prof. m arta rencz, g abor tak acstakacs/electronics/02.pdf · semiconducting materials the basics...

Semiconducting materials The basics of solid state physics PN-junctions and diodes Calculations with diodes

Electronics – The basics of semiconductor physics

Prof. Marta Rencz, Gabor Takacs

BME DED

17/09/2015

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The basic properties of semiconductors

Range of conductivity

[Source: http://www.britannica.com]

Semiconductors’ conductance is between that of conductorsand insulators

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The basic properties of semiconductors

They conduct current and have a negative thermalcoefficient (NTC), which means that their conductivityincreases when temperature rises.

This is exactly the opposite behaviour of metals.

At the moment semiconductors are the basic materials ofelectronic devices.

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The most important semiconductors

The most important semiconductors:Monocristalline or single-crystal materials:

Semiconductor elements: Si (silicon), Ge (germanium)They are used in integrated circuits and semiconducting devices.Compound semiconductors: GaAs (gallium arsenide), GaAsP(gallium arsenide phosphide)They are used to create LEDs.

Amorphous semiconductors: amorphous Si mainlyTFTs, solar cells are made of them.Organic semiconductors: OLEDs (Organic LEDs)

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The band structure I.

The electron’s energy is a quantized quantity – there arecertain energy levels that are allowed for electrons, the rest ofthe levels are forbidden.

When electrons take part in a system (atom or a crystallineconsisting of many atoms), every electron has to be at adifferent level. The electrons take energy levels very close tothe allowed levels – thus in large systems the electrons takeplace in energy bands that are separated by band gaps.

The energy bands of electrons in alarge insulator/semiconductorstructure.

The bands are shown in grey, theband gaps are white.

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The band structure II.

Conductance band:electrons that can movefreely.

Valence band: electronsthat take part in bonds andthus are bound to atoms.

From the viewpoint of conductance the important bands:

The highest band that contains electrons (valence band).

The band above the valence band, which is almost empty(conductance band).

The band gap between them.

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Insulators and conductors

Conductors: the valence band and the conductance bandsoverlap.

Insulators and semiconductors: there are bandgaps – thewidth of the bandgap (Wg) decides whether a material is aninsulator or a semiconductor.

Si (semiconductor): Wg = 1.12 eV

SiO2 (insulator): Wg = 4.3 eV

1 eV = 0.16 aJ = 0.16 · 10−18 J7 / 37


The charge carriers I.

Electrons: at the bottom of theconductance band,

Holes: at the top of the valence band – ahole is an absence of electron.

Both electrons and holes take part inconduction!

Generation: happens when an electron gets to theconductance band from the valence band.This means that two charge carriers are created: an electron in the conductance

band and a hole in the valence band.

Recombination: the opposite of generation – when anelectron falls back to the valence band.

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The charge carriers II.

Charge and mass of charge carriers

Electrons: have a negative charge and a positive mass.

Holes: have a positive charge and a positive mass (!).This can be explained in solid state physics – we’re not going into such depth.

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The crystal structure of silicon

3D crystal structure (diamond lattice) Simplified, 2D crystal structure

Silicon has four electrons that take part in the bond betweenits atoms.

Density: % = 2.33 gcm3

Lattice constant: a = 0.543 nm

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The intrinsic silicon

If the temperature is above 0 K, someelectrons become thermally activatedand get into the conductance band.

Intrinsic charge carrier concentration

ni = pi = 1010/cm3

ni: electron concentration (1/cm3)pi: hole concentration (1/cm3)

The charge carrier density is very low: a cube with edgesof 10µm contains 10 electrons.

The crystalline is doped in order to increase the chargecarrier density.

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Doping

A small number of atoms of a different kind is injectedinto the crystal structure.

This is done in a way that the dopants are placed on positionswhere normally Si atoms are located.

Typical doping density: 1015 − 1019/ cm3 – this is indeeddoping and not alloying (the density is very low).The atom density of silicon is 5 · 1022/cm3, so a typical doping of 1017/cm3

means that two atom is changed to a dopant out of every one million, which

leaves us with a purity of 99.9998 %.

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The n-type semiconductors

Donor dopants: dopants that inject atoms that have one extraelectrons at their valence band (P (phosporus), As (arsenic), Sb(antimony)).The extra electron is easier to raise into the conductance band, because it cannot take

part in a strong bond. Thus its energy level is in the band gap, close to the

conductance band.

Electrons are the majoritycharge carriersHoles are the minority chargecarriers

donor concentration: Nd

electron concentration: nnhole concentration: pn

Concentrations in n-type Si

nn ' Nn

nn > pn

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The p-type semiconductors

Acceptor dopants: dopants that inject atoms that have one lesselectrons at their valence band (B (boron), Al (aluminium), In(indium)).Less electrons result in extra holes, that are easier to bring down to the valence band,

because they cannot take part in a strong bond. Thus their energy level is in the

band gap, close to the valence band.

Electrons are the minoritycharge carriersHoles are the majority chargecarriers

acceptor concentration: Na

electron concentration: nphole concentration: pp

Concentrations in p-type Si

pp ' Np

pp > np

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Drift current I.

When a semiconductor is placed into an electric field, theelectrons start to drift in the opposite direction of the field.

No external field External field is present

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Drift current II.

Drift current is the movement of charge carriers due to anexternal electric field.

Drift velocity is the speed of the charge carriers in the driftcurrent:

Drift velocity

vd = −µn · Evd = µp · E

where

vd: is the drift velocity

µn: is the mobility of electrons (Si: µn = 1500 cm2

V s )

µp: is the mobility of holes (Si: µp = 475 cm2

V s )

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Diffusion current

Diffusion current: is the movement of charge carriers due to aninhomogeneity in their density.The movement is due to thermally induced movement of the electrons that is always

present at temperatures above 0 K.

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The pn-junction: a semiconductor diode I.

A pn-junction is a monocrystalline transitional area where ap-type and an n-type semiconductor is next to each other.

The diode is a device that consists of one single pn-junction.

The figure is distorted: the n-type layer is muchshallower in reality.

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The pn-junction: a semiconductor diode II.

We will be concerned with the area at the center of thestructure (physical distortions at the borders result in specialeffects that we’re not dealing with).

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Most important properties of the diode

When a forward voltage is applied to it, its current is anexponential function of the voltage.

I ' exp(

VVT

)Forward direction: the p side is at a higher potential.In the reverse direction its current is very low and isindependent of the voltage:I ∼ 10−12A/mm2

The current-voltage characteristic of the diode:

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The electrostatic conditions in the pn-junction

The majority carriers at theproximity of the junction diffuseacross the junction to the otherside.This is because there are a lot of electrons on

the n side, and a lot of holes on the p, while

each side has a very low density of the

minority charge carriers. There is a huge

gradient in the densisty of charge carriers.

This results in a depleted area / space charge region – anarea at the junction which is empty of majority charge carriers.

The dopants left by their extra electrons/holes becomecharged ions that create an electric field, which preventsfurther diffusion by generating a drift current of minoritycarriers in the opposite direction.

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The operation of the diode

Equilibrium: the diffusion of the majority carriers is inequilibrium with the drift current of the minority carriers(I = 0).

Forward direction: the forward voltage lowers the electricfield of the dopant ions thus increasing the drift current of themajority carriers (big IF ).

Reverse direction: the reverse voltage enlarges the electricfield of the dopant ions thus lowering the diffusion current ofthe majority carriers and increasing that of the minoritycarriers moved by the drift current (small IR).

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The characteristic equation of the ideal diode

The characteristic equation of the ideal diode

I = I0 ·(

eVVT − 1

)where

I0 is the reverse current (saturation current) of the diode(I0 ' 10−14..15 A)VT is the thermal voltage:

VT =kT

q' 26mV |T=293K

This is clearly a non-linear device – its characteristicequation is exponential.

In the forward direction the current is an exponentialfunction of the voltage.The current is multiplied by ten at every increase of the voltage by 60 mV.

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The characteristic equation of a real diode

Due to secondary effects the equation in the forwarddirection:

I = I0 ·(

eV

m·VT − 1

)where m is the ideality factor (a.k.a. quality factor oremission coefficient) – it represents several secondary effectsand ranges from 1 to 2.

In the reverse direction: thereverse current of the diode startsto increase steeply with the voltageat the breakdown voltage (VBR).

If the diode’s current is limited byexternal means, the breakdownstate does not harm the structure.

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The application of the breakdown voltage

As a very small change in the reverse voltage results in a bigchange in the reverse current at the breakdown state, it canbe used to stabilize voltage.

The diode is placed in a negative feedback configuration.

Zener diode: special diode created to serve as a voltagestabiliser in the breakdown state.

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The operating point of a diode I.

The characteristic equation of a diode gives all thevoltage-current pairs that a diode can have.

In operation the diode usually works at a certain operatingpoint, i.e. at one of the voltage-current pairs of its equation.

This point is determined by the elements surrounding thedevice.

DC analysis: the calculations performed to find the DCoperating point of a non-linear device.

The quantities describing the DC operating point are usuallydenoted with capital letters (V, I).

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The operating point of a diode II.

The KVL for the circuit is:

Vdd = I ·RL + VD

which gives the equation of a line:

I =Vdd − VDRL

The red line is called the load line – it is the characteristicequation of the other element in the circuit (RL) as afunction of the diode’s voltage. 27 / 37


The operating point of a diode III.

The operating point is at the intersecion of the twofunctions.

If the graphical representation of the equations is given, this iseasy to find.

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The approximation of the operating point I.

voltage

current

Vd

We take advantage of the fact that the exponential function isvery steep.

The diode is substituted:

with a voltage source when it is switched on,with an open circuit when it is switched off.

The value of the voltage source (VD) can be looked up inthe datasheet of the diode (VD ' 0.7 V).

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The approximation of the operating point II.

We assume that the diode is switchedon.

The terminals of the resistor:

left-hand side: supply voltage (Vs),right-hand side: the voltage of the diode(VD).

According to Ohm’s law:

I =Vs − VDRl

If Vs = 5 V, VD = 0.7 V, Rl = 1 kΩ then

I =5− 0.7

103= 4.3 mA.

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Small-signal analysis I.

It is important to investigate what happens when there aresmall changes in the input voltage – e.g. when the supplyvoltage changes slightly during operation.

For small changes the exponential function can beapproximated with a linear equation around the operatingpoint.

In terms of the electric model, this means that the diode issubstituted with its differential resistance.

The differential resistance

rd =∂V

∂I=m · VTI

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Small-signal analysis II.

The differential resistance

rd =∂V

∂I=m · VTI

I in the equation of the differential resistance is theoperating point current.

Thus the value of the differential resistance has a very strongdependence on the operating point.

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The small-signal operation of diodes I.

Let’s investigate what happens when small changes occur atthe equilibrium state.

Changes around the operating are usually denoted with lowercase letters.

Vs = Vs0 + vs · sin (ωt)

If the changes are small, the diode’s voltage and current aresinusoidal functions around the operating point.

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The small-signal operation of diodes II.

The calculation is performed in three steps:1 the DC operating point is determined,2 the AC analysis is performed by substituting the non-linear

device with its small-signal model and calculating the effects ofthe changes on this model,

3 the two results are added.

DC analysis – equilibrium AC analysis – small changes

It is important that only small changes can be calculated thisway!

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The small-signal operation of diodes III.

The calculation of the small signal operation:The small-signal changes:

i =vs

Rl + rd

andv = rd · i =

rdRl + rd

vt

If Rl = 1 kΩ, Vt = 5 V and vt = 1 V:The differential resistance:

rd =VTI

=26 mV

4.3 mA= 6 Ω

The change (amplitude) of the diode’s current:

i =1

1.006 k' 1mA

the change (amplitude) of the diode’s voltage:v = 6 Ω · 1mA = 6 mV 35 / 37


The Zener diodes I.

Supply voltages can be stabilized usingZener diodes.

Consider the circuit on the left.

Let’s find the voltage and current of the Zener diode.Vin = 12 V, R = 150 Ω and VBR = 3.3 V.

As the input voltage is larger than the breakdown voltage:the diode is in the breakdown state.

I ' Vin − VBR

R=

12− 3.3

0.15= 60 mA

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The Zener diodes II.

How much does the output voltage change if the inputchanges by 1 V?

The differential resistance is: 3 Ω.

vout = vin ·rd

rd +Rt=

3

153= 20 mV

Thus the change at the input is reduced to 1/50 of its value!

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prof. m arta rencz, g abor tak acstakacs/electronics/02.pdf · semiconducting materials the basics...

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