Analytical Formulas for the Drain Current of Silicon Nanowire MOSFET

Transcription

1 Analytcal Formulas for the ran urrent of Slcon Nanowre MOSFE Outlne Kenj Nator Fronter esearch enter oyo Inst. ech. acground ompact model ervaton and characterstcs of analytc models Summary

2 acground Ⅰ MOSFEs are downszed nto nanometer scale. 3-m structures le FnFEs and r-gate MOS are actvely nvestgated. Nanowre NW FEs attract wde attenton for the excellent controllablty aganst SE. We have proposed a compact O y z x Source ate ran model of NW MOSFE. ut complex numercal analyss s requred n the dran current evaluaton. ross secton

3 acground Ⅱ Analytcal formulas are requred for the crcut applcaton of NW MOSFE e.g. W V W I ox V Vt V, I ox V Vt L L As the analytcal formula for nano-mosfe, Lundstrom s formula, e.g. s well-nown, I eff V V but only the saturaton current s gven, no lnear regon. Objectve erve the analytcal dran-current models of NW MOSFE, manageable wth handy calculators. t v nj

4 Startng pont s the ompact Model Essentals of arrer scatterng n ompact Model Lnear Potental Approx. : Electrc Feld E ~ Source F Elastc acscatt. Vx hannel Intal Elastc Zone x Elastc acscatt. +Optcal Phonon Emsson Optcal Phonon * Energy elax Zone ransmsson Probablty : x o ran

5 ésumé of the ompact Model. ln r t r ox ox ox S qv ompact Model stll requres a complex calculaton. ln qe qex qe m AA Electrostatcs requrement arrer dstrbuton n Subbands d f f g q I S,,, S t Q q V V, nv ox exp exp exp d d g q Q S S

6 Is the ompact Model a relable startng pont? ompact Model esult Numercal smulaton Jn et al. V =.V V =.V V =.3V IEEE rans.e.. vol. 55, pp. 77, 8. ran urrent [A] ran as [V] [ NW evce parameters : =5 nm, L=5 nm, t ox =.8 nm, =3 cm /Vs ] Agreement s satsfactory. We trust the model and derve analytc formulas

7 asc Approxmatons hc cylndrcal NW s assumed, promsng n realstc applcaton. + r ross secton r r,, x A j rexp n exp x jn x j x mr m Subband summaton s smplfed. n Average transmsson coeffcent s sorted out. f, f, d f, f, d S S Energy ntegraton s smplfed. x dn

8 wo approaches are possble. [] Full-degeneracy Model urrent A ran as V 3K V =.V 3K V =.4V 3K V =.7V 3K V =.V 4K V =.V 4K V =.4V 4K V =.7V 4K V =.V Ferm strbuton emperature dependence s not clear n allstc characterstcs. Actual temperature dependence s controlled by carrer scatterng. arrer scatterng s treated separately. We can assume Full-degeneracy n carrer dstrbuton. exp Step strbuton Energy ntegraton s smplfed.

9 Full-degeneracy Model ran current formula ox nv, qv m q q r qv m ln3.5 L V sat V q Vt, V S Vt qv q 4 q 8 V V, I qv 3 q m 3/ 3/ sat q S S V V, I 8 qv 3 q m 3/ sat q sat

10 Full-degeneracy Model evce characterstcs Wre dameter :nm, L=nm SE : not consdered ox =nm, nv : equvalent to.5nm S, = =.54 ev / s - equvalent to =3cm /Vs, =.46 ev / s - ompact Model Full-degeneracy Model ran urrent [ua] 4 3 V-Vt=.V V-Vt=.3V V-Vt=.7V V-Vt=.V ran urrent [ua] 4 3 V-Vt=.V V-Vt=.3V V-Vt=.7V V-Vt=.V ran as [V] ran as [V] Smlar, but the compact model current s larger! he analytc approxmaton mght be too coarse n a thn devce wth the nm dameter!

11 wo approaches [] oltzmann Statstcs Model Ferm strbuton S When Ferm emperature F ~ carrer dstrbuton s Non-degenerate. Actual dstrbuton : egeneracy Non-degeneracy We can assume oltzmann statstcs as opposte to []. exp oltzmann strbuton exp Energy ntegraton s analytcally executed.

12 , Q nv ox eff nj t eff qv qv v V V I exp exp m v nj ln3.5 L m qv qv oltzmann Statstcs Model ran current formula

13 oltzmann statstcs Model evce characterstcs Wre dameter :nm, L=nm SE : not consdered ox =nm, nv : equvalent to.5nm S, = =.54 ev / s - equvalent to =3cm /Vs, =.46 ev / s - Full-degeneracy Model 6 oltzmann Statstcs Model ran urrent [ua] 4 3 V-Vt=.V V-Vt=.3V V-Vt=.7V V-Vt=.V ran urrent [ua] 4 3 V-Vt=.V V-Vt=.3V V-Vt=.7V V-Vt=.V ran as [V] ran as [V] Smlar characterstcs, oltzmann Model slghtly larger.

14 Saturaton urrent expresson Full-degeneracy Model I sat eff V Vt vnj deg oltzmann statstcs Model I sat eff V V t v th v nj deg 4 3m V V qr Injecton velocty of degenerate carrers v th eff m hermal njecton velocty x t oth models are reduced to the Lundstrom Formula I eff V V t v nj wth an approprate njecton veocty.

15 Summary Analytc dran current models of NW FEs, manageable wth handy calculators, are derved. he models meet the case of thc NW FEs, promsng n realstc applcaton. he models represent both lmts of the full-degeneracy of carrers and the oltzmann dstrbuton. he both cases provde a smlar magntude of dran current, whch are smaller than the ompact Model result n a nm dameter devce. he saturaton dran current s reduced to the Lundstrom formula n both cases.