1. Scattering Rates for Confined Carriers Dragica Vasileska Professor Arizona State University

2. Outline General comments on matrix element calculation Examples of scattering rates calculation –Acoustic phonon scattering –Interface roughness scattering – dominant scattering mechanism in nanoscale MOSFETs

3. 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: Transition rate from a state k || in a subband n into a state k || ’ into a subband n’ Matrix element for scattering between state k || in a subband n into a state k || ’ into a subband n’

4. Acoustic Phonon Scattering The matrix element for acoustic phonon scattering in the bulk phonon approximation is: Restricts to longitudinal modes only

5. After integrating over the phonon coordinates, the matrix element for scattering between states k || and k || ’, in subbands n and m becomes: I nm (q z )

6. In the elastic and equipartition approximation, the total scattering rate out of state k is of the form: 2D DOS function Effective extent of the interaction in the z-direction For infinite well:, L is the well width

7. Interface-Roughness Scattering oxide p-type SC n+n+ n+n+ S D G x y L W z 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.

8. V S = 0 ECEC E FS E FD = E FS - V D VDVD x=0 V(x) x Square-Law Theory The charge on the gate is completely balanced by Q N (x), i.e: Total current density in the channel:

9. Integrating the current density, we obtain drain current I D : Effective Mobility Effective electron mobility, in which interface-roughness is taken into account.

10. Mobility Characterization due to Interface Roughness High-resolution transmission electron micrograph of the interface between Si and SiO 2 (Goodnick et al., Phys. Rev. B 32, p. 8171, 1985) 2.71 Å 3.84 Å

11. Bulk samples Si inversion layers Phonon Coulomb Interface-roughness Interface Roughness

12. 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 specu- lar and 50% to be diffusive In k-space treatments of interface roughness, the pertur- bing potential is evaluated from:

13. The matrix element for scattering between states k || and k || ’ is: F nm Random variable that is characterized by its autocovariance function which is obtained by averaging over many Samples – R(r) When the random process is stationary, the autocorrelation function depends only upon the difference of the variables r 1 and r 2.

14. If R(r) is the autocorrelation function, then its power spectral density is S(q || ) and the transition rate is: For exponential autocorrelation function we have: Δ = Roughness correlation length L = Ruth Mean Square (r.m.s.) of the roughness

16. 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

17. No exponential is forever…. But we can delay forever…. Gordon E. Moore