1. Prof. Jang-Ung Park (박장웅)
Chapter 2.
Current Flow
&
Capacitance
Prof. Jang-Ung Park (박장웅)

2. Current, Electric Field, Voltage
Electric field : E = - grad V = -V
Electric field strength :
The direction of electric field : from positive potential to negative.
Electric current : the flow of electric charge (electron / hole) when electric field is applied.
I (coulombs / second) (C / s) = qnvdA
q: the charge on electrons (-1.610-19C)
n: the number of electrons per unit volume
vd: the drift velocity
A : area
J: current density (= I / A)

3. Electrical Conductivity ()
In the macroscale (Ohm’s law),
In the quantum scale,
: Because ions are attached to the lattice sites, ions have no net translational motion.
 Negligible contribution to the current.
: The current is due to the drift motion of the conduction electrons under electric field.
steady-state
force to conduction electron : 
dv/dt=0
vd: average drift velocity
m*: effective mass of electron
: relaxation time
E: electric field strength
: mobility
force exerted by E field
friction force due to the collision

4. Electrical Conductivity ()
drift current:
relaxation time ()?? 
vth: thermal velocity
l: distance between two successive collisions (mean free path)
The thermal energy absorbed by electrons is converted to kinetic energy:
vth: ~ 105 m/s
vd: 10-2 m/s m/s
 Comparison between vd and vth

5. High-Field Effects (hot carrier effects)
In Ohm’s law,
vd: average drift velocity
m*: effective mass of e-
: relaxation time
E: electric field strength
vr: ~ 106 m/s
vd ~ thermal velocity (vth)
vth: ~ 105 m/s (due to the lattice scattering)
high-field effect (Ohm’s law is not valid)
: used to maximize vd (device speed)
Movie (the temperature of the current CPU processors > 350oC)

6. Sheet Resistance (Rs)
 Integrated circuits are composed of thin films of various materials, and it is convenient to define the resistance of a thin conductive sheet as sheet resistance.
unit of Rs : “/square” or “/”
( = L/W)
Measurement of R and Rs
[ two point probe method ]
: The contact resistance btw the sample and probes are included in the measured, total resistance value.
: When we don’t know the contact resistance, it is difficult to find the resistance of the sample.

7. Contact Resistance (Rcontact)
Measurement of Rcontact
[ Transfer length method using two point probe ]
sample L
metal probe
contact interface
: Rcontact is independent on L.
: Values of Rtotal at different L are measured using two probes.
Rtotal
2Rcontact L

8. Sheet Resistance (Rs) Measurement of R and Rs
[ four-point probe method ]
sample
: Because voltage probes are connected in parallel to current probes, sample’s resistance is directly measured without considering the contact resistance.
For a very thin layer (S » t),
For R=V/2I,
(k =  / ln2 = 4.53)

9. Diffusion and Drift of Carriers; Built-in Fields
Diffusion processes
The net motion of electrons due to diffusion is in the direction of decreasing electron concentration.
: not dependent on the carrier concentration but the carrier gradient dn/dx (minority carriers can determine J).
Dn: diffusion constant of electron
Dp: diffusion constant of hole
If an electric field is present in addition to the carrier gradient,
drift
diffusion

10. Diffusion and Drift of Carriers; Built-in Fields
If an electric field is present in addition to the carrier gradient,
(at Room Temperature) : Einstein Relation