1. Sung June Kim kimsj@snu.ac.kr http://nanobio.snu.ac.kr
Chapter 3.
Carrier Action
Sung June Kim

2. Subject What relation do mobility of electron and resistivity have?
Contribution of diffusion of excess carrier in current
Carrier injection and carrier action
direct gap semiconductor and indirect semiconductor
recombination lifetime
quasi-Fermi level
How does current change while considering diffusion and recombination ? 2

3. Contents
Drift
Diffusion
Generation-Recombination
Equations of State

4. Definition-Visualization
Drift
Definition-Visualization
Charged-particle motion in response to electric field
An electric field tends to accelerate the +q charged holes in the direction of the electric field and the –q charged electrons in the opposite direction
Collisions with impurity atoms and thermally agitated lattice atoms
Repeated periods of acceleration and subsequent decelerating collisions
Measurable quantities are macroscopic observables that reflect the average or overall motion of the carriers
The drifting motion is actually superimposed upon the always-present thermal motion.
The thermal motion of the carriers is completely random and therefore averages out to zero, does not contribute to current

5. (a) Motion of carriers within a biased semiconductor bar; (b) drifting hole on a microscopic or atomic scale; (c) carrier drift on a macroscopic scale
Thermal motion of carrier
Drift Current
I (current) = the charge per unit time crossing a plane oriented normal to the direction of flow

6. Expanded view of a biased p-type semiconductor bar of cross-sectional area A

7. The vd is proportional to E at low electric fields, while at high electric fields vd saturates and becomes independent of E

8. where for holes and for electrons, is the constant of proportionality between vd and E at low to moderate electric fields, and is the limiting or saturation velocity
In the low field limit

9. Mobility
Mobility is very important parameter in characterizing transport due to drift
Unit: cm2/Vs
Varies inversely with the amount of scattering
Lattice scattering
Ionized impurity scattering
Displacement of atoms leads to lattice scattering
The internal field associated with the stationary array of atoms is already taken into account in
, where is the mean free time and is the conductivity effective mass
The number of collisions decreases   varies inversely with the amount of scattering

10. Room temperature carrier mobilities as a function of the dopant concentration in Si

11. Temperature dependence
Low doping limit: Decreasing temperature causes an ever-decreasing thermal agitation of the atoms, which decreases the lattice scattering
Higher doping: Ionized impurities become more effective in deflecting the charged carriers as the temperature and hence the speed of the carriers decreases

12. Temperature dependence of electron mobility in Si for dopings ranging from < 1014 cm-3 to 1018 cm-3.

13. Resistivity
Resistivity () is defined as the proportionality constant between the electric field and the total current per unit area
=1/: conductivity

14. Resistivity versus impurity concentration at 300 K in Si

15. Band Bending
When exists the band energies become a function of position
If an energy of precisely EG is added to break an atom-atom bond, the created electron and hole energies would be Ec and Ev, respectively, and the created carriers would be effectively motionless
motionless

16. E-Ec= K.E. of the electrons
Ev-E= K.E. of the holes
The potential energy of a –q charged particle is

17. By definition,
In one dimension,

18. Definition-Visualization
Diffusion
Definition-Visualization
Diffusion is a process whereby particles tend to spread out or redistribute as a result of their random thermal motion, migrating on a macroscopic scale from regions of high concentration into region of low concentration

19. Diffusion and Total Currents
Diffusion Currents: The greater the concentration gradient, the larger the flux
Using Fick’s law,
Diffusion constant

20. Total currents +
Total particle currents

21. Relating Diffusion coefficients/Mobilities
Einstein relationship
Consider a nonuniformly doped semiconductor under equilibrium
Constancy of the Fermi Level: nonuniformly doped n-type semiconductor as an example
= 0

22. To prove that (Under equilibrium, the EF should be continuous.)
Under thermal equilibrium, there are no charge movements between two materials,
Fig. Junction of two materials in equilibrium state. Actually, there is no electron movement, Fermi level should be constant hrough the entire material.
The rate of electron movement from material 1 to material 2 is in proportion to
Also, the rate of electron movement from material 2 to material 1 is in proportion to
Under equilibrium condition, these terms are equal. So,
f1(E) = f2(E) and EF1 = EF2
That is, dEF /dx = 0.

23. Under equilibrium conditions
&
= -
Electron diffusion current flowing in the +x direction “Built-in” electric field in the –x direction  drift current in the –x direction
Einstein relationship:
Nondegenerate, nonuniformly doped semiconductor,Under equilibrium conditions, and focusing on the electrons,

24. With dEF/dx=0,
Substituting
Einstein relationship is valid even under nonequilibrium
Slightly modified forms result for degenerate materials

25. Definition-Visualization
Recombination-Generation
Definition-Visualization
When a semiconductor is perturbed  an excess or deficit in the carrier concentrations  Recombination-generation
Recombination: a process whereby electrons and holes (carriers) are annihilated or destroyed
Generation: a process whereby electrons and holes are created
Band-to-Band Recombination
The direct annihilation of an electron and a hole  the production of a photon (light)

26. R-G Center Recombination
R-G centers are lattice defects or impurity atoms (Au)
The most important property of the R-G centers is the introduction of allowed electronic levels near the center of the band gap (ET)
Two-step process
R-G center recombination (or indirect recombination) typically releases thermal energy (heat) or, equivalently, produces lattice vibration

27. Near-midgap energy levels introduced by some common impurities in Si

28. Generation Processes
thermal energy > EG  direct thermal generation
light with an energy > EG  photogeneration
The thermally assisted generation of carriers with R-G centers

29. Impact ionization
e-h is produced as a results of energy released when a highly energetic carrier collide with the lattice
High field regions

30. Momentum Considerations
One need be concerned only with the dorminat process
Crystal momentum in addition to energy must be conserved.
The momentum of an electron in an energy band can assume only certain quantized values.
where k is a parameter proportional to the electron momentum

31. Si, Ge
GaAs

32. Photons: light
Phonons: lattice vibration quanta
Photons, being massless entities, carry very little momentum, and a photon-assisted transition is essentially vertical on the E-k plot
The thermal energy associated with lattice vibrations (phonons) is very small (in the meV range), whereas the phonon momentum is comparatively large. A phonon-assisted transition is essentially horizontal on the E-k plot. The emission of a photon must be accompanied by the emission or absorption of a phonon.

33. R-G Statistics
It is the time rate of change in the carrier concentrations (n/t, p t) that must be specified
B-to-B recombination is totally negligible compared to R-G center recombination in Si
Even in direct materials, the R-G center mechanism is often the dominant process.

34. Indirect Thermal Recombination-Generation
n0, p0 …. carrier concentrations when equilibrium conditions prevail
n, p ……. carrier concentrations under arbitrary conditions
n=n-n0 … deviations in the carrier concentrations from their equilibrium values
p=p-p0 …
NT ………. number of R-G centers/cm3

35. A specific example of ND=1014 cm-3 Si subject to a perturbation where n= p=109 cm-3
Although the majority carrier concentration remains essentially unperturbed under low-level injection, the minority carrier concentration can increase by many orders of magnitude.
excess carrier

36. The greater the number of filled R-G centers, the greater the probability of a hole annihilating transition and the faster the rate of recombination
Under equilibrium, essentially all of the R-G centers are filled with electrons because EF>>ET
With , electrons always vastly outnumber holes and rapidly fill R-G levels that become vacant  # of filled centers during the relaxation NT
The number of hole-annihilating transitions should increase almost linearly with the number of holes
Introducing a proportionality constant, cp,

37. Generateion rate: p/t|G depends only on # of empty R-G centers
The recombination and generation rates must precisely balance under equilibrium conditions, or p/t|G = p/t|G-equilibrium=- p/t|R-equilibrium , so
The net rate
An analogous set of arguments yields

38. cn and cp are referred to as the capture coefficients
cpNT and cnNT must have units of 1/time
These are the excess minority carrier life times.

39. Continuity Equations Equations of State
Carrier action – whether it be drift, diffusion, indirect or direct thermal recombination, indirect or direct generation, or some other type of carrier action – gives rise to a change in the carrier concentrations with time
Let’s combine all the mechanisms

42. Minority Carrier Diffusion Equations Simplifying assumptions:
One-dimensional
The analysis is limited or restricted to minority carriers
.—no drift current term- so it is called Diffusion Equations.
The equilibrium minority carrier concentrations are not a function of position
Low level injection
Indirect thermal R-G is the dominant thermal R-G mechanism
There are no “other processes”, except possibly photogeneration

43. With low level injection,
 GL=0 without illumination

44. The equilibrium electron concentration is never a function of time

45. Simplifications and Solutions
Steady state
No concentration gradient or no diffusion current
No drift current or  no further simplification
No thermal R-G
No light

46. is set equal to zero.
Therefore, the solution is

49. Sample Problem No.1 : A uniformly donor-doped silicon wafer maintained at room temperature is suddenly illuminated with light at time t=0. Assuming , , and a light-induced creation of electrons and holes per throughout the semiconductor, determine for
Solution :
Step1 – Si, T=300K, , and
at all points inside the semiconductor.
Step2 -
with uniform doping, the equilibrium and values are the same everywhere throughout the semiconductor.

50. Step3 – Prior to t=0, equilibrium conditions prevail and
illumination -> increase.
excess carrier number ↑ -> thermal recombination rate ↑
∴ light generation and thermal recombination rates must balance under steady state conditions, we can even state or
if low-level injection prevails.
Step4 –
the boundary condition

51. is not function of position, the diffusion equation becomes an ordinary differential equation and simplifies to
The general solution is
Applying the boundary condition yields
and

52. Step5 – A plot of the solution equation is shown below.
Note that, starts from zero at t=0 and eventually saturates at after a few

53. Epilogue – Light output from the stroboscope can be modeled to first
Epilogue – Light output from the stroboscope can be modeled to first order by the pulse train pictured below.

54. The upper equation approximately describes the carrier build-up during a light pulse.
However, the stroboscope light pulses have a duration of
compared to a minority carrier lifetime of
With , one sees only the very first portion of the Fig transient before the light is turned off and the semiconductor is allowed to decay back to equilibrium.
The light-off expression derivation closely parallels the light-on sample problem solution.
Exceptions -> at the beginning of the light-off = at the end of the light-on.

55. Sample problem 2: As shown below, the x=0 end of a uniformly doped semi-infinite bar of silicon with ND=1015 cm-3 is illuminated so as to create pn0=1010 cm-3 excess holes at x=0. The wavelength of the illumination is such that no light penetrates into the interior (x>0) of the bar. Determine pn(x).

56. Solution: at x=0, pn(0)= pn0= 1010 cm-3 , and pn  0 as x  
The light first creates excess carriers right at x=0
GL=0 for x>0
Diffusion and recombination
As the diffusing holes move into the bar their numbers are reduced by recombination
Under steady state conditions it is reasonable to expect an excess distribution of holes near x=0, with pn(x) monotonically decreasing from pn0 at x=0 to pn0= 0 as x  
Can we ignore the drift current? E ? Yes
because 1>Excess hole pile-up is very small , here
2> Also the majority carriers redistribute in such a way to partly cancel the minority carrier charge. Thus use of diffusion eq. is justified.

57. So---Under steady state conditions with GL=0 for x > 0
The general solution
where
exp(x/Lp)   as x  
With x=0,

58. Diffusion Lengths Supplemental Concepts
minority carrier diffusion lengths
LP and LN represent the average distance minority carriers can diffuse into a sea of majority carriers before being annihilated
The average position of the excess minority carriers inside the semiconductor bar is

59. Quasi-Fermi Levels
Quasi-Fermi levels are energy levels used to specify the carrier concentrations under nonequilibrium conditions
The convenience of being able to deduce the carrier concentrations by inspection from the energy band diagram is extended to nonequilibrium conditions through the use of quasi-Fermi levels by introducing FN and FP

60. Equilibrium conditions
Nonequilibrium conditions