Transcript 3. Lecture 2 - School Of Electrical & Electronic Engineering

Ideal Diode Model
•Let’s begin with an ideal diode and look at its characteristics
Real PN Junction Diode I-V
Characteristic
•Typical PN junction diode I-V
characteristic is shown on the right.
–In forward bias, the PN junction
has a “turn on” voltage based on
the “built-in” potential of the PN
junction. The turn on voltage is
typically in the range of 0.5V to
0.8V
–In reverse bias, the PN junction
conducts essentially no current until
a critical breakdown voltage is
reached. The breakdown voltage
can range from 1V to 100V.
Breakdown mechanisms include
avalanche and zener tunneling.
Current Equations
•The forward bias current is closely approximated by
 / nV
T
i  I S  e
 1
where VT  kT / q


where VT is the thermal voltage (~25.8mV at room temp T= 300K or 27C )
k = Boltzman’s constant = 1.38 x 10-23joules/kelvin
T = absolute temperature
q = electron charge = 1.602 x 10-19coulombs
n = constant dependent on structure, between 1 and 2 (we will assume
n = 1)
IS= scaled current for saturation current that is set by diode size
–Notice there is a strong dependence on temperature
–We can approximate the diode equation for i>> IS
i  I S e / nVT
•In reverse bias (when v << 0 by at least VT), then
i  I S
•In breakdown, reverse current increases rapidly…a vertical line
Mobile Carriers
•Now let’s look at physical mechanisms from which the current
equations come.
–We’ve seen that holes and electrons move through a semiconductor
by two mechanisms –drift and diffusion
–In equilibrium, diffusion current (ID) is balanced by drift current
(IS). So, there is no net current flow. Drift current comes from
(thermal) generation of electron-hole pairs (EHP).
Band Diagrams
•When the P-type material is contacted
with the N-type material, the Fermi
levels must be at equilibrium.
•Band bending: The conduction and
valence bands “bend” to align the Fermi
levels.
•Electrons diffuse from the N-side to the
P-side and recombine with holes at the
boundary. Holes diffuse from the P-side
to the N-side and recombine with
electrons at the boundary. There is a
region at the boundary of charged atoms
–called the space-charge region (also
called the depletion region b/c no mobile
carriers in this region)
•An electric field is created which results
in a voltage drop across the region –
called the barrier voltage or built-in
potential
What happens when P-type meets
N-type?
Holes diffuse from the p-type into the n-type, electrons diffuse from the ntype into the p-type, creating a diffusion current. The diffusion equation is
given by
dn
J n  qD n
dx
where Dn is the diffusion constant
Once the holes [electrons] cross into the n-type [p-type] region, they
recombine with the electrons [holes].
•This recombination “strips” the n-type [p-type] of its electrons near the
boundary, creating an electric field due to the positive and negative bound
charges.
•The region “stripped” of carriers is called the space-charge region, or
depletion region.
•V0is the contact potential that exists due to the electric field.
E x   
dV
dx
•Some carriers are generated (thermally) and make their way into the
depletion region where they are whisked away by the electric field, creating a
drift current.
E-field and Built-in Potential
•Diffusion is balanced by drift due to
bound charges at the junction that
induce an E-field.
•Integrating the bound charge density
gives us the E-field
Ε x  
1
 r o
  xdx
x

Integrating the E-field gives the
potential gradient
Ε x   
dV
dx

x
V  x    E  x dx

Junction Built-In Voltage
•With no external biasing, the voltage across the depletion region is:
Vo  VT ln
N AND
ni2
–Typically, at room temp, V0 is 0.6~0.8V
•How does V0 change as temperature increases?
–Note that there is no measurable potential difference between
the n-type and p-type materials of pn junction when in
equilibrium. The electrochemical potentials (Fermi levels) are the
equal.
Width of Depletion Region
•The depletion region exists on both sides
of the junction. The widths in each side is a
function of the respective doping levels.
Charge-equality gives:
qx p AN A  qxn AN D
xn
xp

NA
ND
•The width of the depletion region can
be found as a function of doping and the
built-in voltage…
2 s  1
1 
Wdepl  xn  x p 


Vo
q N
N

A
D
s is the electrical permittivity of
silicon = 11.7ε0 (where ε0 = 8.854E14 F/cm)
Pn Junction in Reverse Bias (1)
•As the depletion region grows, the capacitance across the diode
changes.
Wdepl  xn  x p 
2 s
q
 1
1 

Vo  VR 

 NA ND 
–Treating the depletion region as a parallel plate capacitor…
C jo
Cj 
VR
1
Vo
Reverse Bias (2)
•Reverse bias: apply a negative
voltage to the p-type, positive to
n-type.
•Increase the built-in potential,
increase the barrier height.
•Decrease the number of carriers
able to diffuse across the barrier.
•Diffusion current decreases.
•Drift current remains the same
(due to generation of EHP).
•Almost no current flows. Reverse
leakage current, IS, is the drift
current, flowing from n to p.
Reverse Breakdown
•Zener Breakdown: The bands bend so much that carriers
tunnel through the depletion region. This will occur in heavily
doped junctions when the n-side conduction band appears
opposite the p-side valence band.
•Avalanche Breakdown: carriers have enough energy to ionize
an electron-hole-pair (EHP), creating more highly energetic
carriers, which collide to form more EHPs, which creates…
pn Junction in Forward Bias (1)
•Now let’s look at the condition where
we push current through the pn junction
in the opposite direction.
–Add more majority carriers to both
sides shrink the depletion region
lower V0diffusion current increases
•Look at the minority carrier
concentration…
–lower barrier allows more carriers to
be injected to the other side
•Note that np0= ni2/NA and
pn0= ni2/ND
–This comes from two equations…
EF Ei / kT
p  n eEi EF / kT
n  ni e
i
•The forward bias voltage causes excess minority carriers to be
injected across the junction.
pn xn   pnoeV / VT
•The distribution of excess minority hole concentration in the ntype Siis an exponentially decaying function of distance from xn
 x ' / L p
p n  x'  p no   p n  x n   p no e
–where Lp is the diffusion length (steepness of exponential decay)
and is set by the excess-minority-carrier lifetime, p. The average
time it takes for a hole injected into the n region to recombine
with a majority carrier electron
L p  D p p
–The diffusion of holes leads to the following current density vs. x
JP  q
Dp
Lp


p no eV / VT  1 e
 x  xn  / L p
•In equilibrium, as holes diffuse away, they must be met by a constant
supply of electrons with which they recombine. Thus, the current must
be supplied at a rate that equals the concentration of holes at the edge
of the depletion region (xn). Thus, the current due to hole injection is:
Dp
JP  q
pno eV / VT  1
Lp
•Current due to electrons injected into the p region is…
Jn  q
•Combined…




Dn
n po eV / VT  1
Lp
 Dp
 V /V
Dn

I  qA
p no 
n po  e T  1
 Lp

Ln





I  I s eV / VT  1  I s eV / VT

Minority Carrier Concentration and
Current Densities in Forward Bias
•Current is due to the
diffusion of holes and
electrons. Current is
dominated by holes or
electrons depending on
the relative doping of
NA vs. ND
•Is NA> ND or NA<ND in
this example?
Forward Bias (2)
•Forward bias: apply a positive
voltage to the p-type, negative to
n-type.
•Decrease the built-in potential,
lower the barrier height.
•Increase the number of carriers
able to diffuse across the barrier
•Diffusion current increases
•Drift current remains the same
•Current flows from p to n
Review of Biasing
•Applying a bias adds or
subtracts to the built-in
potential.
•This changes the diffusion
current, making it harder or
easier for the carriers to
diffuse across.
•The drift current is
essentially constant, as it is
dependent on temperature.
Photodiodes
•Diodes have an optical generation
rate. Carriers are created by shining
light with photon energy greater than
the bandgap.
•Photodetector: should have large
depletion widths and long diffusion
lengths (minority carrier lifetimes) so
that photo generatedEHPscan be
collected and swept across the
junction.
•Solar Cell: operating in the fourth
quadrant generates current, though
small.
Light Emitting Diodes
•When electrons and holes combine,
they release energy.
•This energy is often released as heat
into the lattice, but in some
materials, known as direct bandgap
materials, they release light.
•Engineering LEDs can be difficult,
but has been done over a wide range
of wavelengths.
•This illustration describes the
importance of the plastic bubble in
directing the light so that it is more
effectively seen.
Diode Circuits
•Look at the simple diode circuit below. We can write two equations:
I D  I s eVD / VT
ID 
V DD  V D
R
Diode Small-Signal Model
•Some circuit applications bias the diode at a DC
point (VD) and superimpose a small signal (vd(t))
on top of it. Together, the signal is vD(t),
consisting of both DC and AC components
–Graphically, can show that there is a translation
of voltage to current (id(t))
–Can model the diode at this bias point as a
resistor with resistance as the inverse of the
tangent of the i-v curve at that point
iD t   I S eVD d / VT  I S eVD / VT ed / VT
iD t   I D ed / VT
–And if vd(t) is sufficiently small then we can expand
the exponential and get an approximate expression
called the small-signal approximation (valid for vd<
10mV)
  
I
i D t   I D 1  d   I D  id  id  D  d
VT
 VT 
–So, the diode small-signal resistance is…
rd 
VT
ID
•Perform the small signal analysis of the diode circuit biased with VDD
by eliminating the DC sources and replacing the diode with a small
signal resistance–The resulting voltage divider gives:
d  S
rd
R  rd
•Separating out the DC or bias analysis and the small-signal
analysis is a technique we will use extensively