Download presentation
Presentation is loading. Please wait.
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
Similar presentations
© 2025 SlidePlayer.com. Inc.
All rights reserved.