Incomplete Notes Intro & Materials1 Solid State Electronics Introduction Ronan Farrell Recommended Book: Streetman, Chapter 3 Solid State Electronic Devices

Intro & Materials 1

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Solid State Electronics

IntroductionRonan Farrell

Recommended Book:Streetman, Chapter 3Solid State Electronic Devices

Intro & Materials 2

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Why S.S.E.

• To learn why semiconductor devices work, the physics of the devices

• To gain an understanding of how we develop equations to explain their behaviour.

• Analog electronics is the base of all electronics.

• Analog electronics design is highly dependent on the models that you use

• Understanding function and modeling is essential, especially knowing the strengths and weaknesses of the models.

Intro & Materials 3

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Course Outline• Semiconductor Materials• PN Junctions• Field Effect Transistors (FET)• Bipolar Junction Transistors (BJT)• Manufacturing Technology

It is important to review the relevant sections of last year’s Physics course.

Intro & Materials 4

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Doping

Principle of doping is that intrinsic semi-conductors, group 4 elements, are mixed with some group 3 and 5 elements.

Group 3 elements, for example Boron, have one less outer electron than Group 4, and thus appear to form a “hole” in the lattice structure

Group 5 elements, for example Arsenic, have one more outer electron than Group 4, and thus appear to provide a “free” electronin the lattice structure

Intro & Materials 5

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Movement of Carriers

It is the movement of carriers, such as holes and electrons that allow current to flow.

In the semiconductor, only electrons actually move. Electrons are always moving due to thermal excitation. The more heat in a substance, the more energetic the electrons.

This movement is random if there is no external effects.

Intro & Materials 6

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Movement of Carriers

Holes do appear to move however. In substances with gaps in the electron bonds between atoms, electrons jump from one lattice position to another.

However as there is only a fixed number of electrons, when they move, they fill one hole but leave another. Taken from one perspective, it appears as if the hole has moved.

Intro & Materials 7

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Number of Carriers

The number of carriers in a system, holes or electrons is the major factor in determining the conductivity of the material as well as determining the behaviour of semiconductors under different circumstances.

Most of the early part of this course will be based on determining the number of carriers and hence the movement of these carriers under different conditions.

Intro & Materials 8

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Course Notation

ND Number of donor atoms in N-type

semiconductor, these provide free

electrons.

NA Number of acceptor atoms in P-type

semiconductor, these provide free

holes.

pn Number of electron carriers in P-type

material (minority carriers).

pp Number of hole carriers in P-type

material (majority carriers).

nn Number of electron carriers in N-type

material (majority carriers).

np Number of hole carriers in N-type

material (minority carriers).

Intro & Materials 9

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Fermi-Dirac Distribution

The fermi-dirac distribution function determines the energy that individual electrons have within a solid. The distribution function gives the distribution of electrons over a range of allowed energy levels at thermal equilibrium.

kTEE

EfFexp1

1)(

Intro & Materials 10

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Fermi-Dirac Distribution

Above zero degrees Kelvin, there is a probability that free electrons and holes will exist due to thermal generation.

Note: At zero degrees Kelvin, the distribution is square, and the value of the cutoff defines the Fermi-Energy Level.

Electrons

Holes


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Fermi Energy Level

The Fermi Energy Level is a mathematical value that allows us, using the Fermi-Dirac distribution, to determine the number of free carriers in a material.

The Fermi-energy level is the energy level at which the probability of an electron being in that energy level is ½ at equilibrium.

An another important criteria for EF is that at absolute zero, all the energy levels below EF are full, and all energy levels above it are empty.


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EF and the Conduction/Valence Bands

In an intrinsic semiconductor, there are equal numbers of hole carriers in the Valence Band as there are free electrons in the Conduction Band.

This means that the equi-probable energy level is half-way between the two bands.

Conduction Band (free electrons)

Valence Band (free holes)


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Fermi Energy Level and Doping

The Fermi Energy level is fixed for any piece of material but can move depending on the material and doping.


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Intrinsic Semiconductor

• In the intrinsic case, EF is in the middle of the two bands and the Fermi-dirac function extends in and out of the conduction and valence bands.

• The number of free electrons in the conduction band is matched with the number missing from the valence band, thus holes match electrons.

• With increased temperature, the number of holes and free electrons increases.


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N-type Semiconductor

• In the N-type semiconductor, the fermi energy level EF has been shifted up in value, closer to the conduction band.

• The shift up has occurred because there is an increased number of free electrons in the conduction band, these free electrons have arisen from the doping.

• As the fermi-dirac distribution retains the same shape irrespective of EF, this means that there are more free electrons in the conduction band that there are holes in the valence band.

• This is why we say that the substance is N-type with electron majority carriers.


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P-type Semiconductor

• Similarly in the P-type semiconductor, the fermi energy level EF has been shifted down in value, closer to the valence band.

• The shift down has occurred because there is an increased number of holes in the valence band, these holes have arisen from the doping.

• As the fermi-dirac distribution retains the same shape irrespective of EF, this means that there are more holes in the conduction band that free electrons in the valence band.

• This is why we say that the substance is P-type with holes the majority carrier.


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Fermi Energy Level

Notes on the Fermi-Energy Level

• The Fermi energy level is a mathematical value that corresponds to a real world value.

• Consider the statement, “EF is the energy level at which, at absolute zero, all the energy levels below it are empty.” then with increased electrons from N-type doping, the Fermi level will rise as there are more levels required to take the extra electrons. With P-type doping there are less electrons, therefore less energy levels are required, and hence EF will drop.

• It is worth re-stating that in a material, the fermi-energy level is the same throughout that material. This is essential for understanding semiconductor junctions.


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Calculating the Number of Carriers

Last year, the Fermi-Dirac Distribution was used to calculate the number of available carriers in a substance. This is most helpful in that it is possible to develop a relationship between the holes and electron carriers in an intrinsic substance for any temperature:

This was covered last year, but it is important to understand how the equation is arrived at, so that you can understand why it is so and very importantly when it is not applicable.

2ioo npn


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The number of free electrons in an intrinsic semiconductor is given by the integral over the conduction band of all the available energy levels by the Fermi-Dirac Distribution function for electron energy levels:

dEENEfncE

o )()(

The conduction band energy levels range from the bottom of the conduction band, Ec,

to infinity.

You can expect that most of the free electrons will be at the lower energy levels as no electron will have infinite or near infinite energy. This is represented by the Fermi-Dirac Distribution tending to zero at infinity.


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Let us assume that all the free states are at the bottom of the conduction band and number Nc, at energy Ec

Graphically, the function f(E)N(E) is zero at all points other than when E=Ec, thus the integral is just the product of these two terms at this energy value.


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So assuming that the Fermi-Energy level is several kT below the conduction band (Ec), about 100mV, which it generally is, the Fermi-Dirac distribution can be simplified:

Thus no is given by


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A similar argument could be made for the holes, except using the valence band energy level Ev

If we define Ei to be the Fermi level in an intrinsic material, we get


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The more normal way that this is expressed is that

But in an intrinsic material,

And so we can now proceed to develop the important relationship

2ioo npn


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Multiplying no and po together you get

However from the earlier equations, it can be easily shown that

therefore

2iiioo npnpn


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This is a very important equation, and one that we will use a lot. It relates the number of free holes and electron carriers to the intrinsic values in the equilibrium state, irrespective of how the holes or electrons where created.

Note this is a temperature dependent equation as ni

2 changes with temperature.

Note: It is not applicable to cases when carriers are being injected, such as active circuitry.

2ioo npn


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Example: nopo=ni2

A Si sample is doped with 1017 As atoms/cm3. What is the equilibrium hole concentration p0 at 300K and where is the fermi energy level with respect to the intrinsic level.


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Example: nopo=ni2

A Si sample is doped with 1017 As atoms/cm3. What is the equilibrium hole concentration p0 at 300K and where is the fermi energy level with respect to the intrinsic level.


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Drift Current

Drift current is due to the movement of majority carriers through the material under the influence of an electric field.

It is dependent on such things as mean free time, the time between collisions between the carrier and the lattice, and the effective mass of the carrier.

However these are all wrapped up into a coefficient called mobility, .

We will be deriving a relationship between carrier numbers and resistivity and conductivity.


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Drift Current

In general, current density is proportional to the electric field applied.

where n is the conductivity of the material due to electron carriers and is given by

And n is the mobility of these carriers

which is dependent on the material as


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Drift Current

In each material there is a similar contribution from the holes, giving

where n is the conductivity of the material due to hole carriers and is given by

Now is always positive, but the hole and electron charges are opposite, but that’s fine, in an electric field they’ll move in opposite directions, so their current contributions combine, giving

Current flows from high voltage to low voltage, as expected.


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Drift Current

The mobility of the carriers is a majority factor in determining the conductivity, the current carrying potential, of the material. This is one of the reasons why GaAs is a faster material for devices than Silicon

Also note that increasing the number of carriers would also help, which is true as doping reduces resitivity.

Note: Holes have lower mobilities than electrons. This will be later shown to be critical in transistors, as N-channels operate faster than P-channel devices.

nptot npq

tottot

1

conductivity

resistivity


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Diffusion Current

Free electrons and holes act like a gas in terms of diffusion. They will tend to move from regions of high concentration to those low concentration until all is even.

Diffusion operates by the principle of random thermal motion, except that for any given space between two regions of different concentration, there’ll be more carriers arriving from the high concentration area that the lower region.

The flow of carriers produces a diffusion current. This is very important where we inject carriers into a piece of semiconductor, either at a junction, by light excitation, or by an external circuit.


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Diffusion Current

Consider two slices of the material of width l. Assume that there is a concentration gradient of electron carriers.

Consider now the junction between the two slices. The movement of the electron is random so it’s equally likely to move left or right.


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Diffusion Current

If the slice is narrow enough, smaller than the mean free path (a term will discuss more later), then the probability of an electron crossing the junction plane is 50%.

This applies to both sides of the junction, so the net movement of electrons, the electron flux density n, can be determined.

NOTE: lt is the electron mean free path, tt is the electron mean free time,


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Diffusion Current

Crudely, the current in electrons is given by


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Diffusion Current

Converting the original equation to differential form, consider the concentration as a factor of distance from one side with a small change in distance x


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Diffusion Current

Normally, the constant coefficient is renamed the diffusion coefficient, and then this equation matches the standard gas diffusion equation in form.

The equation states that movement is positive for negative concentration slope, or simple, carriers flow from high concentration to low concentration


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Diffusion Current

To convert it to the electrical equivalent current, the flux density needs to be multiplied by the charge of the carrier, thus

similarly

Both carriers move from regions of high concentration to those of low, but the difference in sign (due to the charge) will mean that electrical current is in the opposite direction,


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Einstein Equation

There are two current mechanisms in a semi-conductor, diffusion and drift.

However these currents can be in opposite directions to each other.

In a piece of semi-conductor which is in equilibrium, there is no current of either holes or electrons. Therefore both currents, diffusion and drift, if any exists, must be equal to and opposite to each other.


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Einstein Equation

This can be visualised as follows:

In equilibrium, the charges are equally distributed, there is no concentration gradient, there is no electric field, or voltage difference.

Assume a charge carrier moves by thermal motion, the driver of diffusion. This means that there is no longer a balance of charge.

More charge at one spot compared to another means that there will be an electric field set up.

This forces currents to drift in the opposite direction, maintaining the equilibrium.


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Einstein Equation

So setting the current to zero, we get

Now from before, we have an equation for the number of free electrons in a semi-conductor


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Einstein Equation

But by partial derivatives

Therefore

So adding it all together we get


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Einstein Equation

Now the electric field is defined as the rate of change of voltage

Now in equilibrium, the Fermi Energy level is constant in a material, indicating that the carrier concentration is equal and occupying lowest available energy levels, thus if this is so then,

Now we can use the halfway point between the energy bands as a reference, Ei. This is because the shape of the bands remains the same, shifted up and down with voltage.


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Einstein Equation

Therefore

Finally

q

kTD

n

n

Similarly

q

kTD

p

p

The ratio of Mobility and the Diffusion Constant for a material and carrier type is a constant. Very important for determining the diffusion constant and mobilities from experiment.


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Carrier Recombination

One of the most important characteristics of semiconductor behaviour is the manner in which electrons and holes recombine.

In an isolated block of semiconductor, a doped substance has only one form of carrier so recombination is not a major issue.

In cases where the carriers are generated optically or by thermal generation, and thus creating both holes and electrons, where there is the possibility of holes and electrons meeting, then recombination becomes very important.


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The probability of an electron and hole recombining is a matter of probability.

In recombination, an electron leaves the conduction band and joins a hole in the valence band. The loss in energy is emitted as a photon. The frequency of the emitted light is dependent on the energy gap of the material, different materials have different energy gaps and thus emit different “colours”, for example blue, green, red LED’s.

The probability of a recombination occurring is proportional to the number of electrons and the number of holes available to recombine, with some constant of proportionality. (If no holes or no electrons then no recombination


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The probability of a recombination occurring is proportional to the number of electrons and the number of holes available to recombine, with some constant of proportionality.

The net change in electrons in the conduction band is the thermal generation rate less the recombination rate

The argument that we are going to present here is valid for all cases of direct recombination but we will take the case of optically generated carrier pairs in an isolated piece of semiconductor.

)()()( 2 tptnn

dt

tdnrir


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The maths will be confusing, but the aim is to try to understand the physics of what is happening and understand the approach being taken to come to the final result.

Assume a flash of light or heat which generated a quantity of electron-hole pairs, at time t=0, giving instantaneous hole and pair concentrations of n(t) and p(t)

n and p are equal as they are recombining with each other.


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Now if the number of excess carriers is small compared to the number of carriers, then the rate of decay will be low and the square of a small number can be ignored.

Now we know for any semiconductor

Giving


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Now if we consider doped materials then the number of holes will be much greater than electrons, or visa-versa. So assuming p-type, then we can ignore the no contribution.

Integrating this WRT time


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Where p is the number of excess carriers injected at the start.

This is normally rewritten as

Where p is the recombination lifetime of the hole carrier in p-type material. A more generic expression is


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Where p is the recombination lifetime of the hole carrier in p-type material. A more generic expression is

0

1

prp

This is valid for all materials, and equivalently for holes and carriers.

The most relevant use of the carrier lifetime is the lifetime of a minority carrier in a doped semiconductor, so it will be the lifetime of an electron in a P-doped material, or a hole in an N-doped material.


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Note, in an isolated piece of silicon where there is no optical generation of carriers, the thermal generation rate matches the recombination rate in equilibrium.


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Notes:• The assumption was made that the

number of excess carriers was small. When this is not true, the simplifications made cannot be used and the longer form of the expressions must be maintained

• However the assumption is valid for most applications and gives insight into the behaviour of the system.

• This expression is only valid when there is no excess carriers. Solely for use in equilibrium stable conditions.

• Any lattice defects will increase the probability of recombination. This is a manufacturing issue which we won’t examine here.

2ioo npn


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Before we noticed how the number of carriers in a material affected its conductivity. The ability to generate excess carriers provides the facility to modify a materials conductivity.

This is the manner in which photo-diodes work. The semiconductor is specially doped to facilitate excess carrier generation by light. As the material is exposed to light, the number of carriers increased significantly, lowering the resistance.


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Continuity Equation

In the diffusion current equations we developed earlier, we ignored the fact of recombination. This needs to be considered. From this analysis we’ll get a very important value, the diffusion length, the distance an excess carrier will diffuse prior to recombining.

You won’t be expected to learn the derivation of these equations off by heart, but it’s good to see how the maths lets us arrive at useful physical values.

You will be expected to know how to use the results.


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Continuity Equation

Consider a cross-sectional area of a semiconductor;

p

pp

xxx

p

x

xxJxJ

qt

p

)()(1

Rate of buildup = increase in hole concentration

less recombination rate


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Continuity Equation

p

pp

xxx

p

x

xxJxJ

qt

p

)()(1

Taking this to the limits, we get the corresponding partial derivative equation

Where there is solely diffusion current, negligible drift, Jp can be replaced by the diffusion current equation we derived earlier, thus


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Continuity Equation

If a steady state distribution of excess carriers is maintained, and in many cases this can be a realistic expectation. In this case the time derivatives become zero


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Diffusion Lengths

The diffusion length is important in that it is the “average distance a carrier will diffuse before recombination”. This will be very important in later discussions.

It is however an exponential decay so that some carriers may travel further than this distance.

Documents

Incomplete Notes Intro & Materials1 Solid State Electronics Introduction Ronan Farrell Recommended Book: Streetman, Chapter 3 Solid State Electronic Devices