Space charge limited current

1
Space charge limited current

• ch2 of current injection in solid
• ???

2
contents

• Assumption
• Perfect insulator
• Trap free insulator
• Insulator with trap
• conclusion

3
assumption

• One carrier current.
• Neglect diffusion current
• Electrode serves as a reservoir contact.
• Contact plays no role in determination of JV.
• Exceptions
• small cathode-anode spacing
• Low applied voltage
• Ignore spatial variation
• Concern only the average quantities and the
  relations connecting them.

4
Whats need to determine JJ(V)?

• Relation between free and trapped charge
  concentration
• Free carrier drift velocity
• Electric field intensity.

5
General formula

• J current density (A/cm2)
• ? average, injected, free charge concentration
  (e/cm3)
• e static dielectric constant
• ? average drift velocity of a free electron
  (cm/sec)
• Q total injected free charge, per unit area,
  between cathode and anode (C/cm2)
• C capacitance per unit area
• L spacing between electrodes
• µ free electron drift mobility

6
Perfect insulator

• no trap in gap.
• Negligible concentration of free carriers in
  thermal equilibrium.

7
Functional dependence of J on voltage

• e- drift in conduction band of solid is marked by
  frequent collisions with the thermal vibrations,
  the impurities, and the structural defects in the
  solid.

8
Trap free square law on logJ-logV
9
Thermionic vacuum diode

• Childs law

10
Functional dependence of J on voltage

• e- move freely without collisions so that their
  drift velocity is related to the potential
  difference they have moved through.

11
Trap free insulator with thermal-free carriers

• A possible source of thermal free carriers may be
  a set of donors so shallow that they are not
  effective as electron traps
• Ohms law

12
Functional dependence of J on voltage

• Low voltage-gtOhms law
• Qe(nin0)L en0LCV !CVapp
• Intrinsic n0 does not depend on applied voltage
• High voltage-gt trap-free square law
• Qe(nin0)L eniLCVapp
• Cross over (Vx)-gt average, injected, excess, free
  e- concentration ni becomes comparable to the
  thermally generated concentration n0
• Qe(nin0)Le2n0L

13
Ohms law on logJ-logV
14
Insulator with traps

• Additional relationship is required to connect
  ?and ?t

15
Relation between ?and ?t
16
Remark!

• The presence of a moderate electric field will
  not affect these elementary microscopic processed
  of electron capture and thermal re-emission.
• ??Fermi level ?Et, Ec??(Et?????F???),nt/n?????????
  ,?????thermal equilibrium!
• cf Thermal equilibrium, steady state, trasient

17
Shallow trap case

• Definition F ltEt
• Remark
• Vx will be 1/? times larger than trap free case.
• If multiple sets of traps, consider the set with
  smallest ?

18
Deep trap case

• Definition FgtEt
• Vapp?-gt ni?-gt F0?F-gt fill trap
• VTFL the voltage required to fill the set of
  deep traps.
• First, injected ni fills into traps. nn0 until
  VappVTFL
• Then, injected extra carriers exist at conduction
  band. n begins increasing.
• when nnin02n0, we expect Vapp still be roughly
  VTFL.

19
Trap fill limited law on logJ-logV
20
example
21
conclusion

• Current voltage characteristics of SCLC is
  contained in a triangle in the logJ-logV plane.
• If F is far away from Et and Ec, a moderate
  voltage will not affect nt/n ratio.