The transport models as per their applicability and origin are categorized in Table 1.4. In the case of semi-classical formalism, the transport models for all limits are obtained from the simplification of Boltzmann transport equation (BTE) equation. In the classical limit, hydrodynamic, drift-diffusion and six moments equations are obtained from the moments of the Boltzmann transport equation (BTE), while BTE should be solved by the Monte Carlo method or the direct method to obtain the distribution function for quasi-ballistic channel description. These semiclassical models are usually developed by taking ensemble average of the particle motion, which are no longer capable of describing the carrier transport in nanoscale transistor. The transports in the nanoscale transistor are usually contributed by the quantum effects, such as 1) interference effects, 2) quantum mechanical tunneling, and 3) quantization in the inversion layer [84]. There are many quantum correction terms available to catch the essence of quantum behavior of electrons in the BTE equation, but the outcome of these models is very specific to particular condition and will not be able to capture the full quantum
transport effects [85]. For an example, the Wentzel-Kramer-Brillouin (WKB) approximation is widely used to incorporate tunneling through potential barriers of irregular shapes. The major drawback of this approximation is that it assumes well defined energy-momentum relation in device like the BTE does, whereas this is not true when the device lacks of translational symmetry in transport direction, especially at non-equilibrium condition.
Table 1.4: Classifications of transport models
Formalism Classical limit Quasi-ballistic limit Ballistic limit Semiclassical -drift-diffusion
-hydrodynamic -six moments equation
-MC∗ simulation -direct calculation
-analytic solution -moment equation
Quantum- Mechanical
-density-gradient -effective potential
-NEGF∗∗
-Winger function Pauli master equa- tion
-ballistic NEGF -Schrödinger equa- tion
∗: Monte-carlo;∗∗ Non-equilibrium Green’s function.
The full quantum simulation can be developed by direct solving of Schr¨odinger equation self- consistently with Poisson equation. However, this approach is either poorly efficient from the compu- tational point of view or only valid for idealized and simplified structures. A more efficient and simple way is to develop the quantum transport by solving the Schrodinger equation within non-equilibrium Green’s function formalism (NEGF). The NEGF essentially solves Schrödinger equation with open boundary conditions and provides a sound basis for quantum-mechanical simulations that is needed for nanoscale devices. The NEGF is a very powerful technique to treat the quantum law of motion (tunneling and interface) by simply solving Green function and the dissipative scattering process by in- troducing Buttiker probes. Moreover, the computational time requirement for solving NEGF equation depends on the size of the device, rather than the whole system which is practically unsolvable.
The NEGF formalism was independently developed by Keldysh [86] and Kadanoff-Baym [87] to solve the non-equilibrium problems in statistical physics and it is based on the contour-ordered Green function, which was first introduced by Martin and Schwinger [88]. The NEGF approach is based on rigorous many body approach and is derived from the main assumptions: 1) a single particle approach and 2) a mean-field approximation. Therefore, NEGF approach cannot properly describe strongly correlated transport in the devices. NEGF theory has demonstrated its usefulness for simulating nanoscale transistors from conventional Si MOSFETs [89], MOSFETs with novel channel materials,
such as CNT [90], nanowire [91], graphene [92], molecular transistors [93], etc. The objective of NEGF is to get many several one-particle Green functions. In the steady-state condition, the governing equations for the retarded(Gr), lesser(G<) and greater(G>) Green’s functions can be written [94] as
[E−H(r1)]Gr(r1, r2;E)− Z
drΣr(r1, r;E)Gr(r, r2;E) =δ(r1−r2), (1.5) G>/<(r1, r2;E) =
Z dr
Z
dr′Gr(r1, r;E)Σ>/<Ga(r2, r′;E), (1.6) where, H(r) is the single electron Hamiltonian, Ga is the advance Green’s function, which is the Hermitian conjugate of retarded Green’s function,Σ> andΣ< are the self-energy functions related to interactions. The main parameters of interest for transport is to obtain the retarded Green function from equation 1.5, which contains the information on the propagation of the electron wave function excited by the delta function source in space-time.
In the numerical solution, the matrix representation of the Green’s functions is used. The dis- cretization in real space basis is done by representing r1 and r2 with row and column indices of the matrices. The governing key equations describing non-equilibrium transport within a semiconductor device are presented in matrix form as follows
Gr(E) = [EI−H(E)−Σ(E)]−1, (1.7) G<(E) =Gr(E)Σ<(E)Ga(E), G>(E) =Gr(E)Σ>(E)Ga(E) (1.8) In these equations,G< andG> are correlation functions specifying electron and hole density spec- tra, respectively. Σ<andΣ>are the in-scattering and out-scattering self energies for electron and hole distribution functions. After the self-consistent solutions are obtained for the correlation functions, the electron and hole density spectrums, and the terminal current density spectrum can be evaluated as
n(r) =−i Z dE
2πG<(E), p(r) =i Z dE
2πG>(E) (1.9)
Ir(E) = q
hT race[Σ(r, E)<G(r, E)>−Σ(r, E)>G(r, E)<] (1.10) Where, q is the elementary charge constant, and h is the Plank constant. In the ballistic simulation, both source and drain contacts are assumed to be in equilibrium state, but maintained at two different chemical potentials. The electron-electron interaction and electron defection from defects are incor- porated by adding potential in the single electron Hamiltonian matrix, H(r), while the interactions
between the contact reservoirs and the transport carrier system are measured by self-energy matrices ,Σ. On the other hand, in dissipative transport consideration, the carrier scattering are treated by introducing Büttiker probes in the channel region. The Büttiker probes are treated as reservoirs, Σ, similar to the source and drain. The chemical potential have to be computed self-consistently, to ensure that current at the scattering contacts is zero. On the other hand, the phonon-electron scattering is treated by Born approximation, and only the self-energies involving one phonon processes (absorption or emission) are included as higher order processes contribute less to the interaction.
Since the main purpose of this thesis is to make a reliable simulation framework for the nano-scale graphene-based FETs, NEGF formalism is used. As considered device has channel length smaller the electron mean free path and operated at low voltages, we are concentrating on the ballistic NEGF formalism in this work.