1) p represents the number of holes present. We know that,

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1 ECE650R : Reliability Physics f Nanelectrnic Devices Lecture 13 : Features f FieldDependent NBTI Degradatin Date : Oct. 11, 2006 Classnte : Saakshi Gangwal Review : Pradeep R. Nair 13.0 Review In the last six classes, we have talked abut NBTI as an exaple f CMOS degradatin phenena. We saw that NBTI degradatin is characterized by se unique tiedependence and teperaturedependent features, i.e.! Tie dependence " pwer law " recvery " frequency independence! Teperature dependence " dispersive transprt " hpping transprt Ging ahead we will nw lk at the Field dependence f N IT Field Dependence In general, NBTI is studied at high fields t shrten test tie, but perated at lwer vltages. Mrever, at any given tie in an IC, different PMOS transistrs experience different vltage transients depending n the lcal lgic activity. This akes it highly iprtant fr ne t knw hw the degradatin depends n Electric field. We have seen that, kf ( ) ( H k ) r NIT = N D t Here k r is assciated with the passivatin f neutral H and thus it desn t depend n the electric field. Als D H is assciated with the neutral hydrgen s again it is unaffected by the field. Field dependence here ces nly thrugh k f. k f can be written as: kf = k p T e Ε ( F Ε a x)/ kbt σ 1) p represents the nuber f hles present. We knw that, p q = Cx ( Vg Vt ) S, pα Ex n 2) Once the hles are captured by the SiH bnds, they have t tunnel apprxiately 1.52 A int the bnd. This dependence ces thrugh T which is the Transissin cefficient.

2 Hence T varies expnentially with E x, i.e. T e γ T Ε α x 3) If an Electric field is applied (thrugh the gate vltage) then the ptential barrier gets reduced by an aunt ae x. As shwn in Fig. 1, the barrier height is nw E F ae x SiO 2 U E FO ae x x Fig 1: Variatin f Ptential with respect t distance We will nw lk at hw this happens. Our purpse is t eventually knw what is a in ters f aterial prperties. Si H bnd behaves as a stretched peranent diple in the presence f an electric field. This is because f the difference in electrnegativities f Silicn and Hydrgen at. The peranent diple ent due t this can be written as: P = Q r Q is the excess inic charge r is the average separatin between SiH ats. Nw let us cnsider an islated diple separated by a distance r 0. Si H Electric Field x x+ Fig 3: Shift f equilibriu psitin in a SH diple under Electric Field This can be seen as a spring with a spring cnstant γ which stretches under the influence f frce due t Electric field and stps eventually. Fr the Si at, the net frce equatin can be written as: γ ( x x + ) Q ELc = 0 (1) Fr the H at, net frce equatin can be written as: γ ( x + x ) + Q ELc = 0 (2) Slving Equatin (1) and (2), we get

3 ( + ) x x = Q ELc / γ = Thus, ttal plarizatin f an islated lecule wuld be: P Q r Q = + + ELc γ (3) Part I Part II Part I Due t the peranent diple present Part II Electrnic cpnent due the Electric field dependence (which has stretched the bnd) Change in ptential due t plarizatin can be written as: U = P ELc In the case f SiH bnd, cpnent II is alst negligible and hence U = Q r ELc (4) Fr Fig 1, U = aex (5) = Fr (4) and (5), ae = Q r E Lc (6) x T find ut E Lc in ters f E x, cnsider the SiO bnds inside the insulatr SiO 2. These are als Plarized Diples. T begin with, there are equal nubers f psitive and negative diple centers and hence the ttal charge is balanced. In this case, the electric field inside the seicnductr is E x. When a bx f dielectric is cut ut, then there is an ibalance f charges. An electric field, E lc develps inside the cut ut part. When a cnstant gate vltage is applied (V G ), Q = CV G Q air = ( ε /t x )V G Q x (inside the dielectric SiO 2 ) = ( Kε /t x )V G K is the dielectric cnstant Since (by definitin) V G is cnstant, E x = E air = V G / t x Applying Gauss s law (in the blue bx, fig ), ne can write: Si SiO E lc (Air) E x ε E Lc = E x ε x where ε x = K ε

4 E Lc = K E x (7) S field inside a cut ut bx is actually larger than the dielectric field. This is cntrary t what ne bserves when a dielectric is intrduced between a parallelplate capacitr. In that case, the ttal charge n the plate is assued cnstant as ppsed t the Si SiO 2 assuptin that the (gate) vltage is held fixed here Nw when a bx is cut ut at the interface, E Lc = KE x E x Here, K ay vary anywhere between the dielectric cnstant f SiO 2 t that f Si. Fr equatins (6) and (7), a = Q r K S when a cnstant gate vltage is applied, the field prduced at the interface is (KE x ). When islated, dilute, nninteracting SiH bnds are placed within the bx, its barrier fr dissciatin is reduced by aex (Fig. 1). As such, at higher fields, the sae teperature can break re bnds. S far, we have seen the tie, field and teperature dependence f the interface traps. This was the physical explanatin fr hw des the degradatin behave n an average. Nw we will lk at the statistical deling f this behavir Statistical Mdel When a large nuber f transistrs are cnsidered, then N IT can be taken t be having a Gaussian prfile (rand distributin). The black curve in fig. 3a depicts the ean value fr N IT at any tie t while the blue and red curves depict the nrally distributed N IT. The curve saturates at N since the greatest value fr the nuber f traps can nly be the axiu nuber f SiH bnds, which is N. Suppse, is the fractin f brken bnds deterining failure f the transistr. Theretically, this shuld deterine the lifetie liit fr the transistrs. Hwever, due t variatin in threshld vltage ang the transistrs acrss the design and reductin in current, the circuit ight fail uch earlier. Taking 100pp (1/10,000 part failing) t be the failure liit, the real lifetie fr the transistrs turns ut t be uch lwer (as shwn in the Fig. 3b ) Here is the statistical deling f Fielddependent NBTI degradatin. Suppse there are N sites f SiH bnd

5 Prbability that ut f N, i bnds are brken: N! i W () N i = p q i!( N i) ( N i) p: average failure = <N IT >/N = At n /N S the value f p is nt a cnstant rather it changes with tie. Fractin f ttal transistrs that will fail: N F = WN() i i= N lnn IT F lnt F = 100 pp Real life tie Theretical life tie If N is very large and is sall then the binial distributin can be apprxiated t an expnential ne. N i N i F N( N 1)...( N i 1) pq = + i! i= N i N ( ) (1 p N i F = Np ) / i! as N i= N N i Np i NIT F ( Np ) e / i! N e < > = = < > / i! N i = i = IT lnt Fig (3a) N IT variatin with t (lg plt) (3b) Cuulative failure prbability distributin (F) with t When N beces very large, this reaches nral distributin Key Pints The degradatin due t field is re r less like that f a Rand Dpant Fluctuatin (RDF). At any given tie, there is a variatin in the nuber f brken SiH bnds ang the transistrs. This wuld result in a variatin in the threshld vltage f the transistrs acrss the design. An interesting way t study single electrn behavir is by bserving Single trap physics: If ne applies stress in a cntrlled anner fr a certain nuber f secnds such that nly ne SiH bnd is brken then ne can get a transistr with a single trap. This device can nw be used t study hw a single electrn respnds t ther electrns under the influence f agnetic field. A lt f ther interesting spintrnics related studies can be dne based n cntrlled NBTI degradatin. The fluctuatin in threshld vltage due t NBTI degradatin is a huge issue fr sall transistrs like SRAMs and DRAMs. These transistrs are typically f the rder f a few naneters in width and length. T get an apprxiate idea f the nuber f SiH bnds alng the interface fr such sall devices, let us lk at the fllwing calculatin. Silicn at density is f the rder f 5E12/c 2. If W=L=10n fr the device under cnsideratin, the ttal nuber f interface bnds will be 5E12 10e7 10E7 = 5.

6 If such less nuber f SiH bnds are present, then the fluctuatin in the device threshld vltage due t a fluctuatin in the nuber f brken bnds culd be severe Cnclusin In this and the previus few lectures we have talked abut the physics f NBTI degradatin fr a theretical pint f view. But anther bigger prble is abut hw t easure N IT. In the next few lectures we will lk at the varius ethds f easureent f the nuber f interface traps.