Week 3, Lectures 6-8, Jan 29 Feb 2, 2001 EECS 105 Microelectronics Devices and Circuits, Spring 2001 Andrew R. Neureuther Topics: M: Charge density, electric field, and potential; W: Capacitance of pn junction; F: MOS threshold voltage Reading for week: HS Ch 3. Light on math of a) the 2 sided model (3.4.2) and b) the MOS capacitance (3.9.2) Version 1/20/01
Depletion Regions: pn and MOS ANALYSIS Charge Density Elec. Field Potential p Charge Density hole diffusion electron diffusion hole drift electron drift - Electrical Field Potential ρ ξ V + ψ 0 -W 1 W 2 n Distance PHYSICAL RESULTS (a) Current flow. (b) Charge density. (c) Electric field. (d) Electrostatic potential. Capacitance (Small Signal) Carrier density versus potential (eponential) MOS threshold voltage physical basis for the many terms
Physics of Electrostatics E = The electric field E diverges from positive charge and the strength is inversely proportional to the dielectric constant ε (and hence will be discontinuous at boundaries between materials). The potential has a slope that is the negative of the electric field. ρ ε φ( ) E( ) = d
Eample: Sheet Charge Geometry Electrostatic Solution Charge density ρ 0 () Equal area For charge neutrality Electric Field E o () Sheet of positive charge Electric Field Sheet of negative charge Potential φ 0 ()
Eample: Multiple Sheets Geometry Electrostatic Solution Charge density ρ 0 () Electric Field E o () At first sheet see Spike Step Sheet of positive charge Electric Field Sheet of negative charge Potential φ 0 () Change in slope
Eample: Uniform Distribution Geometry Sheet of positive charge Electric Field Electrostatic Solution Uniform distribution of negative charge Charge density ρ 0 () Electric Field Potential E o () φ 0 () Equal area For charge neutrality At edge see Change in slope Parabolic region
Eample: Materials Boundary Geometry Oide ε r = 3.9 ε o Sheet of positive charge Electric Field Sheet of negative charge Electrostatic Solution Silicon ε r = 11.7 ε o Charge density ρ 0 () Electric Field Potential E o () φ 0 () Equal area For charge neutrality At boundary see Step by factor of 3.9/11.7 Change in slope by factor of 11.7/3.9
Eample: MOS Prototype Geometry GATE OXIDE See Fig. E3.3b in tet SUBSTRATE
Charge: MOS Prototype Equal area For charge neutrality Has both a boundary and a distributed charge density GATE OXIDE SUBSTRATE
Field and Potential: MOS Prototype Discontinuity by factor of 3.9/11.7 Change in slope by factor of 3.9/11.7 Parabolic shape X d increases as sqrt (V B )
Eample: pn Prototype hole diffusion electron diffusion Charge Density p Charge Density hole drift electron drift - ρ + n Distance (a) Current flow. Equal area for charge neutrality (b) Charge density. Elec. Field Electrical Field ξ (c) Electric field. E MAX = qna p = qn d n Potential Potential V ψ 0 -W 1 W 2 p n (d) Electrostatic potential. Parabolic dependence of n and p on potential φ o
One-Sided Approimation See Figure in tet pp 126
Potential from Carrier Density 0 = qnµ n E o + qd n (dn o /d) dn o /d = (-µ n /D n )n o E o = (-kt/q) n o (-dφ o /dt) dφ o = (kt/q)dn o /n o = V TH dn o /n o φ o () φ o ( o ) = V TH ln[n o ()/n o ( o )] Reference Level: φ o ( o ) = 0 when n o ( o ) = n i Logarithmic!
Carrier Density from Potential φ o () = V TH ln[n o ()/n i ] n o () = n i ep [φ o ()/ V TH ] φ o () = -V TH ln[p o ()/n i ] p o () = n i ep [-φ o ()/ V TH ] When the potential varies with position in the silicon these carrier densities n o () and p o () will also change with position. Eponential!
Eample of Potential and Carriers Doping N a = N d = 10 16 Holes need 720 mev to jump the barrier φ p = -V TH ln[n a /n i ] = (0.026 mv) ln[10 16 /sqrt(2)10 10 ] = -360 mv φ o () = V TH ln[n d /n i ] =(0.026mV) ln[10 16 /sqrt(2)10 10 ] = 360 mv Barrier 720 mv
Contact Potentials