EECS 105 Spring2005 Lecture 2 R. T. Howe EECS 105 Spring2005 Lecture 2 R. T. Howe Lecture 2 Thermal Equilibrium Rapid, random of holes and at 7 • Last time: “thermal velocity” vth = 10 cm/s with collisions every τ = 10-13 s. – Course introduction c – Start: semiconductor properties of Si Apply an E and charge carriers

accelerate … for τc • Today : zero E field – Drift velocity v – Drift current th (hole case) – Resistivity and resistance; the IC resistor positive E aτ c v x th Dept. of EECS University of California, Berkeley Dept. of EECS University of California, Berkeley

EECS 105 Spring2005 Lecture 2 R. T. Howe EECS 105 Spring2005 Lecture 2 R. T. Howe Drift Velocity and Mobility Mobility vs. Doping in Silicon at 300 K

 F   qE   qτ  v = a ⋅τ =  e τ =  τ =  c E dr c   c   c    mp   mp   mp 

vdr = µ p E

For electrons:

“default” values: Dept. of EECS University of California, Berkeley Dept. of EECS University of California, Berkeley

1 EECS 105 Spring2005 Lecture 2 R. T. Howe EECS 105 Spring2005 Lecture 2 R. T. Howe Velocity Saturation Drift (Holes) Hole case: drift velocity is in same direction as E hole drift current density

J dr p v dp E

x The hole drift current density is:

J dr = q p µ E p p

Dept. of EECS University of California, Berkeley Dept. of EECS University of California, Berkeley

EECS 105 Spring2005 Lecture 2 R. T. Howe EECS 105 Spring2005 Lecture 2 R. T. Howe Drift Current Density (Electrons) Resistivity case: drift velocity is in opposite direction as E Bulk silicon: uniform doping concentration, away electron drift from surfaces current density

J dr n-type example: in equilibrium, no = Nd . n v dn When we apply an electric field, n = Nd . E dr Jn = x Jn = qµnnE = qµn Nd E

The electron drift current density is: Conductivity σn = dr -2 -1 -2 Jn = (-q) n vdn units: Ccm s = Acm Resistivity ρn =

Dept. of EECS University of California, Berkeley Dept. of EECS University of California, Berkeley

2 EECS 105 Spring2005 Lecture 2 R. T. Howe EECS 105 Spring2005 Lecture 2 R. T. Howe Ohm’s Law Sheet Resistance • Current I in terms of J n • IC resistors have a specified thickness – not • V in terms of electric field under the control of the circuit designer • Eliminate t by absorbing it into a new parameter: the sheet resistance

ρL  ρ  L   L  R = =    = Rsq   Wt  t W  W 

– Result for R Dept. of EECS University of California, Berkeley Dept. of EECS University of California, Berkeley

EECS 105 Spring2005 Lecture 2 R. T. Howe EECS 105 Spring2005 Lecture 2 R. T. Howe Using Sheet Resistance Idealizations • Ion-implanted (or “diffused”) IC resistor • Why does current density Jn “turn”? • What is the thickness of the resistor? • What is the effect of the contact regions?

Dept. of EECS University of California, Berkeley Dept. of EECS University of California, Berkeley

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