Full 3D Quantum Transport Simulation of Atomistic Interface Roughness in Silicon Nanowire FETs
SungGeun Kim, Abhijeet Paul, Mathieu Luisier, Timothy B. Boykin, Gerhard Klimeck
aa r X i v : . [ c ond - m a t . m e s - h a ll ] F e b Full 3D Quantum Transport Simulation of AtomisticInterface Roughness in Silicon Nanowire FETs
SungGeun Kim, Abhijeet Paul,
Student Member, IEEE , Mathieu Luisier, Timothy B. Boykin,
Senior Member,IEEE , and Gerhard Klimeck,
Senior Member, IEEE
Abstract —The influence of interface roughness scattering (IRS)on the performances of silicon nanowire field-effect transistors(NWFETs) is numerically investigated using a full 3D quantumtransport simulator based on the atomistic sp d s ∗ tight-bindingmodel. The interface between the silicon and the silicon dioxidelayers is generated in a real-space atomistic representation usingan experimentally derived autocovariance function (ACVF). Theoxide layer is modeled in the virtual crystal approximation (VCA)using fictitious SiO atoms. h i -oriented nanowires with dif-ferent diameters and randomly generated surface configurationsare studied. The experimentally observed ON-current and thethreshold voltage is quantitatively captured by the simulationmodel. The mobility reduction due to IRS is studied througha qualitative comparison of the simulation results with theexperimental results. Index Terms —Si gate-all-around nanowire transistors, atom-istic, full-band simulations, interface roughness scattering
I. I
NTRODUCTION A S the dimensions of the planar metal-oxide-semiconductor field effect transistor (MOSFET)are approaching the nanometer scale, their performancesbecome limited by the short channel effects (SCE), anincreasing OFF-state current, and a poor electrostatic controlof the channel through a single-gate contact.The nanowire (NW) field-effect transistor has been identi-fied as a promising candidate to overcome these issues andbuild the next generation switch [1], [2]. Several experimentalstudies have shown that NWFETs have excellent electricalcharacteristics [3]–[5], especially in a gate-all-around (GAA)configuration that provides a superior channel electrostaticcontrol [6], [7]. Among other advantages, GAA NWFETsexhibit the smallest SCE as compared to other multi-gate(MG) structures [8], [9]. Despite the better electron transportproperties of III-V materials, Si has remained the material ofchoice to fabricate GAA NWFETs due to its well-understood
This work has been supported by NSF grant EEC-0228390 that funds theNetwork for Computational Nanotechnology, by NSF PetaApps grant numberOCI-0749140, and by the Nanoelectronics Research Initiative (NRI) throughthe Midwest Institute for Nanoelectronics Discovery. The authors acknowl-edge the support of the MSD Focus Center, one of six research centers fundedunder the Focus Center Research Program (FCRP), a Semiconductor ResearchCorporation entity, computational resources from nanoHUB, and followingsupercomputer resources: Coates and Steele (RCAC), Kraken (NICS), Jaguar(NCCS), and Ranger (TACC).SungGeun Kim, Abhijeet Paul, Mathieu Luisier, and Gerhard Klimeck arewith Network for Computational Nanotechnology, School of Electrical andComputer Engineering, Purdue University, West Lafayette, IN 47907, USA;email: [email protected] B. Boykin is with Electrical and Computer Engineering Dept.,The University of Alabama in Huntsville, Huntsville, AL 35899, USA; e-mail: [email protected] characteristics and its compatibility to the conventional CMOStechnology [10], [11].In line with experimental efforts, simulation and model-ing approaches have been developed to help design betterperforming NWFETs and understand their physical behavior.Until very recently, drift-diffusion (DD) and Monte Carlo(MC) approaches have been successfully used to simulate largedevices. However, at the nanometer scale it is necessary to gobeyond DD and MC models and use a quantum mechanicalapproach that captures the wave nature of electrons [12].Nano-scale devices are characterized by 3D material varia-tions at the atomistic scale so that an atomistic descriptionof the simulation domain has become indispensable. Non-parabolicities and anisotropies of the bandstructure stronglyaffects the transport characteristics of electrons in NW tran-sistors [13], [14]. Furthermore, the strong confinement of elec-trons in nano-devices lifts degeneracies due to valley splitting[15]. These effects cannot be captured by the effective massapproximation. Therefore, it is inevitable to go beyond theeffective mass approximation and to use a full-band approachsuch as the nearest-neighbor empirical tight-binding (ETB)model [16].With the recent development of OMEN [17], a 3D full-bandquantum transport simulator based on the atomistic sp d s ∗ TB model [18], it has been possible to explore nano-scaleNWFETs. However, the computational burden of such anapproach is so intense that large computing resources and anefficient parallelization scheme of the work load is required[19].The focus of this paper is on interface roughness scattering(IRS) that is, in conjunction with electron-phonon [20] andimpurity scattering [21], crucial to understand the electricalcharacteristics of ultra-scaled NWFETs [22]. At the nanometerscale, the details of the NW surface become more important[23], [24]. In the ballistic transport limit, the normalized ON-current of NWFETs is expected to increase as the diameterdecreases. Experimental data shows that the ON-current in-creases with decreasing diameter until 3 nm [22]. However,decreasing the diameter below 3 nm reverses the trend and re-duces the current density. This behavior has been attributed toelectron-phonon scattering and interface roughness scattering[22]. Here we examine this experimental reasoning throughmodeling and simulations.In traditional MOSFETs, IRS has been identified as oneof the most important scattering mechanisms [25]. Because ofthe strong confinement of carriers in inversion layers that pushthem close to the semiconductor-oxide interface, the effective mobility of MOSFETs is limited by IRS at high electric fieldsor at high electron densities [26], [27].IRS in classical MOS type devices is treated via perturba-tion theory [28]. It is represented by an IR-limited mobilitydepending on the root-mean-square (RMS) of the roughnessamplitude and its correlation length [29]. Early 1–D quantumdevice simulations of IRS in the NEGF formalism also useda perturbation treatment [30]. In 3D nanowire devices, anexplicit IR representation is feasible and could represent thedevice- to- device fluctuation. Recent theoretical studies haveshown that mode-mixing due to IRS is reduced in NWFETscompared to conventional MOSFETs [31]–[33] and the po-tential fluctuation throughout the NW dominates IRS in smallcross-section NWs [33]. However, these studies have beenlimited to a continuum representation of rough surfaces with h i as transport direction and cross-sections larger than 3 × .Atomistic full-band simulations offer a more realistic repre-sentation of rough surfaces for any crystal directions. Atom-istic simulations show that the ON-currents and the mobilitiesof NWFETs are reduced significantly by IRS when the di-ameter of NWs is below 3 nm [34], [35]. These simulationshave been so far limited to ultra-narrow NWs without anyrepresentation of the SiO layer, i.e. hard wall boundaryconditions are applied to the silicon surface and the electronsdo not penetrate into the oxide layer.In this work, IRS is investigated in GAA < > NWFETswith different diameters (2 nm – 4 nm) through an atomisticrepresentation of the rough surface. Contrary to previousstudies, the SiO layers are modeled explicitly and taken intoaccount not only in the Poisson equation, as in Ref. [36], butalso in the quantum transport calculation. The oxide layersare modeled in the virtual crystal approximation (VCA) wherefictitious SiO atoms are used. Disorder is introduced througha disordered spatial representation of Si and “SiO ” atoms.The inclusion of the wavefunction penetration into SiO allowsus to investigate the ON-current and mobility reduction dueto IRS in NWFETs and to compare simulation results to theexperimental work in Ref. [22].The paper is organized as follows: In Section II, thesimulation approach is introduced with an emphasis on thegeneration of rough surfaces in circular nanowires. In SectionIII, the simulation results are presented and discussed.II. M ETHOD
A. Interface roughness model
The statistical properties of the interface between Si andSiO is characterized by the autocovariance function (ACVF) C ij = h s ( r i ) s ( r j ) i = ∆ m e −√ | r i − r j | /L m , (1)where ∆ m is the root mean square (RMS) of the interfaceheight and L m the correlation length of the rough surface.Here, s ( r i ) is a random number representing the amplitudeof the rough surface at position r i where a Si atom is locatedand connected to SiO atoms as shown in Fig. 1(a). Thequantity s ( r i ) can be defined as the distance between theposition of the Si atom and the reference plane. This model (a) Height (nm) (b)Fig. 1. (a) Atomistic representation of the cross-section of a perfect nanowirealong the h i direction. (ref.: reference plane, C ij / C ′ ij : ACVF for Si/SiO , r i / r ′ i : position vector of i -th Si/SiO atom) (b) 2D surface profile generatedfor a nanowire of diameter nm with ∆ m = 0 . nm, L m = 0 . nmfor the NWFET structure presented in Fig. 3 ( y ′ is a coordinate along thecircumference direction.) has been developed from an experimental study [37] wherethe measurement on the roughness amplitude is performed atan atomistic level. It is used in most of the simulation studieson IRS in NWFETs [31]–[33], [36]. Notice that the √ termin the numerator of the exponent in Eq. (1), which is notpresent in Ref. [36], is included to correctly fit the model tothe experiment as in Ref. [37].From Eq. (1), one can calculate s ( r i ) for each Si atom atthe interface following the procedure in Ref. [38]. Then, s ( r ′ i ) for the SiO atoms at the interface should be calculated byaveraging the s ( r i ) of Si atoms that are connected to a SiO atom located at r ′ i as shown in Fig. 1(a). This introduces anerror into the ACVF. For example, for a correlation length L m = 0 . nm, the error due to this approximation is ap-proximately . only (see Appendix A for a more detailedderivation).After s ( r i ) and s ( r ′ i ) are calculated for the atoms locatedat the first Si-SiO interface layer as shown in Fig. 1(a), the s ( · ) for atoms at the second layer are calculated in the samemanner. Then, Si atoms are replaced by SiO atoms and viceversa depending on the value of s ( · ) and a selection criterion.The criterion is described as follows: let d ( r i ) be the distancefrom the atom considered to the reference plane located inbetween Si and SiO atoms. Then, if s ( r i ) > d ( r i ) , the Siatom at r i should be replaced by SiO and if s ( r ′ i ) < − d ( r ′ i ) ,the SiO atom at r ′ i should be replaced by a Si atom.The rough interface of a NW of diameter 3 nm and length15 nm is projected to a two dimensional (2D) plane and plottedin Fig. 1(b). The rapid short-range fluctuation of the surfaceprofile is the characteristic of the chosen exponential modelrather than a Gaussian model [37].At this point, it is necessary to verify that the rough interfaceprofile generated with the algorithm explained above has thesame statistical properties as the theoretical ACVF (TACVF)presented in Eq. (1). It is not clear in the literature [31]–[33], [39] whether the generated rough interfaces are correctrepresentations of theoretical ACVF or not. Only in Ref.[40], the one dimensional (1D) ACVF of a single sample iscompared to its theoretical value and it is shown that the firstfew coefficients of the ACVF from the lowest order agree withthe TACVF.In this paper, 2D ACVFs of the generated rough interface −3 x (nm) C i j (a) TACVFACVF1ACVF2 0 1 2 305101520 x 10 −3 (b)y’ (nm) C i j TACVFACVF1ACVF2
Fig. 2. (a) The average of ACVFs for 100 samples to x direction, and (b)to the circumference direction ( y ′ ) are displayed. The halves of the graphs(a,b) to the negative direction are omitted because of their symmetry. All thecalculations are done for NWs with diameter 3 nm and gate length 15 nm(see Fig. 3). are compared to the TACVF after averaging ACVFs of 100samples. The ACVFs can be obtained either from s ( r i ) (labeled ACVF1) or from the position of the rough interfaceatoms (labeled ACVF2). ACVF1 should agree with TACVFand ACVF2 with ACVF1 and consequently with TACVF. Thefollowing equation is used for 2D ACVF calculation [41] C ij = 1 N x N y N x − i − X m =0 N y − j − X n =0 ( f m,n − ¯ f )( f m + i,n + j − ¯ f ) ,(2)where f m,n is the amplitude of the rough surface perturbationprojected onto the regular grids, and ¯ f is the average value of f m,n . N x and N y are the numbers of grid points on the x and y ′ axis (see Fig. 3 for a reference), respectively.Since it is not easy to directly compare 2D ACVFs with eachother, cutting sections of 2D ACVFs through the origin to x and y ′ direction are compared in Fig. 2. The ACVF1 and theACVF2 agree with TACVF for both x and y ′ direction whenthey are averaged over the entire rough NW samples. Thisvalidates the algorithm used for generation of rough interfacesin this work. The importance of the correct representation ofinterface roughness can be illustrated in the effect of the RMSvalue (= p ACVF (0 , ) on the drain current from Fig. 11(d). B. SiO model Previous attempts to model oxide materials in the tight-binding framework have been made by assuming a well-defined crystallographic structure of SiO such as β -quartz,tridymite [42], or β -cristobalite [43]. All these approacheshave in common a sp description of SiO up to secondnearest-neighbor interactions for the oxygen atoms. Theirapplication is limited to one-dimensional Si-SiO -Si structureswith a semiconductor-oxide interface perpendicular to the < > crystal axis only.As an alternative and to go beyond one-dimensional < > transport we introduce a simplified, but multi-functional modelto describe silicon dioxide. Fictitious SiO atoms are consid-ered and treated in the virtual crystal approximation. They are Fig. 3. The simulated structure of NWFET: A circular NW with varyingdiameter ( D Si =2, 2.5, 3, and 4 nm) is used. The default values of the otherparameters are displayed in the figure. arranged as a diamond crystal with the same lattice constant a = 0 . nm as the Si atoms. A nearest-neighbor sp tight-binding model is used to describe the electrical properties ofthe SiO . The band gap and conduction band offset are fittedto the experimental values with the relations E g = 8 . eV and ∆ E CB = 3 . eV with respect to Si [44] as well as an electroneffective mass m e = 0 . m [45]. A detailed description ofthe model is presented in Appendix B.III. R ESULTS
The NWFET structure considered in this work is illustratedin Fig. 3. ∆ m = 0 . nm and L m = 0 . nm are used as defaultparameters which are experimentally measured in Ref. [37] fora 2D Si/SiO interface. Crystal orientation h i is selectedas the transport direction because of its superior transportcharacteristics compared to other crystal orientations [13] andthe highest immunity to interface roughness scattering [36].Two important parameters the diameter of the NW channel D Si (depicted in Fig. 3) and ∆ m are varied to study theirimpact on the NW performances. The diameter of silicon NWchannel is varied from 2, 2.5, 3, to 4 nm and the RMS ofthe roughness amplitude is varied from 0.14, 0.2, to 0.3 nm.Strain is not considered in this work. The calculated I D – V G characteristics of perfect and rough NWFETs of 100 sampleswith a diameter of 2 nm are shown in Fig. 4.Fig. 4 presents two effects of IRS on I D – V G . The first effectis the positive shift of the threshold voltage and the secondeffect is the reduction of the ON-current. The positive shiftof threshold voltage is ascribed to the fact that the diameterof rough NWs throughout the channel is reduced (on average)from the perfect softwall NW due to rough interfaces [46]and that the density of states (DOS) in the energy spectrumis raised as discussed in Ref. [31]. The amounts of thresholdvoltage shift on average are 49.2, 12.5, 8.1, and 10.0 mV forrough NWs with diameter 2, 2.5, 3, and 4 nm. The standarddeviation σ of the threshold voltage shift is 13.6, 10.1, 5.5,and 2.0 mV for each diameter from 2 to 4 nm.The impact of IRS on the ON-current is measured by thereduction of the ON-current ( I perf − I scatt ) /I perf × where I perf is the drain current for perfect NWs with softwall BC and I scatt is the drain current for rough NWs. The ON-currents of −6 −5 −4 −3 −2 −1 V G (V) I D ( m A / µ m ) V D =0.6VD Si =2 nmt SiO =1 nmD T =4 nmI crit Perfect wire (softwall)Perfect wire (hardwall)Rough wire (softwall)Rough wire (hardwall)0 0.2 0.412 V G −V th (V) I D ( m A / µ m ) Fig. 4. I D vs V G for NWFETs of D Si = 2 nm and total diameter D T = D Si + t SiO × nm. I crit (= D Si × − A ) is the critical current for thethreshold voltage calculation. The gate work function for this simulation ischosen to be 3.95 eV. V D is set to 0.6 V. The error bars indicate the standarddeviation of the current in each bias point. rough NWs are reduced from perfect NWs by 20.5 % ( D Si =2 nm), 16.5 % ( D Si = 2 . nm), 9.0 % ( D Si = 3 nm), and 3.8% ( D Si = 4 nm) due to IRS on average.The effect of hardwall/softwall BC on the threshold voltageis noticeable as shown in Fig. 4. It can be explained fromFig. 5 where the electron density is plotted through the centerof the hardwall/softwall perfect NWs at ( z = 0 , x = L G / .An increase of the carrier concentration near the interfacebetween Si and SiO layer in the softwall NW reducesthe effective oxide thickness leading to the decrease of thethreshold voltage.The standard deviation of the threshold voltage due to IRSin the hardwall rough NWs (for 100 samples) for diameter 2nm is calculated to be 14.3 mV, that is, 5% increase comparedto that in the softwall rough NWs. The effect of BC on the −2 −1 0 1 2024681012 x 10 y (nm) E l ec t r o n d e n s i t y ( c m − ) V G =0 VV D =0.6 VHardwall NWSoftwall NW 0.8 0.9 11.522.53 x 10 y (nm) Fig. 5. Electron density along y -axis in the middle of the channel at ( z = 0 , x = L G / of a nanowire transistor with hardwall or softwall BC inthe Si and SiO interfaces. The inset describes the Si-SiO interfaces around y = 1 nm. V t h ( V ) Perfect wireRough wireIR+Phonon[20]Experiment [22] 2 2.50.850.90.951 D Si (nm) Si (nm)) I o n ( m A / µ m ) Experiment [22]IR+Phn[20]+R S ( ∆ rms =0.2 nm) IR+Phn[20]+R S ( ∆ rms =0.14 nm) Phn[20]Rough Wire ( ∆ rms =0.14 nm) Perfect wire
Fig. 6. (a) The threshold voltage and (b) the ON-current at V G − V th = 0 . Vvs diameter for the perfect and rough NWFETs. The experimental ON-currentand the source series resistance R S are extracted from Ref. [22]. The resultswith phonon scattering are extracted from Ref. [20]. The inset in (a) magnifiesthe threshold voltage trend in the square box. ON-current is even larger. The ON-current reduction in thehardwall NWs is 30% which is 9.5% higher compared to20.5% reduction in the softwall NWs. The standard deviationof the ON-current in the hardwall NWs is 0.179 mA/ µ mcompared to 0.141 mA/ µ m in the softwall NWs, that is, 26.7%increase. As the effective cross-section of the hardwall NWsis smaller than that of the softwall NWs, the fluctuation of theconduction band edge due to IRS is larger in the hardwall NWsand hence the calculation using the hardwall BC overestimatesthe effects of IRS.The threshold voltage and the ON-current for NWFETs withdifferent diameters are plotted in Fig. 6. The threshold voltageis calculated using the constant current method [31], [47] witha critical current I crit (= D Si × − A ) . The threshold voltageshift due to phonon scattering is extracted from Ref. [20]. Be-cause phonon scattering reduces the drain current for the wholerange of the gate bias, the threshold voltage is also shiftedto achieve the same critical current. The threshold voltagescalculated for the perfect NW and the rough NWs have similarvalues and slopes when the diameter of the NWs is larger than2.5 nm, and show similar trends as the experimental valuesextracted from Ref. [22]. However, as the diameter decreasesbelow 2.5 nm, the slope of the V th – D Si curve for the perfect L Rough wireT*(f L −f R ) E ( e V ) Ef L Rough wireEf L Rough wireEf L Rough wireEf L Rough wireEf L Rough wireRoughPerfect x (nm) log (D(E,x))Sub1Sub2Sub3Sub4 Ec0 10 25 35−0.500.51 Fig. 7. (Left) the current spectra (=transmission × ( f L − f R )) with the Fermilevel in the left contact Ef L and (right) the density of states in a log scaleacross a rough nanowire FET resolved in energy E and longitudinal coordinate x at the ON-state ( V G − V TH ∼ . V) overlapped with the conduction bandedge E C . NW deviates from the experimental values. This implies thatto get a consistent trend or quantitatively matched thresholdvoltage, one needs to include IRS in the transport calculationwhen the diameter of NW is scaled below 2.5 nm.The drain current calculated at V G − V th = 0 . V, i.e., theON-current is shown in Fig. 6(b). The ON-current reductiondue to IRS at a diameter smaller than 3 nm is significantand partly explains the decrease of the ON-current in theexperimental results [22].The ON-current with phonon scattering is calculated fromthe ballisticity in Ref. [20] of a very similar structure whereonly the doping density in the source/drain was different.In Ref. [20], the ballistic current is calculated using theelectrostatic potential extracted from the phonon scatteringsimulation to exclude the contribution of the phonon scatteringin the source/drain extension region.Since fully atomistic NEGF calculations are extremely com-putationally intensive, requiring thousands of CPUs for a sin-gle I-V, we cannot afford to perform statistical sampling withIRS. We therefore extract the effect of phonon scattering viathe ballisticity obtained in Ref. [20]. However, the simulationsin Ref. [20] are done for h i -oriented NWs whereby oursimulations are conducted for h i -oriented NWs. From thefact that h i -NWs have approximately 20% larger ballistic-ity than h i -NWs when phonon scattering is included [48],the ballisticities of h i -NWs are approximated from those of h i -NWs. The relationship /T IR+Phn = 1 /T IR +1 /T Phn − isused where the transmission T is calculated from the ballistic-ity factor β using the relation T IR/phn = 2 β IR/phn / (1 + β IR/phn ) .The ON-current reduction due to phonon scattering exceedsthat of IRS, and the drain current including both phonon andIR scattering (with series resistance) are in accordance with theexperimental results. Therefore, phonon scattering should alsobe included in the transport calculation to get a quantitativeprediction for the ON-state characteristics of NWFETs.Fig. 7 conveys detailed information about the effect of IRSon the current spectra and the DOS. IRS causes reflection ofelectrons and reduces the current spectra in the energy rangewhere the peak is located in the perfect NW. A localizationeffect of the DOS, indicated by two circles, due to the D =0.6 VV G =0.6 VE (eV) T r a n s m i ss i o n Perfect wireRough wire(highest current)Rough wire(lowest current)0 0.2 0.4 0.60123V G (V) I D ( m A / µ m ) Fig. 8. (Top) the electron density profile through the rough NW withhigest/lowest drain current, (bottom) transmission through the perfect NW,the rough NW with the highest/lowest drain current at V G = V D = 0 . V ,and (inset) I D − V G plot. fluctuation of the conduction band edge E C in the channelcauses this reduction of the current [34], [46]. The currentspectra and the DOS plot also shows that the relevant energyregion of the electron transport includes subbands that arehigher than the second subband. This supports that a full-band model is necessary to correctly understand and modelthese devices.The transmission curves for the perfect NW, the rough NWwith the lowest/highest drain current in Fig. 8 shed light onthe impact of IRS on the electron transport characteristics ofthe nanowire transistor. The decrease in transmission due toIRS throughout the whole range of energy is observed. Forthe rough NW with the smallest drain current shows morelocalization effect in real space compared to the rough NWwith the highest drain current, as can be seen from the top Si (nm)) B a lli s t i c E ff i c i e n c y ( β s a t ) Phonon [20]IR( ∆ m =0.14 nm)+Phn [20] IR( ∆ m =0.14 nm)IR( ∆ m =0.2 nm)IR( ∆ m =0.3 nm) Fig. 9. The ballistic efficiency β sat which is the ballistic ON-current dividedby the average ON-current with IRS, the phonon scattering (extracted fromRef. [20]), or both the IR and the phonon scattering included. ~D Si2 (a)5 ⋅ ⋅ D Si (nm) µ I R / phn ( c m / V s ) µ IR µ phn Si (nm) µ e ff ( c m / V s ) (b) µ ball µ eff,IR µ eff,phn µ eff,IR+phn Experiment [22]
Fig. 10. (a) The IR-limited mobility µ IR is calculated for the electron density N s = 3 . × cm − from the mean of the effective mobility including IRS µ eff,IR and the ballistic mobility µ ball through the Matthiessen’s rule µ − IR = µ − eff,IR − µ − ball . (b) The ballistic and effective mobility calculated for gate length L G = 30 nm are compared to the experimental results which are extractedfrom Ref. [22]. Effective channel length is taken into account by reducingthe channel length below the gate length according to the experimental result[22]. figure in Fig. 8.To further understand the importance of IR, the ballisticityis calculated for rough NWs with D Si = 2 nm after ∆ m isincreased to 0.2 nm and 0.3 nm. It is then compared to theballisticity including phonon scattering extracted from Ref.[20]. As shown in Fig. 9, the impact of IR on the ballisticity issmaller than that of phonon scattering for all diameters, but itbecomes comparable to the phonon effects as ∆ m increasesto 0.2 nm and starts to exceed the phonon effects when ∆ m = 0 . nm.As the ON-state characteristics of a transistor are alsorelated to their effective mobility [49], [50], the electronmobility reduction due to IRS at low drain bias should beunderstood in relation with the ON-current reduction at highdrain bias. The low-field effective mobility in NWFETs iscalculated from the electron density and the conductance usingthe expression [32], [33] µ eff = G lin L ch qN ch (3)where G lin is the conductance in the linear region i.e. at lowdrain bias ( V D = 5 mV in this work) and N ch is calculatedby integrating the electron density in the subsection of thechannel under the gate where the electron density is nearlyuniform [33] as illustrated in Fig. 11(c).Assuming that the ballistic mobility is the effective mobilityfor a perfect NW, the IR-limited mobility µ IR can be calculatedfrom the Matthiessen’s rule, namely, µ − IR = µ − eff,IR − µ − ball where µ eff,IR is the effective mobility for rough NWs. The IR-limitedmobility µ IR plotted in Fig. 10(a) is calculated from the meanof the effective mobility. It decreases as the diameter of theNW decreases. This is expected because IRS increases as thediameter decreases since the electrostatic potential fluctuatesmore in NWs with smaller diameter [51] and it is consistentwith the ON-current behavior in Fig. 6(b). The rate at whichthe average of µ IR decreases is approximately D Si which ismuch smaller than D Si for small electron density as describedin Ref. [51].To calculate the phonon-limited mobility, the resistance ofthe nanowire transistor is calculated for the same structure (a) ~ ∆ m−2 ∆ m (nm) µ I R / phn ( c m / V s ) µ phn µ IR ∆ m (nm) µ e ff ( c m / V s ) µ ball µ eff,IR µ eff,IR+phn
10 15 20 25246810 x 10 V G =0.5 VV D =0.05 V(c) x (nm) E l ec t r o n d e n s i t y ( c m − ) Perfect NWRough NW ∆ m (nm) I o n ( m A / µ m ) Perfect wireRough wire
Fig. 11. (a) The IR-limited mobility µ IR compared with the phonon-limitedmobility µ phn , (b) the total effective mobility including only IR or both the IRand the phonon scattering through Matthiessen’s rule compared to the ballisticmobility µ ball , (c) 1D electron density along the channel with the vertical barsdefining the subsection of the channel for mobility calculation, and (d) the IR-limited ON-current compared with the ballistic current are displayed. All thefigures are plotted for the nanowires with diameter D Si =2 nm. The electrondensity is set to N s = 3 . × cm − except for (c) where the gate/drainvoltage is fixed. as shown in Fig. 3, but for several different channel lengthof nanowires (15, 22.5, and 30 nm). To remove the effectof the source and drain region, the phonon-limited mobility iscalculated from the slope of the resistance [48]. The calculatedphonon-limited mobility shown in Fig. 10(a) does not show amonotonic behavior with respect to diameter. It instead showsa peak at 2.5 nm and a decreasing trend as diameter increasesto 3 nm. It can be explained by the argument that the highersubbands with a larger effective mass and scattering rates startto be populated with electrons as the diameter increases [52].Fig. 10(b) shows the effective mobility calculated for therough and perfect NWs compared to experimental results.To compare the simulation results with experimental results,we consider the fact that the ballistic mobility increasesproportionally to the channel length and that the effectivechannel length is smaller than the gate length as describedin Ref. [22]. The impact of IR on the effective mobility isstill very small at diameter 4nm, but it increases as diameterdecreases.The discrepancy between the experimental results [22] andthe simulation results is noticeable even after phonon-limitedmobility is included in the total effective mobility throughMatthiessen’s rule. We attribute this difference to the differentelectron density at which the effective mobility is calculated inthe experiment and our simulation. In the experiment, supplyvoltage V DD used is higher than that used in our simulation sothat the effective mobility at the ON-state in the experimentis smaller than that in the simulation. In addition, the surface quality of experimentally fabricated NWs may be differentthan that of the 2D plane surface which is adopted for oursimulation. Nonetheless, IR scattering is crucial to understandthe decreasing effective mobility in NWs of smaller diameter.Fig. 11(a) describes the importance of the quality of theinterfaces on the electron mobility in NWFETs for different ∆ m . The mean of the IR-limited mobility decreases with ∼ ∆ − m and it becomes comparable to the phonon limitedmobility when ∆ m = 0 . nm. This indicates that poor surfacequality can reduce the electron mobility significantly.When ∆ m increases, not only does the effective mobilitydecrease on average as in Fig. 11(b), but also the standarddeviation of the effective mobility increases. The standarddeviations of the effective mobility with respect to the meansof the effective mobility are given in percentages of 11.5 %( ∆ m =0.14 nm), 17.7 % ( ∆ m =0.2 nm), and 32.4 % ( ∆ m =0.3nm). The standard deviations of the ON-current in Fig. 11(d)are, in percentages, 7.37 % ( ∆ m =0.14 nm), 9.09 % ( ∆ m =0.2nm), and 21.6 % ( ∆ m =0.3 nm) from the mean of ON-current.The variability of the ON-current due to IRS is smaller thanthat of the effective mobility. In conclusion, the quality ofthe interface between Si and SiO is important in reducingthe variability of the ON-current and the effective mobility inNWFETs. IV. C ONCLUSION
The effect of interface roughness between Si and SiO onthe performance of NWFETs has been studied through anatomistic 3D full-band simulation. The SiO layer is modeledthrough VCA and its impact on the drain current is found tobe significant. IRS is important in determining the thresholdvoltage of NWFETs. The experimentally observed [22] trendof nonlinear threshold voltage slope below 2.5nm is capturedby the simulation. The significant reduction of the ON-currentis observed in NWs with diameter 2 nm with more than 20 %reduction.Though the phonon scattering is found to be an importantscattering mechanism in NWFETs, IRS cannot be ignored.IRS has a significant effect on both the ON-current and theeffective mobility specially when the diameter decreases downto 2 nm. As the effect of IRS depends on the surface quality inNWFETs, the interface roughness scattering is a crucial factorto predict the device performance of NWFETs.A PPENDIX AC ALCULATION OF A UTOCOVARIANCE FUNCTION FOR S I O The random variables describing the rough SiO surface,denoted as s ( r ′ i ) , must be calculated from the Si atoms becausethey are inter-correlated with each other. Let us assume thatone SiO atom is connected to two Si atoms whose random E s E p V ssσ V spσ V ppσ V ppπ SiO -4.8 1.83 -2.27 3.9265 5.4655 -0.3140TABLE I SP T IGHT - BINDING PARAMETERS FOR THE
VCA S I O MODEL . variables are s ( r i ) and s ( r i +1 ) . Then we approximate s ( r ′ i ) by averaging the two random variables as s ( r ′ i ) = s ( r i ) + s ( r i +1 )2 . (4)This approximation induces an error in the autocovarianceof the SiO atom ( C ′ ij ) as shown in the following calculation: C ′ ij = (cid:10) s ( r ′ i ) s ( r ′ j ) (cid:11) = (cid:28)(cid:20) s ( r i ) + s ( r i +1 )2 (cid:21) (cid:20) s ( r j ) + s ( r j +1 )2 (cid:21)(cid:29) ∼ = ∆ m e −√ | r i − r j | /L m (cid:16) e √ ∆ r/L m + e −√ ∆ r/L m (cid:17) = C ij " (cid:0) √ ∆ r/L m (cid:1) , (5)where Eq. (1) and the relationship ∆r = r i +1 − r i ∼ = r j +1 − r j are used. Then, Eq. (5) can be further approximated with aTaylor expansion up to the second order as C ′ ij ∼ = C ij (cid:20) x (cid:21) (6)with x = √ ∆r /L m . If ∆r ≈ . nm and L m = 0 . nm,the difference between C ij and C ′ ij due to this approximationis x / ∆r / (2 L m ) ≈ . .A PPENDIX BS I O2 MODEL AND RESULTS
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