1. PHYSICS OF SEMICONDUCTOR DEVICES
There are two classes of materials (solids): conductors and insulators.
Conductor: The small D.C. conductivity at 0K is non-zero.
Insulator: The small D.C. conductivity at 0K is zero.
All insulators have finite conductivity at T > 0K.
Semiconductors are those insulators with conductivity that is not too small at room temperature (300K).

2. Block Theorem
In a crystalline solid, atoms are arranged in a periodic fashion. Silicon, for example, is a diamond structure, whereas GaAs is a zinc-­blende structure.
With single electron approximation, time independent S.E. becomes:
with V(r) a periodic function corresponding to the lattice.
The solution of such S.E. is called the Block function, and has the form:

3. The solution of the Schrodinger Equation of a crystalline semiconductor gives the following results:
The allowed electronic states are lumped together to form bands.
There is a bandgap.
At finite temperature, there are electrons and holes in the semiconductor.

4. The most important aspect of the block functions is that Ek formed into bands:
In the case of insulator (semiconductor), there is a bandgap between the valence band and the conduction band.

5. Brief Summary of Block Theorem
For a single electron in a “rigid” lattice, the time independent S.E. again is again:
Let TR be an operator that moves the wave function
by , a lattice constant.
i.e.,
Now

6. since
i.e.,
In other word, the two operators commute. A general QM theory states that there is a set of common eigenfunctions for two commuting operators.

7. If

8. Write

9. periodic in the lattice
In general,
with
i.e.,
periodic in the lattice

10. Born-Von Karman boundary condition
(Block’s theorem)
But
i.e.,

11. number of unit cells in the crystal
allowable
vectors in a primitive cell
number of unit cells in the crystal

12. Consequence of the Block Theorem
there is an infinite solution for each indexed by n.

13. ( = a reciprocal vector)
with proper choice of the index
is periodic with K

14. Wave Packet
Using block-waves, we construct a wave packet to localize the electrons. This is necessary since the Block waves are extended states of the stationary Schrodinger Equation. In order to treat a wave packet, we require time-dependent perturbation theory. The results are that, with a wave packet centered at k,

15. Where (1/m)ij is the (1/effective mass) tensor.
and
In reality we have imperfection in the crystal and lattice vibration. Thus there will be scattering, and the above equation will only be followed between scattering events. In general, the transport is governed by the Boltzmann transport equation.

16. Fermi-Dirac Distribution
Boltzmann Equation
Before going into Boltzmann Equation, we need to discuss f.
Fermi-Dirac Distribution
In Equilibrium
where m is the electro-chemical potential, defined as

17. m is also called the Fermi-energy EF .
In a solid with En(k)
The total number of electrons in a band is therefore

18. Converting S into integration
Since
i.e.,

19. Splitting the integration over surfaces of constant energy, and then integrate over all energy in the band, we have:
That is, the total number of electrons between E(k) and E(k) +dE is

20. i.e.,
Near the conduction band minimum, the constant energy surface is approximated by an ellipsoid, and it is very simple to obtain N(E).
Let n(E) = total number of states inside an ellipsoid with constant energy surface of E

21. But
for an ellipsoid with the proper choice of principal axes, and the principal radii are:
i.e.,

22. Since
i.e.,
where MC is the number of equivalent minimum valleys (6 for silicon).

23. For electron density in the conduction band, we have:if
NC is the effective density of states of conduction band.

24. In general
Defining
which is Fermi integral of order j

25. What about valence bands?
Key: a full band does NOT conduct
(velocity cancelled pair-wise and therefore average v=0)
Conduction of a current happens when some electron in the valence band is missing. Thus, we introduce an accounting principle, called a “hole” º an electron state that is empty.
i.e.,
and

26. where
and
For silicon, we actually have two valence bands (+ a split-off band):

27. Ignoring the split off band,
To make sure everything is consistent, we have hole transformation:

28. In summary, we have
Now, let’s look at the Boltzmann Equation. Under the following two assumptions:
relaxation time approximation
(1)
(2)
f is essentially the same as f0

29. Then the Boltzmann Equation is equivalent to the device equations:
Actually, if the system is not far from equilibrium,

30. For the purpose of this course, it is sufficient to use these equations except when the electric field is high. In that case, the carrier distributions are no longer Maxwellian, and velocity overshoot can happen; or when dE/dx is larger, we must abandon even Boltzmann transport.
It is important to distinguish the effective masses.
Conduction effective mass is the arithmetic mean of the “masses”
For silicon, it turns out that for conduction minimum, it is a circular ellipsoid.

31. For silicon conduction band, with

32. Extrinsic Semiconductor
To change the conductivity (as well as the dominant carrier type), we introduce into the seminconductor donors and acceptors.
NA º acceptors concentration
ND º Donors concentration

33. Under charge neutrality
What about the interaction between the electrons and the holes. Since they are from orthogonal block waves, their interaction is small!
Concept of quasi-fermi energies: Efn, Efp
i.e., the electrons in the conduction band are in quasi-equilibrium with an electro-chemical potential mfn, while the holes in the valence band are in equilibrium with mfp (although mfn¹ mfp).

34. This concept makes sense when the relaxation time between the electrons and holes is large and is in general true.
Recall
ni2 is the pn product of non-degenerate semiconductor sample at equilibrium.

35. ED-EF > 3kT, EF -EA > 3kT
In such case, the semiconductor is non-degenerate (i.e. pn=ni2), and

36. But
with

37. Generation and Recombination of Excess Carriers
Electrons and holes are not at equilibrium (with each other). Processes to restore equilibrium
Note: Electrons are in equilibrium among themselves with relaxation time in the order of ~ femto-second.

38. (a) Band-to-Band (radiative recombination)o EC EV hν

39. At equilibrium i.e., Net recombination: Assume quasi-charge neutrality

40. For low level injection (i.e. )
We can write:
with
i.e., t ↓as dopant concentration ↑.
For direct band-to-band recombination, t is small (~1-100 pS)

41. B. Auger recombination:
This is a process involving three particles.
Since three particles involved Þ C is small.
In heavily doped material,
say N+ material, n is large Þ
Auger can be important

42. C. Shockley-Read-Hall (SRH) recombination (“trap-assisted” recombination)
For indirect gap materials , say silicon, to the first order, band-to-band recombination is not possible. hν
Impossible!

43. (a) e- capture; (b) e- emission; (c) hole capture; (d) hole emission.Ev EC
Single trap level
recombination
process
(a) e- capture;
(b) e- emission;
(c) hole capture;
(d) hole emission.

44. (b) is number of filled traps:
Similarly

45. Net rate of recombination of e-
Net rate of recombination of p

46. Rate of change of the number of e- in the traps:

47. By detailed balance, at equilibrium:
i.e.,
At equilibrium

48. gives

49. with
Similarly, with
we have

50. \We have
In non-equilibrium, these two equations are still correct, but we do not know fT.
Let us look at the special case of steady-state:

52. With
we have

53. If we write
and
is the capture cross-section

54. Since
with

55. Defining
and
we have

56. Quasi-neutral region recombination
Quasi-neutrality:
and
Assume
can ignore its charge contribution

57. Define t :
(i.e., if the excitation is removed)
Since
we have

58. n- type; low level injection (Dn <<no).
(Note: no>>po)
Assume
(i.e., ET is not too close to the band edges)

59. Similarly, for p-type material
we have

60. Einstein Relationship
We have
and

61. That is
Comparing with

62. For non-degenerate n-type semiconductor,
Similarly, for p-type semiconductor,

63. Carriers’ Drift Mobility
m is determined by the scattering lifetime t(E):
Two types of carrier scattering:
1. phonon scattering
2. impurity scattering (ionized impurity)
(Matthiessen’s rule)

64. Typically, it was found that for silicon:
and

65. m versus Temperature

66. m versus doping concentration

67. Irvin’s curves

68. To summarize:
and
True for e not too large.

70. The rate that e- gain energy from the E-field is:
With very high E, the effective T of e- is high. The rate e- loss energy to the lattice can be drived to be:
where Te is the “temperature” of the electron gas, is speed of sound in the semiconductor and is the mean free path between collision.

71. tm : mean time between scatterings
In steady-state, energy gain=energy loss,
with
tm : mean time between scatterings
But
and

72. Accounting for the angle of scattering:
With for phonon scattering:

73. i.e.,
and

74. If If

75. Basic Semiconductor Device Equations
Electron (hole) conduction eqn.
Electron (hole) continuity eqn.
Poisson’s eqn.

76. Also, we have
and

77. Simple Example n-type Also
Creating excess carriers Dn=Dp=Dpn(0) at x=0.
Also

78. In steady state,
i.e.,

79. Minority carriers are much simplier to deal with!

80. So, the cont. eqn for p becomes:
Boundary conditions
for very long sample
Solution for very long sample:
where

81. Solution for small W (W<Lp):
In general
Boundary conditions

82. Surface recombination
s is called surface recombination velocity. Typically <1 cm/sec for well passivated surface
Consider minority carriers Þ Diffusion only

83. We have in the steady state:
Boundary conditions
i.e., at x=0

84. General solution (with the B.C. for is:
Applying B.C. for x = 0

85. as expected