Long-channel MOSFET IV orrections Three MITs of the Day The body ect and its influence on long-channel V th. Long-channel subthreshold conduction and control (subthreshold slope S) Scattering components (impurity, phonon and roughness) in interface universal mobility and their F dependence hapter 4 1
Substrate Biasing (Body Effect) When we derive the threshold voltage and the square law of MOSFET, we have always assume that the MOS and the source junction has the same electrical potential (by forcing V BS 0) When V BS 0, the quasi-fermi levels are split across the pn junction. If source is kept as the reference ground, strong inversion means ψ s 2 ψ B -V BS. Sign convention: when the source/bulk diode is reverse-biased, the MOS at the source end needs more work for inversion (nmos threshold will be more positive, and pmos threshold will be more negative) nmosfet G S n + n + hapter 4 2 B D V BS 0 now
Body-ect correction on MOSFET voltage: V V + 2ψ V ± V GB threshold FB B BS 2ε ε qn 2 ( ± ψ mv ) si 0 B B BS upper sign: nmos, p-type substrate, Ψ B >0, V BS <0 for reverse bias lower sign: pmos, n-type substrate, Ψ B <0, V BS >0 for reverse bias In summary, with all ects we have discussed until now (V th for LONGchannel large MOSFETs), V GS that can turn on the channel is: γ 0 hapter 4 3 ( ± ψ m ) 2εsiε0qNB 2 B VBS QF V + 2ψ ± th FB B t Q γ t 0 it S t Body Effect Lumped into V th xρ ρ Q ( ϕ ) +ΔΦ ( x) dx ( x) dx QM ( ϕ ) S Remember the reference potential here is source and (V GS ) through PDE
Subthreshold hannel onduction Subthreshold current estimation by diffusion There are only drift, diffusion and G-R mechanisms for classical carrier transport. In the square-law theory, we have neglected diffusion current in above-threshold currents. G-R will be considered later with parasitic and hot-carrier ects. V DS Saturation V DS (V GS -V t )/m log 10 (I D ) Subthreshold Linear diffusion drift V DS 0.1V V t V GS V GS hapter 4 4
The Subthreshold Slope S (mv/decade) The subthreshold current is one of the the MOST important concern in scaled device design, since it determines the standby power and data retention time in dynamic logic/memory. Typically we need I ON /I OFF > 10 7-10 9. We need either to increase the voltage swing or improve subthreshold slope S (ΔV GS to decrease the channel current by a decade) to maintain this ratio. A small S is usually desirable, so that smaller V DD swing can be applied (low-power, low-voltage MOS design). However, this also implies the sensitivity of current to V th variation is larger. Let s first assume only diffusion in subthreshold (this can be examined with the exact-charge theory and is usually a good enough estimation for estimating the subthreshold slope. It is NOT accurate enough for weak inversion). We will use ψ S to denote the surface potential as before. We will use n-mosfet as an example. hapter 4 5
Diffusion urrent and the apacitive Divider Diffusion current only: dn J D qdn qd dx J n( S) D qn L n p0 qψ s / KT p0 e ; n( D) D n e n n( S) n( D) L n p0 e q( ψ V s DS )/ KT qψ S / KT qvds / KT ( 1 e ) np0 n N ψ S Q S 2 i ch V G OX Si(dep ) If V DS larger than a few kt, J D will have very weak dependence on V DS (This is only true for long-channel devices) Perturb the gate charge again to obtain the charge distribution. However we are now discussing the depletion and weak inversion region (there are few channel carriers being inverted yet). V G Si( dep) ( + 1) ψ s hapter 4 6
The Exponential urrent: (Dis)advantages at room temperature: Si qvgs /(1 ) KT p0 D + n qv / KT qn DS JD (1 e ) e L d(log 10 Ids ) 1 Si S ( ) (1 + ) 60mV dv G Si 1+ m 1 This S is like the ideality factor in the diode analysis that measures the exponential relations of current with respect to the applied voltage with the thermal voltage as the normalization factor. Subthreshold MOSFET has actually been used very early. Generations of inexpensive electronic watches have utilized the subthreshold regions of MOSFET to make logic and memory devices operation speed can be very low very low power consumption Subthreshold behavior is seriously affected by the short-channel ects (SE) and will be discussed in the next chapter. hapter 4 7 + dm
MOSFET Interface Universal Mobility The physical origins of MOSFET interface mobility are: phonon scattering ph (acoustic and optical, strong temperature dependence impurity scattering imp (ionized (attractive and repulsive; screened and unscreened), neutral, interaction with deep traps, etc.) surface roughness scattering sr (orientation dependent) surface oulombic scattering and remote oulombic scattering (RS) carrier-carrier scattering MOSFET interface mobility by Mathiensen s Rule: 1 1 1 1 + + interface ph imp sr hapter 4 8
attractive screened impurity scattering Scattering omponents surface roughness scattering phonon scattering everywhere remote oulomb scattering in gate interface or inside the gate dielectric screened impurity scattering repulsive screened/unscreened impurity scattering carrier-carrier scattering when n is very high In addition to knowing all detailed scattering mechanisms for determining mobility, practical knowledge of functional dependence of average channel mobility on measurable terminal characteristics will greatly help device modeling and technology prediction. An expression of average channel mobility at low V DS that is independent of t, N sub and V BS will be valuable. hapter 4 9
Effective Vertical Field F to Define Field-ect Mobility: FE ( L / W ) V g DS m V D 0 g m I V D GS Empirically, it has been found that when is plotted again an ective normal field F, there exists a universal relationship independent of the substrate bias, doping concentration, and gate ide thickness. df dx x 1 ε ε si 0 ρ( x) 1 1 F Q + Q si inv B εsiε0 2 x F F 1 1 Q ε ε 2 holes hapter 4 10 si 0 1 1 Q ε ε 3 si 0 inv inv + Q + Q B B for for electrons
harge omponents in F Q Q inv B int erface int erface n( x, The average field in the channel should take only 1/2 or 1/3 of the inversion charge ect N Intuitively, this can come from the fact that the inversion carrier has a finite thickness, and the average field felt by the surface roughness should be smaller than F si. The bulk charge should be included as a whole since W T is much larger than the thickness of the inversion layer. imp from the impurity scattering (2D screened oulomb) should increase with F since screening is stronger with larger vertical field. ph from the phonon scattering (acoustic and optical combined) should have a very mild depedence on F. sr from the surface roughness scattering should decrease with F since the carriers are pushed strongly to the interface. hapter 4 11 A ( x, y) dx y) dx
Universal Mobility: vs. F ph F 0.3 η η > 0 imp F N B increases T Lattice increases Surface Roughness increases γ sr F η 1.0 ~ 2.6 1 1 imp + 1 ph + 1 sr F The universal mobility curve should be independent of t, N B, V BS (included into F ), W, L and to the first order The universal mobility depends on temperature, degree of surface roughness, surface orientation, and for sure, materials. hapter 4 12
Empirical Form of Universal Mobility 1 + θ + bulk 0.3 1F θ2 F bulk 1 + θ V V + θ V ( ) γ GS th b BS After the physical understanding of the universal mobility, we have successfully linked the physical ects with the terminal characteristics. The universal mobility will replace the 0 we have used for low V DS, and we can then add the correction from velocity saturation accordingly. hapter 4 13