RC 22539 (W0208-025) August 9, 2002
IBM Research Report
Ballistic FET Modeling using QDAME: Quantum Device
Analysis by Modal Evaluation
Steven E. Laux, Arvind Kumar, Massimo V. Fischetti
IBM Research Division
Thomas J. Watson Research Center
P.O. Box 218
Yorktown Heights, NY 10598
Almaden - Austin - Beijing - Haifa - India - T. J. Watson - Tokyo - Zurich
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Ballistic FET Modeling using QDAME:
Quantum Device Analysis by Modal Evaluation
Steven E. Laux, Arvind Kumar, and Massimo V. Fischetti
Wepresent an algorithm for self-consistent solution ofthe Poisson and Schr¨ odinger equations in two spatial dimensions with open
boundary conditions to permit current flow. The algorithm works by discretely sampling a device’s density of states using standing
wave boundary conditions, decomposing the standing waves into traveling waves injected from the contacts to assign occupancies,
and iterating the quantum charge with the potential to self-consistency using a novel hybrid Newton-Broyden method. A double-gate
FET is simulated as an example, with applications focused on surface roughness and contact geometry.
The quantum ballistic limit, in which electrons propagate as waves without scattering, is an important one for understanding
device behavior as feature sizes are scaled to lengths comparableto the electron wavelength. Although simple analytical transport
models can provide useful insight in this regime , , a complete understanding for realistic device geometries and operating
conditions requires a self-consistent solution of the Poisson and Schr¨ odinger equations with open boundary conditions to permit
the flow of current. Self-consistent solution of the Poisson and open-boundary condition Schr¨ odinger equations is complicated
due to issues such as discretization of the energy continuum, wavefunction normalization, the presence of quasi-bound states,
population of the electron states by the reservoirs, and finally numerical iteration to convergence, .
In this paper we introduce QDAME, an efficient and physically intuitive algorithm for solving the Poisson and open-boundary
condition Schr¨ odinger equations self-consistently for devices in two spatial dimensions with ellipsoidal parabolic energy bands,
employingaformulationgeneralizedfromtheone-dimensionalframeworkinRefs. ,. Incontrasttoearlierapproachesbased
on the nonequilibrium Green’s function , , QDAME is based on the Quantum Transmitting Boundary Method (QTBM) 
which, through use of a finite-element mesh, is capable of modeling devices of arbitrary geometry. Although in its present
embodiment QDAME finds purely ballistic solutions, we envision that it may eventually be used as part of a Master equation
approach that includes scattering, as in Refs. , . After a discussion of the basic theory underlying QDAME, we present
simulation results from a double-gate FET as an example.
The algorithm begins with an initial guess for the potential, which can be obtained either from a classical solution or a solution
at a previous bias point. The solution for the electron charge in the potential is found using a modified version of the QTBM,
in which a two-dimensional device geometry is connected to infinitely long leads which supply and receive current in discrete
transverse modes (i.e., confined in the transverse direction and plane wave in the longitudinal direction) as illustrated in Fig.
1(a). Just as periodic boundary conditions are used to count states in a bulk solid, we impose standing wave boundary conditions
at the device-lead interfaces, giving rise to an eigenvalue problem whose solutions are a discrete sampling of the continuum of
energies allowed by the open boundary conditions. For each inequivalent alignment of the conduction band masses (e.g., 3 times
for (100) Si), we generate two such discrete samplings – which we call the device “normal modes” – by imposing zero-value and
zero-normal-derivativestanding-wave boundary conditions on the wavefunction at the device-lead interfaces:
sine : Ψ(s)
n = 0 =⇒ eigenvalues E(s)
cosine : ? ηi• ∇Ψ(c)
n = 0 =⇒ eigenvalues E(c)
where ? ηidenotes the outward normal at lead i. Although either Eq. (1) or Eq. (2) by itself is a complete discretized sampling of
and its normal derivative at the device-lead interfaces.
Because the normal mode states cannot carry current, we must decompose each normal mode eigenfunction Ψ(s,c)
states we call “traveling eigen-components” Φ(s,c)
The authors are with the IBM Semiconductor Research and Development Center (SRDC), T. J. Watson Research Center, P.O. Box 218, Yorktown Heights, NY
10598 USA (email: firstname.lastname@example.org).
the device DOS, we must use both conditions (as a minimum) to be able to represent well both the electron distribution’s value
where the sum is taken over all injecting leads i. Analogous to a scattering state in quantum theory, a traveling eigen-component
is an injection “experiment” from a single lead i, containing a specific superposition of transverse modes of the lead i, at
the eigen-energy E(s,c)
of the sine (s) or cosine (c) normal mode. For each transverse mode p in lead i, the energy of the normal
can be expressed as the sum of a transverse and a longitudinal component
Given the normal mode energy and the injection coefficients, the traveling eigen-components can be computed using a modified
version of the QTBM, as explained later. The fact that either Eq. (1) or Eq. (2) gives rise to a complete discretized sampling of
the device’s DOS together with the assumption that the leads all inject incoherently establishes a normalization condition for the
traveling eigen-componentsmaking up a normal mode:
where the integration extends over the device domain Ω and the factor 1/2 arises because we use two complete samplings.
The QTBM must be modified to use the supercell boundary conditions shown in Fig. 1(b), where the solution in one device
is effectively “mirrored” across the device-lead interface. This modification is necessary for two reasons. First, coupling of
the device to infinitely long leads results in solutions which are dominated by unphysical lead-device resonances; in contrast
the sampling of the DOS is device-driven, not lead-driven, when supercell boundary conditions are used. Second, the use of
supercell boundary conditions allows us to treat quasi-bound states in the device, corresponding to energies classically forbidden
in the leads, analogously to propagating states: the traveling waves comprising the longitudinal part of the solution in the leads at
classically allowed energies simply become growing and decaying exponentials at classically forbidden energies.
Figure 2 illustrates two example “cosine” normal modes and their traveling eigen-components for the SOI MOSFET shown
schematically in the inset of Fig. 2(a). Well below the top of the energy barrier, the normal modes are localized on either the
sourceorthe drainside, andarepredominantlycomposedofjust onetravelingeigen-componentwhichis almostentirelyreflected;
fig. 2(a) illustrates such a case where the normal mode is localized on the drain side and the drain eigen-component consists of
injection in primarily the lowest two transverse modes of the drain lead. In contrast, the normal mode illustrated in Fig. 2(b) at an
energy well above the barrier peak involves a substantial mixing of the traveling eigen-componentsinjected from both leads.
The total electron density is the sum of contributions from all ladders arising from the different possible alignments of the
conduction band masses; for simplicity we concentrate here on the electron density from a single conduction band valley. The
electron density n(x,y) is the sum of the electron densities from the sine sampling n(s)(x,y) and cosine sampling n(c)(x,y),
found by summing over all traveling eigen-componentdensities multiplied by a thermal occupancy factor at temperature T:
Here the thermal occupancy factor is a sum of occupancy factors ρp,i(ki
fractional weight of each transverse mode p in lead i making up the traveling eigen-component. We use a drifted Fermi-Dirac
occupancy factor ρp,i(ki
lead and device , . Since the Poisson equation is solved with boundaryconditions that enforce charge-neutralityat the leads,
both important conditions of flux conservation and charge-neutralitynear the contacts are met.
Once the quantum charge is computed for a given input potential, it must be iterated with the Poisson equation to obtain
a self-consistent solution. A major stumbling block in this outer loop, which requires solution of the changes in potentials
and drift momenta at the leads, is in the evaluation of the Jacobian matrix describing the nonlocal variation of the quantum
charge with respect to potential. Figure 3 compares convergence of a typical problem using three different approaches: the
Broyden method , , which constructs an approximate Jacobian based on the previous iterations, the Newton method with
the derivative of the charge evaluated semiclassically, and a hybrid method combining these two techniques. While the Newton
method with semiclassical derivative is seen to offer marked improvementover the Broyden method, the best results are obtained
by a novel hybrid Newton-Broyden approach in which successive guesses to the solution are chosen based on a combination of
the approximate semi-classical derivative evaluated at the present guess together with the guidance based on the solution history
provided by the Broyden method.
ηare respectively the transverse confinement energy of mode p and the longitudinal effective mass in lead i.
) with coefficients c(s,c)
D,T,E) for transverse mode p given by
F− E − (¯ h2/2mi
zis the out-of-plane effective mass in lead i, Ei
η is defined by Eq. (4), and the drift momentum ¯ hki
Fis the Fermi level in lead i, F1/2is the Fermi-Dirac integral of order
Dat lead i is found from the requirement of current continuity between
III. EXAMPLE: DOUBLE-GATE FET
here as insulator contacts, although treatment as current-carrying leads to model gate leakage is straightforward. Figures 4(b-c)
show the self-consistent conduction band edge and electron density. Although qualitatively similar to their classical counterparts,
distinct quantum-mechanical features are clearly visible. Of particular note is the seamless approach in the contact regions to
the source and drain leads, arising from the a priori setting of the lead potentials by charge neutrality. Figure 5 shows transfer
characteristics for double gate structures of various effective channel lengths Leff(separation between metallurgical junctions)
and silicon thicknesses tSi. For well-chosen Leff and tSi, a nearly ideal subthreshold swing is obtained, as anticipated, while
short-channel effects become apparent for less optimal values. Figure 6 shows the evolution of the electron density from the
subthreshold regime through the on-state (points (A)-(D) in Fig. 5): the electron density goes from being concentrated near the
channel center in subthreshold to having peaks close to the oxide interfaces well above threshold .
Inthis high-fieldregime,we expectroughnessalongthe gateoxideinterfaceto playan importantrole. Althoughrealistic rough-
ness also varies in the out-of-plane direction, we can gain qualitative insight into how roughness affects the device characteristics
by using a one-dimensional roughness potential; we choose one having an exponential autocorrelation function with parameters
∆ and Λ characterizing the roughness height and correlation length, respectively . In contrast to the smooth distribution in
Fig. 6, Fig. 7 shows the electron density in a device with roughened Si-SiO2interfaces characterized by ∆ = 0.24 nm and
Λ = 0.72 nm. A comparison of the current density distribution between a smooth versus rough case shows that the roughness
not only redirects the current but also degrades it. Figure 8(a) compares transfer characteristics for two devices with roughened
Si-SiO2interfaces having different roughness parameters with those of a smooth device. As expected, the current degradation
increases with increasing gate voltage and is more pronounced in the device with greater roughness. Note that we expect some
variation in transfer characteristics over many devices characterized by the same roughness parameters but having different real-
izations of the roughness; in Fig. 8(b) we show an average over five different realizations having the same roughness parameters
so that the error bars give an idea of the variation expected over a large ensemble of devices.
Since QDAME is not restricted to rectilinear meshes, we can look at how the contact geometry affects the approach of the
current into the device. Figure 9 compares the current density distribution in a thin (3 nm) double-gate FET attached to contact
regions that are as wide as the Si channel (straight case), that taper gently away from the channel (tapered case), and that have
an abrupt transition (dog bone case); the source (drain) lead is the full width of the left (right) extreme of the device (5 nm wide,
including the oxides, for the straight case; 9 nm wide for the tapered and dog bone cases). Not surprisingly, the current has a
similar distribution in the contact region as in the channel for the straight case. In the tapered case, the current enters and exits the
channel following paths that hug the angled edges of the contact regions. In sharp contrast, the abrupt edges in the dog bone case
result in strong back-reflections which cause the transmitted current to be largely concentratedat the center of the contact regions.
IV. CONCLUSION AND FUTURE WORK
equations in two spatial dimensions and demonstrated its utility through relevant examples. Problems left for future work include
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Figure 1: Schematic of device geometry solvable using QTBM (a) with infinitely long leads; (b) with supercell boundary condi-
tions imposed at device-lead interfaces.
Figure 2: Normal modes (top) decomposed into traveling eigen-components from the source (middle) and drain (bottom) leads
for the SOI MOSFET shown in the inset at an energy (a) well below the top of the barrier; (b) well above the top of the barrier.
Figure 3: Comparison of convergence behavior using three different numerical techniques: Broyden method, Newton method
with charge derivative evaluated semiclassically, and hybrid Newton-Broyden method.
Figure 4: Double-gate nFET used in simulations: (a) schematic; (b) conduction band edge; and (c) electron density.
Figure 5: Transfer characteristics for double-gatenFETs of varying effective channel length Leffand Si thickness tSi. The oxide
thickness is tox= 1 nm and the drain-source bias is 0.1 V. Note A-D refer to Fig. 6.
Figure 6: Four contours of electron density (2 contours/decadestarting from 1.26×1018cm−3) for the bias points shown in Fig.
5 (Leff= 25 nm, tSi= 10 nm, tox=1 nm).
Figure 7: Contours of electron density in a double-gate nFET with interfaces roughened as shown using parameters ∆ = 0.24
nm, Λ = 0.72 nm. Here Leff= 25 nm, tSi= 10 nm, tox=1.5 nm, the gate-source bias is 0.4 V, and the drain-source bias is 0.1
Figure 8: (a) Transfer characteristics for smooth device and devices roughened with two different sets of roughness parameters.
Here Leff= 25 nm, tSi= 10 nm, tox=1.5 nm, and the drain-source bias is 0.1 V. (b) Transfer characteristics for smooth device
and average of five devices with interfaces characterizedby the same roughnessparameters but having different realizations of the
roughness. Error bars give an idea of variation expected over a large ensemble of devices.
Figure 9: Device layouts (above) and contours of current density magnitude (below) for double-gate nFETs having straight, ta-
pered, and dog bone contact geometries. In the layouts, regions of piecewise constant doping are shaded and oxide regions are in
black. In the contours, dark (light) represents areas of low (high) current density.
0 102030 405060 7080
Newton (classical charge deriv.)
Error in Potential (V)
-0.3-0.2-0.1 0.00.1 0.20.3
60 mV/dec slope
Drain Current (A/cm)
Gate Voltage (V)
Double Gate FETs
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Drain Current (A/cm)
Gate Voltage (V)
? ?=2A, ? ?=25A
? ?=2.4A, ? ?=7.2A