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Nov 2, 2013 (4 years and 12 days ago)

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Transistors Appendix

Transistors are scalable electronic switches, made from doped silicon. Silicon forms a crystal,
and have no free electrons at low temperature (around 0K). At higher temperatures (300K),
thermal energy is sufficient to release some electrons from their covalent bonds. The
availability of conducting electrons is, however, limited (more quantitative) because the band
-
gap energy is high relative to a good conductor (copper).

Dopant :

Phosphorus

Boron

Means :

Add electrons to

conduction band

Remove electrons from
valence shell, holes
created

Creates :

N
-
type material

P
-
type material

N
-

and P
-
type doped silicon are the components necessary to build switching devices, like
diodes and transistors. Dopant atoms can integrate with silicon’s crystal lattice, and create
additional holes or conducting electrons. Because phosphorus has five valance electrons, an
additional weakly bound electron is present when it integrates. Thermal energy frees this
electron, producing a large number of conducing electrons in this material, known as N
-
type.
Tri
-
valent Boron creates a hole upon integration, producing P
-
type material.

N

P

The diode device architecture fuses an N and a P
-
junction together. The NP junction with no
voltage applied establishes an equilibrium such that the force of diffusion, which draws
electrons into the P
-
type material, is opposed by the force of the electric field, which draws
electrons back into the N
-
type material. This “equilibrium” electric field represents a potential
difference around 0.76V (proven, p. 18 of
Electronic Circuit Design and Analysis
).

N

P

Si

P

Si

Si

Si

Si

B

Si

N

P

Si

P+

Si

Si

Si

Si

B
-

Si

Diffusion

Field

Diffusion

N

P

Si

P+

Si

Si

Si

Si

B
-

Si

+

-

Sets up a field

Establishes current equilibrium

Diffusion

The N
-
region contains a higher concentration electrons than the P
-
region. Electrons diffuse
from the N to P
-
type material.

Relative concentration of minority carrier in P
-
type material: electrons

Diffusion “force”

Relative concentration of majority carrier in N
-
type material: electrons

Si

P

Si

Si

Si

Si

B

Si

P
-
type

N
-
type

Diffusion establishes a charge separation, which sets up an electric field. The field exerts a
force on the electrons opposite the direction of diffusion.

P
-
type

N
-
type

Diffusion “force”

Si

P
+

Si

Si

Si

Si

B
-

Si

Electric
force
(on electrons) due to equilibrium field

Electrons

Electrons

Electric
field
(oriented from positive to negative), ~0.75V

Diffusion continues and the charge separation continues to grow in magnitude, until the force
of the electric field due to the charge separation equally opposes the force of diffusion. At this
point, there is no net current flow across the junction, and the electron concentration at either
side of the junction reaches an equilibrium value.

Diffusion “force”

P
-
type

N
-
type

Electrons : equilibrium

Electric
force
(on electrons)

Electric
field
(oriented from positive to negative) , ~0.75V

In reverse bias voltage, the fields are oriented in the same direction, and the magnitude of the
electric field in the space charge region increases above the equilibrium value. This holds back
electrons, and no current flows. With forward bias, the net result is that the eclectic field at
the junction is lower than the equilibrium value, and electrons diffuse.

N

P

Si

P+

Si

Si

Si

Si

B
-

Si

-

+

Applied

Field

Reverse Bias

Equilibrium

Field, ~0.75V


Force

Net Field > Equilibrium

Reverse
-
bias voltage increases net field at the junction, opposing diffusion. Charges are drawn
away from the junction by the field.

P
-
type

Net electric force due to fields

N
-
type

Diffusion “force”

Electrons

Electric
field
(oriented from positive to negative)

Applied electric field from reverse
-
bias voltage polarity

Electrons drawn from P
-
type material near junction

In reverse bias voltage, the fields are oriented in the same direction, and the magnitude of the
electric field in the space charge region increases above the equilibrium value. This holds back
electrons, and no current flows. With forward bias, the net result is that the eclectic field at
the junction is lower than the equilibrium value, and electrons diffuse.

N

P

Si

P+

Si

Si

Si

Si

B
-

Si

Equilibrium

Field

+

-

Applied

Field

Net diffusion

Forward Bias

Net Field < Equilibrium

Forward
-
biased voltage reduces net field at the junction, reducing the width of the depletion
region. The force of diffusion dominates.

P
-
type

Net
electric force due to fields

N
-
type

Diffusion “force” exceeds the net electric force, electrons diffuse
across junction, where electron concentration is above equilibrium


Electrons diffuse from junction into the bulk

Electrons

Electric
field
(oriented from positive to negative)

Applied electric field from forward
-
bias voltage polarity

The output is highly responsive to increases in the forward
-
biased input, above the “cut in”
voltage threshold ~0.7V. At zero or reverse bias input voltage, the output current is the very
small saturation value, around 5*10^(
-
14)A. Because the forward bias voltage is included in
the exponent, the exponential terms increases with respect to the input voltage. At the cut in
voltage ~0.7V, the input voltage driven exponential term begins to dominate the very small
reverse bias saturation current term, resulting in the exponential behavior. Physically, this
means that diffusion is unrestricted once the forward bias voltage establishes a field that fully
counters the built
-
in equilibrium field.

Two
-
state:

Non
-
linear response

at “on” threshold

Sensitive:

Only 0.7V “on” threshold

Voltage input

Transfer function

Input: voltage

V forward bias

V reverse bias

Device schematic

Output: current

Current

No current

I = Is*[ e^(Vin/Vt)
-
1 ]

Is = Reverse bias current, 5*10^(
-
14)A

Vt = Thermal voltage = 0.026

Vin

I

The diode current and voltage are given by the intersection between the circuit load line and
the diode performance curve. This intersection, or Q
-
point, gives the DC voltage and current
for the forward
-
biased diode in the circuit: this intersection identifies the feasible diode
operating conditions that also satisfy the energy balance of the circuit.

R

Vd

KVL (energy balance):

Vs=IR+Vd

I=
-
(1/R)*Vd+Vs/R

Voltage

Source

Diode

Vd

V=IR

Energy balance:

I=
-
(1/R)*Vd+Vs/R

Vd = Vs

I=Vs/R

A diode circuit can make a switch. If the output is defined as the voltage drop across the
resistance, then output is zero in the reverse bias condition (red). This is because the diode is
reverse biased in this condition, resulting in no circuit current and no resistor voltage drop.

R

Supply : 5V

R

+

-

-

+

Voltage drop

0.7V

4.3V,

I=4.3/R

0V,

I=0

5V

I

Time

Switch voltage

Polarity forward to
reverse bias

Zero output

A diode circuit can make an AND gate. If both inputs are high (5V), then there is no potential
difference across either diode (no voltage difference between input and supply). Neither diode
is forward biased, no current flows. Since there is no current in the circuit, there is no voltage
drop across the resistance and output voltage is 5V (high). If either input is low, then the diode
is forward biased and the voltage drop across the diode will be ~0.7V. Thus, ~4.3V will drop
across the resistance, resulting in a 0.7V (low) output.

Vi (1)

Vi (2)

R

V (o)

Supply : 5V

I

A diode circuit can make an OR gate. If either input is high (5V), the corresponding diode is
forward biased with a 0.7V drop. The output voltage is 4.3V, and the resistance determines the
output current (4.3V/R).

Vi (1)

Vi (2)

R

V (o)

I

A diode in the circuit will not switch the output because the applied field from the supply
voltage will over
-
ride the applied field from the input voltage. The diode is forward biased
with respect the supply voltage , so current will always flow through the circuit.





N










P






Input Voltage

+

-

Supply Voltage, V

10V

Resistance, R

Voltage

Out =
Always High

P+

Si

Equilibrium

Field

Net diffusion

+

-

Supply Voltage, V

10V

Adding a second diode establishes a three
-
terminal device in which the voltage across the first
diode is set by the input and unaffected by the supply. The second diode is reverse biased with
respect to the supply voltage, which shields the first diode. However, no current will flow
through the outer circuit because the second diode is reverse biased.

+

-

Input Voltage

Resistance, R

Voltage

Out =
Always Low

Current through first diode
controlled by input voltage

Second diode is reverse biased with
respect to supply, so no current

An NPN junction produces the effect of two opposing diodes in the circuit.





N









N










P






+

-

Supply Voltage, V

10V

No current flows in the outer circuit because all electrons entering the P
-
type region exit
through P
-
region lead, and because no current can flow across the reverse biased PN junction
Electrons diffuse from N to P
-
type material, recombine with holes, and exit through the
conductor in the P
-
type material.

+

-





N









N










P






-

Input Voltage

Resistance, R

Voltage

Out =
Always Low

Base current

Applied field

from input voltage

Equilibrium

Field

Force

Net diffusion

Equilibrium

Field

Applied field from
supply voltage

Reverse Bias

Forward Bias

The electron concentration across the P region varies from high at the forward biased junction
through which electrons are passing to low at the reverse biased junction, at which no current
is flowing. Transistors are designed to allow diffusion of electrons across this concentration
gradient, from the emitter across the base and into the collector, thus completing the outer
circuit. This is done by making the base thin. The thin base leads to a sharper concentration
gradient, and reduces the likelihood of recombination.





N









N










P






Make base thin

Force

Net diffusion

+

-

Supply Voltage, V

10V

In a bi
-
polar junction NPN transistor, electrons are injected into and diffuse through the base
to the collector, completing the outer circuit. Many of the electrons will
not
recombine with
holes in the base (P
-
region) for two reasons. First, the emitter is heavily doped and the base is
lightly doped. Second,
the P
-
region is thin
. Because there is a concentration gradient across
the region, electrons diffuse towards the reverse-biased base-collector junction. The electrons
will be captured by the strong electric field at this junction, and will flow into the collector.

+

-





N

Collector









N

Emitter










P

Base






-

Input Voltage

Resistance, R

Voltage

Out =
Contollable

Ib

Equilibrium

Field

Force

Net diffusion

The transistor can be de
-
coupled into two parts, first one being the base
-
emitter, which
functions like a diode. Diode performance can be determined (along with the base current) by
the load line intersection with the diode performance curve.

Rb

B

E

I (base)

Vbase

Vbe

Load line

Ib

Energy balance:

Ib=
-
(1/Rb)*Vbe+Vbase/Rb

Intersection gives
Ib and Vbe

+

-

Device

Transfer Function

The current exiting the collector, Ic, is determined by the voltage across base
-
emitter junction
(the input) only. This is because electron injection to the base from the emitter is the
limiting
factor
on the current through the circuit, and base voltage control the degree of electron
injection. As a result, Ic is independent of the reverse
-
bias voltage polarity across the BC
junction: if, for example, Rc decreases, the voltage drop across the transistor, Vt, will increase
(energy balance). The voltage drop across the BE junction is fixed at 0.7V, for a forward biased
diode. Thus, the reverse bias voltage across the BC junction will have to increase. However,
this has no affect on the output current. The collector current is related to the base current by
a factor B, or gain, which is between 50


200 for transistors. The above design is a common
emitter (emitter is the common connection) bi
-
polar junction transistor.

Rb

+

-

B

E

C

-

+

Rc

I (base)

Vt

Energy balance:

Vsource = Vt +IcRc

Vbase

Vsource

Vbe

Ib

Vout

Two elements are necessary for the shared emitter, bi
-
polar junction NPN transistor. First,
there must be reverse bias voltage polarity across the base
-
emitter junction to capture
diffusing electrons in the base. This is represented on the x
-
axis. The output is weakly
dependent upon the degree of reverse bias, reflected by weak slope of each line. But the
output will drop off rapidly as if the voltage across the transistor is too low to maintain reverse
bias across BE junction. Second, there must be forward
-
biased input voltage. As the input
voltage increases (represented by each curve), the output current increases. The lower line
represents zero input voltage, and output is zero, as expected.

Vt

Various inputs (Vbe)

Energy Balance:


(1/Rc)Vt+Vsource/Rc=Ic

Vt must be > Vbe, else
there is no reverse bias

across the base
-
collector

B

E

C

In (Vbe)

Vt

Out (Ic)

Output

Off

On