#### Scattering Rates for Confined Carriers Dragica Vasileska Professor Arizona State University Outline • General comments on matrix element calculation • Examples of scattering rates calculation – Acoustic phonon.

Download Report#### Transcript Scattering Rates for Confined Carriers Dragica Vasileska Professor Arizona State University Outline • General comments on matrix element calculation • Examples of scattering rates calculation – Acoustic phonon.

Scattering Rates for Confined Carriers Dragica Vasileska Professor Arizona State University Outline • General comments on matrix element calculation • Examples of scattering rates calculation – Acoustic phonon scattering – Interface roughness scattering – dominant scattering mechanism in nanoscale MOSFETs Matrix Element Calculation • Suppose we want to calculate the scattering rate out of state k|| in a subband n. • For that purpose, we will use Fermi’s Golden Rule result: Snm (k|| , k ) ' || 2 ' || M (k|| , k ) Transition rate from a state k|| in a subband n into a state k||’ into a subband n’ E E ' nn ' 2 Matrix element for scattering between state k|| in a subband n into a state k||’ into a subband n’ i k|| k||' r 1 2 * M k|| , k d re dz m ( z ) H q ( R) n ( z ) A ' || Acoustic Phonon Scattering • The matrix element for acoustic phonon scattering in the bulk phonon approximation is: Restricts to longitudinal modes only H ( R) ac q , 2MN q q eq aq eiqR aq e iqR where q (q|| , qz ), R (r , z ) After integrating over the phonon coordinates, the matrix element for scattering between states k|| and k||’, in subbands n and m becomes: M k|| , k||' ac 1 Nq 2 V q 2 1/ 2 1 2 k|| k||' q dz m* ( z )e iqz z n ( z ) Inm(qz) S nm k|| , k ' || 2 2 q 2ac 2 V q 1 Nq 2 1 2 I nm (qz ) 2 k|| k||' q En Ek|| Em Ek ' q || • In the elastic and equipartition approximation, the total scattering rate out of state k is of the form: 2D DOS function 2 k T 1 1 B ac g 2 D En Em Ek || 2 n (k|| ) m vs Wnm 1 dz n2 ( z ) m2 ( z ) Wnm • Effective extent of the interaction in the z-direction 2 nm 1 • For infinite well: , L is the well width Wnm L Interface-Roughness Scattering Gradual Channel Approximation • This model is due to Shockley. • Assumption: The electric field variation in the direction parallel to the SC/oxide interface is much smaller than the electric field variation in the direction perpendicular to the interface. z G W S x D y oxide n+ L p-type SC n+ dFy dFx dx dy Square-Law Theory • The charge on the gate is completely balanced by QN(x), i.e: Q N ( x ) Ctot VG VT V ( x ) VS= 0 EFS x=0 V(x) x VD EC EFD= EFS - VD • Total current density in the channel: dn dV J n qn n F ( x ) qDn qn n dx dx negligible • Integrating the current density, we obtain drain current ID: Effective Mobility W yc ( x ) 0 0 I D dz dV W dx dV dy qn( x, y ) n ( x, y ) dx yc ( x ) qn( x, y ) n ( x, y )dy 0 QN ( x ) eff dV dV QN ( x) eff W CoxW eff VG VT V ( x ) dx dx Effective electron mobility, in which interface-roughness is taken into account. Mobility Characterization due to Interface Roughness 2.71 Å 3.84 Å High-resolution transmission electron micrograph of the interface between Si and SiO2 (Goodnick et al., Phys. Rev. B 32, p. 8171, 1985) Interface Roughness Phonon 17 N A 7 10 cm 500 0 3 2 Mobility [cm2/V-s] 1000 Mobility [cm /V-s] Coulomb 1500 10 10 16 10 17 10 -3 Doping [cm ] 18 s ) depl N 300 200 100 15 -1 (aN + bN 400 experimental data uniform step-like (low-high) retrograde (Gaussian) 12 10 13 10 -2 Inversion charge density N [cm ] s Interface-roughness Bulk samples -1/3 s Si inversion layers Mathematical Description of Interface Roughness • In Monte Carlo device simulations, interface-roughness is treated in real space and approximately 50% of the interactions with the interface are assumed to be specular and 50% to be diffusive • In k-space treatments of interface roughness, the perturbing potential is evaluated from: V ( z ') z ' z r V ( z ) V ( z ) V r z H ( R) eE ( z ) r V ( z ) eE ( z ) r • The matrix element for scattering between states k|| and k||’ is: Fnm i k|| k||' r 1 2 M k|| , k e ( z ) E ( z ) m ( z )dz d r (r )e A 2 i k|| k||' r1 r2 ' 2 2 1 2 2 M k|| , k|| e Fnm 2 d r1 d r2 e (r1 ) (r2 ) nm A ' || nm * n Random variable that is characterized by its autocovariance function which is obtained by averaging over many Samples – R(r) M k|| , k ' || 2 nm e F 2 2 nm i k|| k||' r1 r2 1 2 2 d r1 d r2e (r1 )(r2 ) 2 A When the random process is stationary, the autocorrelation function depends only upon the difference of the variables r1 and r2. If R(r) is the autocorrelation function, then its power spectral density is S(q||) and the transition rate is: S (k|| , k ) ' || 2 2 e Fnm 2 S (q|| ) A ( E E ') For exponential autocorrelation function we have: R(r ) 2e r / L S q|| 2 L2 1 12 L2 q||2 Δ = Roughness correlation length L = Ruth Mean Square (r.m.s.) of the roughness Conventional MOSFETs: Scaling MOSFETs Down When we scale MOSFETs down, we reduce the oxide thickness which in turn leads to increased: - gate leakage due to direct tunneling - more pronounced influence of remote roughness No exponential is forever…. But we can delay forever…. Gordon E. Moore