ensemble monte carlo study of channel quantization in a 25-nm n-mosfet

1864 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 47, NO. 10, OCTOBER 2000

Ensemble Monte Carlo Study of ChannelQuantization in a 25-nm n-MOSFET

S. C. Williams, K. W. Kim, Senior Member, IEEE, and W. C. Holton, Fellow, IEEE

Abstract—We develop a self-consistent, ensemble Monte Carlodevice simulator that is capable of modeling channel carrier quan-tization and polysilicon gate depletion in nanometer-scale n-MOS-FETs. A key feature is a unique bandstructure expression for quan-tized electrons. Carrier quantization and polysilicon depletion areexamined against experimental capacitance–voltage (C–V) data.Calculated drain current values are also compared with measuredcurrent-voltage data for an n-MOSFET with an effective channellength (Le� ) of 90 nm. Finally, the full capabilities of the MonteCarlo simulator are used to investigate the effects of carrier con-finement in a Le� = 25 nm n-MOSFET. In particular, the mech-anisms affecting the subband populations of quantized electronsin the highly nonuniform channel region are investigated. Simula-tion results indicate that the occupation levels in the subbands area strong function of the internal electric field configurations andtwo-dimensional (2-D) carrier scattering.

Index Terms—MOSFETs, simulation, quantization.

I. INTRODUCTION

A S MOSFETs are scaled into the deep submicron regime,device designers utilize ultrathin gate dielectrics and high

channel doping to maximize drain current while minimizingshort-channel effects, such as punchthrough. While the latestsemiconductor roadmap plans for the introduction of metalgates into preproduction by the year 2008 [1], polysilicon cur-rently remains the gate material of choice in CMOS fabrication.When using ultrathin gate oxides, the depletion width in thepolysilicon gate can be comparable to the oxide thickness. Tominimize the impact of poly depletion on device performance,the polysilicon doping is typically increased in each succeedingtechnology generation. Even when using a relatively highpolysilicon doping, the finite capacitance associated with thepoly depletion causes the overall gate capacitance to decrease[2]–[5], and calculated current–voltage (I–V) curves differ fromtheir measured values [3].

This design with high channel doping also results in a largeelectric field perpendicular to the oxide that produces a two-dimensional (2-D) electron gas near the Si/SiOinterface. Inaddition to polysilicon depletion, the quantum effect has beenidentified as a critical factor for predicting inversion layer ca-pacitance for both n-type [5]–[9] and p-type [10] MOSFETs.

Manuscript received February 11, 2000. This work was supported in part bythe Office of Naval Research and the Center for Advanced Electronic MaterialsProcessing, established at North Carolina State University by the National Sci-ence Foundation. The review of this paper was arranged by Editor D. Ferry.

The authors are with the Department of Electrical and Computer Engi-neering, North Carolina State University, Raleigh, NC 27695-7911 USA(e-mail: [email protected]).

Publisher Item Identifier S 0018-9383(00)07779-0.

Channel quantization has also been experimentally shown tohave a strong effect on both the MOSFET threshold voltage[11] andI–Vcharacteristics [12]. As a result, modeling both gatepolysilicon depletion and the quantum mechanical effect is im-portant to predict the electrical behavior of these transistors.

Many different schemes have been devised to model carrierquantization in the channel of deep submicron MOSFETs (see,for example, [12]–[15]) using moment based simulators such asMEDICI [16]. While moment based modeling tools are usefulfor obtaining terminal characteristics in small devices, they donot provide fundamental insight into the complex mechanismsof electron transport under nonuniform channel conditions. Incontrast to moment based simulators, the Monte Carlo solu-tion to the Boltzmann transport equation considers all relevantphysical mechanisms and can predict device performance withaccuracy. The ensemble Monte Carlo method typically treatseach electron as a classical point charge (i.e., 3-D electron) thatmoves through the device and interacts with its environmentincluding other simulated electrons. Moreover, this treatmentcan be readily modified to simulate the confinement of elec-trons in one or more spatial dimensions. To our knowledge,only one group [17] so far has developed an ensemble MonteCarlo tool capable of modeling quantized electrons in MOS-FETs. The treatment of 2-D carriers in [17] features a non-parabolic bandstructure and is focused heavily on details of thescattering mechanisms in quantized inversion layers. However,the mechanisms affecting subband occupation in highly nonuni-form channel configurations have not been previously well doc-umented.

In this paper, we present a self-consistent, ensemble MonteCarlo simulator that is capable of modeling the quantization ofchannel carriers in nanometer-scale n-MOSFETs. This simula-tion tool also accounts for depletion in the polysilicon gate ma-terial. Utilizing this simulator, we investigate n-MOSFETs withsub-100 nm channel lengths. First, the models for carrier quan-tization and polysilicon depletion are examined against exper-imental capacitance–voltage (C–V) data. Calculated drain cur-rent values are also compared with measuredI–V data for ann-MOSFET with an effective channel length (L) of 90 nm.Finally, the full capabilities of our Monte Carlo simulator areapplied to study the effects of carrier confinement in a Lnm n-MOSFET. In particular, subband populations of quantizedelectrons in the channel region are analyzed. Simulation resultsindicate that the occupation levels in the subbands are a strongfunction of the internal electric fields and 2-D carrier scattering.

This paper is organized as follows. In Section II, a briefoverview of the key components of the Monte Carlo simulatoris given. The differences between our treatments and those

0018–9383/00$10.00 © 2000 IEEE

WILLIAMS et al.: MONTE CARLO STUDY OF CHANNEL QUANTIZATION 1865

of [17] are discussed when appropriate. Section III presentsnumerical results and discussion for nanometer-scale n-MOS-FETs. A brief conclusion follows at the end.

II. M ODELS

A self-consistent Monte Carlo simulator was developed pre-viously by the authors to model deep-submicron n-MOSFETs[18]. This program ignores the spatial quantization of carriers;in other words, all electrons in the simulation are treated as clas-sical, or 3-D, point-like charges. The Monte Carlo model em-ploys a realistic silicon band structure for the two lowest con-duction bands, and contains all relevant aspects of 3-D elec-tron transport including interactions with other 3-D electrons,acoustic and intervalley phonons, and ionized impurities alongwith impact ionization. A numerical solution to the Poissonequation is coupled to the Monte Carlo method and recalcu-lated on a femptosecond time scale to provide a self-consistent,dynamic electric field distribution. Additionally, interface scat-tering due to the roughness of the Si/SiOinterface was consid-ered using a combination of specular and diffusive scattering as3-D particles hit against the interface [19]. The percentage ofdiffusive scattering events is chosen by matching the predictedcurrent to the measured current for a particular device processtechnology. After the percentage of diffusive scattering is cal-ibrated, Monte Carlo simulations are expected to provide rea-sonable estimates of MOSFET performance.

To enable simulation of 2-D carriers, several major compo-nents have been added to this simulator [18]. These modifica-tions are summarized in the following sections. In Section II-A,a unique analytical bandstructure model is presented. The equa-tions used to model quantized carrier scattering due to bulkphonons are discussed in Section II-B. Finally, in Section II-C,a framework that allows quantized and classical (pointlike) par-ticles to simultaneously coexist within the simulator is detailed.

A. Bandstructure Model

To simulate 3-D carriers, the Monte Carlo simulator employsa silicon bandstructure calculated using an empirical pseudopo-tential method and stored in a look-up table listing energy andderivative information against corresponding wavevectors. Inthe case of the 2-D electron gas in the channel of a nanometerscale MOSFET, the particles are confined along the dimensionoriented perpendicular to the oxide interface and are restricted todiscrete energy values by the solutions to the Schrödinger equa-tion. For the other two dimensions along which the channel elec-trons are not confined, the wavevectors and, therefore, the corre-sponding particle energies, may assume a continuum of values.To enable the solution of the Schrödinger equation along theone confining direction, while at the same time permitting boththe evaluation of unrestricted energy values along the other twodirections and the 2-D density of states, an analytical expres-sion for the bandstructure is required in lieu of a tabulated solu-tion. Several widely used analytical models exist to describe thebandstructure of silicon (see [20] for examples). However, dueto the complex shape of the actual bands, these models usuallyrepresent the bandstructure for energies no larger than about 50meV. A new bandstructure model that fits the tabulated data over

Fig. 1. Dispersion relations in the [100] and [101] directions for theE valleyexpressed using the current work [(1)], the nonparabolic band expression [17],and the tabulated empirical pseudopotential data. The nonparabolic expressionis spherically symmetric and therefore applies to both directions. The dispersionis shown for positive~k values away from the conduction band minimum atk =

(0; 0:85; 0).

a larger energy range will allow for more realistic simulation ofthe 2-D particle ensemble.

To obtain a suitable analytical model, a unique, two-stage sta-tistical method has been developed to fit a polynomial expres-sion to the empirical pseudopotential data. In the first stage ofthe method, a regression analysis is performed to find the ini-tial parameter estimates using only the available tabulated en-ergy and momentum information. The regression analysis deter-mines which terms from a fourth order polynomial expressionprovide a best fit to the available bandstructure data. A simplesecond-order (parabolic) model with no cross terms in, , or

is selected in the regression. In the second stage, the deriva-tive information is incorporated to hone the parameter estimatesand achieve a more complete model reflecting all knowledgegained from the pseudopotential calculations. A full descriptionof the regression procedure is available in [21].

The statistical method is applied to the pseudopotential datadescribing the portion of the first conduction band with energyup to 200 meV. The resulting analytical energy dispersion is

(1)

The regression analysis provides the following effective massvalues: , , and .The goodness of fit for the polynomial expression is .For a comparison, the values for the effective masses in the par-abolic approximation to the bandstructure are

and . Nonparabolicity may be intro-duced into the parabolic model by using a nonparabolicity pa-rameter, , which is usually taken as eV for silicon.

In Fig. 1, the values of energy calculated using the bandstruc-ture in the current work [(1)] and using the analytical band ex-pression from [17] (referred to as nonparabolic), and taken fromthe tabulated empirical pseudopotential data, are plotted versus

along the [100] and [101] directions in the irreducible zone.The energy range in Fig. 1 extends well beyond 200 meV, which


is the maximum energy used to fit our band model to the pseu-dopotential data. As expected, the bandstructure of (1) matchesthe pseudopotential data along both directions for energies upto 200 meV . Above 200 meV, the cur-rent work continues to match the [100] data fairly well. How-ever, along the [101] direction, (1) overestimates the energy fora given value.

The expression in (1) is used to obtain solutions to theSchrödinger equation for the valleys of the first conductionband. Since the effective masses are all nonequal, quantizationalong one spatial direction results in three pairs of equiva-lent valleys. A distinct set of subband energies is associatedwith each pair of (degenerate) valleys. The solutions to theSchrödinger equation are directly determined from the valueof the effective mass along the direction of confinement.Throughout this paper, the subscript that labels a particularvalley, , corresponds to the subscript of the effectivemass, , that lies along the quantization direction. As anexample, the pair of degenerate valleys whose effective mass,

, lies along the quantization direction arelabeled as the valleys. For these valleys, the masses alongthe unconfined directions ( and in this case) definethe time evolution of the 2-D electron energy as a function ofwavevector. The and valleys are expressed in a similarnotation.

In the Monte Carlo simulator, the devices are defined on arectangular mesh where thedirection is aligned parallel tothe oxide interface and the direction is oriented perpendic-ular to the oxide. The Poisson equation is solved on this meshwhich is characterized by relatively large spacings between the

- and -mesh nodes. The-mesh spacings (1.5 nm) are uni-form across the entire device. Moreover, the-mesh spacings(1.0 nm) are uniform in the channel region where the accuracyof carrier transport is most crucial. The Schrödinger equationis solved along the direction (i.e., 1-D) at each-node usingan extremely closely-spaced-mesh to satisfactorily resolve de-tails of the electron wavefunctions. These details are critical fordetermining the overlap integrals in the 2-D scattering rates.However, the Schrödinger equation mesh retains the samedirection spacing as the Poisson equation mesh. The electronwavefunctions are periodically translated from the Schrödingerequation mesh to the Poisson equation mesh to insure that the2-D charge in the channel is properly assigned whenever thePoisson equation is solved. A self-consistent solution to theSchrödinger and Poisson equations is periodically found at reg-ular time intervals over the course of the simulation. Details ofthis procedure are given at the end of Section II-C.

B. Scattering Processes for Quantized Electrons

Interactions with intravalley acoustic and intervalley phononsare the dominant scattering mechanisms for electron transportin silicon. In this work, treatment of 2-D electron scatteringwith phonons follows Price [22] and Ridley [23]. Our emphasis,which differs from [17], is to include important scattering mech-anisms using simple yet sufficiently rigorous treatments. For thefollowing rate calculations, it is assumed first that the quantizedelectrons interact with bulk (3-D) phonon modes, and secondthat the deformation potential is isotropic. Moreover, in this

treatment, phonon scattering is left unscreened following [17](these authors argue that static screening may result in an exces-sive reduction of the strength of the electron-phonon interaction,while dynamic screening is not important in situations whereelectron-phonon collisions play a major role; hence, screeningis altogether ignored). Lastly, intravalley scattering with opticalphonons in Si is relatively insignificant for electrons sufficientlyclose to the bottom of the -valley minima, and therefore thisprocess is not considered.

Under these assumptions, and in the elastic, equipartition ap-proximation, the intravalley acoustic phonon scattering rate invalley from subband to subband is given as [24]

(2)

where is the 2-D density of states calculated for eachvalley, and is the amplitude of the subband wavefunction fromthe solution to the Schrödinger equation as a function of.A wide range of values for the acoustic deformation potential,

, have been reported in the literature, includingeV [25], 9.0 eV [26],[27], 9.5 eV [18], 9.9 eV [17], and 12.0 eV[28]. In this work, a value of eV is adopted so thatthe scattering rate of (2) calculated using the bandstructure (1)is in close agreement with the acoustic phonon scattering ratefrom [17].

The intervalley phonon scattering rate from subbandinvalley to subband in valley is given as [24]

(3)

where the index runs over all - and -processes, thetop/bottom sign refers to absorption/emission, is the unitstep function, and is the subband energy from the solutionto the Schrödinger equation. The factor, , expresses thedegeneracy of the final state for the process. Values fordeformation potentials ( and ) and phonon energies( ) chosen for our Monte Carlo simulator are given inTable I, alongside values from other works. Equations (2) and(3) indicate that the scattering rates of the 2-D electron gasdue to acoustic and intervalley phonons depend heavily on thewavefunctions of the 2-D subbands. Finally, the degeneracy ofthe electron gas is accounted for with the last term in (3) [29].For elastic acoustic phonon scattering, the degeneracy factorreduces to unity regardless of the location of the Fermi level.

In Fig. 2, the electron–phonon scattering rates [(2) and (3)]for the first quantized subband in the valley (i.e., withalong the direction of quantization) are calculated for two cases.In the first case, the Schrödinger equation is solved using thebandstructure of (1), while the parabolic band approximation isused in the second case (i.e., , ). These tworesults are shown alongside those from [17]. The 2-D nature ofthe scattering rates (characterized by strong step-like jumps) ispronounced for electron energies less than about 70 meV dueto the large energy gaps between subbands. Above 70 meV, the


TABLE IPARAMETERS FOR2-D ELECTRON-PHONON SCATTERING. THE INTERVALLEY DEFORMATION POTENTIALS (� ) HAVE UNITS OF10 eV/cm,THE ACOUSTIC

DEFORMATION POTENTIAL (� ) HAS UNITS OF eV, AND THE ENERGY OF THEINTERVALLEY PHONONS(E ) HAS UNITS OF meV

Fig. 2. Phonon scattering rates for 2-D electrons in the first quantized subbandof theE valley as a function of energy. The energy scale is referenced tothe bottom of the subband. The scattering rates are calculated using both theparabolic band and the band model in the current work [(1)]. These results areshown alongside the phonon scattering rate data from [17], which was calculatedusing a parabolic bandstructure expression. To provide an exact solution over theenergy range shown, 40 or more subbands are used for all three cases. The valuesfor the deformation potentials and the phonon energies are given in Table I.

subband levels are more closely spaced in energy and the 2-Dscattering rate becomes 3-D like.

C. Quantum Region

Within this simulator, both 2-D and 3-D carriers are allowedto coexist simultaneously. To distinguish between quantized andunquantized particles, a multidimensional boundary termed thequantum region is defined using both energy and spatial criteria,similar to previous approaches (see, for example, [14] and [17]).This area encloses the channel region where the confining elec-tric field, and therefore the effects of carrier confinement, arestrongest. Inside the quantum region, all particles are treatedas 2-D, and outside the region simulated electrons are treatedclassically. A particle that moves across the quantum regionboundary is transitioned between the quantized and unquantizedstates in a manner that minimizes violation of momentum andenergy conservation.

An appropriate subband level is chosen as the energy cri-teria that defines the quantum region (i.e., the 2-D/3-D transi-tion energy). An adequate number of subband levels below thetransition energy should be included to properly describe the2-D charge distributions and scattering rates. A review of perti-nent works clearly reveals that a minimum of three subbands isneeded to yield realistic 2-D carrier distributions [12], [30], andas many as ten may be needed to adequately describe 2-D scat-tering [17]. In this work, the energy criteria for the 2-D/3-D tran-sition is the fifth subband of the valley ( ). In other words,electrons occupying the first four subbands of thevalley aremodeled as 2-D carriers. Those electrons whose total energy ex-ceeds are treated as classical particles. Further, all electronsin the and valleys occupying subbands whose energylevels fall below are also considered quantized. Simula-tions show that near midchannel in an n-MOSFET under oper-ating conditions where the effects of quantization are strongest,the first two subbands of the valleys and the first three sub-bands of the valleys fall below . A total of nine subbandsis therefore typically included in the regions where the quantummechanical effect is most strongly felt.

The spatial criteria for the 2-D/3-D transitions must also becarefully chosen. Along the direction, the quantum region ex-tends into the source and drain regions, where quantization ofchannel carriers is less pronounced. At each-node within thequantum region, the top boundary in thedirection is located atthe Si/SiO interface. The bottom boundary in thedirection isplaced at the y-node within the substrate where the wavefunc-tion of the subband becomes negligible. Both the bottom ofthe spatial y-boundary and the transition energy depend solelyon the shape of the potential profile and are, therefore, nonuni-form in the direction.

For the case of a 2-D electron inside the quantum region, thetransition to a 3-D state can occur following either free flightduring a given time step or a 2-D phonon scattering event. Freeflight in response to the local electric field can cause the car-rier within the quantum region to attain a total energy that ex-ceeds , or to physically exit the left or right edges of thequantum region. On the other hand, a phonon scattering eventmay increase the total carrier energy to greater thanthrough


intrasubband phonon absorption or intersubband scattering. Toaccount for intersubband scattering into subbands with energygreater than , the Schrödinger equation is solved in all threepairs of valleys for two subbands greater than . These highersubbands are included in the 2-D phonon scattering rate calcu-lations. For the case of a 3-D particle outside of the quantumregion, the transition to a quantized state can occur followingthe inverse of either of the two cases outlined above. The gen-eral method for a 2-D/3-D transition is similar to the approachused in [17].

Within the quantum region, the Schrödinger and Poissonequations are self-consistently solved at regular time intervalsusing an iterative approach [31]. The iterative process iscomputationally expensive and is, therefore, performed onceevery five hundred time steps. Since the potential profile withinthe device is relatively stable after the initial simulation run,the relatively large time interval between iterative solutions isnot expected to introduce significant errors in the final datacollecting runs. After each self-consistent solution converges,the solutions to the 1-D Schrödinger equation in thedirectionare available for each -node in the quantum region. The2-D electron scattering rates which depend on the subbandenergy levels, wavefunctions, and occupation probabilities (fordegeneracy factor) are then recalculated at each-node onceevery 25 femtoseconds.

III. RESULTS AND DISCUSSION

A. MOS Capacitor

In order to evaluate the effectiveness of the models presentedin Section II, a 1-D simulation tool has been developed to modelcapacitance in an MOS structure. The simulator accounts for de-pletion in the polysilicon region as well as the quantization ofelectrons in those subbands with energies less than the transitionenergy, . Subband energies and wavefunctions are providedby self-consistently solving the Schrödinger and Poisson equa-tions. The contribution of the classical tail of the electron con-centration is also calculated following [14]. The results from the1-D capacitance simulator are compared against experimentalC–V data from an n-MOS capacitor [32]. The capacitor has a300 nm thick polysilicon gate that is doped to cm ,and a physical oxide thickness of 4.5 nm. The channel is dopedusing a retrograde profile from a value of cm atthe Si/SiO interface to a maximum of cm over adistance of 65 nm.

The general validity of the approach to simulatingC–Vchar-acteristics from the self-consistent solution to the Schrödingerand Poisson equations has previously been well demonstrated[12], [14], [30], [33]. However, all of these approaches eithersolve the Schrödinger equation in the parabolic approximationor use approximate analytical solutions for the wavefunctionsand subband energies. The purpose of these capacitance simu-lations is to demonstrate that the bandstructure model in (1) iscapable of reproducing experimental results for total gate capac-itance over a range of applied gate voltages.

When the capacitor is biased in inversion, the inversion layerscreens the depletion region charges. In this case, the effects ofpolysilicon depletion and the quantum mechanical shift in the

Fig. 3. Electron density calculated using the 1-D capacitance simulator as afunction of distance from the Si/SiOinterface for three values of applied gatevoltage. The inversion layer centroids are located at a distance of 1.87 nm, 1.60nm, and 1.45 nm from the oxide interface forV = 1.0 V, 2.0 V, and 3.0 V,respectively.

electron distribution introduce capacitances ( and , re-spectively) that are taken in series with the gate oxide capaci-tance ( ). The total gate capacitance, , is therefore givenas

(4)

where , and is the thickness of the polysilicondepletion layer. The term is approximated using a devia-tion capacitance in series with a corrected centroid capacitance[30]. In other words

(5)

The first term on the right-hand side of (5) is the deviation ca-pacitance. This value is a correction term that accounts for thedifference between the actual value of the inversion layer ca-pacitance (where ) and the capacitance dueto the inversion layer centroid [ ] in thestrong inversion limit. The parameter,, is a fitting factor takenas 0.96, is the inversion charge density per unit area, andis defined as . Here, is the inversioncharge density per unit volume as a function of distance,, fromthe Si/SiO interface. In addition to the corrections for the in-version layer capacitance used here [30], an alternative formu-lation is available that entails no fitting parameters [34]. Last,since there is no correction term in (4) for the depletion-layercapacitance, the expression for is valid only for .

In this example, the applied gate biases range from the onset ofweak inversion ( V ) to deep into strong inversion( V). The charge density near the interface of the capac-itor is shown in Fig. 3 for applied gate voltages of 1.0 V, 2.0 V,and 3.0 V. The results of Fig. 3 illustrate that the total charge dis-tributions calculated using the new band (1) are in general agree-ment with previously published results (for example, [14]). Thedisplacement of charge away from the interface has the net effect


Fig. 4. Measured (solid line) and calculated (filled circles) total gatecapacitance,C , as a function of applied gate voltage.

of increasing the effective oxide thickness. On the other hand, aclassical charge distribution is sharply peaked at the Si/SiOin-terface. The resulting equivalent oxide thickness is nearly equalto the physical oxide thickness. Hence,C–Vsimulations using aclassical description for the charge distribution tend to overesti-mate the total gate capacitance. The experimental and simulatedvalues of total gate capacitance are shown in Fig. 4 as a functionof applied gate voltage. For gate voltages greater than 1.75 V, thegate capacitance exhibits the characteristic negative slope due tothe widening of the polysilicon depletion layer. Overall, the cal-culated and measured values show good agreement, indicatingthat our quantization models coupled to a Poisson solution thataccounts for the polysilicon depletion layer performs reasonablywell.

Calculated values for the subband energies and populationsare shown against the applied gate voltage in Fig. 5(a) and (b),respectively. The subband energies increase steadily as the gatevoltage is increased. The occupation level for a given subband,on the other hand, is directly determined from the location of theFermi level relative to the subband energy. At an applied voltageof 1.0 V, the Fermi level is located somewhat below . As thegate voltage is increased to 3.0 V, the Fermi level moves above

but remains below . Correspondingly, the occupationlevel of increases with increasing applied voltage, whilethe populations of the higher subbands decrease.

B. MOS Transistor

The models for channel quantization and polysilicon de-pletion alone are sufficient to describe the charge profilesand subband occupation levels in an n-MOS capacitor. Underthe static conditions present in the capacitor (i.e., no drivingelectric field), the subband populations may be determineddirectly from the location of the quasi-Fermi level. However,for the case of an n-MOS transistor, electric field values canchange relatively quickly over the short length of the channelregion and carriers may experience nonequilibrium transport.In this section, the Monte Carlo simulator is used to modelthe nonuniform channel conditions in nanometer-scale n-MOStransistors.

The predicted drain current from the Monte Carlo simulatoris first calibrated against available experimental data for ann-MOS device with L 90 nm [35] under an applied biasof 2.0 V and 2.0 V. Throughout this paper, thesource and drain junctions are defined at a source-drain dopingof cm . The effective channel length is calculatedas the distance between the source and drain junctions. At themaximum transverse electric field value in the 90-nm design,the small number of confined channel carriers does not signifi-cantly impact the predicted drain current value; therefore, thechannel quantization models are not activated when calibratingthe Monte Carlo simulator. However, the polysilicon depletioneffect is included. Calibration of the simulator is achieved byadjusting the ratio of diffusive to specular interface scatteringfor 3-D electrons. For a reasonably good match with exper-imental data for the 90-nm device, simulations show that a30% value for diffusive surface scattering is necessary. Thepercentage of diffusive scattering events for 3-D carriers foundabove for the 90-nm design is subsequently used for all 25 nmdevice simulations.

To demonstrate the predictive capability of the channel quan-tization and polysilicon depletion models, the Monte Carlo sim-ulator is used to evaluate a bulk n-channel MOSFET with L

nm [36]. For these simulations, electron-electron scatteringis not activated. Since this study is focused primarily on thebehavior of quantized channel electrons, the omission of e-escattering is not expected to affect the averaged qualitative be-havior of the 2-D channel carriers. Key device parameters in-clude an n polysilicon gate structure with a uniform dopingof cm , and a gate oxide thickness of 1.5 nm. Thesource and drain regions have a junction depth of 25 nm and alateral doping gradient of 4 nm/decade. The threshold voltageis 0.25 V, and the applied bias condition for all of the simu-lations are 1.0 V and 1.0 V. The full simulatedI–V characteristics for this device are available in [36]. Thisdevice has a relatively thick oxide and is designed to operateunder a relatively high power supply voltage. However, othercritical device dimensions are scaled for the appropriate tech-nology node [1]. The predicted drain current from our MonteCarlo simulator is 730–750A/ m, which compares to a valueof 404 A/ m using MEDICI [16] and 825–850A/ m from[36] which used the Monte Carlo simulator DAMOCLES [17].For all MEDICI simulations, both field [37] and bulk and sur-face roughness [38] mobility models are activated, and quantummechanical effects are considered [13]. Fermi–Dirac statisticsare also incorporated following the Joyce-Dixon approximation[39]. The Monte Carlo simulator described in the present paperdoes not account for the interaction of 2-D electrons with in-terface roughness (3-D electron scattering against the interfaceis described above) which tends to decrease mobility at highvalues of transverse electric field. Therefore, our drain currentprediction may be somewhat overestimated.

Fig. 6 displays key results for the internal device behavior ofthe 25 nm n-MOSFET. In Fig. 6(a), both the parallel () andperpendicular ( ) components of the channel electric field areshown. Here, a positive value of compels electrons to movefrom left to right in the figure (i.e., from source to drain), while apositive value of tends to confine electrons near the interface.


Fig. 5. (a) Quantized subband energies and (b) subband occupation levels for the lowest four subbands, as a function of applied gate voltage. The capacitorsubstrate is doped using a retrograde profile that varies from a value of3:0 � 10 cm at the Si/SiO interface to a maximum of8:8 � 10 cm over adistance of 65 nm.

Fig. 6. (a) Components of the electric field in the directions parallel (E ) and perpendicular (E ) to the interface, (b) subband energy levels ("), and (c) fractionalsubband occupations (n), as a function of location along the channel in the 25 nm n-MOSFET. The fractional subband occupations for the first subband in eachof the three quantized valleys are denoted as" , " , and" . The occupation levels for the 3-D particles that occupy the first and second conduction bandsare given asn andn , respectively. The subband energy levels are referenced to the conduction band (CB) energy at the Si/SiOinterface. The sourceand drain junctions are defined at a source-drain doping of2 � 10 cm [36], and are located at approximately 73 nm and 98 nm, respectively. The effectivechannel length is calculated as the distance between the source and drain junctions. These locations also correspond to the left and right boundariesof the quantumregion, respectively. Near the drain junction, well over 90% of the channel charge is located in the 3-D conduction bands.

As can be seen from the figure, has a large positive valueover most of the channel region, and diminishes in strength closeto the source and drain junctions. Similarly, the values ofare small on the source side of the channel. Nearer the drain,however, rapidly increases in strength and peaks at a location2 nm to the left of the drain junction. Such a rapid increase in thedriving electric field results in quasi-ballistic transport for the3-D electrons in the pinch-off region. Between the location ofpeak and the drain junction, the parallel electric field quicklyloses strength but maintains a relatively large positive value.

Fig. 6(b) shows the energy level of the first subband in eachof the three quantized valleys ( , , and ) within thequantum region. Since the next highest subband level (, notshown) lies above in energy across the entire channel re-gion, nearly all of the quantized electrons are expected to oc-cupy the three lowest-lying subband levels under low values of

[see Fig. 5(b)]. The subband level corresponding to the tran-sition energy between quantized and unquantized states ()

is also shown. The bottom of all the subbands are referencedto the conduction band (CB) energy at the Si/SiOinterface.The driving force acting on the quantized channel particles isdirectly related to the change in subband energy over distance.Therefore, under the channel configuration for this device, 2-Dparticles with a position less than about 78 nm experience avery small drift force to the left (toward the source). To theright of this point, the driving force accelerates the 2-D parti-cles toward the drain with continually increasing strength. Theforce exerted on the 2-D particles in the channel due to the thespatially varying subband energies is qualitatively similar to theforce acting on the 3-D carriers due to. Correspondingly, ourMonte Carlo simulations predict that the quantized electrons ex-perience quasi-ballistic transport near the drain junction similarto the 3-D electrons.

Finally, Fig. 6(c) illustrates the simulated fractional subbandoccupation versus location along the channel for the three lowestquantized subbands as well as for the 3-D particles that oc-


cupy the first and second conduction bands ( and ,respectively). Near the source and drain regions, nearly all ofthe electrons in the simulation occupy the 3-D bands. Withinthe channel region, however, the strong confining perpendicularcomponent of the electric field quantizes most of the inversionlayer electrons. This general trend in subband occupation in thesource, channel, and drain regions has been previously reported[12] using a drift-diffusion simulator coupled to the Schrödingerequation. However, the results from our Monte Carlo simula-tions incorporating 2-D effects indicate that the subband occu-pations in the channel region of a nanometer-scale n-MOSFETunder operating conditions may be largely affected by 2-D scat-tering and quasi-ballistic transport. In the remainder of this sec-tion, the effect of these phenomena on subband occupation arebriefly investigated.

As previously noted, nearly all of the electrons in the sourceregion occupy the first two 3-D conduction bands. As particlesinject from the source into the channel and experience the strongconfining perpendicular electric field, nearly all of the channelelectrons become strongly quantized. Consequently, the popula-tion of 3-D carriers drops sharply. Our simulations predict thatthe fractional subband occupation for 2-D electrons is highestfor carriers in the subband, followed by and indecreasing order, as shown in Fig. 6(c). On the source side ofthe channel, the electron quasi-Fermi level resides very closeto , but below the and energy levels. Electronsare therefore expected to primarily occupy the lowest-lying sub-band followed in decreasing order by the next two subbands.

The injected electrons slowly propagate across thesource-side of the channel. Since the driving forces inthis region are relatively small, the 2-D electrons tend to remainat low kinetic energies since they are in near-equilibrium withthe lattice. However, for a given value of small electric field, thedistribution of quantized carriers in the valley may achievea somewhat larger value of average energy than carriers in the

or valleys due to the difference in curvature of the bands.As a result, the higher-energy particles in the subbandundergo more intervalley scattering events that tend to reducethe population of while, at the same time, increasing theoccupations of and . Quantized electrons in the groundstate of the valley tend to remain in the ground state afteran intervalley scatter due to the relatively large overlap factorwhen compared to a transition into a higher subband. This trendis observed in Fig. 6(c) for locations less than about 85.5 nm.

For locations greater than about 85.5 nm, the value forquickly falls and the separation between subband energy levelscorrespondingly decreases. In other words, the 2-D/3-D transi-tion level approaches the quantized subband energies. Further,the value of rapidly increases and the 2-D electrons experi-ence quasi-ballistic transport. Due to the higher subband energy,quantized carriers in cross the 2-D/3-D transition energy ata location of about 87 nm while the carriers in the lower twosubbands have not yet crossed the energy transition point. Asa result, the occupation level of drops quickly beyond thispoint in the channel while the populations of the andsubbands remain relatively high. Correspondingly, the occupa-tion of the 3-D carriers in the first conduction band begins togrow. Shortly beyond this location, the quantized populations

of the and then the subbands quickly fall as the ma-jority of the 2-D carriers in these bands convert to 3-D carriers.This is the region where impact ionization begins to occur for“hot” 3-D electrons. The location at which nearly all of the car-riers populate the 3-D bands lies about 3.5 nm to the left of thedrain junction. At the drain junction, the population of 3-D car-riers in the first conduction band dips slightly before recoveringto a value near unity. This may be due to some back-scatteringof low-energy 3-D carriers into the quantized subbands.

IV. CONCLUSION

In this work, a self-consistent, ensemble Monte Carlo sim-ulator that incorporates models for channel quantization andpolysilicon depletion is developed to study nanometer-scaleMOS devices. Good agreement with experiment is observed forcapacitance simulations, and our drain current predictions fora 25 nm n-MOSFET match well with results from establishedsimulation tools. The Monte Carlo tool is used to examineoccupation of the subbands by 2-D electrons under highlynonuniform channel conditions. Simulation results suggestthat the subband populations are highly influenced by 2-Dcarrier scattering near the source end of the channel and byquasi-ballistic transport near the drain side of the channel.

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S. C. Williams, photograph and biography not available at the time of publica-tion.

Ki Wook Kim (M’88–SM’93) received the B.S. degree in electronics engi-neering from Seoul National University, Korea, in 1983, and the M.S. and Ph.D.degrees in electrical engineering from the University of Illinois, Urbana, in 1985and 1988, respectively.

Since graduation, he has been with North Carolina State University, Raleigh,most recently as Professor of electrical engineering. He is experienced inthe Monte Carlo method and other simulation techniques, and has a strongbackground in theoretical semiconductor physics. His current research interestsrange from nanoscale MOSFET modeling to quantum information processingand communication.

W. C. Holton (F’82), photograph and biography not available at the time ofpublication.

ensemble monte carlo study of channel quantization in a 25-nm n-mosfet

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