Quantum Corrections to the ’ Atomistic ’ MOSFET Simulations

We have introduced in a simple and efficient manner quantum mechanical corrections in 
our 3D ’atomistic’ MOSFET simulator using the density gradient formalism. We have 
studied in comparison with classical simulations the effect of the quantum mechanical 
corrections on the simulation of random dopant induced threshold voltage fluctuations, 
the effect of the single charge trapping on interface states and the effect of the oxide 
thickness fluctuations in decanano MOSFETs with ultrathin gate oxides. The introduction 
of quantum corrections enhances the threshold voltage fluctuations but does not 
affect significantly the amplitude of the random telegraph noise associated with single 
carrier trapping. The importance of the quantum corrections for proper simulation 
of oxide thickness fluctuation effects has also been demonstrated.


INTRODUCTION
The scaling of MOSFETs in integrated circuits is reaching the stage where the granularity of the electric charge and the atomicity of matter start to introduce substantial variation in the characteristics of the individual devices and has to be included in the device simulations.The variation in number and position of dopant atoms in the active region of the decanano MOSFETs introduces significant variations in the device characteristics [1].At the same time the thickness of the gate oxide becomes equivalent to several atomic layers with a typical interface roughness of the order of 1-2 atomic layers [2].This will introduce more than 50% variation in the oxide thickness within an individual transistor and will make the transis- tors microscopically different in terms of oxide thickness pattern as well.The trapping/detrapping of individual charges at the interface also will have a dramatic effect on the current in such devices [3].
The statistical variations in the decanano devices shift the paradigm of the numerical device simulations.It is no longer sufficient to simulate a single device with continuous doping distribution, uniform oxide thickness and unified dimensions to represent one macroscopic design.Each device is microscopically different at the level of dopant *Corresponding author.Tel.: +44 141 330 5217, Fax: +44 141 330 5236, e-mail: a.asenov@elec.gla.ac.uk A. ASENOV et al. distribution, oxide thickness and gate pattern, so an ensemble of macroscopically identical but microscopically different devices must be charac- terised.The simulation of a single device with random dopants, oxide thickness and gate pattern variation requires essentially a 3D solution with fine grain discretisation.The requirement for statistical interpretation transforms the problem into a four dimensional one where the fourth dimension is the size of the statistical sample [4].
At the same time the increase in doping con- centration and the reduction in the oxide thickness in decanano MOSFETs results in a strong quanti- zation in the inversion layer and a corresponding threshold voltage shift and oxide capacitance degradation [5].However traditionally the 3D simulation studies of random dopant fluctuation effects [4, 6-8] use drift-diffusion (DD) approxima- tion and do not take into account quantum effects.
Until recently [9] it was unclear to what extent the quantum effects may enhance or reduce the varia- tions in the device characteristics associated with random dopant, and oxide thickness fluctuation and the effects associated with trapping/detrapping of individual interface charges.
In this paper we study the influence of the quan- tum effects in the inversion layer on the parameter fluctuations in decanano MOSFETs.The quan- tum mechanical (QM) effects are incorporated in our previously published 3D 'atomistic' simulation approach [4] using a 3D implementation of the density gradient (DG) formalism.This results in a 3D, QM picture which incorporates the vertical inversion layer quantization, lateral confinement effects associated with the current filamentation in the valleys of the potential fluctuation, and eventually tunnelling through the sharp potential barriers associated with individual dopants.

IMPLEMENTATION OF DG APPROACH IN 3D 'ATOMISTIC' SIMULATIONS
The DG model is an approximate approach for introducing quantum mechanical corrections into the macroscopic drift-diffusion approximation by considering a more general equation of state for the electron gas, depending on the density gradient.It has been demonstrated in [10] that, to lowest order, the quantum system behaves as an ideal gradient gas for typical low-density and hightemperature semiconductor conditions.Quantum corrections have been included in the DD simula- tions by introducing an additional term in the carrier flux expression: Fn n#nV DnVn + 2#nV bn V (1) where bn h/(12qm,), and all other symbols have their usual meaning.One possible approach [12] to avoid the discretisation of fourth order derivatives when using (1) in multidimensional numerical simulations is to introduce a generalised electron quasi-Fermi potential qSn, as follows: Thus the unipolar DD system of equations with QM corrections, which in many cases is sufficient for MOSFET simulations, becomes: V. (eVb) -q(p n + Nz N2) (3)   kT n V2V/ n b + ln--(4) where , bn and v/ are independent variables.
Compared to the conventional DD simulations the DG approach increases the number of equations by one for each type of carriers.However we have restricted our simulations to low drain voltage which allows us to disentangle Eqs. ( 3) and (4) from Eq. ( 5) by considering a quasi-constant quasi-Fermi level.First we solve self-consistently the 3D Poisson equation (3) for the potential and Eq. ( 4), which can be considered as a DG approximation of Schr6dinger's equation, for the electron concentration.Standard boundary conditions are used for the potential in the Poisson equation (3) with zero bias applied at the source and drain contacts.Dirichlet boundary conditions are applied to the electron concentration in the DG equation ( 4) at the contacts and Si/SiO2 interface introducing charge neutrality and vanish- ing small values respectively, and Neumann boundary conditions are applied at all other boundaries of the solution domain.
Knowing the electron concentration from the selfconsistent solution of Eqs. ( 3) and (4), and following the procedure described in [4] we extract the current from the resistance of the MOSFET by solving the drift approximation of Eq. ( 5): Before moving to 3D atomistic simulations the DG approach has been carefully calibrated for continuous doping against rigorous 1D full band Poisson-Schr6dinger simulations presented in [5].
By using effective mass m*= 0.19 mo an excellent agreement has been achieved between the DG simulations and the comprehensive Poisson- Shr6dinger solution in respect of the QM thresh- old voltage shift [9].Even more important for this study is the good agreement between the electron distributions in the inversion layer obtained using the two simulation techniques and illustrated in Figure 1.
(6) 3. EXAMPLES in a thin slab near the Si/SiO2 interface engulfing the inversion layer charge.Dirichlet boundary conditions are applied for the 'driving' potential V at the source and drain contacts with V=0 and V= VD respectively and Neumann boundary conditions are applied at all other boundaries of the slab.We have demonstrated in [4] that at low drain voltage this approach is in excellent agree- ment with the full self-consistent solution of the DD equations.
1E+18 l S. Jal!epalli e .t al. [5]    We illustrate the fruitfulness of the DG approach in several 3D 'atomistic' simulation examples including random dopant fluctuation, single charge trapping at the Si/SiO2 interface and oxide thickness fluctuation in ultrathin gate oxides.

Random Dopant Fluctuations
A typical result of the atomistic simulation of a 30 50 nm 2 n-channel MOSFET with oxide thick- ness tox-3 nm and a junction depth xj-7 nm is outlined in Figure 2. The uniform doping concentration in the channel region ND--5 1018 cm -3 is resolved down to individual dopants using fine grain discretisation.The number of dopants in the random dopant region of each individual transistor follows a Poisson distribu- tion.The position of dopants is chosen at random and each dopant is assigned to the nearest gridnode.More complex doping profiles in the random dopant region of the device may be introduced using a rejection technique.
Current criterion IT--lO-SWeff/Leff [A] is used to estimate the threshold voltage.Typically, samples of 200 microscopically different transis- tors are simulated for each combination of macroscopic design parameters, in order to extract the average threshold voltage and its standard deviation crVT.The corresponding thin (1 nm) and thick (1.5 nm) oxide strips in the direction parallel to the channel.The whole change in the oxide thickness occurs at the Si/SiO2 interface.The threshold voltage depend- ence on the period d of the superlattice, calculated classically and with QM corrections, is illustrated in Figure 6.Completely opposite behaviour is observed with the two types of simulations.The classical results show reduction in the threshold voltage with the reduction of d while the quantum mechanical results show an increase in the threshold voltage.Figure 7 offers the explanation of the observed behaviour.The top of the figure illustrates the Si/SiO2 interface followed by two equiconcentration contours obtained from classical and DG simulations and the potential distribution at the bottom.In the classical simulations there is an increase in the carrier concentration near the edges of the well associated with the increase of the potential there.Such increase in the potential near the corners is well known in such geometries and is the origin of the inverse narrow channel effect in trench isolated devices.The increasing contribution to the current from the corners, when the period of the superlattices decreases, results in a reduction of the threshold voltage in the classical case.
However, due to the small depth of the trenches (0.5nm) the QM charge distribution can not follow the local increase in the potential in the corners and the QM maximum in the charge concentration is in the middle of the wells.This is causing an increase in the threshold voltage when d becomes smaller.

CONCLUSION
The DG approach provides relatively simple means to include quantum correction in the 3D DD 'atomistic' MOSFET simulation.Applicable at relatively low carrier density it can be used in the DD framework to simulate the QM threshold voltage shift and to investigate the effect of the dopant and oxide thickness fluctuations on the variation of the threshold voltage.We have found that the inclusion of the QM corrections enhances the random dopant induced threshold voltage fluctuations in decanano MOSFETs.The enhance- ment is more than 50% in devices with gate oxide below 5 nm.In the same time we have not observed significant impact of the QM corrections on the RTS amplitudes in similar devices.We have demonstrated that the QM corrections may affect the simulations of effects associated with oxide thickness fluctuations in MOSFETs with ultrathin gate oxides and further investigations in this area are in progress.
FIGUREComparison of the 1D charge distribution ob- tained from DG and full band Poisson-Schr6dinger simula- 17 tions for acceptor concentration NA=5 x 10 cm-, oxide thickness tox 4 nm, inversion charge density 1.67 x 1011 cm- and vertical field 3.05 V/cm.