In this letter, we explore the bandstructure effects on the performance of
ballistic silicon nanowire transistors (SNWTs). The energy dispersion relations
for silicon nanowires are evaluated with an sp3d5s* tight binding model. Based
on the calculated dispersion relations, the ballistic currents for both n-type
and p-type SNWTs are evaluated by using a semi-numerical ballistic model. For
large diameter nanowires, we find that the ballistic p-SNWT delivers half the
ON-current of a ballistic n-SNWT. For small diameters, however, the ON-current
of the p-type SNWT approaches that of its n-type counterpart. Finally, the
carrier injection velocity for SNWTs is compared with those for planar
metal-oxide-semiconductor field-effect transistors, clearly demonstrating the
impact of quantum confinement on the performance limits of SNWTs.Comment: 12 pages, 4 figure

Ghosh, Avik

Klimeck, Gerhard

Lundstrom, Mark

Rahman, Anisur

Wang, Jing

English

arXiv

Jing Wang

Anisur Rahman

Avik Ghosh

Gerhard Klimeck

Mark Lundstrom

Crossref

Performance evaluation of ballistic silicon nanowire transistors with atomic-basis dispersion relations

In this letter, we explore the band structure effects on the performance of ballistic silicon nanowire transistors sSNWTsd. The energy dispersion relations for silicon nanowires are evaluated with an sp3d5s* tight binding model. Based on the calculated dispersion relations, the ballistic currents for both n-type and p-type SNWTs are evaluated by using a seminumerical ballistic model. For large diameter nanowires, we find that the ballistic p-SNWT delivers half the ON-current of a ballistic n-SNWT. For small diameters, however, the ON-current of the p-type SNWT approaches that of its n-type counterpart. Finally, the carrier injection velocity for SNWTs is compared with those for planar metal-oxide-semiconductor field-effect transistors, clearly demonstrating the impact of quantum confinement on the performance limits of SNWTs

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Purdue UniversityPurdue e-PubsOther Nanotechnology Publications Birck Nanotechnology Center2-25-2005Performance evaluation of ballistic silicon nanowiretransistors with atomic-basis dispersion relationsJing WangPurdue University - Main CampusAnisur RahmanPurdue University - Main CampusAvik GhoshPurdue University - Main CampusGerhard KlimeckPurdue University - Main Campus, gekco@purdue.eduMark LundstromPurdue University - Main CampusFollow this and additional works at: http://docs.lib.purdue.edu/nanodocsThis document has been made available through Purdue e-Pubs, a service of the Purdue University Libraries. Please contact epubs@purdue.edu foradditional information.Wang, Jing; Rahman, Anisur; Ghosh, Avik; Klimeck, Gerhard; and Lundstrom, Mark, "Performance evaluation of ballistic siliconnanowire transistors with atomic-basis dispersion relations" (2005). Other Nanotechnology Publications. Paper 114.http://docs.lib.purdue.edu/nanodocs/114Performance evaluation of ballistic silicon nanowire transistorswith atomic-basis dispersion relationsJing Wang,a! Anisur Rahman, Avik Ghosh, Gerhard Klimeck, and Mark LundstromSchool of Electrical and Computer Engineering, Purdue University, West Lafayette, Indiana 47907sReceived 26 October 2004; accepted 10 January 2005; published online 25 February 2005dIn this letter, we explore the band structure effects on the performance of ballistic silicon nanowiretransistorssSNWTsd. The energy dispersion relations for silicon nanowires are evaluated with ansp3d5s* tight binding model. Based on the calculated dispersion relations, the ballistic currents forboth n-type andp-type SNWTs are evaluated by using a seminumerical ballistic model. For largediameter nanowires, we find that the ballisticp-SNWT delivers half the ON-current of a ballisticn-SNWT. For small diameters, however, the ON-current of thep-type SNWT approaches that of itsn-type counterpart. Finally, the carrier injection velocity for SNWTs is compared with those forplanar metal-oxide-semiconductor field-effect transistors, clearly demonstrating the impact ofquantum confinement on the performance limits of SNWTs. ©2005 American Institute of Physics.fDOI: 10.1063/1.1873055gWith the rapid progress in nanofabrication technology,silicon nanowiressSiNWsd with small diameterss,20 nmdhave been synthesized and extensively studied for potentialapplications in nanoelectronics.1–5 Among them, the siliconnanowire transistorsSNWTd has attracted broad attention asa promising structure for future integrated circuits due to itsexcellent scaling capability and compatibility with Si-basedelectronic technology.1,5–7 Device theory stipulates that tomaintain a good electrostatic gate control, the diameter of aSNWT should be reduced as the gate length of the transistorscales down. For this reason, SNWTs with ultrasmall diam-eterss,5 nmd are needed when the gate lengths enter thesub-10 nm regime.7,8 Due to strong quantum confinementsQCd in ultrathin SiNWs, atomic band structure effects9–11are expected to play an important role on their devicecharacteristics.In this letter, we theoretically explore the impact ofbandstructure, nanowire diameter and carrier typesn- orp-typed on the performance limits of SNWTs. First, the en-ergy dispersionsE–kd relations for SiNWs with differentwire widths se.g., 0.5–6.8 nmd are calculated by using ansp3d5s* semiempirical tight-binding approach.10–12A seminu-merical ballistic model13 is then employed to evaluate theballistic current–voltagesI –Vd characteristics of bothn-typeand p-type SNWTs based on the calculatedE–k relations.Finally, the carrier injection velocity for the simulated SN-WTs is compared with that for the corresponding metal-oxide-semiconductor field-effect transistorssMOSFETsd.Silicon nanowires with various cross-section shapes andtransport orientations have been synthesized by different ex-perimental groups.1–5 In this work, we focus on one specificSiNW structure with one particular transport orientation as afirst step in exploring bandstructure effects in small SNWTs.The inset of Fig. 1 shows the cross section of the simulatedSiNWs. The shape of the cross section is rectangular, trans-port is along thef100g direction, and the faces of the rect-angle are along the equivalentk100l axes. The Si body thick-ness,TSi, is assumed to be equal to the wire width,WSi si.e.,TSi=WSi=Dd. A unit cell of the SiNW crystal consists of fouratomic layers along thex stransportd direction and has alength of a0=5.43 Å. In the nanowire Hamiltonian, eachatom is modeled using 20 orbitals—thesp3d5s* basis withthe spin–orbital coupling. The orbital-coupling parametersused in this work are from Ref. 12, which have been opti-mized by Boykinet al. to accurately reproduce bulk Si prop-ertiessband gap, effective-masses, etc.d. A hard wall bound-ary condition for the wave function is used at thesemiconductor–oxide interfaces and the dangling bonds atthese interfaces are pacified using a hydrogenlike terminationmodel of thesp3 hybridized interface atoms.14 As demon-strated in Ref. 14, this technique successfully removes all theinterface states from the band gap.sIt should be mentionedthat the wave function penetration into the oxide may reducethe wire band gap and increase the effective wire width, butit should not affect the qualitative conclusions of this letter.dFigure 1 shows theE–k dispersion relations for thesimulated SiNWs withsad D=1.36 nm andsbd D=5.15 nm.The bulk conduction band of Si has six equivalentD valleyslocated near the X points in the Brillouin zone. For a nano-wire with the transport axis alongf100g, four of the sixequivalentD valleyssi.e., f010g, f01̄0g, f001g, andf001̄gd areprojected into theG point in the one dimensional Brillouinzone to form the conduction band edge. The other twoD valleys si.e., f100g and f1̄00gd, located at kx= ±0.815·2p /a0= ±1.63p /a0 in the bulk Brillouin zone, arezone-folded to the pointskx= ±0.37p /a0 in the wire Bril-louin zone and become off-G states. A similar phenomenon isobserved by Koet al.10 and Zhenget al.11 in a f100g orientedrectangular wire with equivalentk110l confinement direc-tions. As in a Si quantum well, the degeneracy of the four-fold G valleys in af100g oriented nanowire can be lifted bythe interaction between the four equivalent valleys, which isso called “band splitting.”15 It is clear from Fig. 1 that thecorresponding band splitting is more evident in the thinnerwire sD=1.36 nmd than in the thicker wiresD=5.15 nmd,analogous to the band splitting observed in Si quantumwells.15Figure 2 shows the conduction and valence band edgesssolid with circlesd for the simulated wires with a wire widthranging from 1.0 nm to 6.8 nm. It shows that the wire bandgap is enlarged by QC and the increment isroughly inverselyproportional to the square of the wire width.10,11 The dashedadElectronic mail: jingw@purdue.eduAPPLIED PHYSICS LETTERS86, 093113s2005d0003-6951/2005/86~9!/093113/3/$22.50 © 2005 American Institute of Physics86, 093113-1Downloaded 26 Apr 2005 to 128.211.171.180. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jspline in the upper plot is for the conduction band edge calcu-lated by a simple “particle in a box” model with the bulkeffective masses. The difference between the effective-masssEMd curve and the tight binding result becomes evidentwhenD,3 nm, which shows that the band structure effectscan be important in Si nanowires with small diameters.9–11After obtaining the tight bindingE–k relations of thesimulated SiNWs, theI –V characteristics of the correspond-ing SNWTs can be explored by a seminumerical ballisticmodel.13 The model captures three-dimensional electrostat-ics, quantum capacitance16 and bias-charge self-consistencyin ballistic field-effect transistorssFETsd. sSource-to-draintunneling is not considered in this model.d In the past, thismodel was used to evaluate theI –V characteristics of SiMOSFETssRef. 13d and high electron mobility transistors17with parabolic energy bands and Ge MOSFETs with numeri-cal E–k relations.18 To be concise, we do not indicate thedetails of this model but refer the readers to publishedreferences.13,17,18 sThe Matlab® scripts of this model areavailable.19dTo compare the device performance ofn-type vsp-typeSNWTs, we adjust the gate work functions to achieve thesame OFF-currents for the two structures at each wire widthfsee Fig. 3sad for an example ofD=1.36 nmg, then the ON-currents for both thenFET and pFET are compared. Theinset of Fig. 3sad plots the ratio of thepFET ON-current tothat for thenFET at the same wire width. The result showsthat for a large wire width, a ballisticpFET delivers aboutone-half the ON-current of a ballisticnFET. For a smallerwire width, however, the ON-current of the ballisticpFETapproaches that of itsn-type counterpart. To explain this ob-servation, we plot the average carrier velocitysunder highdrain biasd vs gate bias for then-type/p-type SNWTs withD=1.36 nm andD=5.15 nm fFig. 3sbdg. Interestingly, thecarrier velocity fornFETs decreases while that forpFETsincreases when the wire width is reduced. The origin of thesetrends involves different physics:1d Due to the nonparabo-licity of the four D valleys in the bulk Si conduction band,the projectedG valleys in the wires exhibit a larger transporteffective mass at a smaller wire widthfsee the inset of Fig.3sbdg, where stronger QC occurs. Since the electron thermalvelocity is inversely proportional to the square root of thetransport effective-mass,13,18 the average electron velocity inn-type SNWTs decreases as the wire width scales down.s2dFor the valence band of the wires, our simulations show thatthe curvature of the valence band edgeshole effective-massdis insensitive to the wire width. However, with increasingwire width, more and more higher subbands with largertransport hole effective-masses become populatedfs e Fig.1sbdg, effectively lowering the average hole velocity inp-type SNWTs.It is of great interest to compare SNWTs vs planar MOS-FETs. Previous studies7,8 show that the SNWT obtains a bet-ter gate control as well as a larger threshold voltage variationthan the planar MOSFET due to its stronger QC. In thiswork, we compare the performance of SNWTs vs planarMOSFETs in terms of carrier injection velocity.13 Figure 4shows the average carrier velocitiessunder high drain biasdfor the simulated SNWTssD=1.36 nm,5.15 nmd and theplanar MOSFETs with comparable cross sectionssTSi=1.36 nm,5.15 nmd. sThe E–k relations for the planarMOSFETs are calculated with the same tight binding ap-proach as used in the nanowire calculation.d The normalizedFermi level,hF, is defined ashF=sEF−ECd /kBT for nFETsand hF=sEV−EFd /kBT for pFETs, whereEF is the Fermilevel andECsEVd is the conductionsvalenced band edge. Theresults show that for the same normalized Fermi level thep-SNWT sdashed lined displays a,20% lower hole velocityas compared to thep-MOSFET sdashed line with opencirclesd at both DsTSid=1.36 nm andDsTSid=5.15 nm. Forthe n-type FETs, however, the SNWTssolid lined obtains ahigher electron velocity than the planar MOSFETssolid linewith closed circlesd at DsTSid=5.15 nm while a lower one atDsTSid=1.36 nm. An explanation of this interesting observa-tion requires an understanding of the role of QC on theelec-tron velocity in a Si nanowire/thin-film.Generally speaking, QC affects electron velocity in twoopposite ways. First, QC lifts the sixfold degeneracy of theDvalleys in bulk Si so that the unprimed valleyssi.e., f010g,f01̄0g, f001g, and f001̄g for a f100g oriented nanowire, orf001g and f001̄g for thin-films with a f001g confinement di-FIG. 1. The energy dispersion rela-tions for the simulated Si nanowirestructures withsad D=1.36 nm andsbdD=5.15 nm. The inset shows a sche-matic diagram of the nanowire crosssectionsTSi=WSi=Dd. For the thinnerwire sD=1.36 nmd, strong band split-ting is observed at theG point in theconduction band.FIG. 2. The conductionsupperd and valenceslowerd band edgessolid withcirclesd at G point vs D. The dashed line in the upper plot is for the con-duction band edge calculated by the effective-masssEMd approach with asimple “particle in a box” model.093113-2 Wang et al. Appl. Phys. Lett. 86, 093113 ~2005!Downloaded 26 Apr 2005 to 128.211.171.180. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsprectiond with a relatively small transport effective-massmt=0.19me in bulk Sid display a smaller conduction band mini-mum and acquire a higher electron occupancy. As a result,the averageelectron velocity is increased by this valley-splitting effect caused by QC. Second, when the wire widthor the film thickness is small, strong QC increases the trans-port effective massfsee the inset of Fig. 3sbdg and conse-quently decreases the average electron velocity. From theresults shown in Fig. 4, we conclude that whenDsTSid isrelatively large se.g., 5.15 nmd, the valley-splitting effectdominates, and the wire obtains a larger electron velocity dueto the additional QC in the width direction. WhenDsTSid issmallse.g., 1.36 nmd, however, the unprimed valleys are wellseparated from the primed valleys in both the wire and thethin-film, and the effective-mass-raising effect dominates.Thus, the wire displays a smaller electron velocity than thethin-film due to stronger QC.In summary, by using ansp3d5s* tight binding model, weexplored the energy dispersion relations off100g orientedrectangular Si nanowires with a wire width up to 6.8 nm.Based on theseE–k relations, we calculated the ballistic cur-rents for bothn-type andp-type SNWTs with the use of aseminumerical ballistic FET model. We found that for verysmall diameterssD,1.5 nmd, ballistic p-SNWTs approachthe performance ofn-SNWTs. The carrier injection velocityfor SNWTs was compared with those for planar MOSFETs,and we observed thatp-SNWT displays a,20% lower holevelocity than thep-type planar MOSFETfat both DsTSid=1.36 nm andDsTSid=5.15 nmg. For nFETs, however, dueto the effects of quantum confinement, the SNWT displays ahigher electron velocity than the planar MOSFET when thewire width sor film thicknessd is relatively large fe.g.,DsTSid=5.15 nmg while a lower one whenDsTSid is smallse.g., 1.36 nmd. In short, the band structure effects play animportant role in nanowires with small diameters and shouldbe seriously considered when evaluating the performancelimits of SNWTs.This work is supported by the Semiconductor ResearchCorporationsSRCd, the Microelectronics Advanced ResearchCorporation sMARCOd focus center on Materials, Struc-tures, and DevicesMSDd and the National Science Founda-tion sNSFd Network for Computational NanotechnologysNCNd. The authors would like to thank Professor SupriyoDatta at Purdue University for the useful discussions.1Y. Cui, Z. Zhong, D. Wang, W. U. Wang, and C. M. Lieber, Nano Lett.3,149 s2003d.2D. D. D. Ma, C. S. Lee, F. C. K. Au, S. Y. Tong, and S. T. Lee, Science299, 1874s2003d.3Y. Wu, Y. Cui, L. Huynh, C. J. Barrelet, D. C. Bell, and C. M. Lieber,Nano Lett. 4, 433 s2004d.4L. Samuelson, M. T. Bjork, K. Deppert, M. Larssonb, B. J. Ohlssonc, N.Panevaet al., Physica EsAmsterdamd 21, 560 s2004d.5H. Majima, Y. Saito, and T. Hiramoto, Tech. Dig. - Int. Electron DevicesMeet. 733s2001d.6J. Wang, E. Polizzi, and M. Lundstrom, J. Appl. Phys.96, 2192s2004d.7J. Wang, E. Polizzi, and M. Lundstrom, Tech. Dig. - Int. Electron DevicesMeet. 695s2003d.8J. Wang, P. Solomon, and M. Lundstrom, IEEE Trans. Electron Devices51, 1366s2004d.9X. Zhao, C. Wei, L. Yang, and M. Y. Chou, Phys. Rev. Lett.92, 236805s2004d.10Y. J. Ko, M. Shin, S. Lee, and K. W. Park, J. Appl. Phys.89, 374 s2001d.11Y. Zheng, C. Rivas, R. Lake, K. Alam, T. B. Boykin, and G. Klimeck,IEEE Trans. Electron Devicess ubmittedd.12T. B. Boykin, G. Klimeck, and F. Oyafuso, Phys. Rev. B69, 115201/1–10s2004d.13A. Rahman, J. Guo, S. Datta, and M. Lundstrom, IEEE Trans. ElectronDevices 50, 1853s2003d.14S. Lee, F. Oyafuso, P. von Allmen, and G. Klimeck, Phys. Rev. B69,045316s2004d.15T. B. Boykin, G. Klimeck, M. A. Eriksson, M. Friesen, S. N. Coppersmith,P. Von Allmenet al., Appl. Phys. Lett. 84, 115 s2004d; A. Rahman, G.Klimeck, M. Lundstrom, N. Vagidov, and T. B. Boykin, Jpn. J. Appl.Phys., Part 1ssubmittedd.16S. Luryi, Appl. Phys. Lett.52, 501 s1988d.17J. Wang and M. Lundstrom, IEEE Trans. Electron Devices50, 1604s2003d.18A. Rahman, G. Klimeck, and M. Lundstrom, Tech. Dig. - Int. ElectronDevices Meet. 139s2004d.19http://nanohub.orgFIG. 3. sad The I –V curves for then-type/p-type SNWT withD=1.36 nmand the ratio of thepFET ON-currentto thenFET’s vsD sinsetd. The oxidethickness is assumed to be 1 nm andthe temperature is 300 K.sbd The av-erage carrier velocitiessunder highdrain biasd vs gate bias for then-type/p-type SNWTs withD=1.36 nmand D=5.15 nm. The inset shows thedependence of the transport effectivemasssin the conduction bandd at theGpoint on D. The carrier velocities forthenFET and thepFET show differenttrends with decreasingD due to differ-ent physics.FIG. 4. Average carrier injection velocitysunder high drain biasd vs normal-ized Fermi level,hF, for the simulated SNWTs and planar MOSFETs.DsTSid is equal to 1.36 nmsleftd and 5.15 nmsrightd for the SNWTssplanarMOSFETsd.093113-3 Wang et al. Appl. Phys. Lett. 86, 093113 ~2005!Downloaded 26 Apr 2005 to 128.211.171.180. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp

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Performance Evaluation of Ballistic Silicon Nanowire Transistors with
  Atomic-basis Dispersion Relations

Performance Evaluation of Ballistic Silicon Nanowire Transistors with Atomic-basis Dispersion Relations

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