Carrier concentration vs inverse
Temperature
Compensated
semiconductors
Charge
Neutrality
Motion in solids
Electron motion in semiconductor
scattering
The mobility defined in Eq. (3–40a) can be expressed as the average
particle drift velocity per unit electric field.
Effect iof doping and temperature on mobility
Conductivity of Intrinsic and Extrinsic Semiconductors
For Si μn = 0.135 m2/Vs, μp = 0.048 m2/Vs;
for Ge μn = 0.39 m2/Vs, μp = 0.19 m2/Vs.
Conductivity of Extrinsic Semiconductors
Typical carrier densities in intrinsic & extrinsic semiconductors
Si at 300K, intrinsic carrier density ni = 1.5 x 1016/m3
Extrinsic Si doped with As → typical concentration 1021atoms/m3:
Majority carriers n0 = 1021 e/m3 ; Mass action law: ni2 = n0p0
Minority carriers: p0 = (1.5×1016)2/1021 = 2.25 x 1011 holes/m3
Conductivity:
Majority carriers: σn = 1021x0.135x1.6x10-19 (e/m3 ) (m2 /Vs) (A⋅s C) =0.216 (Ω
cm)-1
Minority carriers: σp = 2.25x10-11 x 0.048 x1.6x10-19 = 0.173x10-10 (Ω cm)-1
Conductivity total σtotal = σn + σp ≈ 0.216 (Ω cm)-1
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Modulators of
conductivity
Just reviewed how conductivity of a semiconductor is affected by:
Temperature – Increasing temperature causes conductivity to
increase
• Dopants – Increasing the number of dopant atoms (implant dose)
cause conductivity to increase.
• Holes are slower than electrons therefore n-type material is
more conductive than p-type material.
• These parameters are in addition to those normally affecting
conducting material,
Cross sectional area 🡹 Resistance 🡹
Length 🡹 Resistance 🡹
Silicon Resistivity Versus Dopant Concentration
1021
1020
Dopant Concentration (atoms/cm3)
1019
1018
1017
n-type p-type
16
10
1015
1014
101310-3 10-2 10-1 100 101 102 103
Electrical Resistivity (ohm-cm)
Redrawn from VLSI Fabrication Principles, Silicon and Gallium Arsenide, John Wiley & Sons, Inc.
Summary
Intrinsic semiconductors
Doped semiconductors
n-type p-type
Hall Effect
pn Junction
The interface separating the n and p
regions is referred to as the
metallurgical junction.
For simplicity we will consider a step
junction in which the doping
concentration is uniform in each
region and there is an abrupt change
in doping at the junction.
Initially there is a very large density
gradient in both the electron and
hole concentrations. Majority carrier
electrons in the n region will begin
diffusing into the p region and
majority carrier holes in the p region
will begin diffusing into the n
region. If we assume there are no
external connections to the
semiconductor, then this diffusion
process cannot continue
indefinitely.
pn Junction
This cannot occur in the case of the charged particles
in a p-n junction because of the development of space
charge and the electric field ε. As electrons diffuse from the n region,
positively charged donor atoms are left
behind. Similarly, as holes diffuse from the p
region, they uncover negatively charged
acceptor atoms. These are minority carriers.
The net positive and negative charges in the
n and p regions induce an electric field in the
region near the metallurgical junction, in the
direction from the positive to the negative
charge, or from the n to the p region.
The net positively and negatively charged regions
are shown in Figure. These two regions are
referred to as the space charge region (SCR).
Essentially all electrons and holes are swept out of
the space charge region by the electric field. Since
the space charge region is depleted of any mobile
charge, this region is also referred to as the
depletion region
Density gradients still exist in the majority carrier concentrations at each edge of the space charge
region. This produce a "diffusion force" that acts the electrons and holes at the edges of the space
charge region. The electric field in the SCR produces another force on the electrons and holes
which is in the opposite direction to the diffusion force for each type of particle. In thermal equilib
rium, the diffusion force and the E-field (ε) force exactly balance each other.
pn Junction – built-in potential barrier
No applied voltage across pn-junction
The junction is in thermal equilibrium
—the Fermi energy level is constant
throughout the entire system. The
conduction and valence band energies
must bend as we go through the space
charge region, since the relative
position of the conduction and valence
bands with respect to the Fermi energy
changes between p and n regions.
Electrons in the conduction band of the n region see a potential barrier in trying to move
into the conduction band of the p region. This potential barrier is referred to as the built-in
potential barrier and is denoted by Vbi (or V0). The built-in potential barrier maintains
equilibrium between majority carrier electrons in the n region and minority carrier electrons
in the p region, and also between majority carrier holes in the p region and minority carrier
holes in the n region. The potential Vbi maintains equilibrium, so no current is produced by
this voltage.
The intrinsic Fermi level is equidistant from the conduction band edge through the junction,
thus the built-in potential barrier can be determined as the difference between the intrinsic
Fermi levels in the p and n regions.
pn Junction An applied voltage bias V appears across the
transition region of the junction rather than in the
neutral n and p region. Of course, there will be some
voltage drop in the neutral material, if a current flows
through it. But in most p-n junction devices, the
length of each region is small compared with its area,
and the doping is usually moderate to heavy; thus the
resistance is small in each neutral region, and only a
small voltage drop can be maintained outside the
space charge (transition) region. V consider to be
positive when the external bias is positive on the p
side relative to the n side.
The electrostatic potential barrier at the junction is
lowered by a forward bias Vf from the equilibrium
contact potential V0 to the smaller value V0-Vf. This
lowering of the potential barrier occurs because a
forward bias (p positive with respect to n) raises the
electrostatic potential on the p side relative to the n
side. For a reverse bias (V=-Vr ) the opposite occurs;
the electrostatic potential of the p side is depressed
relative to the n side, and the potential barrier at the
junction becomes larger (V0 + Vr ).
The electric field within the transition region can be
deduced from the potential barrier. We notice that the
field decreases with forward bias, since the applied
electric field opposes the buid-in field. With reverse
bias the field at the junction is increased by the
applied field, which is in the same direction as the
equilibrium field.
Physics of the Depletion Region
• When n and p type material are placed in contact with
each other, the electrons diffuses into the p-type region in
order to equalise the Fermi levels.
• This loss of electrons from the n-type material leaves the
surface layer positively charged.
• Similarly the p-type material will have a negatively
charged surface layer.
• Thus an electric field is established which opposes the
diffusion of electrons when the Fermi levels are equal
(dynamic equilibrium is established)
The diffusion current crossing a unit area (the
current density) is the particle flux density
multiplied by the charge of the carrier:
From equation 3-43 the drift
Current density due to an
electric field
Is jx=q(nμn+ pμp)εx
(Example
5,1)
Equilibrium Fermi Levels
EXPLANATION OF PREVIOUS GRAPH
The charge density within W is plotted in Fig. 5–12b.
Neglecting carriers within the space charge region, the
charge density on the n side is just q times the
concentration of donor ions Nd , and the negative
charge density on the p side is - q times the
concentration of acceptors Na. The assumption of
carrier depletion within W and neutrality outside W is
known as the depletion approximation.
(Example
5,2)
