A simple quantum mechanical treatment of scattering in nanoscale transistors

We present a computationally efficient, two-dimensional quantum mechanical simulation scheme for modeling dissipative electron transport in thin body, fully depleted, n-channel, silicon-on-insulator transistors. The simulation scheme, which solves the nonequilibrium Green’s function equations self consistently with Poisson’s equation, treats the effect of scattering using a simple approximation inspired by the “Buttiker probes,” often used in mesoscopic physics. It is based on an expansion of the active device Hamiltonian in decoupled mode space. Simulation results are used to highlight quantum effects, discuss the physics of scattering and to relate the quantum mechanical quantities used in our model to experimentally measured low field mobilities. Additionally, quantum boundary conditions are rigorously derived and the effects of strong off-equilibrium transport are examined. This paper shows that our approximate treatment of scattering, is an efficient and useful simulation method for modeling electron...


I. INTRODUCTION
Metal oxide field effect transistors ͑MOSFETs͒ with channel lengths in the 10 nm regime have been recently demonstrated by several research groups. 1,2Design considerations to yield devices with desirable channel lengths have been theoretically explored in Refs.3-5.The smallest devices use a silicon-on-insulator ͑SOI͒ geometry in which electron transport occurs in a thin silicon film which is sandwiched between two insulators.Device physics in such structures has been examined mainly in the pure ballistic limit and in some cases by invoking simple one-dimensional ͑1D͒ approximations.Real devices typically operate below the ballistic limit as a result of scattering.Therefore, our primary objective in this article is to describe a computationally efficient, quantum mechanical treatment of scattering in n-channel MOSFETs based on the nonequilibrium Green's function formalism. 6,7cattering within a device can be due to several reasons.Microscopically, electrons are confined within a very narrow channel and the channel is sandwiched between insulators in SOI devices.These insulator surfaces are never perfectly smooth, and the semiconductor lattice is never defect free.Therefore electrons in the channel experience surface roughness scattering.Both, channel carrier densities and impurity concentrations are typically very large.Therefore there is significant electron-electron and electron-impurity scattering within the device.Moreover, devices typically operate at relatively high temperatures ͑greater than 300 K͒ resulting in strong phonon-electron interactions.All these mechanisms need to be considered for an accurate treatment of scattering.Currently, solutions to the Boltzmann equation using Monte Carlo methods offer the best platform for modeling the effects of scattering in detail.Although comprehensive in their treatment of scattering, these simulation platforms do not treat quantum effects rigorously.As critical device dimensions are shrunk, quantum effects begin to manifest themselves strongly.These effects need careful consideration because they affect critical device performance metrics.However, a simulation platform based on quantum mechanics, which also treats the effects of each scattering mechanism individually is not computationally viable.Therefore an approximate treatment of scattering, which in addition captures the effect of source-to-channel tunneling and includes quantum effects which affect the threshold voltage and gate capacitance is clearly useful for device design.
Within the nonequilibrium Green's function ͑NEGF͒ formalism, a detailed treatment of the various scattering processes ͑e.g., surface roughness, phonon, impurity etc.͒ is possible.However, due to the large computational cost involved in a detailed simulation of a MOSFET, scattering is treated approximately using the concept of Bu ¨ttiker probes.Although introduced phenomenologically by Bu ¨ttiker, 8,9 a comparison with the detailed NEGF scattering treatment provides insights and allows us some control over the approximation.Bu ¨ttiker probes are used to simulate the effect of scattering due to all possible mechanisms and including the effect of degeneracy.Our scattering model is a one parameter fitting model and the parameter that we use is analytically related to an equivalent mobility ͑Appendix B͒.From a design perspective, it is important to be able to calibrate the parameters used in our quantum mechanical model to mimic an equivalent low field mobility because the low field mobility can be measured experimentally.Such a calibration enables us to relate our results to those obtained from conventional simulation tools, thus serving as a benchmark which can be used to validate simpler models ͑based on the density gradient and effective potential treatments͒. 10,11It also helps fit our parameters to measured device data in a meaningful fashion.
Irrespective of the technique used to simulate scattering, the essential physics of scattering has to be captured. 12This physics can be summarized by considering the on-state of a MOSFET with electrons thermally injected from the source, undergoing scattering in the channel and being collected by the drain.Scattering can occur anywhere inside the device, but for a fixed potential profile, only those scattering events occurring in the low field region near the source have the largest effect on the on-current.Although the scattering rate in the high field region near the drain can be very large, electrons scattering near the drain end are forced by the high electric fields to leave the channel, having little chance making it back to the source.Therefore scattering in this region does not degrade the on-state current directly ͑although it does do so indirectly through its effect on the self-consistent potential profile͒.We demonstrate that our transport model does capture this essential physics of scattering.This article describes the numerical methods used to treat the effects of dissipative transport with specific emphasis on relating Bu ¨ttiker probe strengths to low field mobilities and on the essential physics of scattering.Quantum boundary conditions are derived and quantum mechanical features of the simulation results are highlighted.The article is divided into the following sections: Sec.II presents the solution scheme.Section III presents simulation results obtained by applying the scattering model to a nanoscale, double-gate ͑DG͒, n-channel MOSFET.Section IV compares two different versions of the scattering model with regards to the essential physics and Sec.V summarizes key findings.

II. THEORY
The simulated device structure is shown in Fig. 1.A uniform rectangular grid with a grid spacing of a, along the x direction and b along the z direction is used.Note that we restrict our focus to the intrinsic device and account for the large source/drain ͑S/D͒ reservoirs to which the device is coupled, using open boundary conditions ͑no x dependence of the potential͒.The Fermi levels at the ends of the intrinsic device are specified by the applied voltage.The width ͑y dimension͒ of the device is assumed to be large and all potentials ͑including the scattering potential͒ are assumed to be translationally invariant along the width ͑W͒.A single band effective mass Hamiltonian is used to model carrier transport.In our modeling scheme, the scattering potential is treated as a perturbation to the ballistic device Hamiltonian.Therefore we briefly review the steps common to both ballistic and dissipative transport models.

A. The ballistic solution
We begin by solving a 1D, z directed effective mass equation for each vertical slice along x ͑Fig.1͒, to obtain a set of eigenenergies and eigenfunctions ͑modes͒ along the gate confinement direction.The equation that is solved is where m z * is the electron effective mass in the z direction, V(x,z) the electrostatic potential, q the electron charge, and ⌿ i (x,z) and E i (x), the wave function and eigenenergy for mode i at slice x, respectively.Note that the simulation domain in the confinement direction can be extended to include the insulator regions.Each vertical slice has a width a and within each slice, all quantities are assumed to be a constant in the x direction.
We then expand the three-dimensional ͑3D͒ effective mass Hamiltonian for the device in terms of ␦(x ϪxЈ)⌿ i (x,z) and e jk j y /ͱW.The plane wave function, e jk j y /ͱW, represents the device width and the quantum number k j , corresponds to the eigenenergy, where m y * is the electron effective mass in the y direction.
We use plane waves to represent the device width because all potentials are assumed to be invariant along y.The over all wavefunction in this orthonormal basis is where ⌽ ˜i(xЈ), is the expansion coefficient of ⌿ i (xЈ,z) and the summation over i runs over all the modes.If we assume that the shape of a mode does not change as we move from the source to the drain (‫ץ‬⌿ i (x,z)/‫ץ‬xϭ0) and invoke the orthogonality criterion ͓͐⌿ i *(x,z)⌿ j (x,z)dzϭ␦ i j ͔, the 3D effective mass Hamiltonian reduces to a set of decoupled 1D Hamiltonians ͑one for each mode i and k j ) of the form where E is the total electron energy and E L ϭEϪE k j is the longitudinal or channel directed electron energy.Equation ͑3͒ is the decoupled mode-space transformation of the 3D effective mass Hamiltonian.It is the starting point for simulating both ballistic and dissipative transport in thin body, fully depleted, SOI transistors.A detailed expansion of the 3D Hamiltonian and the validity of the decoupled modespace solution has been presented in Ref. 13. Simplified versions of the decoupled mode-space solution scheme have been extensively used by several authors to model electron transport in both SOI and carbon nanotube transistors in the literature. 3,5,14The decoupled mode-space solution is a computationally efficient method to model electron transport in ultrathin SOI transistors because only a few low energy modes are occupied in these devices even at room temperature.High energy modes which are unoccupied, can be safely ignored without any loss of accuracy.Note that the 1D Hamiltonian is coupled to the electrostatic potential through the Poisson equation.The mode-space solution is therefore obtained self consistently.The NEGF formalism is not only an effective method to simulate the ballistic MOSFET, but also allows for scattering to be included either in detail or approximately.

B. The scattering model
In the ballistic limit, there are only two reservoirs connected to a device, namely, the source and drain contacts.These contacts inject carriers into and extract carriers from the intrinsic device while conserving the current ͑net current at the source contact equals the net current at the drain con-tact͒.In the presence of scattering, Bu ¨ttiker probes can be used to model dissipative transport phenomenologically within the transistor. 8,15These Bu ¨ttiker probes perturb the Hamiltonian of the device in a manner similar to the source and drain contacts and can also be viewed as reservoirs coupled to the device.However, the fundamental difference between the real S/D reservoirs and those represented by Bu ¨ttiker probes is that the probes can only change the electron energy/momentum and not the electron number within the device.This implies that one can view a Bu ¨ttiker probe as extracting electrons from the device, perturbing the energy/momentum of those electrons and reinjecting an equal number back into the system with a different energy/ momentum distribution.The coupling energy between the device and the probes can be adjusted to vary the scattering strength smoothly from zero ͑ballistic transport͒ to a high value ͑diffusive transport͒ as illustrated in Appendix A.
The Fermi energy characterizes how a reservoir exchanges carriers with the device.Since Bu ¨ttiker probes extract and inject electrons into the system, they have an associated Fermi energy that should be adjusted to achieve carrier conservation within the device.Carrier conservation at each scattering center ͑zero probe current͒ guarantees current continuity through the transistor even in the presence of scattering.
To include the effects of dissipative transport, we start with the 1D Hamiltonian ͓Eq.͑3͔͒ in the transmission direction ͑x͒ for mode i.This Hamiltonian is discretized on a finite difference grid to obtain a tridiagonal matrix of the form

͑4͒
where N X is the number of grid points and t x , the coupling energy between adjacent grid points along the channel.The coupling energy is

͑5͒
where a is the grid spacing and m x,i * , the electron effective mass in the x direction for mode i.We attach semi-infinite, 1D, Bu ¨ttiker probes to each mode i, as shown in Fig. 2. Two unknowns, namely the bandwidth and the band center, define the density of states within a probe.These parameters cannot be chosen arbitrarily since they affect the Bu ¨ttiker probe approximation.However, a comparison with the detailed NEGF scattering formalism ensures that the approximation gives physically reasonable results when the density of states within the probes coincides with that of the device.This is discussed further in Sec.III.Each probe is characterized by a coupling energy U p and by a Fermi level p ͑Fig.2͒.The quantity that we are interested in is the retarded Green's function for mode i within the intrinsic device region ͓the corresponding Hamiltonian is the matrix within the box in Eq. ͑4͔͒.The Green's function, at a specific longitudinal energy (E L ϭEϪE k j ), is Note that Eq. ͑4͒ represents an infinite 1D Hamiltonian ͑because the S/D contacts are infinite͒.However, all of the effects associated with coupling a finite device to infinite S/D reservoirs and the effect of scattering within the device can be accounted for by introducing an appropriate self-energy matrix, which is denoted by ⌺ i (E L ) ͑refer to Appendix A for details of the self-energy calculation͒.The self-energy concept allows us eliminate the semi-infinite reservoirs and work solely within the device subspace whose dimensions are much smaller.It is FIG.
2. The profile of a generic mode is illustrated along with the placement of Bu ¨ttiker probes.Note that the Fermi level of the S/D contacts is fixed, while that of the probe is adjusted to preserve the number of carriers.The probe self energy is related to the coupling strength.
Ϫt x e jk x,N X a ΅ ,

͑7͒
where jϭͱ(Ϫ1).On including the self-energy matrix ͓Eq.͑7͔͒, the size of the discrete Green's function matrix for mode i is N X 2 .Since Bu ¨ttiker probes represent isolated scattering centers, the self-energy matrix is diagonal and has nonzero entries only at those points where a probe has been introduced.Note that the self energy for a probe is proportional to U p ͓Eq. ͑7͔͒.Just as t x represents the coupling strength between adjacent points within the device, U p represents the coupling strength between points in the device and points in the probe.If this energy is large, it implies that an electron in the device can easily scatter into the probe.When U p tends to zero, there is no coupling between the device and the probes and electrons ballistically traverse the intrinsic device region.An analytic relation between the probe coupling energy and the mean free path is presented in Appendix B. It should be noted that although we use 1D probes, the nature of the probes can be easily modified by adjusting the probe self-energy.This may provide a useful technique for mimicing scattering processes in other systems.
Once the retarded Green's function is evaluated, electron density and current due to injection from the S/D contacts and all of the Bu ¨ttiker probes can be easily computed.We define a new quantity in terms of the self-energy as ⌫ i ϭ j(⌺ i Ϫ⌺ i † ). 9,7This broadening function describes the electron exchange rate between the active device and all of the reservoirs to which the device is coupled.The state spectral functions due to injection from the S/D and all probes for mode i is where n runs over all the reservoirs ͑including the S/D͒.Note that A i n is a matrix with the same size as G i , and that its diagonal entries constitute the local density of states ͑LDOS͒ due to injection from reservoir n Conceptually, the spectral function is proportional to perturbation strength ⌫ i n and propagates through the entire domain according to ͉G i mn ͉ 2 .Since G i mn ͑with a running index m͒ is the nth column of G, one does not need to calculate the entire G ͑computationally expensive͒ in order to obtain the spectral function.Only those columns corresponding to S/D contacts or Bu ¨ttiker probe positions need to be calculated.Thus it is clear that in the ballistic case, we need only the first and last columns of G, while with scattering turned on everywhere, the entire G needs to be evaluated.Transmission between any two reservoirs labeled m and n is Knowing the LDOS ͓Eq.͑9͔͒, the two-dimensional ͑2D͒ electron density at node m, for mode i, including the effect of all scattering centers and the S/D is where n is the reservoir index that runs over all the probes and the S/D, f the Fermi-Dirac function for reservoir n and ͱm y,i * /2ប 2 E k j the transverse mode state density for subband i ͑including spin degeneracy͒.Since the spectral function, (A i n ) mm depends on the longitudinal energy alone ͓Eq.͑8͔͒, it can be moved out of the integral in Eq. ͑11͒ which reduces to where the argument of the F Ϫ1/2 function has been normalized by k B T ͑for an analytic form of F Ϫ1/2 , refer to Ref. 16͒.The net 2D electron density at node m is obtained by summing contributions from all modes and valleys.In a similar fashion, the net current at reservoir m including contributions from all reservoirs ͑n͒, modes ͑labeled by i͒ and valleys is Note that the index n includes the S/D contacts as well and that the transmission between nodes m and n is as specified by Eq. ͑10͒.Also note that while the Fermi level of the S/D contacts is fixed by the applied voltage, the Fermi level of the Bu ¨ttiker probes has to be determined from current continuity.Current continuity requires that the net current at each probe equals zero.This implies that or for each E L at each probe ͑probe indices are denoted by m͒. 17 Equation ͑14͒ imposes a set of constraints on the Fermi potentials of the probes.These constraining equations are solved for the probe Fermi levels.Note that Eq. ͑14͒ provides two options to ensure that the net current at each scattering center equals zero.Both of these options are examined in this paper as they represent different phenomenological treatments of scattering.In the first model, we assume that the net current at each scatterer, when integrated over all E L equals zero i.e., ͐ Ϫϱ ϱ I m ͓E L ͔dE L ϭ0.This requirement implies that electrons from all modes are fully thermalized at each probe according to the corresponding probe Fermi potential and temperature ͑the local distribution is Fermi-Dirac͒.It also implies that the probe Fermi potentials are just position and not energy dependent.We will refer to this treatment of scattering as the energy relaxed probe model because the longitudinal energy of the electrons within the device is relaxed due to scattering.In the second model we assume that the current at each scatterer equals zero for each E L .Therefore, Eq. ͑14͒ is trivially satisfied.However, in this model, the Fermi level of the Bu ¨ttiker probes are both position and longitudinal energy dependent because we obtain a set of chemical potentials for the probes at each E L ͑Note that although we refer to the chemical potential of the probe as a Fermi potential for mathematical convenience, the distribution locally is not Fermi-Dirac͒.We will refer to this model as the phase breaking scattering model because the channel directed energy of the electrons is not relaxed although the channel directed momentum is relaxed.Irrespective of our choice of the scattering model, it should be noted that carrier populations in different modes are mixed as a result of scattering because the net current at a probe includes a sum over all modes.Therefore both models capture the effect of intervalley scattering.
In case of the energy relaxed model, the position dependent probe potentials can be adjusted iteratively using Newton's method because the constraining equations are nonlinear ͑due to the integration over E L ).The Jacobian matrix for the Newton iteration scheme is numerically evaluated as ͓Eqs.͑10͒ and ͑13͔͒ and the corrections to the probe potentials during the solution searching iterations as Note that the S/D Fermi levels are specified by the applied voltage, therefore the size of the Jacobian in Eq. ͑15͒ is (N X Ϫ2) 2 although the summation index n in Eq. ͑15͒, includes the S/D contacts.For the phase-breaking model, we directly solve for the occupancy function F Ϫ1/2 ( m ϪE L ), at each longitudinal energy using a linear solution scheme of the form AF Ϫ1/2 ϭB because there is no energy integral to be dealt with.Here, A and B, which are E L dependent are In Eq. ͑17͒, the summation index n runs over all the reservoirs including the S/D contacts for the diagonal terms and over the probes alone for the off-diagonal terms of A, while index m runs over the probes alone ( s is the source Fermi level and d the drain Fermi level͒.We mainly focus on the energy relaxing probe model in this article because this model seems to capture the essential physics of scattering more accurately within a nanoscale transistor when compared to the phase breaking model ͑Sec.IV͒. Once the probe distribution functions have been evaluated using either Eqs.͑15͒ and ͑16͒ or Eqs.͑17͒, the net 2D charge density (n i ) for mode i can be calculated by integrating Eq. ͑11͒ over E L .The 3D charge density at each node of our 2D real-space grid is obtained by multiplying n i with the corresponding distribution function ͉⌿ i (x,z)͉ 2 /b, and by summing over all modes ͑i's͒ and conduction band valleys.Since the eigenvalue problem is solved exactly in the z direction ͑Fig.1͒, quantum effects associated with confinement are accurately treated within our modeling scheme.This 3D density is used to solve Poisson's equation for a potential, and the self-consistent process repeated till convergence is achieved.

III. RESULTS
The simulated device structure ͑Fig.1͒ is an ultrathin body, fully depleted, symmetric, dual gate n-MOSFET with the S/D regions doped at 10 20 /cm Ϫ3 and an intrinsic channel.The gate length is 10 nm and there is no gate-to-S/D overlap.The S/D extensions are 10 nm and the junctions are abrupt.In order to highlight quantum effects and the effect of dissipative transport, we choose a thin silicon body ͑1.5 nm͒, which exhibits single mode occupancy.The oxide thickness ͑1 nm͒ and power supply voltage (V DD ϭ0.4 V) are set based on the International Technology Roadmap for Semiconductors ͑ITRS͒. 18We adjust the gate work function for both the top and bottom gates to obtain a ballistic off-current of 10 A/m, consistent with the ITRS requirement.Gate oxides are treated as infinite potential barriers for electrons in all of our simulations.For scattering simulations, the input parameter is a position dependent low field mobility ͑Appendix B, provides an analytical relation between the mobility and scattering strength͒.0][21] As the silicon film thickness is reduced, there are two competing effects which affect the mobility.Increased surface roughness scattering reduces the low field mobility, while confining all of the electrons to the unprimed bands raises the low field mobility. 22Therefore, as an approximation, we assume that these effects cancel, and that a low field mobility of 200 cm 2 /V s is a reasonable value to use in the channel region ͑our channel charge is also ϳ10 13 /cm Ϫ2 in the on state͒.For the S/D extensions which are highly doped, we use a doping dependent mobility model that yields a value of 55 cm 2 /V s at a donor doping concentration of ϳ10 20 /cm Ϫ3 .
In order to highlight quantum effects and present a general picture of how the Bu ¨ttiker probe models work, we compare internal quantities with and without scattering within our model device.Figure 3 shows the self-consistent current spectrum versus longitudinal energy ͓Eq.͑13͔͒ in the on state (V GS ϭV DS ϭ0.4 V) from both, the energy-relaxing ͓Fig.3͑a͔͒ and the phase-breaking scattering models ͓Fig.3͑b͔͒.The ballistic current is superposed on each of the figures for comparison.In the ballistic limit, electrons enter the device from the source and leave though the drain.Both, the number and the energy of the electrons are conserved throughout the device.This is clear from Fig. 3͑a͒, which indicates that the ballistic current spectra ͑light lines͒ at the source and drain ends are symmetric ͑while reading this plot, it should be noted that source injected current is positive for electrons entering the device, while the drain collected current is negative for electrons leaving the device͒.If we compare this ballistic spectra to the one obtained from the energy relaxed Bu ¨ttiker probe model, we observe that the source injected current is reduced in magnitude, and the drain collected current no longer mirrors the source current distribution.This is because this model strongly relaxes the channel directed energy of the electrons injected from the source.These electrons leave the device with lower longitudinal energies because they lose energy due to scattering.The phase breaking model, ͓Fig.3͑b͔͒ on the other hand, preserves the longitudinal energy of electrons, thus resulting in a symmetric spectrum ͑similar to the ballistic case͒.It should be noted that although the current spectra are symmetric, the current magnitude is reduced when compared to the ballistic case, because back scattered electrons lose their directed momentum due to scattering.We will mainly focus on results obtained using the energy relaxing model for the rest of this section.A more detailed comparison between the energy and phase relaxing scattering models is deferred to Sec.IV of this article.
Figure 4 compares the LDOS ͓Eq.͑9͔͒ and the 2D charge density spectra ͓Eq.͑11͔͒ versus longitudinal energy, with and without scattering (V GS ϭV DS ϭ0.4 V).Note that Fig. 4 is plotted for a qualitative comparison only and that white areas in the figure represent a high density.The first mode is superposed on each of the plots in Fig. 4 ͑dotted white line͒ to indicate the effective potential energy of electrons.In the ballistic case, there is no phase relaxation within the device.Therefore, states injected from the drain end of the device undergo reflections and interfere strongly to the right of the source-to-channel barrier.This interference results in coherent oscillations in the LDOS as seen in Fig. 4͑a͒ ͑left͒.When scattering is turned on throughout the device, phase information of the electrons within the device is randomized and the energy levels are broadened when compared to the ballistic case.Therefore, all of the interference effects are washed out as seen in Fig. 4͑a͒ ͑right͒.Note that both, the ballistic LDOS and the LDOS with scattering, exhibit nonzero values below the source-to-channel barrier resulting in source-to-channel tunneling.In the ballistic case, the charge density spectrum ͑the square root of the charge is plotted to resolve low charge densities at the drain end͒ can be resolved into two components; one due to source injection and the other due to drain injection.The source injected charge propagates from the source to the drain without any energy relaxation, resulting in a ballistic peak in the charge spectrum at the drain end, as seen in Fig. 4͑b͒ ͑left͒.The drain injected charge on the other hand, is completely reflected by the source-to-channel barrier in the on state.When scattering is turned on, the longitudinal energy of the electrons is relaxed ͑in case of the energy relaxing model͒ and the source and drain populations can no longer be distin-FIG.3. ͑a͒ The current spectrum from the energy relaxed scattering model is compared against the ballistic limit in the on-state (V GS ϭV DS ϭ0.4 V).Note that the drain current spectrum is relaxed in energy.͑b͒ The current spectrum from the phase relaxed scattering model is compared against the ballistic limit, in the on state.The source and drain spectra are identical in the presence of scattering, because this model relaxes channel directed momentum only.
guished.Also, any coherencies in the charge spectrum are also washed out as seen in Fig. 4͑b͒ ͑right͒.
Next, we look at the full range I DS versus V GS characteristics for the model device with and without scattering in Fig. 5͑a͒.Current in the off-state (V GS ϭ0, V DS ϭ0.4 V), is comprised of two components; source-to-channel tunneling, for longitudinal energies below the source-to-channel barrier, and diffusion, for energies above the barrier.Scattering, in general, broadens the LDOS as discussed earlier.Therefore, the tunneling current component is increased in the off-state as a result of scattering, when compared to the ballistic limit.However, the detailed NEGF scattering model indicates that the scattering rate should be proportional to the LDOS at each energy. 9Therefore, by choosing the energy band of the Bu ¨ttiker probes to coincide with the local energy band within the device ͓Eq.͑7͔͒, we ensure that the broadening in the LDOS due to scattering and thus the increase in the tunneling current is not uncontrolled.This is particularly important when modeling low power devices where the off-current is dominated by tunneling.
The diffusion current component reduces when scattering is turned on, because the degenerate thermal injection velocity is reduced when compared to the ballistic limit. 23herefore, the cumulative effect of reduced diffusion and increased tunneling in the presence of scattering, is that the ballistic off current is always higher than the off current in the presence of scattering.This is clearly seen from Fig. 5͑b͒, where we plot the off current versus channel length for channel lengths down to 5 nm.The ballistic off current sets an upper limit on the leakage current and provides a fairly accurate picture of the subthreshold behavior as transistors are scaled to smaller dimensions.
The Fermi level of each scatterer is adjusted to conserve current in the presence of scattering ͓Eqs.͑15͒ and ͑16͔͒.This quantity, which is analogous to the quasi Fermi level computed in semiclassial models, has a clear physical interpretation.Unlike the quasi Fermi level ͑which is derived from the charge instead of the current͒, the Fermi level of the Bu ¨ttiker probes is an actual representation of how the potential drops from the source to the drain.Figure 6͑a͒, plots this quantity in the linear region (V DS ϭ10 mV) of operation.In the off state, the channel resistance is high.Therefore all of the applied voltage drops in the channel region of the device.As the gate voltage increases, the channel conductivity increases, and the voltage dropped in the channel is reduced.This leads to a flattened Fermi potential profile in the channel and large voltage drops in the S/D regions.In the ballistic limit, there is no mechanism for internal voltage drop and all of the V DS is dropped across the contact/device interface, resulting in a finite ballistic current.Note that the source FIG. 4. ͑a͒ The local density of states in the on-state, from the ballistic ͑left͒ and energy relaxed scattering models ͑right͒.The first mode is also plotted ͑dotted line͒.Note that coherent oscillations in the LDOS are washed out when scattering is turned on.͑b͒ The charge density spectrum from the ballistic ͑left͒ and energy relaxed scattering models ͑right͒, in the on state.In the ballistic limit, the source and drain injected populations can be clearly identified.When scattering is turned on, these populations are mixed.
͑drain͒ Fermi potential represents the Fermi energy of source ͑drain͒ injected carriers.In the off state there is very little current flow within the device and near equilibrium conditions prevail at both, the source/device and the drain/device interfaces.However, as the device is turned on, the distribution at the source/device interface ͑and the drain/device in-terface͒ is strongly off equilibrium in order to maintain a large current flow.This leads to the observed discontinuity in the Fermi potential at the contact/device interface as seen in Fig. 6͑a͒.
In the linear region, it is possible to derive the sheet resistivity versus position along the channel, from the probe Fermi levels using 24 sh ϭ ‫‪x‬ץ/ץ‬ The derived resistivity, at low drain and high gate voltage (V DS ϭ10 mV, V GS ϭ0.4 V) is plotted in Fig. 6͑b͒.The sheet resistivity can be divided into the following regions: ͑1͒ quantum contact resistance, ͑2͒ S/D extension resistance, ͑3͒ tip resistance, and ͑4͒ channel resistance.Note that the gate modulates the channel resistance and a fraction of the tip resistance only.Therefore, in our device ͑with a channel mobility of 200 cm 2 /V s͒, the on current is primarily limited by the S/D parasitic resistances ͑S/D mobility is just 55 cm 2 /V s͒. Figure 7͑a͒ plots the I DS versus V DS characteristics in the on state (V GS ϭV DD ϭ0.4 V).For a ballistic off current of 10 A/m, the simulated on current in the presence of scattering, is only 50% of the ballistic limit due to S/D and tip parasitics.To reinforce this point, we plot the on current as a function of channel mobility in Fig. 7͑b͒.The channel mobility is progressively increased from zero to very high values.It is clear from Fig. 7͑b͒, that the on current saturates at ϳ55% of the ballistic limit, and does not increase with increasing channel mobility because of the parasistic resistances.It is expected that transistors with a double-gate geometry will yield twice the on current when compared against those with a single gate ͑SG͒ geometry for the same level of off current.This expectation will definitely be met in the ballistic limit.However, if series resistance limits device performance, the performance of a DG MOSFET is degraded to a greater extent than a transistor with a SG geometry.Therefore, the performance benefit expected from DG tran- sistors may not be met.Our model enables us compare DG and SG MOSFETs from the ballistic to the diffusive limit and serves as a valuable design tool to evaluate device design and performance.It captures the essential physics of ballistic and dissipative transport in nanoscale, SOI transistors.

IV. DISCUSSION
In this section we assess the behavior of the energy and phase relaxing scattering models in order to ascertain their applicability to modeling electron transport within a MOSFET.To understand the physics of scattering that is captured by each model, we compare their performance to an analytic theory presented in Ref. 23, where the essential physics of scattering is outlined.We choose to compare the energy and phase relaxing scattering models nonself consistently, in order to avoid the complicated behavior associated with self-consistent electrostatics. 25Self consistency is important, and we discuss it briefly at the end of this section.Scattering can occur anywhere inside the device, but as shown in Ref. 23, those scattering events that occur in the low field region near the source have the largest effect on the on current.To understand why the importance of backscattering reduces from the source to the drain, we consider a single electron injected from the source into the channel with a total energy, E ͑note that E L is its longitudinal energy͒.Figure 8, tracks this electron in 2D momentum space ͑initial state is the solid arrow in Fig. 8͒.Now assume that this electron undergoes a single elastic, isotropic scattering event in the channel ͑final state is the dotted arrow in Fig. 8͒.For this electron to make it ballistically back to the source, its longitudinal energy should be greater than ⑀ max ͑Fig.2͒.It is clear from Fig. 8 that only a small cone of electrons have enough longitudinal energy to backscatter into the source and that this cone reduces as we move towards the drain (⑀ max increases, and the dotted circle approaches the solid circle in Fig. 8͒.Therefore, scattering near the source affects the current more strongly than near the drain.
Figure 9 shows the effect of scatterer placement on the device performance from both, the energy and phase relaxing FIG. 7. ͑a͒ I DS vs V DS characteristics from the ballistic ͑solid line͒ and energy relaxed scattering models ͑dashed line͒ for V GS ϭ0.4 V.The oncurrent in the presence of scattering is ϳ50% of the ballistic limit.͑b͒ The on current vs channel mobility is plotted to indicate that the ultimate performance of our device is primarily controlled by device parasitics.FIG. 8.A pictorial representation of the essential physics of scattering ͑also refer Fig. 2͒.The fraction of the scattered electrons that can surmount the source-to-channel barrier and make it back into the source, reduces as we move towards the drain.These carriers, whose total energy is E, are delineated by the cone.FIG. 9.The current as a function of the scatterer number ͑placed only in the channel region͒, which is progressively increased from the drain to the source is plotted from both, the energy relaxing ͑solid line͒ and the phase relaxing scattering models.This plot indicates that the energy relaxed model captures the essential physics of scattering in transistors.models.This figure is generated by fixing the potential profile ͑ballistic on state͒ and increasing the number of scatterers progressively from the drain towards the source.When the scatterer number is zero, both models yield the ballistic limit.But as the scatterer number is increased, the energy relaxing scattering model shows very little change in current initially and a linear drop thereafter.When scatterers are placed near the end of the channel, and scattering treated using the energy relaxing model, electrons scatter near the drain and lose much of their longitudinal energies.This makes it difficult for a backscattered electron to make it back to the source.Therefore, the current is initially unaffected.However, as the number of scatterers increases towards the source, electrons scattering very near the source-to-channel barrier do backscatter back into the source as they have not dissipated enough longitudinal energy, thus reducing the current.The phase breaking model shows distinctly different results.The transmitted current decreases with increasing number of scatterers irrespective of their location.This is because, in this model, electrons can reverse direction without losing their channel directed energy.Therefore, scattering anywhere in the channel can reflect electrons back into the source.It should be noted that our qualitative argument considered the effect of elastic scattering alone, and that the addition of any other scattering mechanism serves to relax the longitudinal energy even more.This further reduces the probability ͑non-self-consistently͒ of an electron backscattering all the way to the source from the drain end of the device.Therefore, an assessment of the two models clearly indicates that the energy relaxing scattering model, which captures the essential physics of scattering, is the better model to simulate dissipative transport in nanoscale transistors.
Having selected the energy relaxing scattering model, we use it to examine the importance of scattering near the source and drain ends of the device including self-consistent effects.We divide the device into two halves and consider two cases; In the first case, scattering ͑mobility of 100 cm 2 /V s͒ is turned on in the first half of the device while the second half is ballistic, and in the second case scattering is turned on in the second half of the device while the first half is ballistic.Our self-consistent simulation results using the energy relaxing model indicate that irrespective of the region where scattering is turned on, the on current is significantly degraded ͑ϳ43% in the first case and ϳ33% in the second case͒ when compared to the ballistic limit.To understand this behavior, we plot the self-consistent subband profile and 2D charge density for both cases in Fig. 10.When scattering is introduced only in the first half of the device, the on current is strongly degraded when compared to the ballistic limit due to backscattering of electrons at and around the top of the source-to-channel barrier ͑Fig.10͒.Scattered electrons can easily re-enter the source region as they still preserve most of their channel directed energy and hence reduce the net current.When scattering is turned on in the second half of the device alone, the reduction in on current is due to a complicated interplay between self consistency and the effect of scattering. 25The ballistic stream of source injected electrons entering the second half of the device undergoes scattering.
Once reflected, if the mean free path of these electrons is comparable to the channel length ͑mean free path is ϳ10 nm for a mobility of 100 cm 2 /V s͒ a fraction of these scattered electrons makes it back to the source.This is the first mechanism that reduces the on current.The second mechanism is because the electrons reflected in the second half of the device result in an increased 2D electron density in the channel as shown in Fig. 10 ͑dotted line͒.Self consistency causes the subband potential to float to higher energies in the channel and also broadens the potential profile from the source to the drain.An increased source-to-channel barrier combined with a broadened potential profile ͑which increases the probability of a backscattered electron to re-enter the source as seen from Fig. 8͒ causes the on current to decrease further when scattering is turned on the second half of the device.This self-consistent behavior of the on current as a function of scatterer placement clearly indicates that scattering is important not only at the source but throughout the channel in nanoscale transistors.This behavior of the on current, when the channel length is comparable to or shorter than the mean free path, has also been observed when scattering is treated rigorously using the Green's function formalism by Ref. 25.The energy relaxed, Bu ¨ttiker probe based scattering model thus captures all the essential physics of scattering within a MOSFET, including self-consistent effects in a simple and elegant fashion.

V. SUMMARY
We presented a computationally efficient method to quantum mechanically treat the effects of dissipative transport in thin body, fully depleted, SOI transistors including the effect of degeneracy.The proposed model is single parameter model, and the quantum mechanical parameter was related to a low field mobility using simple analytical expressions.We then applied our model to treat the effects of dissipative transport in an ultrathin body ͑1.5 nm͒, DG, FIG. 10.The profile of the first mode and the 2D electron density in the on state with scattering turned on in the first half of the device ͑solid lines͒ and with scattering turned on in the second half of the device ͑dashed lines͒.Note that the potential drops in the source or the drain only when scattering is turned on.Also note that turning on scattering only in the second half of the device increases the 2D electron density in the channel.
n-MOSFET.In doing so, quantum effects that are observed in nanoscale transistors, the role of scattering, effect of parasitics and the treatment of open boundaries were highlighted and discussed.We also presented and discussed the essential physics of scattering in relation to the energy and phase relaxing scattering models and showed that the energy relaxing model was better suited for modeling electron transport in transistors.
Our scattering model is a one parameter model, and the parameter we use can be analytically related to a low field mobility.Such a relation, enables the use of our model to explore physics and device design issues in nanoscale transistors because it can be calibrated to experimentally measured mobility data.This scattering model is also very useful because the detailed NEGF formalism can be used to motivate changes to the nature of the Bu ¨ttiker probes to mimic the effects of scattering in different material systems such as carbon nanotubes and molecules by appropriately changing the probe self energy.Our model provides an excellent tradeoff between increased computational cost and the physics of scattering that needs to be captured in devices of the future.

ACKNOWLEDGMENTS
The authors would like to thank the Semiconductor Research Corporation for funding this project.The authors are indebted to Zhibin Ren ͑IBM͒ whose PhD thesis laid the foundation for the work presented in this article.Useful discussions with J. Rhew and P. Damle from Purdue University, M. Anantram from NASA, and D. Jovanovic from Motorola are also appreciated.R. Venugopal is supported by an Intel PhD Fellowship.

APPENDIX A: THE SELF-ENERGY CALCULATION FOR THE BU ¨TTIKER PROBES
To illustrate the self-energy calculation which accounts for the device leads and the Bu ¨ttiker probes, we consider the effect of coupling the Hamiltonian for mode i ͓Eq.͑4͔͒ to a probe at site m ͑Fig.2͒.It is always possible to number the nodes within the mode ͑i͒ and the probe, such that the Hamiltonian for the mode plus the probe can be expressed as

͑A2͒
Note that the Hamiltonian representing the probe at site m has the same form as the Hamiltonian for mode i and that the potential within the probe is assumed to be a constant ͑fixed by the potential at node m for mode i͒.Therefore, the diagonal elements of the probe Hamiltonian repeat themselves.The choice of E i (m) and t x to be the same in the probe and the device ensures that the DOS spectrum within the probe coincides with that within the device.The reason for this choice is explained in Sec.II.The Green's function can in turn be partitioned as The matrix block we are interested in is G i,device as we do not care about the Green's function within the probe.Using Eq. ͑A3͒ G i,device can be expressed in terms of known quantities as 7 where the probe self-energy matrix is

͑A5͒
Note that for evaluating the matrix product in Eq. ͑A5͒, we only need the first element of the inverse of the infinite matrix associated with the probe.Also, note that the diagonal blocks of this infinite matrix are repeated due to invariance of the potential within the probe ͓Eq.͑A2͔͒.Using this property, and partitioning the matrix as shown in Eq. ͑A5͒, a closed form expression for the first element of the inverse ͑denoted by g i,probe ) of the infinite matrix can be obtained as Once g i,probe has been solved for, we have Note that only node m of the device couples to the probe.Therefore the self energy for the probe ͓Eq.͑A7͔͒ has a single nonzero entry that perturbs the (m,m)th diagonal entry of the mode-space Hamiltonian.By replacing U p m , with t x,i in Eq. ͑A7͒ we can easily obtain the self-energy for the S/D leads in a similar manner.

APPENDIX B: THE RELATION BETWEEN THE BU ¨TTIKER PROBE SELF ENERGY AND THE LOW FIELD MOBILITY
In order to relate the probe self energy to a mobility, we first relate it to a mean free path, i .To do so, consider a single mode ͑we drop the subscript i for convenience͒ with a uniform potential, which is coupled to S/D contacts and N identical Bu ¨ttiker probes, each of which has the same coupling strength, U p .The spacing between adjacent probes is a.If a unit amplitude is injected from the source ͑left con-tact͒, the net transmission ͓denoted T(N)] from the source to the drain can be expressed as where T p is the net transmission across a single probe.Note that Eq. ͑B1͒ has been derived by invoking the additive property of (1ϪT p )/T p . 9 The net transmission across a single probe, in terms of the transmission into the probe ͑denoted by T p in ) is The reason T p does not equal T p in , is because each probe isotropically reinjects electrons into the system in order to conserve charge.Substituting Eq. ͑B2͒ into Eq.͑B1͒ and solving for 1/T(N) we obtain where L is the distance from the source to the drain.The transmission into a single probe is obtained analytically using Eq.͑10͒.In the weak scattering limit (U p ϳ0, therefore T p in ϳ0), the final form of the mean free path ͓from Eq. ͑B3͔͒ in terms of the lattice spacing a is Once the mean free path is obtained, it can be related to the diffusion coefficient through Schokley's relation, which inturn is related to the low field mobility through Einstien's relation. 23The relation between the low field mobility and the mean free path is mobilityϭͱ q2 2k B Tm x * ϫ F Ϫ1/2 ͑ ϪE mode ͒F Ϫ1 ͑ ϪE mode ͒ F 0 2 ͑ ϪE mode ͒ , ͑B5͒ where the arguments to the Fermi functions have been normalized by k B T and E mode represents the potential energy of electrons in a specific mode.Equations ͑B4͒ and ͑B5͒ relate the low field mobility to the Bu ¨ttiker probe strength.Note that the degeneracy factors in Eq. ͑B5͒ are position dependent because the mode energy is position dependent.Also note that in our analysis, we assumed single mode occupancy.If several modes are occupied, the mean free path can be interpreted as an average mean free path for all modes and the equivalent mobility as an average low-field mobility for all electrons.The Bu ¨ttiker probe strength in this case is adjusted differently for each mode ͑because t x is mode depen-dent͒ to reflect the same average mean free path for all modes.
In order to verify the validity of our derivation, we simulate a uniformly doped resistor ͑thicknessϭ1.5nm, lengthϭ40 nm, Nϭ161, and dopingϭ10 20 /cm3 ) self consistently in the linear response region ͑low V DS ) for different probe strengths ͓or alternatively, mean free paths evaluated using Eq.͑B4͔͒.The low bias conductance extracted from numerical simulations is compared against analytical values obtained using Eq.͑B3͒ in Fig. 11.͓Note that the conductance is directly proportional to T(N).] 9 Based on our analytical expressions ͓Eqs.͑B3͒ and ͑B4͔͒, we expect a linear relationship between the inverse of the conductance and the inverse of the mean free path.It is clear from Fig. 11 that the numerical and analytical values are in close agreement ͑within 10%͒ over a wide range of scattering strengths ranging from a mean free path of ϱ ͑pure ballistic transport͒, down to 5 nm.This plot clearly validates our interpretation of the Bu ¨ttiker probe strength and enables us calibrate quantum mechanical parameters to experimental mobility data in a simple and elegant fashion.The slight discrepancy between the numerical and analytical values is because our analytical expressions are derived in the weak scattering limit.

FIG. 1 .
FIG.1.An ultrathin body DG MOSFET structure with S/D doping of 10 20 cm Ϫ3 and an intrinsic channel ͑channel thicknessϭ1.5 nm͒.A slice of the device within which a 1D, z directed effective mass equation is solved, is also indicated.

FIG. 5 .
FIG.5.͑a͒ I DS vs V GS characteristics from the ballistic ͑solid line͒ and energy relaxed scattering models ͑dashed line͒ for V DS ϭ0.4 V.The off current, from the scattering model is lower despite an increase in the tunneling current due to a broadening in the LDOS below the source-to-channel barrier.͑b͒ The off current vs channel length from the ballistic ͑solid line͒ and energy relaxed scattering models ͑dashed line͒.Ballistic simulations are good enough to evaluate leakage and subthreshold characteristics.

FIG. 11 .
FIG.11.The low bias conductance for a 40 nm resistor is plotted as a function of scattering strength, from analytical ͑solid line͒ expressions and self-consistent simulations ͑dashed line͒.The analytic conductance matches the simulated value to within ϳ10%, indicating that our relation between the Bu ¨ttiker probe strength and the mean free path ͓Eq.͑B4͔͒ is physically correct.