P-N Junction Diode,
Current Components in PN Junction Diode,
Diode Equation,
V-I Characteristics,
Temperature Dependence on V-I Characteristics,
Diode Resistance,
Diode Capacitance,
Energy Band Diagram of PN Junction Diode
PN Junction diode in Equilibrium with no applied Voltage (can be
treated as Open Circuited PN Junction)
In a piece of
sc, if one half is doped by p type impurity and the other half is doped by n
type impurity, a PN junction is formed. The plane dividing the two halves or
zones is called PN junction. As shown in the fig the n type material has high
concentration of free electrons, while p type material has high concentration
of holes. Therefore at the junction there is a tendency of free electrons to
diffuse over to the P side and the holes to the N side. This process is called
diffusion. As the free electrons move across the junction from N type to P
type, the donor atoms become positively charged. Hence a positive charge is
built on the N-side of the junction. The free electrons that cross the junction
uncover the negative acceptor ions by filing the holes. Therefore a negative
charge is developed on the p –side of the junction. This net negative charge on
the p side prevents further diffusion of electrons into the p side. Similarly
the net positive charge on the N side repels the hole crossing from p side to N
side. Thus a barrier sis set up near the junction which prevents the further
movement of charge carriers i.e. electrons and holes. As a consequence of
induced electric field across the depletion layer, an electrostatic potential
difference is established between P and N regions, which are called the
potential barrier, junction barrier, diffusion potential or contact potential,
Vo. The magnitude of the contact potential Vo varies with doping levels and
temperature. Vo is 0.3V for Ge and 0.72 V for Si.
No
Applied Bias (V = 0 V)
At the instant
the two materials are “joined” the electrons and the holes in the region of the
junction will combine, resulting in a lack of free carriers in the region near
the junction, as shown in Fig. 1.5a . Note in Fig. 1.5a
that the only particles displayed in this region are the positive and
the negative ions remaining once the free carriers have been absorbed.
This region of
uncovered positive and negative ions is called the depletion region due to the
“depletion” of free carriers in the region.
Figure 1.5a: No bias Semi Conductor Diode
Figure 1.5b: No bias Semi Conductor Diode without ions
Figure 1.5c Symbol of PN Junction Diode
Figure 1.5d: Physical Representation of PN Junction Diode
The electrostatic field across the junction caused by the positively charged N-Type region tends to drive the holes away from the junction and negatively charged p type regions tend to drive the electrons away from the junction. The majority holes diffusing out of the P region leave behind negatively charged acceptor atoms bound to the lattice, thus exposing a negatives pace charge in a previously neutral region. Similarly electrons diffusing from the N region expose positively ionized donor atoms and a double space charge builds up at the junction as shown in the fig. 1.7a
Figure 1.7a: Diffusion of holes and electrons in P-N Diode
It is noticed
that the space charge layers are of opposite sign to the majority carriers
diffusing into them, which tends to reduce the diffusion rate. Thus the double
space of the layer causes an electric field to be set up across the junction
directed from N to P regions, which is in such a direction to inhibit the
diffusion of majority electrons and holes as illustrated in fig 1.7b. The shape
of the charge density, ρ, depends upon how diode id doped. Thus the junction
region is depleted of mobile charge carriers. Hence it is called depletion
layer, space region, and transition region. The depletion region is of the
order of 0.5µm thick. There are no mobile carriers in this narrow depletion
region. Hence no current flows across the junction and the system is in
equilibrium. To the left of this depletion layer, the carrier concentration is
p= NA and to its right it is n= ND.
Figure
1.7b: Diffusion of holes and electrons in P-N Diode
Barrier voltage
Positive charge present at n-side and negative charge present at p-side of p-n junction acts as barrier between p-type and n-type semiconductor. Thus, a barrier is build near the junction which prevents the further movement of electrons and holes.
Figure 1.8: Indicates barrier potential and depletion width
The negative charge formed at the p-side of the p-n junction is called negative barrier voltage while the positive charge formed at the n-side of the p-n junction is called positive barrier voltage. The total charge formed at the p-n junction is called barrier voltage, barrier potential or junction barrier as shown in Figure 1.8.
The
size of the barrier voltage at the p-n junction is depends on, the amount of
doping, junction temperature and type of material used. The barrier voltage for
silicon diode is 0.7 volts and for germanium is 0.3 volts.
This electric field created by the diffusion process has created a “built-in potential difference” across the junction with an open-circuit (zero bias) potential of
Eo is
the zero bias junction voltage, VT the
thermal voltage of 26mV at room temperature, ND and NA are the impurity concentrations and ni is the intrinsic concentration.
Typically at room temperature the voltage across the
depletion layer for silicon is about 0.6 – 0.7 volts and for germanium is about
0.3 – 0.35 volts. This potential barrier will always exist even if the device
is not connected to any external power source, as seen in diodes.
Let us consider the width of the depletion region in the
junction as shown in Figure 1.8 figure.
The region contains space charge due to the fact that, donors on the N-Side and
acceptors on the P-Side have lost their accompanying electrons and holes. Hence
electric field is established which turns causes a difference in potential is
built up across the junction. Hence space charge finally described as an alloy
junction, the depletion width W is proportional to (VO )1/2
Biased P-N Junction
Forward-Bias Condition (VD>
0 V):
A forward-bias or “on” condition is established by applying the positive potential to the p -type material and the negative potential to the n -type material as shown in Fig. 1.9. The application of a forward-bias potential VD will “pressure” electrons in the n -type material and holes in the p -type material to recombine with the ions near the boundary and reduce the width of the depletion region as shown in Fig. 1.9a . The resulting minority-carrier flow
Figure 1.9: Forward-biased P-N junction: (a) internal distribution of charge under forward-bias conditions; (b) forward-bias polarity and direction of resulting current.
of electrons from the p -type material to the n -type material (and of holes from the n –type material to the p -type material) has not changed in magnitude (since the conduction level is controlled primarily by the limited number of impurities in the material), but the reduction in the width of the depletion region has resulted in a heavy majority flow across the junction. An electron of the n-type material now “sees” a reduced barrier at the junction due to the reduced depletion region and a strong attraction for the positive potential applied to the p-type material. As the applied bias increases in magnitude, the depletion region will continue to decrease in width until a flood of electrons can pass through the junction, resulting in an exponential rise in current as shown in the forward-bias region of the characteristics of Fig. 1.16 Note that the vertical scale of Fig. 1.16 is measured in milli amperes (although some semiconductor diodes have a vertical scale measured in amperes), and the horizontal scale in the forward-bias region has a maximum of 1 V. Typically, therefore, the voltage across a forward-biased diode will be less than 1 V. Note also how quickly the current rises beyond the knee of the curve.
Figure 1.10: Forward biased P-N Junction with flow of charge carriers with resistor.
Figure 1.11: Circuit connection of Forward biased PN Diode
Reverse-Bias
Condition (VD < 0 V):
If an external potential of V volts is applied across the p – n junction such that the positive terminal is connected to the n -type material and the negative terminal is connected to the p -type material as shown in Fig. 1.12 , the number of uncovered positive ions in the depletion region of the n-type material will increase due to the large number of free electrons drawn to the positive potential of the applied voltage. For similar reasons, the number of uncovered negative ions will increase in the p-type material. The net effect, therefore, is a widening of the depletion region.
Figure 1.12: Reverse-biased P-N Junction: (a) internal distribution of charge under reverse-bias conditions; (b) reverse-bias polarity and direction of reverse saturation current.
Figure 1.13: Reverse-biased P-N Junction with resistor
Figure: 1.14 Circuit Connection of Reverse biased PN Diode
This
widening of the depletion region will establish too great a barrier for the
majority carriers to overcome, effectively reducing the majority carrier flow
to zero, as shown in Fig. 1.12a .
The number of minority carriers, however,
entering the depletion region will not change, resulting in minority-carrier
flow vectors of the same magnitude indicated with no applied voltage.
The current that exists under reverse-bias
conditions is called the reverse saturation current and is represented by Is
The reverse
saturation current is seldom more than a few microamperes and typically in µA
and nA, except for high-power devices. The term saturation comes from
the fact that it reaches its maximum level quickly and does not change
significantly with increases in the reverse-bias potential, as shown on the
diode characteristics of Fig. 1.15 for VD<0V. The reverse-biased conditions are depicted in
Fig.1.13b for the diode symbol and P – N Junction. Note, in
particular, that the direction of IS is against the arrow of
the symbol. Note also that the negative side of the applied voltage is
connected to the p -type material and the positive side to the n
-type material, the difference in underlined letters for each region revealing
a reverse-bias condition.
Sometimes this avalanche effect has practical applications in voltage stabilizing circuits where a series limiting resistor is used with the diode to limit this reverse breakdown current to a preset maximum value thereby producing a fixed voltage output across the diode. These types of diodes are commonly known as Zener Diodes
This increase in level is due to a wide range of factors that include
Leakage currents, Generation of carriers in the depletion region and Temperature Sensitivity whereas a 10°C increase in current will result in doubling of the actual reverse current of a diode
Current Components in PN junction Diode :
Drift current
The flow of charge carriers, which is due to
the applied voltage or electric field is called drift current. In a
semiconductor, there are two types of charge carriers, they are electrons and
holes. When the voltage is applied to a semiconductor, the free electrons move towards the positive terminal of a
battery and holes move towards the negative terminal of a battery.
Electrons are the negatively charged particles and holes are the positively charged particles. As we already discussed that like charges repel each other and unlike charges attract each other. Hence, the electrons (negatively charged particle) are attracted towards the positive terminal of a battery and holes (positively charged particle) are attracted towards the negative terminal.
In a
semiconductor, the electrons always try to move in a straight line towards the
positive terminal of the battery. But, due to continuous collision with
the atoms they change the direction of flow. Each time the electron strikes
an atom it bounces back in a random direction. The applied voltage does not stop
the collision and random motion of electrons, but it causes the electrons to
drift towards the positive terminal.
The
average velocity that an electron or hole achieved due to the applied voltage
or electric field is called drift velocity.
The drift velocity of electrons is given by
Vn = µnE
The drift velocity of holes is given by
Vp = µpE
Where vn = drift velocity of electrons
vp =
drift velocity of holes
µn =
mobility of electrons
µp =
mobility of holes
E = applied electric field
The drift current density due to free electrons is given by
Jn= enµnE
and the drift current density due to holes is
given by
Jp = epµpE
Where Jn =
drift current density due to electrons
Jp =
drift current density due to holes
e = charge of an electron = 1.602 × 10-19 Coulombs
(C).
n
= number of electrons
p
= number of holes
Then the total drift current density is
J = Jn + Jp
= enµnE + epµpE
J = e (nµn + pµp)
E
Diffusion current:
The process by which, charge carriers (electrons or holes) in a semiconductor moves from a region of higher concentration to a region of lower concentration is called diffusion.
The region in which more number of electrons is present is called higher concentration region and the region in which less number of electrons is present is called lower concentration region. Current produced due to motion of charge carriers from a region of higher concentration to a region of lower concentration is called diffusion current. Diffusion process occurs in a semiconductor that is non-uniformly doped.
Consider an n-type semiconductor that is non-uniformly doped as shown in below figure. Due to the non-uniform doping, more number of electrons is present at left side whereas lesser number of electrons is present at right side of the semiconductor material. The number of electrons present at left side of semiconductor material is more. So, these electrons will experience a repulsive force from each other.
The electrons present at left side of the
semiconductor material will moves to right side, to reach the uniform
concentration of electrons. Thus, the semiconductor material achieves equal
concentration of electrons. Electrons that move from left side to right
side will constitute current. This current is called diffusion current.
In p-type semiconductor, the diffusion process occurs in the similar
manner.
Both drift and
diffusion current occurs in semiconductor devices. Diffusion current occurs
without an external voltage or electric field applied. Diffusion
current does not occur in a conductor. The direction of diffusion current is
same or opposite to that of the drift current.
Concentration gradient
The diffusion current density is directly proportional to the concentration gradient. Concentration gradient is the difference in concentration of electrons or holes in
a given area. If the concentration gradient is high, then
the diffusion current density is also high. Similarly, if the concentration
gradient is low, then the diffusion current density is also low.
The concentration gradient for n-type semiconductor is given by
The concentration gradient for p-type semiconductor is given by
Where
Jn =diffusion
current density due to electrons
Jp = diffusion current density due to holes
Diffusion
current density
The
diffusion current density due to electrons is given by
Where Dn is
the diffusion coefficient of electrons
Where Dp is the diffusion coefficient of holes
The total current density due to electrons is the sum of drift and diffusion currents.
Jn = Drift current + Diffusion currentThe total current density due to holes is the sum of drift and diffusion currents.
Jp = Drift current + Diffusion current
The total current density due to electrons and holes is given by
J = Jn + Jp
The following figure shows a P-N Junction with a forward bias by an external voltage V as shown in Figure 1.15a. Due to the applied voltage, there exists a potential gradient in P and N materials.
Figure: 1.15a PN Diode by an external voltage V.
Now, the holes from P-region and the electrons from N-region drift towards the junction. The holes drifted from P-region towards the junction enter the N-region where they represent minority carriers. Similarly, the electrons drifted from N-region towards the junction enter the P-region where they represent minority carriers. The minority carriers diffuse away from the junction exponentially with distance as shown following Figure: 1.15b.
Figure: 1.15b Current components in forward-biased unsymmetrical junction.
Their concentration reduces steadily because of recombination with electrons and holes respectively. We know that diffusion current of minority carriers is proportional to the concentration gradient and hence this must also vary exponentially with distance.
Current Components:
Ipn(x) = hole current in N material.
Ipn(0) = hole current at junction (x=0)
Inp(x) = electron current in P material.
Inp(0) = electron current at junction
(x=0)
(The first letter refers to the type of the carrier
and the second to the type of material)
At junction (x=0), the electrons crossing from right
to left constitute a current in the same direction as hole crossing from left
to right.
Thus the total current I at junction is given by
I = Ipn(0) + Inp(0)
The majority (electron) current Inn is
given by Inn(x) = I - Ipn(x)
The majority (hole) current Ipp is given
by Ipp(x) = I – Inp(x)
Quantitative
Theory of PN Diode currents
By using Quantitative theory to derive the expression for the total current as a function of the applied voltage. When a P-N diode is forward biased, then the holes are injected from P-Side into the N-Material. As shown in following figure the several components of hole concentration in N-side of a forward biased diode. It is obvius from the figure the hole concentration decreases exponentially with distance.
Figure: 1.15c Graph between Concentration and Distance
i) The hole concentration in N material is given by
iii) Using Boltzmann relationship of Kinetic theory of gases, it can be established that
This is known as Law of Junction. Here
V = applied voltage, and
VT= Volt equivalent of temperature = KT/q
= T/ 11,600
Where k is Boltzmann Constant.
Total
Diode Current
The total diode current I at x = 0 is given by
I = Ipn(0) + Inp(0),
Ipn(0)
= current caused by holes entering N – region
Inp(0) = current caused by electrons entering P – region
Diode Equation :
In solid-state
physics that the general characteristics of a semiconductor diode can be
defined by the following equation, referred to as Shockley’s equation, for the
forward- and reverse-bias regions for exact demonstration.
IS
is
the reverse saturation current
VD is
the applied forward-bias voltage across the diode
ŋ
is
an ideality factor, which is a function of the operating conditions and
physical construction; it has a range between 1 and 2 depending on a wide
variety of factors ( n =1 will be
assumed throughout this text unless otherwise noted).
The voltage VT is called the thermal voltage and is determined by
Where k is Boltzmann’s constant= 1.38 x 10-23 J/K
T
K is the absolute temperature in kelvins =
273 + the temperature in oC
q is the
magnitude of electronic charge = 1.6 X10-19c
Problem 1 At a temperature of 27°C (common temperature for components in an enclosed operating system), determine the thermal voltage VT
Solution:
= 25.875mV =
The thermal
voltage will become an important parameter in the analysis to follow in this
chapter and a number of those to follow.
V-I Characteristics:
https://youtu.be/wqIcQW7Rzvk
Initially, Eq. (1.2) with all its defined quantities may appear somewhat complex. However, it will not be used extensively in the analysis to follow. It is simply important at this point
to understand the source of the diode characteristics and which factors affect its shape. A plot of Eq. (1.2) with Is= 10 pA is provided in Fig. 1.15 as the dashed line. If we expand Eq. (1.2) into the following form, the contributing component for each region of Fig. 1.15 can be described with increased clarity:
For positive values of VD the first term of the above equation will grow very quickly and totally overpower the effect of the second term. The result is the following equation, which only has positive values and takes on the exponential format e x appearing in Fig 1.16:
Figure: 1.16a Silicon semiconductor Diode Characteristics
Figure: 1.16b Silicon semiconductor Diode Characteristics with exponential representation
Figure: 1.16c Diode Characteristics in Forward Bias and Reverse Bias
Figure: 1.16d Comparison of Ge, Si, and GaAs commercial diodes.
Simply plots the actual response of commercially available units. The total reverse current is shown and not simply the reverse saturation current. It is immediately obvious that the point of vertical rise in the characteristics is different for each material, although the general shape of each characteristic is quite similar. Germanium is closest to the vertical axis and GaAs is the most distant. As noted on the curves, the center of the knee (hence the K is the notation VK) of the curve is about 0.3 V for Ge, 0.7 V for Si, and 1.2 V for GaAs as shown in Figure 1.16d
Temperature Dependence on V-I Characteristics:
Temperature can have a marked effect on the characteristics of a semiconductor diode, as demonstrated by the characteristics of a silicon diode shown in Fig. 1.17. An increase from room temperature (20°C) to 100°C (the boiling point of water) results in a drop of 80(2.5 mV) = 200 mV, or 0.2 V, which is significant on a graph scaled in tenths of volts. A decrease in temperature has the reverse effect, as also shown in the figure: In the reverse-bias region the reverse current of a silicon diode doubles for every 10°C rise in temperature.
Figure: 1.17 Variation in Si diode characteristics with temperature change.
It is not uncommon for a germanium diode with an Io in the order of 1 or 2 A at 25°C to have a leakage current of 100 A - 0.1 mA at a temperature of 100°C. Typical values of Io for silicon are much lower than that of germanium for similar power and current levels. The result is that even at high temperatures the levels of Io for silicon diodes do not reach the same high levels obtained. For germanium—a very important reason that silicon devices enjoy a significantly higher level of development and utilization in design. Fundamentally, the open-circuit equivalent in the reverse bias region is better realized at any temperature with silicon than with germanium. The increasing levels of I with temperature account for the lower levels of threshold voltage, as shown in Fig. 1.11. Simply increase the level of Io in and not rise in diode current. Of course, the level of TK also will be increase, but the increasing level of Io will overpower the smaller percent change in TK. As the temperature increases the forward characteristics are actually becoming more “ideal,”
Problem :
Using the curves of Fig 1.16d:
a. Determine the voltage across each diode at a current of 1 mA.
b. Repeat for a current of 4 mA.
c. Repeat for a current of 30 mA.
d. Determine the average value of the diode voltage for the range of currents listed above.
Diode resistance:
DC
or Static Resistance
The application
of a dc voltage to a circuit containing a semiconductor diode will result in an
operating point on the characteristic curve that will not change with time. The
resistance of the diode at the operating point can be found simply by finding
the corresponding levels of VD and ID as shown in Fig. 1.18
and applying the following Equation:
The dc
resistance levels at the knee and below will be greater than the resistance
levels obtained for the vertical rise section of the characteristics. The
resistance levels in the reverse-bias region will naturally be quite high.
Since ohmmeters typically employ a relatively constant-current source, the
resistance determined will be at a preset current level (typically, a few
milliamperes). In
general, therefore, the higher the current through a diode, the lower is the dc
resistance level.
Typically, the
dc resistance of a diode in the active (most utilized) will range from about 10
Ω to 80Ω
Problem
2 Determine the dc resistance levels for the
diode of following figure. at
a. ID= 2 mA (low level)
b. I D=20 mA (high level)
c. VD=10 V (reverse-biased)
a.
Clearly
supporting some of the earlier comments regarding the dc resistance levels of a
diode. AC or Dynamic Resistance
The dc resistance of a diode is independent of
the shape of the characteristic in the region surrounding the point of
interest.
If a sinusoidal
rather than a dc input is applied, the situation will change completely. The
varying input will move the instantaneous operating point up and down a region
of the characteristics and thus defines a specific change in current and
voltage as shown in Fig.1.18b with no applied varying signal, the point of
operation would be the Q –point appearing on Fig. 1.18b, determined by
the applied dc levels. The designation Q-point is derived from the word quiescent,
which means “still or unvarying.”
Figure 1.18b Defining the Dynamic or ac resistance.
A straight line
drawn tangent to the curve through the Q -point as shown in Fig. 1.18c will define a particular change in voltage
and current that can be used to determine the ac or dynamic
resistance for this region of the diode characteristics. An effort should be
made to keep the change in voltage and current as small as possible and
equidistant to either side of the Q -point. In equation form.
where Δ
signifies a finite change in the quantity
Figure 1.18c Determining the ac resistance at a Q – point.
Diode Equivalent Circuits (Add on Course)
An
equivalent circuit is a combination of elements properly chosen to best
represent the actual terminal characteristics of a device or system in a
particular operating region.
In other words, once the equivalent circuit is defined, the device symbol can be removed from a schematic and the equivalent circuit inserted in its place without severely affecting the actual behavior of the system. The result is often a network that can be solved using traditional circuit analysis techniques.
Diode capacitance:
Transition
and Diffusion Capacitance
Every electronic or electrical device is
frequency sensitive.
That
is, the terminal characteristics of any device will change with frequency. Even
the resistance of a basic resistor, as of any construction, will be sensitive
to the applied frequency. At low to
mid-frequencies most resistors can be considered fixed in value. However,
as we approach high frequencies, stray capacitive
and inductive effects start to play a role
and will affect the total impedance level of the
element.
For
the diode it is the stray capacitance levels that have the greatest effect. At
low frequencies and relatively small levels of capacitance the reactance of a
capacitor, determined by XC = 1/2πfc is usually so high it can be
considered infinite in magnitude, represented by an open circuit, and ignored.
At high frequencies, however, the level of XC can drop to the
point where it will introduce a low-reactance “shorting” path. If this shorting
path is across the diode, it can essentially keep the diode from affecting the
response of the network.
In the p – n semiconductor diode, there are two capacitive effects to be considered. Both types of capacitance are present in the forward- and reverse-bias regions, but one so outweighs the other in each region that we consider the effects of only one in each region. Recall that the basic equation for the capacitance of a parallel-plate capacitor is defined by C = €A/d where € is the permittivity of the dielectric (insulator) between the plates of area A separated by a distance d.
Figure 1.19 Transition and diffusion capacitance versus applied bias for a silicon diode.
Where
τ is the minority carrier lifetime the time is world take for a minority
carrier such as a hole to recombine with an electron in the n -type
material. However, increased levels of
current result in a reduced level of associated resistance (to be demonstrated
shortly), and the resulting time constant (τ = RC), which is very
important in high-speed applications, does not become excessive. In general,
therefore,
The transition capacitance is the predominant capacitive effect in the reverse-bias region whereas the diffusion capacitance is the predominant capacitive effect in the forward-bias region.
Energy Band Diagram of PN Junction Diode:
A p-n junction consists of two semiconductor regions with
opposite doping type as shown in Figure 4.2.1. The region on the left is p-type with an
acceptor density Na, while the region on the right
is n-type with a donor density Nd. The
dopants are assumed to be shallow, so that the electron (Hole) density in
the n-type (p-type) region is approximately equal to the
donor (Acceptor) density.
Figure 1.20: P-N Junction diode with density representation
Cross-section of a P-N Junction
We will assume, unless stated otherwise, that the doped
regions are uniformly doped and that the transition between
the two regions is abrupt. We will refer to this structure as an abrupt p-n
junction. Frequently we will deal with p-n junctions in which one side is
distinctly higher-doped than the other. We will find that in such a case only
the low-doped region needs to be considered, since it primarily determines the
device characteristics. We will refer to such a structure as a one-sided abrupt
p-n junction.
The junction is biased with a voltage Va as shown in Figure 1.21 (a). We will call the junction forward-biased if a positive voltage is applied to the p-doped region and reversed-biased if a negative voltage is applied to the p-doped region. The contact to the p-type region is also called the anode, while the contact to the n-type region is called the cathode, in reference to the anions or positive carriers and cations or negative carriers in each of these regions.
Flatband diagram
Figure 1.21 Energy band
diagram of a p-n junction (a) before and (b) after merging the n-type and
p-type regions
Note that this does not automatically align the Fermi energies, EF,n and EF,p. Also, note that this flatband diagram is not an equilibrium diagram since both electrons and holes can lower their energy by crossing the junction. A motion of electrons and holes is therefore expected before thermal equilibrium is obtained. The diagram shown in Figure 1.21 (b) is called a flatband diagram. This name refers to the horizontal band edges. It also implies that there is no field and no net charge in the semiconductor.
Thermal Equilibrium
To reach thermal equilibrium, electrons/holes close to the metallurgical junction diffuse across the junction into the p-type/n-type region where hardly any electrons/holes are present. This process leaves the ionized donors (acceptors) behind, creating a region around the junction, which is depleted of mobile carriers. We call this region the depletion region, extending from x = -xp to x = xn. The charge due to the ionized donors and acceptors causes an electric field, which in turn causes a drift of carriers in the opposite direction. The diffusion of carriers continues until the drift current balances the diffusion current, thereby reaching thermal equilibrium as indicated by a constant Fermi energy.
Figure 1.22a Energy band diagram of a p-n
junction in thermal equilibrium
Figure 1.22 b Energy Band Structure
While in thermal equilibrium no external voltage
is applied between the n-type and p-type material,
there is an internal potential, fi, which is caused by the work
function difference between the n-type and p-type
semiconductors. This potential equals the built-in potential,
which will be further discussed in the next section.
The built-in potential
The built-in potential in a semiconductor equals the potential across the depletion region in thermal equilibrium. Since thermal equilibrium implies that the Fermi energy is constant throughout the p-n diode, the built-in potential equals the difference between the Fermi energies, EFn and EFp, divided by the electronic charge. It also equals the sum of the bulk potentials of each region, fn and fp, since the bulk potential quantifies the distance between the Fermi energy and the intrinsic energy. This yields the following expression for the built-in potential.
Forward and reverse bias
We now consider a p-n diode with an applied bias
voltage, Va. A forward bias corresponds to applying a
positive voltage to the anode (the p-type region) relative to the
cathode (the n-type region). A reverse bias corresponds to a
negative voltage applied to the cathode. Both bias modes are illustrated with Figure 1.23. The applied voltage is
proportional to the difference between the Fermi energy in the n-type
and p-type quasi-neutral regions.
As a negative voltage is applied, the potential
across the semiconductor increases and so does the depletion layer width. As a
positive voltage is applied, the potential across the semiconductor decreases
and with it the depletion layer width. The total potential across the
semiconductor equals the built-in potential minus the applied voltage.
Figure 1.23 Energy band diagram of a p-n
junction under reverse and forward bias
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