1. Semiconductor Energy Band & Impurity Energy Levels
Eg=1.12 eV
in Si EC EV Ei
Conduction band
Valence band
Donor impurity ED EA
Acceptor impurity
Semiconductor

2. Thermal Equilibrium
- physical quantities are time-invariant
- physical quantities are time-invariant
- every process is balanced by its inverse process
- every process is balanced by its inverse process
- single Fermi energy level (EF) quantifies both electron (n) & hole (p)
concentration
- single Fermi energy level (EF) quantifies both electron (n) & hole (p)
concentration
- EF is flat
- EF is flat
- n p=ni2 – law of mass action ; ni is the intrinsic carrier concentration
- n p=ni2 – law of mass action ; ni is the intrinsic carrier concentration

3. Equilibrium Statistics
Electrons / holes exhibiting the duality (wave-like & particle-like
nature) reside in bands: EC EV Ei
Conduction band
Eg=1.12 eV
in Si
Valence band

4. where
gn(E)dE is the number of electrons between E and E+dE with the 3-D
density of states given by

5. is the Fermi-Dirac distribution function denoting
and
is the Fermi-Dirac distribution function denoting
the electron occupation probability with Fermi energy level, EF, f E EF
0.5 1
T=0 K T1 T2 T3
T1<T2<T3

6. The integration yields
with
denoting the effective density of states at conduction band bottom, and
the Fermi 1/2 –integral.
For non-degenerate case where EC - EF  2kT

7. Similarly,
For non-degenerate case where EF - EV  2kT
with

8. EF in extrinsic semiconductor in equilibrium
Donor atoms are incorporated as
The donor energy level ED lies a few kT below EC; the degeneracy factor
g in Si is 2.
Similarly, for acceptor atoms
The acceptor energy level EA lies a few kT above EV; the degeneracy factor
g in Si is 4.

9. EF is determined by charge neutrality condition with ND, NA as parameters:-1
-0.8
-0.6
-0.4
-0.2
0.2
0.4
0.6
0.8 1
100
200
300
400
500
600
700
800
T (K)
EF-Ei [eV]
Conduction band
Valence band
1014
NA=1012
ND=1012
1016
1018

11. Intrinsic & extrinsic (n- & p-type) semiconductor
- EFi + EFi
- EFi + EFi
where, ni is the intrinsic carrier concentration, and EFi (=Ei) the intrinsic
Fermi energy level determined by n = p, i.e., Ei
and

12. Fermi potential (fFn) in n-type Fermi potential (fFp) in p-typeEC EV Ei
qfFn
EFn EC EV Ei
qfFp
EFp
Fermi potential (fFn) in n-type
Fermi potential (fFp) in p-type

13. In non-degenerate semiconductor,
n p = ni2
in thermal equilibrium (law of mass action).

14. Carrier Concentration
f(E)
N(E)
n(E), p(E) n p E
N-type
Intrinsic
P-type

15. Charge Transport; drift & diffusion
Recall charge conservation / continuity equation; x
x+dx
Jn(x)
Jn(x+dx)
Jn consists of drift & diffusion components:
drift velocity
electron mobility
electron diffusion coefficient
Thus
Continuity equation

16. Similarly, for holes
In 3-D

17. Transport Coefficient
Conceptually, vd is driven by E between collisions:
mean collision time
with electron mobility specified by
Likewise

18. l Non-uniform n / p gives rise to diffusion flux: n(x)
The net # of electrons crossing the x-plane of cross sectional area A
is given in terms of the mean free path ln as
Now, electron flux

19. The mean free path ( ln ) is associated with the thermal velocity (vT) by
where the thermal velocity (vT) is given by equipartition theorem:
Thus,
Similarly,
i.e., one obtains Einstein relation in equilibrium

20. EF in equilibrium
a) Single S/C system
Recall in 1-D that
EC, EV, Ei all represent the electronic potential energy:
electrostatic potential
Since
In equilibrium, Jn = 0 and

21. b) Two S/C systems in equilibrium contact
S/CL
S/CR
FLR(E)
FRL(E)
Now, in view of Pauli exclusion principle
vacant electron states
Transfer matrix
occupied electron states
Thus
and
No net flux leads to fL(E)=fR(E), i.e.
Therefore, Fermi level lines up, EFL=EFR .
In equilibrium EF is flat and lines up.

22. Non-equilibrium and quasi-Fermi levels
A system, when under bias or illumination, is driven away from equilibrium
to non-equilibrium conditions.
In non-equilibrium, quantifying n, p requires respective quasi-Fermi level
(imref), EFn & EFp
The role of EFn , EFp identical to that of EF in equilibrium;

23. In intrinsic S/C under irradiation
Why imrefs?
In intrinsic S/C under irradiation
where the photogenerated component is given via recombination time t
and generation rate g (∝ light intensity).
When gtn >>ni, gtn >>ni, EF should be above Ei for quantifying n & below
Ei for p. Impossible to meet the condition with single EF. EC EV Ei
EFn
EFp
Splitting of two imrefs ∝ light intensity.

24. Shockley Hall Read Recombination: equilibrium vs. non-equilibrium
Recall in equilibrium
If np > ni2 : charge injection
If np < ni2 : charge extraction
In the presence of single intermediate trap level, Et , Et EC EV
e capture
e emission
h capture
h emission
In equilibrium, # of electrons captured is
trap density

25. Away from equilibrium, ec rate is
empty trap sites
occupation probability in non-equilibrium
electron capture cross-section
thermal velocity
Likewise
emission probability
Similarly, for holes

26. en, ep can be extracted from equilibrium condition, i.e. ree= rec
with
Similarly

27. Steady state (S.S.) vs. equilibrium
In S/C uniformly irradiated
with
denoting e/h generation rate via band to band excitation
At S.S.,
i.e. the net recombination of e/h is same
In equilibrium, a process is balanced by its inverse process,
The S.S. condition is automatically satisfied.

28. The S.S. condition reads as
Hence, the S.S. distribution function or the probability of electron
being trapped at Et is given by
Compare this with equilibrium distribution function,

29. Next, the recombination rate is given by
For simplicity, take sn = sp =s and let 1/t = svTNt , obtaining
In equilibrium, n p=ni2, U=0
If n p>ni2 (charge injection), U>0: net recombination
If n p<ni2 (charge extraction), U<0: generation

30. Minority carrier lifetime & surface recombination
In n-type S/C, nn>>pn , ni
tp being the minority carrier lifetime.
Near interface,
effective thickness
vR : recombination velocity

31. p-n JUNCTION DIODE
Definition: p-n junction diode is a rectifier and constitutes a
key element in semiconductor transistors. p n V I V I
FORWARD
REVERSE
BREAKDOWN
Keywords: forward & reverse bias and current, breakdown.

32. p- & n-type semiconductors brought in equilibrium contact to
Operation:
p- & n-type semiconductors brought in equilibrium contact to
form p-n junction.
Under bias, the junction is pushed away from equilibrium and
current flows to restore the equilibrium. p n V
V=VF>0 : Forward bias
V=VR<0 : Reverse bias

33. Equilibrium junction band bending
p- & n-type S/C are brought into equilibrium contact via exchange of e/h.
In equilibrium, EF should line up and be flat, leading to energy band bending.
p-type n-type qf F p n E C V i qf F p n E C V i qf F p n E C V i qf F p n E C V i qf F p n E C V i qf F p n E C V i qf F p n E C V i qf F p n E C V i qf F p n E C V i qf F p n E C V i qf F p n E C V i
qfbi = q (fFn+fFp )
No net flows of electrons and holes, Jndrift= Jndiff , Jpdrift=Jpdiff

34. Electrostatics in depletion approximation
The band bending is supported by the space charge developed in depletion width. p e
-xp xn
-em f
fbi Q x
qNd
qNa n
qfbi E EF EC EV
Specifically, e/h spills over from n/p to p/n regions, leaving behind uncompensated ND+/NA-, viz. space charge, r .

35. Junction band bending under biasx f
(fbi-V) Q p n
V > 0 W EC EV E
q(fbi-V)
EFN
EFP
-xp xn
Under forward bias, V>0
-- p- side is lowered by qV relative to n-side, reducing band bending. Equivalently, n-side is raised by qV relative to p-side
-- p- and n-bulk are preserved as in equilibrium
These two necessitate introduction of EFn , EFp & bias is accommodated via splitting of EFn & EFp, qV=EFn-EFp.

36. Under reverse bias, V<0EC EV Q e f E
q(fbi+|V|)
EFN
EFP
(fbi+|V|) p n
-xp xn W
Under reverse bias, V<0
-- p- side is raised by q|V| relative to the n-side, increasing band bending. Equivalently, n-side is lowered by qV relative to p-side
-- p- and n-bulk are preserved as in equilibrium
These two necessitate introduction of EFn , EFp & bias is accommodated via splitting of EFn & EFp, q|V|=EFn-EFp.
Key words: ND, NA, maximum field (Emax), built-in or contact potential (fbi), depletion width (W )

37. Carrier concentration under non-equilibrium (non-degenerate case)EV E
q(fbi+|V|)
EFn
EFp
-xp xn p n
V =VR < 0
q|V| EC EV E
q(fbi-V)
EFn
EFp p n
V =VF > 0
-xp xn qV
Note that in W :
Forward bias (V>0)
Charge injection
Reverse bias (V<0)
Charge extraction
inducing recombination or generation, i.e. current flows.

38. Ideal current-voltage characteristics: Shockley equation
Shockley assumptions:
The abrupt depletion-layer approximation
The built-in potential & applied voltages are supported by a dipole layer with abrupt boundaries, and outside the depletion layer S/C is assumed neutral.
Low injection level
The injected minority carrier densities are much smaller than the majority carrier densities in equilibrium;
The Boltzmann approximation
It gives simplified boundary conditions
No recombination/generation currents exist in the depletion layer
Therefore, the electron and hole currents are constant through the depletion layer. The total current can be approximated as

39. The continuity equation and boundary conditions in n-region, for example, reads as
where Lp is the hole diffusion length in n-region given by
The general solution is given by
Likewise, one can write

40. The respective diffusion currents are given by
and total current is given by
where
is the saturation current.

41. (ii) the current is dominated by minority diffusion currentp n
V = VF > 0
-xp xn x
n, p
np(x)
np0
pn0 J
pn(x) Jn Jp JF Ln Lp xn
-xp x
np0
pn0
pn(x)
np(x)
|J| Jn Jp JR p n
V = -VR < 0
n, p
Note (i) the condition that I is constant throughout is met by respective majority drift
component
(ii) the current is dominated by minority diffusion current

42. Ideal I-V characteristics: linear & semilog plot-6 -4 -2 2 4 6 8 10 12 14 16 18
qV/kBT
J/JS
FORWARD
REVERSE 2 4 6 8 10
10-1
100
101
102
103
104
|J/JS|
q |V| /kBT
FORWARD
REVERSE
Ideal I-V characteristics: linear & semilog plot

43. Non-ideal current Recombination / generation current
There is recombination (V>0) or generation (V<0) throughout the depletion and the quasi-neutral region near the junction edges
For V>0
For V<0
Generation of e/h in depletion occurs primarily via intermediate trap sites inducing alternating e/h emissions

44. with fitting parameter m between 1~2.
High level injection
The quasi-neutral approximation is no longer valid, and electric field exists outside the depletion region, leading to minority drift current
In this case, the effective voltage drop through the depletion region is smaller than V, and the current expression becomes exp(qV/mkT)
Series resistance
It further reduces the junction voltage drop and the current is better approximated by exp(qV/mkT)
Thus, in general,
with fitting parameter m between 1~2.

46. p-n junction used as LED or diode laser

47. lasing due to feedback

48. p-n junction used as photodiode