where I is the drain current (Id) and V is the gate voltage Vgs (Fig. 1.3). When Vgs ≤ Vth, the current is an exponential function of gatesource voltage. While the current between drain and source should ideally be zero when the transistor is being used as a turned-off switch, there is a weak-inversion current, called subthreshold leakage. In weak inver-sion, the channel surface potential is almost constant across the channel and the current flow is determined by diffusion of minority carriers due to a lateral concentration gradient. The leakage current varies exponentially with gate-to-source Vgsbias as given approximately by:

I ∝ exp Vgs nVT

, (1.1)

where VT is the thermal voltage given by:

VT = kBT

q (1.2)

with kB the Boltzmann constant, T the absolute temperature, and q the elec-tron charge. The subthreshold slope factor n of a long-channel uniformly doped device can be calculated using simple expressions for the gate and bulk capac-itances Cgand Cb, respectively:

n = 1 + Cb Cg Cb= ǫsi wd Cg=ǫox tox (1.3) In the latter equations ǫox and ǫsi denote the dielectric constants of the oxide and silicon, respectively, wdis the depletion width under the channel, and tox is the gate oxide thickness. The exponential subthreshold behavior can be explained by the exponential dependence of the minority carrier density on the surface potential which, itself, is proportional to the gate voltage. The slope of this line is called Subthreshold Slope. The inverse of this slope is usually referred to as Subthreshold Swing SS, given in units (mV/decade) and is given by:

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SS =dlogId dVgs

= nVTln(10) = 2.3nVT≥ 2.3 kBT

q (1.4)

Due to the bulk effect the subthreshold swing of a conventional transistor in bulk technology will always be higher than a certain optimum value which is roughly 60 mV/dec at room temperature, and which can be calculated by set-ting n equal to 1 in Eq. 1.4 which means that the bulk effect is fully suppressed. In a realistic case n will always be larger than 1. Therefore, the actual

sub-Figure 1.4: Energy barrier formed applying a gate voltage Vgfor a conventionale FET. Figure extracted from [23].

threshold swing SS will always be larger than 60 mV/dec depending on how well the channel surface potential can be controlled by the gate contact. A small subthreshold swing is highly desired since it improves the ratio between the on- and off-currents. This requires that the bulk charge in the depletion region under the channel changes as little as possible when the gate voltage varies, therefore Cbshould be small. Any additional bulk charge increases the voltage drop between the channel surface and the bulk contact, thus reducing the impact of the gate voltage on the surface potential.

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Following the conventional scaling rules, shrinking the device geometry, also Vth decreases. As shown in Fig. 1.3, when Vth decreases the current at Vg = 0 V (Ioff) is increased, due to the subthreshold slope limit. Leakage current in the off state (shown in Fig. 1.4) is due to carriers in the hot tail of the equilibrium energy distribution of carriers in the source and to the flowing of electrons through interband tunneling. Since our aim is to increase the Ion/Ioff value, to have the best performance of a device we have to minimize the interband tunneling current contribution.

Figure 1.5: ON and OFF state energy barrier for a TFET. Electrons go through the energy barrier not over it. Figure extracted from [23].

The interband tunnel field effect transistor (TFET) is a gated p − i − n diode. Gate voltage modulates the position of energy bands in the channel and con-trol interband tunneling between the p-doped source and the n-doped drain.

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Band-to-band tunneling is used to filter the energy distribution of charge car-riers in the source. If electrons are the charge carcar-riers, the injected electrons are limited in energy from below by the conduction band in the channel and from above by the valence band in the source (Fig. 1.5). This cuts off the hot Boltzmann tail of carriers in the source and should produce a much sharper turn-on.

1.3 Graphene Geometry

The tridimensional structure of graphite is composed by layers of carbon atoms displaced in hexagonal lattices; two consecutive layers along the z axis are rotated by an angle of π₃. Graphene is a single layer of graphite and therefore is a bidimensional material. In graphite interaction between adjacent layers is weak (Van der Waals) if compared with strong covalent bond between two adjacent carbon atoms in the same layer. The separation between adjacent layers is of 3.35 ˚A while the distance between two carbon atoms at first neighbor is a_C−C = 1.42 ˚A. Therefore, the electronic structure of graphene can be considered a first approximation of that of graphite1_.

Direct space Graphene lattice is composed by regular hexagons as shown in Fig. 1.6. The primitive cell is a rhombus formed by primitive vectors ~t1and ~t2 which can be written in terms of the distance between nearest neighbors a_C−C as: ~ t1 = aC−C √ 3, 0 ~ t2 = √ 3a_C−C 2 1,√3 (1.5) (1.6) The lattice constant is a = |~t1| = |~t2|. The unit cell contains two carbon atoms, indicated in Fig. 1.6 by the letters A and B, which belong to different

1_{Wallace [24] in 1947 was the first to make a Tight Binding calculation for graphene}

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Figure 1.6: Direct lattice and unit cell for graphene. Figure extracted [25] sub-lattices, generated by the base vectors ~d1and ~d2:

~ d1 = ~ t1 3 + ~ t2 3 ~ d2 = 2 ~d1. (1.7) (1.8) Reciprocal space The reciprocal lattice is generated by ~g1and ~g2which are rotated by π/2 with respect to ~t1 and ~t2. Using the definition:

~ gi· ~tj = 2πδij (1.9) therefore: ~ g1 = 2π √ 3aC−C 1, −√1 3 ~ g2 = 0, 4π 3a_C−C . (1.10) (1.11)

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4π

3aC−C. In the BZ, for definition, do not exist two equivalent inner points, for

g g 2 1 Γ M K

Figure 1.7: Brillouin Zone of graphene, reciprocal unit vectors and high sym-metry points Γ, K and M of the irreducible part of the BZ.

translation of reciprocal lattice vectors, but a point on the surface zone can be equivalent to one or more points of the surface zone. For example, the mid point of a face is always equivalent to the mid point of the opposite face, because opposite faces are separated by a reciprocal vector. In Fig. 1.7 the high symmetry points Γ, K and M which delimit the irreducible portion of the first BZ are highlighted. The coordinates are:

Γ = (0, 0) K = _4π 3√3a_C−C, 0 M = _π 3√3a_C−C, π 3aC−C . (1.12) (1.13) Symmetry The space group of monolayer graphene is symmorphic, with point group D6h. D6h is generated by {C6+, σvl, σh} where:

C

+

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σ

l

v is a reflection by the xz plane

σh is a reflection by the horizontal xy plane that contains the graphene

layer.

The symmetry operations which contain the spatial inversion also include the fractional translation. The presence of the temporal inversion is responsible for bands degeneration in the K point of the Brillouin Zone of graphene. In-deed, the absence of this operation, in a equivalent lattice (i.e BN), produces degeneration splitting.

1.4 The Tight Binding Method

Tight Binding (TB) is an approximation to solve the Schr¨oendinger equation with a crystalline periodic potential, to obtain the electronic states. Within this method the crystalline Bloch functions are expanded on a set of orbital atoms of the atoms inside the unit cell. This method provides a reasonable description of occupied states in any type of crystal and often also of the lowest lying conduction states. In the band theory of crystals, one considers the one-electron Scroendinger equation

" _~ p2 2m+ V (~r) # ψ (~r) = Eψ (~r) (1.14) where V ~r + ~tn = V (~r) (1.15)

is the crystalline potential, ~tn are the translational vectors, and the eigenfunc-tions ψ (~r) must be of Bloch type. In the tight binding method, a number of Bloch sums of vector ~k are used for expanding the crystal wave functions of vector ~k in the form:

ψ~k,~r = X i

ci~k Φi~k,~r (1.16) where the coefficients ci~k are to be determined with standard variational methods. The Bloch sum, Φi~k,~r

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the atomic orbital φi, of quantum numbers i and energy Eiof the atom centred in the reference unit cell, and it is given by:

Φi~k,~r = 1√ N X ~ tm ei~k· ~tm_φ i ~r − ~tm (1.17) where N is the number of unit cells of the crystal. For sake of simplicity and for keeping notations, we have begun to consider a simple crystal with only one atom per unit cell.

From the eigenvalue equation 1.14, inserting the expansion of eq. 1.16, and applying the variational principle, crystal eigenvalues and eigenfunctions are obtained from the determinantal equation:

kMij~k − ESij~k k = 0 (1.18) where Mij~k are the matrix elements of the crystal Hamiltonian between the Bloch sums, and Sij~k

are the overlap matrix elements; namely

Mij~k = hΦi~k,~r |H|Φj~k,~ri (1.19) Sij~k = hΦi~k,~r |Φj~k,~r i. (1.20)

The matrix elements in Eq. 1.19 can be evaluated numerically, and frequently some assumptions are performed on the overlap and Hamiltonian matrix ele-ments.The first approximation concerns Sij~k. We assume extremely local-ized atomic orbitals and therefore the overlap between atomic-like functions centred on different atoms becomes negligible. This justifies the assumption that localized atomic orbitals are orthonormal, and so are the corresponding Bloch sums; in this approximation, the overlap matrix Sij~k is taken as the unit matrix δij. It is thus left to estimate the Hamiltonian matrix elements of Eq. 1.19. If we express the crystal potential as sum of spherically symmet-ric atomic-like potentials Vα ~r − ~tn centred at the lattice positions. We thus approximate the crystal Hamiltonian in the form:

H = p~ 2 2m+ X ~ tn Vα ~r − ~tn . (1.21)

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The Hamiltonian matrix elements becomes: Mij~k = X ~ tn Z φ∗i (~r) " _~ p2 2m+ Vα(~r) + V ′_(~r) # φj ~r − ~tn d~r, (1.22) where d~r is the volume element in direct space, and V′_{(~r) denotes the sum} of all the atomic potentials of the crystal except the contribution Vα(~r) of the atom at the origin. Using the properties that the atomic orbital phii(~r) is eigenfunction of the atomic Hamiltonian with energy Eiand the orthonormality of localized atomic orbitals, Eq. 1.22 is:

Mij~k = Eiδij+ X ~ tn ei~k ~tn Z φ∗i(~r) V′(~r) φj ~r − ~tn d~r. (1.23) In the last equation the term ~tn= 0 gives the so called crystal field integrals:

Iij= Z

φ∗i (~r) V′(~r) φj(~r) d~r. (1.24)

If the tails of the neighbouring atomic-like potentials are almost constant in the region where the wave functions φi(~r) extend, the matrix with elements Iij becomes a constant diagonal matrix.

For what concerns the therms with ~tn 6= 0 in Eq. 1.24, we invoke again the localized nature of the atomic orbitals, so that we can limit the sum to a small number of neighbours. Considering nearest neighbor interaction it is frequently used the two-center approximation, where crystal field terms are disregarded. the two-center integrals can be expressed in terms of a small number of independent parameters, which are evaluated either analytically, or numerically, or semi-empirically.

The first TB description of graphene was given by Wallace in 1947 [24]. Carbon atom has an electronic configuration 1s2_2s2_2p2 _{with orbital energies E}

1s = −21.37 Ryd, E2s = −1.29 Ryd and E2p = −0.66 Ryd. Given the energy distance of 1s orbital with respect to the others, in order to reproduce the graphene electronic properties it is sufficient to consider the atomic orbitals φ2s, φ2px, φ2py and φ2pz. In graphene there are three σ bonds originated from the

hybridization sp2_{of the orbital s}2_{and p}2_{in the xy plane, while the other orbital} 2pzis perpendicular to the graphene plane. The wave functions originated by s,

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pxand pyorbitals (σ bands) are even under reflection with respect to graphene plane, while π bands originated from pzorbitals are odd, therefore there is no mixing between σ and π bands and we can study it separately.

Moreover, the most interesting transport properties of graphene involve the electronic states around the Fermi level, hence only those concerning π bands. Given that graphene unit cell has 2 atoms in the position ~d1 and ~d2 (Eq. 1.7) and that for each atoms only pz orbital are considered, there are 2 orbitals in the cell. The Bloch functions corresponding to the 2 atoms in the unit cell are expressed by means of pz atomic orbital as:

Φ1~k,~r = √1 N X ~ tn ei~k·(d~1+ ~tn)φ_{~r − ~}_d 1− ~tn Φ2~k,~r = 1 √ N X ~ tn ei~k·(d~2+ ~tn)φ ~r − ~d2− ~tn (1.25) where N is the number of unit cells in the crystal, ~tn are the primitive vectors (Eq. 1.5) and φ is the atomic orbital. The Hamiltonian matrix elements are given by: M~k = 1 N X ~ tnt~m ei~k·(t~n+ ~d1− ~tm− ~d2)D_φ_{~r − ~}_t m− ~d1 |H|φ~r − ~tn− ~d2 E (1.26) Using first neighbor and two center approximation, the diagonal elements of M~k are equal to the on site energy E2pz, while the off diagonal elements

are: M~k = X ~ RI ei~k·(R~I)D_{φ (~r) |H|φ}_{~r − ~}_R I E (1.27)

with ~RI = ~R1, ~R2, ~R3first neighbor vectors to the atom centered in ~d1. The first neighbor integrals are all equal and are indicated with V (ppπ), where the p indicates that 2 orbitals are of p type and the π that the orbitals are orthogonal to the axis joining their centres. Therefore, the matrix element M~k is:

M~k = V (ppπ) ei~k·(R~1) + ei~k·(R~2) + ei~k·(R~3)

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Using the first neighbor vector coordinates: f~k= eikxa/2 " 2 cos kya √ 3 2 ! + e−i3kxa/2 # . (1.29)

Therefore the equation to solve to obtain the π bands is:

det   E2pz − E Vppπf~k Vppπf∗~k E2pz − E  = 0, (1.30) that is: E~k = E2pz± V (ppπ) r |f~k |2_, _(1.31) where the + sign indicates the conduction band and the − sign indicates the valence band.

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Chapter 2

Graphene bilayer

2.1 Introduction

The progress of CMOS technology, with the pace foreseen by the International Technology Semiconductor Roadmap (ITRS) [26], cannot be based only on the capability to scale down device dimensions, but requires the introduction of new device architectures [27] and new materials for the channel, the gate stack and the contacts. This trend has already emerged for the recent technology nodes, and will hold – probably requiring more aggressive innovations – for devices at the end of the Roadmap.

In the last decade carbon allotropes have attracted the attention of the sci-entific community, first with carbon nanotubes [3] and, since its isolation in 2004, with graphene [1], which has shown unique electronic [4] and physical properties [2], such as unconventional integer quantum Hall effect [5, 6], high carrier mobility [1] at room temperature, and potential for a wide range of applications [7, 8, 9], like nanoribbon FETs [28]. Despite graphene is a zero gap material, an energy gap can be engineered by ”rolling” it in carbon nan-otubes [12] or by the definition of lateral confinement like in graphene nanorib-bons [13]. However, theoretical [16] and experimental [14] works have shown that significant gap in nanoribbons is obtained for widths close to 1-2 nm, which are prohibitive for fabrication technology on the scale of integrated circuits, at

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least in the medium term.

Recently, theoretical models [17, 18, 19] and experiments [29] have shown that bilayer graphene has the interesting property of an energy gap tunable with an applied vertical electric field. Anyway, the largest attainable gap is of few hundreds of meV, which make its use questionable for nanoelectronics applica-tions: limits and potentials of bilayer graphene still have to be shown.

From this point of view, device simulations can greatly help in assessing de-vice performance. Bilayer-graphene FETs (BG-FETs) have been compared against monolayer FETs, by means of the effective mass approximation [30] and Monte Carlo simulations [31] in the ballistic limit, showing really poor potential as compared to ITRS requirements [26]. These approaches, however, did not take into account some of the main specific and important properties of bilayer graphene, such as the possibility of tuning the band gap and the disper-sion relation with the vertical electric field, and dielectric polarization in the direction perpendicular to the 2D sheet. Such problems have been overcome in Ref. [32], using a real space Tight-Binding approach. However, for the limited set of device structures considered, the small band gap does not allow a proper on and off switching of the transistor.

One limitation of detailed physical simulations is that, despite their accu-racy, they are typically too demanding from a computational point of view for a complete investigation of device potential. Analytical approaches could help in this case. One example has been proposed in Ref. [33], but it has se-rious drawbacks, because it completely neglects band-to-band tunneling and the dependence of the effective mass on the vertical electric field, providing a unrealistic optimistic picture of the achievable performance.

In this work, we have developed a semi-analytical model for a bilayer-graphene FET with two gates to study the possibility of realizing an FET by tuning the gap with a vertical electric field. The model has been validated through comparison with results obtained by means of a full 3D atomistic Poisson-Schr¨odinger solver, showing good agreement in the applied bias range [32, 34]. Interband tunneling proves to be the main limiting factor in device operation, as demonstrated by the device analysis performed in the parameter space.

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Chapter 2. Graphene bilayer

2.2 Model

In this section we provide a detailed description of the developed model, which is based both on a top of the barrier model [35] and on the calculation of all the interband tunneling components. In particular we adopt the ballistic transport and the effective mass approximation, whose main electrical quantities, such as the effective mass and the energy gap, have been extracted from the energy bands obtained from a pz-orbital Tight Binding (TB) Hamiltonian. Since we want to address long channel devices, short channel effects have been completely neglected, as well as inelastic scattering mechanisms, which are expected to be negligible in this kind of material [2]. With respect to more accurate atomistic models, the followed approach may underestimate the actual concentration of carriers in the channel, especially for large drain-to-source (VDS) and gate voltages (VGS), when parabolic band misses to match the exact dispersion relation. We however believe that the developed model represents a good trade-off between accuracy and speed.

2.2.1 Effective mass approximation

In order to proceed with the definition of an analytical model based on the effective mass approximation, we first need an expression for the energy bands of bilayer graphene. The top view of the bilayer-graphene lattice structure with carbon-carbon distance a = 1.44 ˚A is shown in Fig. 2.1(a): A1-B1 atoms lay on the top layer, while A2-B2 on the bottom layer. The energy dispersion rela-tion can be computed by means of a pz-Tight Binding (TB) Hamiltonian [24] considering two layers of graphene coupled in correspondence of the overlaying atoms A1 and A2.

The energy dispersion relation reads [17] : E(k) = U1+ U2 2 ± (2.1) r |f(k)|2₊U2 4 + t2 ⊥ 2 ± 1 2 q 4(U2_{+ t}2 ⊥)|f(k)|2+ t4⊥,

where U1 and U2 are the potential energies on the first and second layer, re-spectively, U = U1− U2, t⊥=-0.35 eV is the inter-layer hopping parameter [17],

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-2.5 -K -1 0 1 K 2.5

k

_y

(nm

-1

)

-5 0 5

E(eV)

-K -1.5 -2

k

_y

(nm

-1

)

-1 -0.5 0 0.5 1

E(eV)

U=0.5 eV 1 2 3 4 a) b) c)

Figure 2.1: a) Real space lattice structure of bilayer graphene. The bilayer consists of two coupled hexagonal lattices with inequivalent sites A1, B1 and A2, B2 in the first and in the second sheet, respectively, arranged according to Bernal (A2-B1) stacking. b) Tight-Binding band structure of bilayer graphene for U =U1-U2=0.5 eV. c) Detail of the band structure in correspondence of band minimum kmin: K is the Dirac point.

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Chapter 2. Graphene bilayer k= kxkˆx+ kyˆky and [24]: f (k) = teikxa/2 " 2 cos kya √ 3 2 ! + e−i3kxa/2 # , (2.2)

which is the well known off-diagonal element of the 2×2 graphene pz-Hamiltonian, where t is the in-plane hopping parameter (t=-2.7 eV). In Fig. 2.1(b) the band diagram for U = 0.5 eV is shown. As can be seen, bilayer graphene has four bands, symmetric with respect to the coordinate axis. For large U , the “mexican-hat” behavior in correspondence of the band minima can be observed, as detailed in Fig. 2.1(c).

Let us now consider the third band (Fig. 2.1(b)), which corresponds to the conduction band (same considerations follow for the valence band, i.e. second band) and apply a parabolic band approximation in correspondence of the minimum kmin, which reads [17]:

kmin= s U2_{+ 2t}2 ⊥ U2_{+ t}2 ⊥ U 2vF~ . (2.3)

The dispersion relation can now be expressed as [17] E(k) = Egap 2 + ~2 2m∗(|k| − kmin) 2₊U1+ U2 2 , (2.4) where m∗₌ t⊥(U2+ t2⊥)3/2 2U (U2_{+ 2t}2 ⊥) 1 v2 F ; Egap= U t_⊥ pU2_{+ t}2 ⊥ , (2.5)

vF = 3at_2~ is the Fermi velocity and ~ is the reduced Planck’s constant.

As can be observed in (2.5), the effective mass m∗ _{has a singularity for} U = 0, which is clearly unphysical. In order to avoid such an issue, energy bands in the range U ∈ [0, 0.14] have been fitted with the parabolic expression in (2.4), within an energy range of 2kBT from the band minimum (where kB is the Boltzmann constant and T is the room temperature), and using m∗ _{as a} fitting parameter. In Figs. 2.2(a), 2.2(b), we show, for two different inter-layer potential energies (U =0 eV and U =0.1 eV), the TB energy bands as well as the parabolic bands exploiting the analytical expression in (2.5) and the fitted

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values for m∗_{, respectively. As can be seen, the fitted effective mass manages} to better match the TB band in the specified energy range. In Fig. 2.2(c), we show the fitted effective mass for different U . In particular, for U < 0.14 eV, m∗ _{can be expressed as:}

m∗= 0.09U + 0.043, (2.6)

while for larger values eq. (2.5) recovers.

2.2.2 Electrostatics

Once obtained the expression for m∗_{, the electron concentration n can be} ex-pressed as: n =ν 2 Z +∞ Ec D(E) [f (E − EF S) + f (E − EF D)] dE, (2.7) where f is the Fermi-Dirac occupation factor, EF S and EF D are the Fermi energies of the source and drain, respectively, and ν=2 is band degeneracy. D(E) is the total density of states per unit area (for the complete calculation see the Appendix I), which reads:

D(E) = 1 2π~ 2m∗ ~ + r 2m∗ E − Ec kmin , (2.8)

where Ec is the conduction band edge. If we define:

fn(Ef) = m∗ π~2kBT ln 1 + exp Ec− Ef kBT + kmin√2m∗kBT 2π~ F1/2 Ec− Ef kBT , (2.9)

where F1/2 is the Fermi-Dirac integral of order 1/2, the electron concentration reads:

n = [fn(EF S) + fn(EF D)] . (2.10)

Analogous considerations can be made for the hole concentration p, which reads:

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Figure 2.2: Comparison between energy dispersions obtained by means of the analytical effective mass (dashed-dotted line), fitted effective mass (solid line) and TB Hamiltonian (circle) for an inter-layer potential equal to a) U =0 eV and b) U =0.1 eV. c) Analytical and fitted relative effective mass as a function of the inter-layer potential U . me is the free electron mass. d) αcond(U ) and αval(U ) as a function of the inter-layer potential U .

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where fp(Ef) = m∗ π~2kBT ln 1 + exp Ef− Ev kBT +kmin √ 2m∗_k_B_T 2π~ F1/2 Ef− Ev kBT , (2.12)

and Ev is the valence band edge.

Once n and p are computed, attention has to be posed on how charge dis-tributes on the two layers i.e. on dielectric polarization. To this purpose, we have numerically extracted from TB simulations αval(U ) and αcond(U ), that represent the fraction of the total states in the valence band and of electrons in the conduction band, respectively, on layer 1 [32]. We computed αcond(U ) for a particular bias (U1= −U2= U/2 and EF = 0 eV) and made the assumption that its dependence on the bias can be neglected. As far as αval is concerned, we assumed in our considered bias range, that all electron states in the valence band are fully occupied and therefore f (E) = 1. Fig. 2.2(d), shows αcond(U ) and αval(U ) as a function of the inter-layer potential U . The charge density ρj per unit area on layer j (j=1,2) is expressed as the sum of the polarization charge, electrons and holes and finally reads:

ρ1(U ) = q {[1 − 2αval(U )] Ntot

−n [1 − 2αcond(U )] + pαcond(U )} ; (2.13) ρ2(U ) = q {[2αval(U ) − 1] Ntot

−nαcond(U ) + p [1 − 2αcond(U )]} ,

where q is the electron charge and Ntot is the concentration of ions per unit area.

The considered device structure is a double-gate FET embedded in SiO2. The bilayer graphene inter-layer distance d is equal to 0.35 nm, while two different oxide thicknesses t1and t2 have been considered (Fig. 2.3(a)). An air interface between bilayer graphene and oxide has also been taken into account (tsp=0.5 nm) [36]. For such a system, we can define an equivalent capacitance circuit as in Fig. 2.3(b), where C0=ǫ_d0, C1=

h t1 ǫ1+ tsp ǫ0 i−1 , C2= h t2 ǫ2+ tsp ǫ0 i−1 and ǫ1= ǫ2= 3.9ǫ0, while ǫ0 = 8.85 × 10−12 F/m. VT g and VBg are the top gate

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and back gate voltage respectively, V1≡ −U_q1 and V2≡−U_q2. In Fig. 2.3(c), the flat band diagram along the transverse direction (y axis) is shown. Metal work functions for the back gate and top gate are equal to 4.1 eV [ΦBg=ΦT g=4.1 eV], while the graphene work function (Φgra) is equal to 4.5 eV [37]. EF T g, EF Bg are the Fermi level of the top and of the back gate, respectively.

The conduction band edge inserted in eq. (2.9), can be expressed as:

Ec= ΦBg+ EF Bg− Φgra+

U1+ U2

2 +

Egap

2 . (2.14)

Applying the Gauss theorem, we obtain the following expression: (

C1(VT g− V1) + (V2− V1)C0= −ρ1 C0(V1− V2) + (VBg− V2)C2= −ρ2.

(2.15) Eqs. (2.13) and (2.15) are then solved self-consistently till convergence on V1 and V2is achieved.

2.2.3 Current

Drain-to-source (JT OT) current is computed at the end of the self-consistent scheme. As depicted in Fig. 2.3(d), JT OT consists of three different components: the first is due to the thermionic current Jth over the barrier [35], whereas the second (JT S) and the third (JT D) to band-to-band tunneling. In the same picture, we sketch the conduction band edge ECS (ECD) and the valence band edge EV S (EV D) at the source (drain). Assuming reflectionless contacts, the thermionic current is due to electrons injected from the source with positive velocity vx> 0 and to electrons injected from the drain with vx< 0:

Jth = −q π2~ Z +∞ −∞ dky Z k> x ∂E ∂kxf (E − EF S )dkx+ Z k< x ∂E ∂kxf (E − EF D )dkx , (2.16) where E = Ec+ ~ 2 2m∗ (|k| − kmin) 2 , vx = 1~ ∂E

∂kx is the group velocity and k

> x (k<

x) is the wavevector range for which vx > 0 (vx < 0). For the complete derivation see the Appendix II and Appendix III.

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Φ Bg E_FBg sp t Ev Ec E_i 0 d E_FTg t₁ tsp t2 Φ_Tg Φgra ε₂ 1 ε V CHANNEL DRAIN J_th E_C gap E E J TS J TD W x ∆ E VS FS E EFD E_VD SOURCE ECS ECD S V_Bg n+ n+ t₂ V Tg SiO₂ SiO₂ x V Source Drain 1 t tsp Back gate Top gate DS C C C V V 1 2 V V_Bg 2 0 1 Tg tsp a) b) c) d) y

Figure 2.3: a) Sketch of the considered bilayer-graphene FET: d is the inter-layer distance, t1 and t2are the top and back oxide thicknesses. b) Equivalent capacitance circuit of the simulated device. c) Flat band diagram of the BG-FET along the y direction. d) Conduction and valence band edge profiles in the longitudinal direction; we assume that deep in the source and drain regions, the electric field induced by the gate vanishes and the gap gradually reduces to zero.

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Let us now discuss the band-to-band tunneling current due to the barrier at source(drain) contact, which reads:

JT i = 2 Z ky Z k> x q 1 2π2 1 ~ ∂E ∂kxTi(ky) [f (E − EF S) −f(E − EF D)] ∂kx∂ky i = S, D, (2.17) where S refers to the source and D to the drain, while Ti(ky) is the transmission coefficient at the different reservoirs. The key issue in computing (2.17) is the definition of an expression for Ti(ky), which accounts for band-to-band tunneling process.

We have assumed a non charge-neutrality region of fixed width ∆x at the contact/channel interface and an electric field Ei=(Ec− EF i)/(q∆x) with i = S, D. For what concern the JT S term, electrons emitted with electrochemical potential EF S see two triangular barriers, one at the source junction and one in correspondence of the drain (Fig. 2.3(d)), whose heights are equal to Egap and width Wi = Egap/(qEi). Assuming the same ∆x for both source and drain junctions, the drain barrier is transparent with respect to the source barrier, since, for large VDS, the electric field at the source is smaller than the electric field at the drain barrier: TS(ky) is therefore essentially given by the source junction barrier. Same considerations follow for the other band-to-band tunneling current component JT D, flowing only through the drain-channel contact. In this case ED= Ec− EF D= Ec−EF S_∆xq+qVDS.

Assuming the WKB approximation, the transmission coefficient can be ex-pressed as:

Ti(ky) = e−2 R

[Wb]|Im{kx}|dx_, _{i = S, D,} _(2.18)

where Im{kx} is the imaginary part of kx and is obtained from: ~2

2m∗(|k| − kmin) 2

= qEix − Egap, i = S, D. (2.19) Finally, JT i is computed performing the integral (2.17) numerically.

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2.3 Exploration of the design space

In order to validate our model, we have first compared analytical results with those obtained by means of numerical NEGF Tight Binding simulations [34], considering a test structure with t1= t2=1.5 nm, tsp= 0.5 nm, Φgra= ΦG= ΦBg=4.1 eV, VDS=0.1 V and VBg=0 V. In Fig. 2.4(a)-(b) the electron

con-0.2 0.4 0.6

V

Tg

(V)

0 1e+16 2e+16 3e+16

Charge Density (1/m

2

)

ρ₁num. ρ₂ num. ρ₁an. ρ₂an. 0.2 0.4 0.6

V

Tg

(V)

-0.05 0 0.05

Voltage(V)

V₁ num. V₂ num. V₁ an. V₂ an. -2 0 2

V

Tg

(V)

0 0.05 0.1 0.15

E

gap

(eV)

a)

b)

c)

Figure 2.4: Comparison between analytical and numerical simulation of a) ρ1 and ρ2 and b) V1 and V2 as a function of VT g. VBg=0 V and VDS=0 V. c) Energy gap as a function of top gate voltage, with VBg = 0 V.

centrations (ρ1, ρ2) and the electrostatic potentials (V1, V2) on layer 1 and 2 are shown, as a function of VT g, for VDS=0 V and VBg=0 V. As can be seen, results are in good agreement. Some discrepancies however occur for larger VDS (VDS> 0.2 V ), where the parabolic band approximation misses to repro-duce band behavior for large ky. In Fig. 2.4(c) the energy gap is plotted as a function of VT g. As can be seen, even for large VT g, the biggest attainable Egapis close to 0.15 eV.

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Let us now consider the different contributions of the three current compo-nents (Jth, JT S and JT D) to the total current JT OT (Fig. 2.5(d)). For each of these components, we can define a sort of threshold voltage, above which their contribution is not negligible. In particular, Jth starts to be relevant as soon as Ec ∼ EF S. We then define Vthas the VT g for which Ec= EF S. Similarly, interband current JT S is not zero when Ev≥ECS so we define VT S the top-gate voltage for which Ev=ECS. Finally, JT D is not zero in the energy range ECD < E < EV S: we define VT D> and VT D< the top-gate voltages for which EV S=ECD; thanks to these definitions, we can qualitatively evaluate current contribution by observing the band structure.

Our goal is indeed to obtain the largest value for the Ion/Ioff ratio, and this is only possible if the band-to-band component of the current is suppressed. We have considered three different solutions to accomplish this task: by varying the back gate oxide (t2), by varying the EF S−ECSor EF D−ECDdifference, or by simply varying the back gate voltage. If otherwise specified, ∆x = 0.7 nm, as obtained from TB simulations of an abrupt junction with the same doping of the considered BG-FET. In Fig. 2.5(a)-(b)-(c) the above-defined thresholds are shown for the three considered cases.

As shown in Fig. 2.5(a), back gate oxide thickness has no effect in our case, since the top layer screens the electric field induced by the top gate, as can also be seen from Fig. 2.4(b), where V2 remains almost constant. Fig. 2.5(b) shows thresholds as a function of (EF S− ECS), and therefore as a function of dopant concentration. We observe that for EF S− ECS = 1 eV, VT D> = VT D< , so that JT D is practically eliminated, while JT S increases since VT S becomes larger. In Fig. 2.5(c), we show Vthand VT S, for EF S− ECS = 1 eV , as a function of VBg. Unfortunately the two curves have the same behavior, so that VT Scannot be reduced to values smaller than Vth, or –in other words– we cannot suppress current due to interband tunneling at source contact.

We have then computed the transfer characteristics for VBg = 0V , t1 = t2=1.5 nm, EF S− ECS=1 eV. In Fig. 2.5(e) JT OT is shown. As can be seen, poor Ion/Ioff ratio can be obtained since band-to-band tunneling at source contact is too high as also observed in graphene FET [38]. Reducing E, i.e. T (ky), could lead to a reduction of JT Sand consequently to an improvement of

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Figure 2.5: a) Thresholds as a function of the oxide thickness t2 for VDS= 0.5 V, VBG=0 V, EF S− ECS=0.9 eV. b) Thresholds as a function of EF S− ECS for t2=1.5 nm, VDS= 0.5 V, VBG=0 V. c) Thresholds as a function of VBg for t2=1.5 nm, VDS= 0.5 V and EF S− ECS= 1 eV. d) Total current JT OT and its three components (JT S, JT D, Jth) for VDS=0.1 V, VBg=0 V and EF S− ECS= 0.5 eV . Thresholds are shown along the coordinate axis. e) Total current for t2=1.5 nm, VDS= 0.5 V, EF S− ECS= 1 eV and VBG=0 V and different ∆x.

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Figure 2.6: Simulated band edges when: a) Tunnel current weakly affects JT OT; b) JT S+ JT D is one order of magnitude smaller than the total current; c) Inter band tunneling current is the dominant component.

the Ion/Ioffratio. As can be seen in Fig. 2.5(e), an improved Ion/Ioffis obtained increasing ∆x to 5-10 nm, but it is still lower than the ITRS requirements (104₎ for digital circuits. In Fig. 2.6, we also sketch the simulated band edges for three different cases: a) when tunneling is negligible ( JT OT ≃ JT i for VT g = 2 V), b) when tunneling weakly affect the total current (JT OT ≃ 10(JT S+ JT D) for VT g = 0.4 V) and c) when tunneling represents the predominant component (JT OT ≃ (JT S+ JT D) for VT g = −2 V).

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2.4 Conclusion

We have developed an analytical model for bilayer-graphene field effect tran-sistors, suitable for the exploration of the design parameter space. The model is based on some simplifying assumptions, such as the effective mass approx-imation, but includes all the relevant physics of bilayer graphene. First and foremost, it includes the tunable gap of bilayer graphene with the vertical elec-tric field, which is exploited in order to induce the largest gap, when the device is in the off state. It also fully includes polarization of bilayer graphene in response to a vertical electric field. As far as transport is concerned, it includes the thermionic current components and all interband tunneling components, which are the main limiting factor in achieving a large Ion/Ioff ratio. Sig-nificant aspects of the model have been validated through comparisons with numerical TB NEGF simulations.

Due to the small computational requirements, we have been able to explore the parameter design space of bilayer-graphene FETs in order to maximize the Ion/Ioffratio. Despite applied vertical field manages to induce an energy gap of the order of one hundred meV, band-to-band tunneling greatly affects device performance, limiting its use for device applications. A larger gap must be induced to make bilayer graphene a useful channel material for digital applica-tions, probably by combining different opapplica-tions, such as using bilayer graphene in addition to limited lateral confinement, stress, or doping.

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Chapter 3

Epitaxial graphene on SiC

FET

3.1 Introduction

From the beginning [39, 5], graphene has attracted the attention of the scien-tific community due to its exceptional physical properties, such as an electron mobility exceeding more than 10 times that of silicon wafers [2], and in view of its possible applications in transistors [36] and in sensors [40]. To induce a gap in graphene structures, several methods have been used: lateral confinement in graphene ribbons [36, 41] carbon nanotubes [42], impurity doping [43], or a combination of single and bilayer graphene regions [29, 32, 44]. Unfortunately, they all face different problems.

Carbon nanotubes exhibit large intrinsic contact resistance and are difficult to pattern in a reproducible way; the inability to control tube chirality, and thus whether or not they are metallic or semiconducting, make solid state integra-tion still prohibitive. Graphene nanoribbons [36, 41] allow to obtain a very interesting device behavior [28], but require extremely narrow ribbons with single-atom precision, since a difference of only one dimer line in the width may yield a quasi-zero gap nanoribbon. Bilayer graphene exhibits a gap in

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the presence of a perpendicular electric field, but the range of applicable bias can only induce a gap of 100 − 150 meV, not sufficient to obtain a satisfactory behavior in terms of Ion/Ioff ratio [32].

Graphene on SiC is a two-dimensional material, thus does not require extremely sophisticated lithography, and provides a higher energy gap: for the sake of comparison, a 0.26 eV energy gap would require an armchair nanoribbon of width smaller than 3 nm, or nanotubes with diameter smaller than 2 nm. Epitaxially grown graphene on SiC provide potential for large scale integra-tion of graphene electronics. The first challenge to the use of graphene as a channel material for FETs is to induce a reasonable gap for room temperature operation. Recently Zhou et al. [21] have experimentally demonstrated that a graphene layer, epitaxially grown on a SiC substrate, can exhibit a gap of about 0.26 eV, measured by angle-resolved photo-emission spectroscopy. The gap is probably due to symmetry breaking between the two sublattices forming the graphene crystalline structure, as also confirmed by recent density func-tional calculations [45, 46]. According to the authors of Ref. [21], this method of inducing a gap is very easy and reproducible; in addition, the thickness of graphite grown on SiC can be precisely controlled to be either single- or mul-tiply layered depending on growth parameters [47]. From a manufacturability point of view it is also extremely promising, since it would be highly convenient to prepare an entire substrate of graphene on an insulator and then obtain sin-gle device and integrated circuit through patterning [48].

In this work we present a semi-analytical model of an FET with a channel of epitaxial graphene grown on a SiC substrate, where the band structure, the electrostatics, thermionic and band-to-band tunneling currents are carefully accounted for. On the basis of our model, we assess the achievable device performance through an exploration of the device parameter space, and gain understanding of the main aspects affecting device operation.

3.2 Model

We adopt the Tight Binding (TB) Hamiltonian for single layer graphene on SiC that was proposed by Zhou et al. [21]. The empirical TB valence (−) and

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Chapter 3. Epitaxial graphene on SiC FET

conduction (+) bands of a single epitaxial layer of graphene on SiC, read: E±(kx, ky) = ±pm2+ t2|f(k)|2, (3.1) where t is the in-plane hopping term (2.7 eV), m = 0.13 eV is an empirical potential energy shift between the two non-equivalent graphene sublattices due to interaction with the SiC substrate, and f (k) is the off-diagonal element of the considered Hamiltonian [21]. In the six Dirac points of the graphene Brillouin zone, where f (k) is zero, there is a finite energy gap Eg = 2m, corresponding to the channel conduction minimum ECC = m − qφch and the channel valence maximum EV C = −m − qφch, where q is the electron charge and φch the self-consistent potential in the central region of the channel.

The device under consideration, depicted in Fig. 3.1(a), is a transistor with a channel of epitaxial graphene on a SiC substrate of thickness tsub= 100 nm, with a top gate separated by a SiO2 layer of thickness tox. In Fig. 3.1(b) we have sketched the band edge profiles along the transport direction ˆx, where ECi and EV i respectively represent the conduction and valence band edges in the three different regions denoted by i=S, D, C (Source, Drain, Channel). Source and drain contacts are n+_{doped, with molar fraction α}

D, which translates into an energy difference A between the electrochemical potential µS (µD) and the conduction band edge ECS(ECD) at the source (drain) contact. The analytical relation between αD and A is given by: αD = 1/DR_BZf (ECS− A)∂kx∂ky where D = 2/(√3a2_{/2) is the atomic density per unit area and f is the} Fermi-Dirac distribution function. The potential is set to zero at the source and to Vdsat the drain contact. In the center of the channel φchis imposed by vertical electrostatics. We assume, as usual, complete phase randomization along the channel, which is particularly important because it allows us to neglect the effect of resonances in the presence of tunneling barriers.

Exploiting the Gauss theorem we can write the surface charge density in the central part of the channel as

Q = −Cg(Vg− VF Bt− φch) − Csub(Vsub− VF Bb− φch) , (3.2) where Cg = ǫSiO2/tox (Csub = ǫSiC/tsub) is the capacitance per unit area

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b) a)

Figure 3.1: a) Schematic picture of a graphene on SiC transistor. The grey line between SiO2 and SiC oxide represents the graphene plane acting as device channel. Source (S) and drain (D) contacts are also in graphene. b) Profile band structure along the transport direction. The dashed lines mark the energy region in which it is possible to have thermionic current (Jth), tunneling current from source to channel (JtD) and tunneling current from drain to channel (JtS).

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gate), which we set to −0.4 eV.

The transit time of the device in the channel has been estimated as τt = QthLC

Jth ≈ 10

−16 _{s where Q}_th _{and J}_th _{are the thermionic charge and current,} respectively, LC= 20 nm is the channel length.

In certain spectral regions, for example in the valence band when the device is in the off state, carriers are quasi confined by tunneling barriers, and can dwell in the channel for a much longer time and be subject to some degree of relaxation, even if transport in the conduction band is practically ballistic. It is indeed possible to include the treatment of energy relaxation in semi-analytical models, through the virtual probe approach, as shown in Ref. [49]. And this approach has also been extended to consider carbon-related materials, where the interplay of energy relaxation and tunneling barriers has have been ana-lyzed [50]. Here we are not interested in developing a complete treatment of this problem, instead we employ a minimal model, in which a process, account-ing for carrier thermalization with the contacts, has been included. Our only aim is to provide a slight correction to a canonical ballistic transport model. By increasing the effectiveness of this relaxation process with contacts, we can estimate the limit to the electrostatics imposed by a model with strict ballistic transport in all spectral regions.

In steady-state conditions, considering an infinitesimal element of area dkxdky in the wave-vectors space, charge distribution in the channel is obtained as a balance between two types of charge exchange processes with the contacts: one elastic, and one of thermalization.

We can write the electron charge in the channel as the sum of two contribu-tions: Ne

+ and N−e. N+e = f+eLCn2D (N₋e = f₋eLCn2D) represents the density of forward (backward) going electrons, fe

±(kx, ky) denotes the occupation fac-tors of forward (+) and backward (−) states in the channel and n2D(kx, ky) is the 2-dimensional density of states in the k-space. For each contribution we can write a rate equation in steady-state conditions:

dNe + dt = J + S − J + D+ (1 − TS) f₋evx− (1 − TD) f+evx+ fe S− f+e τS LCn2D= 0 (3.3) dNe − dt = −J − S + JD−− (1 − TS) f₋evx+ (1 − TD) f+evx+ fe D− f−e τD LCn2D= 0(3.4)

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where: fe

D,S= F (E+− qφch− µD,S) is the occupation factor at the drain (D) and source (S) contacts, and F is Fermi-Dirac distribution function.

Let us focus on eq. 3.3 (similar considerations can be made for eq. 3.4): JS+= TSfSevxis the tunneling current component injected from source, JD+ = TDf+evx is instead the drain tunneling current component ejected to the drain, TS,Dare the transmission probabilities from source/drain contacts to channel and vx is the group velocity. (1 − TS) f₋evx and (1 − TD) f+evx are the reflected current components from source and drain barriers, respectively. Thermalization is in-troduced in the last terms of eqs. 3.3 and 3.4: the term in eq. 3.3(eq. 3.4) brings forward(backward) electrons in equilibrium with the source(drain) reservoirs, with characteristic time τS (τD). The steady-state f+e and f−e can be obtained by solving eq. 3.3 and eq. 3.4:

f+e = TD+_τ_D1_ν (1 − TS) fDe + 1 τDν + 1 TS+_τ_S1_ν fe S 1 τSν + 1 1 τDν + 1 − (1 − TS) (1 − TD) , (3.5) f₋e = TS+_τ_S1_ν (1 − TD) fSe+ 1 τSν + 1 TD+_τ_D1_ν fe D 1 τSν + 1 1 τDν + 1 − (1 − TS) (1 − TD) , (3.6) where ν = 2π2vx

LC is the inverse of the crossing time τt. For the sake of simplicity,

in the following we consider τS = τD= τ .

The same reasoning can be applied to derive the hole occupation factors in the channel fh

±.

The charge, to be self-consistently solved with eq.3.2 in order to obtain the channel potential φch, is computed through the integration on the BZ

Q = −_4πq2 Z Z BZ f+e + f−e dkxdky+ + q 4π2 Z Z BZ f+h+ f−h dkxdky, (3.7) where the total current density is expressed as [51]

Jtot = q 4π2 Z Z BZ vx f+e− f−e dkxdky + Z Z BZ vx f+h− f−h dkxdky (3.8)

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The transmission probability TS (TD) of the interband barrier at source (drain) is zero in the source (drain) band gap and 1 when there is no barrier between source (drain) and channel. When a barrier is present TS is computed analyti-cally with the WKB approximation, assuming kyconservation due translational invariance along the y direction:

TS(E, ky) = exp −2 Z x2 x1 |Im(kE,ky x (x))|dx , (3.9)

where x1 and x2 are the classical turning points, and E is the particle kinetic energy. The same approach is repeated for TD.

The potential profile of the barrier connecting each contact and the cen-tral region of the channel is described by an exponential, with characteristic variation length λ, obtained from evanescent mode analysis [52] and assuming that in the barrier region mobile charge has a negligible effect on the potential. Assuming tsub≫ tox> tch we obtain:

λ ≈ tox+ tch 2 2 π, (3.10)

where tchis the effective separation between the interfaces of the SiO2 and SiC layers, for which we assume tch = 1 nm [36]. As long as LC >> λ, the effect of source and drain contacts on the potential is vanishing in the middle of the channel, which is therefore dominated by vertical electrostatics.

3.3 Electrostatics

From analysis of the electrostatics we can gain a better insight of the device performance limitations. In fact gate voltage control upon the channel poten-tial (of which the subthreshold slope S is a measure) is strictly limited by the quantum capacitance Cq of the channel. Device electrostatics can be schema-tized as in Fig. 3.2(a). The differential capacitance seen by the gate is

Ctg= Cg 1 − ∂φch ∂Vg (3.11)

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q

C

g

sub

∂V

g

V

sub

∂

ch

φ

∂

a)

b)

Figure 3.2: a) Equivalent circuit of device electrostatics. b) Quantum capacitance-voltage characteristics for Vds = 0, 0.1, 0.25 V, tox = 1 nm, tsub= 100 nm and αD= 2.4 × 10−2 in case of fully ballistic transport. but, from Fig. 3.2(a), Ctg can also be expressed in terms of capacitances Cg, Csub, and Cq:

Ctg = Cg(Csub+ Cq) Cg+ Csub+ Cq

. (3.12)

From eqs. (3.11) and (3.12) we get the derivative of the channel potential with respect to the gate potential

∂φch ∂Vg

= Cg

Cg+ Csub+ Cq

. (3.13)

The expression of the sub-threshold slope S then turns out to be S = 1 + Csub+ Cq Cg kT q ln(10), (3.14)

from which it is clear that S is an increasing function of Cq, and therefore a large quantum capacitance severely limits device performance.

Fig. 3.2(b) shows the capacitance-gate voltage characteristics for Vds = 0, 0.1 and 0.25 V obtained by solving the Schr¨odinger equation self-consistently with Poisson equation. In the fully ballistic case the quantum capacitance is low for small Vds, indicating a good control of the channel by the gate voltage,

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but, as soon as Vds increases, hole accumulation in the channel occurs and Cq increases, rapidly degrading S (Fig. 3.2(b)). When we include the thermaliza-tion process, hole accumulathermaliza-tion is slightly suppressed by thermalizathermaliza-tion with the source contact. However, the overall effect on the quantum capacitance is practically negligible. We have observed that for τ larger than 10−4 _{ns, the} quantum capacitance basically does not change with respect to Fig. 3.2(b); on the other hand, for τ < 10−4 _{ns C}_q _{decreases with respect to the fully ballistic} case, but the energy relaxation process becomes dominant, approaching a limit where eq. 3.14 loses validity.

3.4 Perspectives for device operation

In order to evaluate the possible performance of the SiC-graphene FET, we have computed the transfer characteristics by varying three device parameters: drain-source voltage Vds= (µS− µD)/q, oxide thickness toxand the donor mo-lar fraction at the contacts αD. We also account for different possible values of energy relaxation time τ . First,in Fig. 3.3, we analyze the trend of the transfer characteristics for different τ for Vds= 0.25 V, tox= 1 nm and αD= 2.5×10−3 (corresponding to A = 0.01 eV). We observe that for τ ≥ 1 ns the transfer char-acteristics are unaffected and identical to the ballistic case (τ → ∞). Reducing the relaxation time under 1 ns, the minimum current increases and the sub-threshold slope remains almost constant since the quantum capacitance of the channel does not change. The introduction of the thermalization process with contacts has mainly two effects in the transfer characteristics: one is a gradual change of the current in the sub-threshold region, the other is an increase of saturation current for τ < 10−4 _{ns or when not-ballistic current becomes} rele-vant. In the most favorable case a sub-threshold slope of 140 mV/dec can be obtained.

In Fig. 3.4 we have highlighted the effect of Vdsand of the doping level of con-tacts. As expected the main visible effect of increasing Vdsis a gradual degrada-tion of the sub-threshold slope, both in the fully ballistic case (Fig. 3.4(a)-(b)) and in the case of relaxation time τ = 10 ps (Fig. 3.4(c)-(d)), from 84 mV/dec to 202 mV/dec. The reason is simply related to the increased accumulation of

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-1

-0.8

-0.6

-0.4

-0.2

V

_gs

(V)

10

-2

10

0

10

2

10

4

J

tot

(A/m)

_{τ= 10}

-6

ns

τ= 10

-5

ns

τ= 10

-4

ns

τ= 10

-3

ns

τ= 10

-2

ns

τ= ∞

Figure 3.3: Transfer-characteristics for τ = 10−2_{, 10}−3_{, 10}−4_{, 10}−5_{, 10}−6 _{ns, for} Vds= 0.25 V, αD= 2.5 × 10−3 and tox= 1 nm.

holes in the channel with increasing Vds, which implies a larger quantum capac-itance of the channel and therefore a reduced control of the channel potential from the gate voltage.

Increasing the doping causes an increase of both the maximum current, due to an improved capacity of the source to inject electrons, and the minimum current. From Fig. 3.4 we draw the indication that by reducing doping at the contacts we improve the current dynamics. As already noted, when the source-drain voltage exceeds the gap of the semiconducting channel (Vds> 0.26 V), the characteristics drastically degrade, since band-to-band tunneling current becomes comparable with the thermionic current, and hole accumulation in the channel inhibits channel control from the gate.

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10

-2

10

0

10

2 A=0.01 eV A=0.05 eV A=0.10 eV A=0.20 eV A=0.30 eV

-0.5

0

10

0

10

2

-1

-0.5

0

~ 84 mV/dec ~ 202 mV/dec a) b) c) d) V_ds=0.1 V V_ds=0.25 V ~ 84 mV/dec ~ 202 mV/dec

J

tot

(A/m)

V

_gs

(V)

τ=∞

τ=10 ps

Figure 3.4: Transfer-characteristics for varying with doping parameter A = 0.01, 0.05, 0.1, 0.2, 0.3 eV corresponding respectively to αD of 2.5 × 10−3, 9.8 × 10−3_{, 2.4 × 10}−2_{, 6.8 × 10}−2_{, 1.3 × 10}−1_{, t}

ox= 2 nm with: a) Vds= 0.1 V, τ = ∞, b) Vds= 0.25 V, τ = ∞, c) Vds= 0.1 V, τ = 10 ps and d) Vds= 0.25 V, τ = 10 ps.

The increase of oxide thickness tox has mainly two effects, which can be associated to a reduction of the capacitive coupling between gate and channel: it increases the sub-threshold slope S (as shown in Fig. 3.5(a)), and the opacity of tunneling barriers (i.e. a larger λ). The former effect is more evident for Vds = 0.25 V, where the quantum capacitance is larger, instead S is almost

Modeling of graphene electron devices

Contents

Introduction

This thesis

Chapter 1

Basic concepts

1.1

Introduction

1.2

FET and TFET

Threshold Voltage

V

60 mV/dec

leakage current

I

Gate Voltage (V

)

Drain Current (Id) LogScale

1.3

Graphene Geometry

1.4

The Tight Binding Method

Chapter 2

Graphene bilayer

2.1

Introduction

2.2

Model

2.2.1

Effective mass approximation

k

(nm

)

E(eV)

k

(nm

)

E(eV)

2.2.2

Electrostatics

2.2.3

Current

2.3

Exploration of the design space

V

(V)

Charge Density (1/m

)

V

(V)

Voltage(V)

V

(V)

E

(eV)

a)

b)

c)

2.4

Conclusion

Chapter 3

Epitaxial graphene on SiC

FET

3.1

Introduction

3.2

Model

3.3

Electrostatics

q

C

C

C

g

sub

∂V

g

V

sub

∂

_{τ= 10}