A Drift-Diffusion Equation for Ballistic Transport in Nanoscale Metal-Oxide-Semiconductor Field Effect Transistors

We develop a drift-diffusion equation that describes ballistic transport in a nanoscale metal-oxide-semiconductor field effect transistor (MOSFET). We treat injection from different contacts separately, and describe each injection with a set of extended McKelvey one-flux equations [Phys. Rev. 123, 51 (1961); 125, 1570 (1962)] that include hierarchy closure approximations appropriate for high-field ballistic transport and degenerate carrier statistics. We then reexpress the extended one-flux equations in a drift-diffusion form with a properly defined Einstein relationship. The results obtained for a nanoscale MOSFET show excellent agreement with the solution of the ballistic Boltzmann transport equation with no fitting parameters. These results show that a macroscopic transport model based on the moments of the Boltzmann transport equation can describe ballistic transport.


I. INTRODUCTION
As transistors are scaled down to their ultimate limit, carrier transport may approach the ballistic limit.This imposes a great challenge for predictive assessments of device performance, especially the on current of transistors, because ͑1͒ commonly used macroscopic transport models assuming collision-dominated transport are expected to lose their validity near the ballistic limit 1,2 and ͑2͒ computational burdens of reliable first-principles simulators ͑Monte Carlo simulators 3 or full Boltzmann solvers 4 ͒ preclude them from routine use for extensive design studies.A macroscopic model derived from the moments of the Boltzmann transport equation ͑BTE͒ capable of describing carrier transport from the diffusive to the ballistic limit would be of great interest.Because conventional moment-based macroscopic models ͑e.g., drift-diffusion, or energy transport models͒ fail in the ballistic limit, 1,2 we turn our attention to McKelvey's oneflux method, 5,6 whose usefulness in qualitatively describing quasiballistic transport was demonstrated in the scattering theory. 7But, the one-flux method is unable to describe carrier acceleration in a high-field region ͑e.g., the channel or the collector of a transistor͒. 8In this article, we extend McKelvey's one-flux method and develop a drift-diffusion equation to describe pure ballistic transport in a nanoscale metal-oxide-semiconductor field effect transistor ͑MOS-FET͒.
We solve the ballistic drift-diffusion equation for the model device shown in Fig. 1 and compare the results to the ballistic BTE solution obtained in Ref. 9. The model device is a 10-nm channel-length, double-gate MOSFET with an ultrathin body thickness of 1.5 nm.The gate oxides are 1.5 nm and both top and bottom gates have a midgap work function.The strong quantum confinement across the thin body provides two simplifying assumptions: ͑1͒ one subband model for which we assume all carriers are accommodated in the lowest subband and ͑2͒ quasi-two-dimensional ͑2D͒ simulation in which we solve one-dimensional ͑1D͒ transport along the body and a 1D Schro ¨dinger equation across the body selfconsistently with a 2D Poisson equation. 9his article is organized as follows.In Sec.II, we examine the assumptions employed in the one-flux method.In Sec.III, we present the set of extended one-flux equations that can describe ballistic transport.In Sec.IV, we convert the extended one-flux equations into a drift-diffusion equation and solve in the ballistic limit for our model device, and compare the results against the solution of ballistic BTE.We then conclude in Sec.V.

II. MCKELVEY'S ONE-FLUX METHOD
Figure 2 illustrates the idea of McKelvey's one-flux method. 5,6The two flux densities, J ϩ (x) and J Ϫ (xϩdx) ͑defined positively͒, incident on a semiconductor slab with thickness dx transmit or reflect with the backscattering probabilities per length, and Ј, respectively, contributing to the outward fluxes J Ϫ (x) and J ϩ (xϩdx).This is described by the one-flux equations, 5,6 which are given as where the backscattering probabilities per length are and The low-field backscattering probability per length 0 is associated with the low-field mobility 0 through Shockley's relation, 6 and T L is the lattice temperature.See Appendix A for a brief derivation of Eqs.͑2.1a͒-͑2.1d͒.The backscattering coefficients and Ј consist of two terms: ͑1͒ backscattering by actual scattering ( 0 ) and ͑2͒ backscattering by reflection from an opposing electric field ( x ) as shown in Fig. 3.In Eqs.͑2.1c͒ and ͑2.1d͒, the effect of actual scattering is assumed to be symmetric under a low-field condition.By specifying the incoming fluxes at the boundaries, we can solve Eqs.͑2.1͒ for J Ϯ .However, to solve them self consistently with the Poisson equation, we need to know the carrier densities n Ϯ associated with J Ϯ , which can be obtained by specifying the average velocities of Ϯ streams,

͑2.2͒
As in conventional moment equations, the average velocities are obtained in two ways ͑1͒ by solving the next order moment equations or ͑2͒ by employing hierarchy closure approximations. 10The one-flux method closes the hierarchy by assuming ͗ x ͘ Ϯ to be the average velocity of hemi- Maxwellian, i.e., Thanks to its flux description, the one-flux method successfully describes carrier transport from the diffusive to the ballistic limit when there is negligible acceleration, e.g., transport across a thin base. 11However, the one-flux equations are derived from the zeroth moments of the BTE, 8 in which the numbers of carriers in streams do not change due to acceleration.Thus, the field terms describe only backscattering and are associated with hemi-Maxwellian distributions in nondegenerate conditions ͑see Appendix A͒.The effect of acceleration such as velocity overshoot should be cast into hierarchy closure approximations on ͗ x ͘ Ϯ , which is missing in Eq. ͑2.3͒.In conclusion, the one-flux method is valid under nondegenerate conditions with negligible acceleration.

A. Separating injections, closure approximations, and degenerate statistics
Figure 4 shows types of transport in a nanoscale MOS-FET.In the channel region after the source barrier, the ϩ stream from the source (J S ϩ ) is accelerated by the electric field whereas the injection from the drain contact (J D Ϫ ) is decelerated and backscattered.Different populations experience different types of transport in the same region.Therefore, it is natural to treat the source-injected fluxes separately from the drain-injected fluxes and then describe each injection with the one-flux equations as illustrated in Fig. 5.In other words, we solve one set of flux equations for the source-injected carriers and another for the drain-injected carriers.The total carrier density and the net current are the sum of the quantities obtained for each injection.The idea of separating injections is based on the linearity of the ballistic BTE, which we can solve for the source injection and the drain injection separately and then obtain the final distribution as a superposition of the two.This approach enables us to apply different macroscopic approximations ͑scattering parameters and hierarchy closure assumptions͒ to sets of one-flux equations that describe different populations.It is obvious that we can use the original one-flux equations for streams under deceleration ͑e.g., drain-injected fluxes under bias͒, but new approximations are required to describe streams under acceleration.Another modification required for a nanoscale MOSFET is to include degenerate carrier statistics because they may affect the on current. 9In the ballistic limit, we can implement degenerate statistics into each injection separately.
In conclusion, we will extend McKelvey's one-flux method by ͑1͒ treating carriers injected from the source and drain separately, ͑2͒ introducing hierarchy closure approximations for the streams under acceleration, and ͑3͒ including degenerate carrier statistics.

B. Ballistic one-flux equations with degeneracy for nanoscale MOSFETs
In the ballistic limit ( 0 ϭ0) with degenerate carrier statistics, the backscattering coefficients become ͑see Appendix B for a derivation.͒ and .

͑3.1b͒
For 2D electrons in the lowest subband of our model device, the degeneracy factors are where F Ϫ1/2 and F 1/2 are Fermi-Dirac integrals of order Ϫ1/2 and 1/2, and the normalized energies are in which ϩ and Ϫ are the Fermi levels associated with ϩ and Ϫ stream, respectively, and E S (x) is the lowest subband energy of our model device.The Fermi levels depend on whether a stream comes from the source or from the drain contact. 9or the source injection, we solve Eqs.͑2.1a͒ and ͑2.1b͒ with the backscattering coefficients in Eqs.͑3.1͒, where the normalized energies are where S is the Fermi level of the source contact and x max is the position of the top of the source barrier.In the region x max ϽxϽL where x Ͻ0 in Fig. 4, Ϫ (x) can be any finite value because there is no source-injected negative stream in the ballistic limit.Thus, Eq. ͑3.4b͒ causes no error in the ballistic limit.In a similar way, we solve Eqs.͑2.1a͒ and ͑2.1b͒ for the drain injection with where D is the Fermi level of the drain contact.
To solve Eqs.͑2.1a͒ and ͑2.1b͒, we need to specify ͑1͒ hierarchy closure assumptions on the average velocities of source and drain injections ͗ x ͘ S Ϯ and ͗ x ͘ D Ϯ , and ͑2͒ boundary conditions for source and drain injections.

C. Ballistic hierarchy closure approximations for streams under acceleration
Figure 6 depicts the development of ballistic peaks along the channel of the model device, 9 and Fig. 7 shows the corresponding average velocity profiles under bias.Figure 6 implies that in the ballistic limit, the distribution function approaches a Dirac delta function, and can be written as ͑assuming parabolic band structure͒ where and E max is the maximum of E S (x) ͑Fig.4͒ and m t * is the transverse effective mass of ellipsoidal valleys of Si.As shown in Fig. 7, in the limit where the ␦-peak approximation applies, the distribution in Eq. ͑3.6͒ yields where ᐉ is the order of moments.Consequently, Eq. ͑3.8͒ allows us to terminate the hierarchy of macroscopic moments.This is analogous to Baraff's maximum anisotropy closure for the spherical harmonics expansion of the distribution function. 12 build ballistic closures for ͗ x ͘ S Ϯ and ͗ x ͘ D Ϯ based on the above argument.In 0ϽxϽx max , only J D Ϫ experiences acceleration and in x max ϽxϽL, only J S ϩ does.Hence, the following closure assumptions can apply: and x max ϽxϽL .

͑3.9d͒
The degenerate thermal velocity ˜T() for a twodimensional electron gas ͑2DEG͒ is given as 13 ˜T͑ ͒ϭͱ There are two things worth noting in Eqs.͑3.9͒.First, the closures in Eqs.͑3.9a͒ and ͑3.9d͒ reduce to near-equilibrium closures at x max because E max ϪE S (x)ϭ0 capturing the injection limit properly, but become high-field ballistic closures satisfying Eq. ͑3.8͒ where the accelerated carriers develop to a ␦ peak away from x max .Second, ͗ x ͘ S Ϫ in x max ϽxϽL and ϩ in 0ϽxϽx max are not specified because in the ballistic limit, J S Ϫ and J D ϩ do not exist in the respective regions.

D. Boundary conditions
To solve Eqs.͑2.1a͒ and ͑2.1b͒ self consistently with the Poisson equation, the carrier densities and flux densities of the streams coming into the device should be specified.Those quantities are directly obtained from the distributions given at the contacts.For the source injection, the boundary distributions of the incoming fluxes are at the source contact, and at the drain contact if perfect absorbing contacts are assumed. 9In Eqs.͑3.10͒, the total energy of a 2D electron in the x -y plane ͑due to the vertical confinement in z in our model device͒ is

͑3.11͒
For the drain injection, the boundary distribution given at the drain contact is and at the source contact, it is Then, integrating Eqs.͑3.10͒ and ͑3.12͒, we obtain the boundary conditions, which are for the source injection, and for the drain injection, respectively.
In principle, we solve Eqs.͑2.1a͒ and ͑2.1b͒ with Eqs.͑3.13͒ for the source injection, and with Eqs.͑3.14͒ for the drain injection.However, we convert the extended one-flux equations and the boundary conditions into a drift-diffusion form and use Scharfetter-Gummel discretization method. 14here the quantities J and N are defined as

A. Conversion into drift-diffusion equation
The thermal velocity T given in Eq. ͑2.3͒ just defines the unit of N. Note that J denotes the net flux density but that N is not the actual carrier density but simply represents the sum of J Ϯ in the unit of carrier density.Thus, Eq. ͑4.2͒ is not restricted to the carriers moving at a fixed velocity T .Using Eqs.͑2.1c͒ and ͑2.1d͒ with the implementation of degenerate statistics, Eq. ͑4.2͒ can be expressed as a driftdiffusion equation The equivalent mobility M and diffusivity ⌬ are defined as where .

͑4.5d͒
Although Eq. ͑4.5a͒ is in a drift-diffusion form, it describes spatial variation of flux densities due to transmission and reflection as in the original one-flux equations.In the nearequilibrium diffusive limit, Eq. ͑4.2͒ reduces to a driftdiffusion equation and the equivalent mobility and diffusivity become the low-field mobility and corresponding diffusivity.6 However, when 0 ϭ0, Eqs.͑4.5͒ describe ballistic transport.Equation ͑4.5c͒ is an equivalent Einstein relation under degenerate conditions, which originates from the equilibrium Fermi-Dirac distribution associated with field backscattering ͑see Appendix B͒.For degenerate bulk semiconductors, the Einstein relation is 15 which would yield the degeneracy factor for the 2D carriers in our model device as where n is the carrier density.However, the degeneracy factors in Eq. ͑3.2͒ imply that Equation ͑4.8͒ reduces to Eq. ͑4.7͒ when the degeneracy associated with the average velocities is independent of position, which is true in uniform bulk semiconductors.

B. Boundary conditions
Using Eqs.͑4.3͒ and ͑4.4͒, the boundary conditions for Eqs.͑4.1͒ and ͑4.5a͒ are expressed as, at the source contact and at the drain contact where J ϩ (0) and J Ϫ (L) are given in Eqs.͑3.13͒ for the source injection and Eqs.͑3.14͒ for the drain injection.We solve Eqs.͑4.1͒ and ͑4.5͒ for the source injection using the Scharfetter-Gummel discretization technique 14 with the boundary conditions given by Eqs.͑3.13͒ and ͑4.9͒ to obtain N S (x) and J S (x), i.e., J S ϩ and J S Ϫ and then using the closure assumptions in Eqs.͑3.9a͒ and ͑3.9b͒, we get the carrier densities of source injection n S ϩ and n S Ϫ because J S Ϯ ϭn S Ϯ ͗ x ͘ S Ϯ .Similarly, we solve Eqs.͑4.1͒ and ͑4.5͒ for the drain injection using Eqs.͑3.14͒ and ͑4.9͒ to obtain J D ϩ and J D Ϫ and then n D ϩ and n D Ϫ .The total carrier density is then and the total flux net density is

C. Results
The I D -V DS characteristics and injection quantities versus drain bias ͑injection velocity and injection charge density at the top of the barrier͒ are plotted in the left column of Fig. 8.They are in excellent agreement with the solution of the ballistic BTE 9 showing that the closure approximations in Eqs.͑3.9͒ do not cause error in capturing the injection limit.We then plot profiles of internal quantities ͑the first subband profile, internal velocity, and charge density versus position͒ in the right column of Fig. 8. Also, they show excellent overall agreement with the solution of the ballistic BTE except the tolerable errors in the low-field region after the barrier where the delta-peak assumption in Eq. ͑3.8͒ is not valid.Thus, the results show that a macroscopic transport model based on the moments of the BTE can describe carrier transport in the ballistic limit.

V. CONCLUSION
We derived and solved a drift-diffusion equation in the ballistic limit for a nanoscale MOSFET.The equation is developed extending McKelvey's one-flux method by ͑1͒ treating carrier injection from the source and drain separately, ͑2͒ introducing hierarchy closure approximations for the streams under acceleration, and ͑3͒ including degenerate carrier statistics involving a properly defined Einstein relation.The results show that a moment-based macroscopic transport model can describe ballistic transport in excellent agreement with the solution of the ballistic BTE with no fitting parameters.
The development of the ballistic macroscopic equation was relatively simple because ͑1͒ we were able to treat injection from the source and drain separately due to the lin-earity of the ballistic BTE and ͑2͒ the hierarchy closure approximations were independent of scattering.When scattering is present, however, those two simplifications should be reexamined.First, carrier-carrier scattering under degenerate conditions may couple the source and the drain injections ͑the BTE becomes nonlinear͒ weakening the assumption of separating injections in principle.Second, the scattering and the accelerating electric field brings complicated transport because the strength of scattering depends on the carriers' kinetic energy, which again depends on the interplay of the accelerating field and the impeding scattering.Describing these phenomena in an acceptable error range still remains the most difficult challenge in developing a macroscopic model valid in the high-field quasiballistic regime.
where A is a normalization area of the y-z plane.Note that f (0, p y , p z ) represents stationary carriers in the x direction, which are generated by the decelerating electric field x ͑see Fig. 3͒.Hence, it is reasonable to assume that the distribution associated with a stream under deceleration remains in a near-equilibrium shape as in thermionic emission.In nondegenerate conditions, a hemi-Maxwellian, f ϳexp(ϪE/k B T L ), can be assumed.Then, from Eq. ͑A3͒, we obtain The averages of the scattering term of the BTE are expressed as where n Ϯ are carrier densities in Ϯ streams and Ϯ are corresponding macroscopic relaxation times associated with backscattering.Although Eq. ͑A7͒ can be derived rigorously from the scattering integral of the BTE, the following phenomenological explanation verifies Eq. ͑A7͒.The averages of the BTE over ⍀ Ϯ yield the rate equations for carrier densities n Ϯ .Thus, the backscattering of ϩ stream decreases n ϩ with the rate of n ϩ / ϩ but the backscattering of Ϫ stream increases n ϩ with the rate of n Ϫ / Ϫ .Scattering that causes a carrier to remain in the same stream does not appear because it does not change the number of carriers.The sign Ϯ in Eq. ͑A7͒ results from the continuity condition 1 since scattering neither creates nor destroy carriers ͑we exclude explicit generation or recombination of carriers͒.To make Eq.͑A7͒ compatible with the one-flux equations, we define scattering mean-free-paths for Ϯ streams as Ϯ ϵ1/ Ϯ ϭ͗ x ͘ Ϯ Ϯ .

APPENDIX B: DERIVATION OF DEGENERACY FACTORS
We derive Eqs.͑3.1a͒ and ͑3.1b͒.In degenerate semiconductors, the near equilibrium distribution f associated with streams under deceleration can be assumed to be Using the property, ‫ץ‬ f /‫ץ‬E ϭϪ ‫ץ‬ f ‫ץ/‬ Ϯ , we can see that

FIG. 1 .
FIG. 1.The model, ultrathin-body double-gate MOSFET with 10 nm channel length.The source and drain regions are doped at 10 20 cm Ϫ3 .

FIG. 6 .
FIG. 6. Cross-sectional shapes of off-equilibrium distributions along the longitudinal direction ( x ) at different locations in the device at V GS ϭV DS ϭ0.6 V ͑from Ref. 9͒.The source injection ͑solid lines͒ develops into a ballistic peak approaching a delta function.