chapter_3

Chapter 3:Energy Bands and Charge Carriers in Semiconductors

• A perfect semiconductor material with no impurities or defects is called intrinsic

• No charge carriers at 0K

• At higher temperatures, EHP’s are generated and are the only charge carriers in the material

• Energy to break a bond and create EHP:

• Carrier concentrations:

• ni :

Intrinsic Material

• If a steady-state carrier concentration is maintained, there must be recombination

• Recombination:

• At equilibrium: ri = gi

• Recombination and generation are T dependent and are proportional to the equilibrium concentration of holes and electrons

Intrinsic Material

• It is possible to generate carriers by introducing impurities into the crystal, also known as doping

• If a crystal is doped, it can be altered so it has a majority of either electrons or holes

• There are two types of doped semiconductors:

• n-type: mostly electrons

• p-type: mostly holes

• When a crystal is doped such that n0 and p0 are different from ni , the material is said to be extrinsic

Extrinsic Material

• When impurities or defects are introduced into the crystal, additional levels are created in the energy band structure within the bandgap

• Impurities from column V are donor impurities because they donate electrons to conduction band

• Semiconductors doped with donors are called:

• Impurities from column III are acceptor impurities because they accept electrons from the valence band

• Semiconductors doped with acceptors are called:

Extrinsic Material

• To calculate semiconductor electrical properties it is necessary to know the number of charge carriers / cm3

also known as the carrier concentration

• Need to know the distribution of carriers over available energy states

• Electrons in solids obey the Fermi-Dirac statistics:

• f(E) is called the Fermi-Dirac distribution function and gives the probability that an available energy state at E will be occupied by an electron at temperature T

• k is Boltzmann’s constant = 8.62 ×10-5 eV/K = 1.38×10-23

J/K

• EF is called the Fermi level

Carrier Concentration

• For E = EF , the occupation probability =

• At 0K, every available energy state is filled with an electron up to EF

• At 0K, all states above EF are empty

• As T increases, f(E) above EF increases

• Probability that a state ΔE above EF is filled =

• Fermi level is reference point in calculations of electron and hole concentrations

Fermi Dirac distribution

• If there is no available state at E, there is no possibility of finding an electron there

• For intrinsic material, EF must apply lie in the middle of the bandgap

• N-type material, EF close to EC

• p-type material, EF close to Ev

Fermi Dirac distribution

• Fermi distribution function can be used to calculate carrier concentration if the densities of available states in valence and conduction band are known.

• In conduction band:

• N(E) is the density of states (cm-3)

• No. of electrons / unit volume in dE is product of density of states and probability of occupancy

• N(E) in conduction band increases with electron energy

• But f(E) becomes extremely small for large energies

• Very few electrons occupy energy states above the conduction band edge

• Same fore holes in the valence band

Electron and Hole Distribution at Equilibrium

• Represent all electron states in conduction band by effective densities of states Nc :

• Assuming EF lies several kT below Ec :

• n0 :

• Nc :

• mn* : density of states effective mass

for electrons


• For Si, mn* = 0.067m0

• Concentration of holes in valence band (p0):

• Probability of finding an empty state at Ev :

• Assuming EF larger than Ev by several kT:

• po =

• Nv =

• Equations for n0 and po are valid for both intrinsic and extrinsic materials


• For intrinsic material, EF lies near the middle of the bandgap at a level Ei

• ni = pi =

• Product of n0 and p0 is a constant for particular material and temperature:

• nopo =

• nipi =

• ni = Si at room T, ni = 1.5 ×1010

• nopo =

• Another way of writing no and po :

• no = po =


• Example: A Si sample is doped with 1017 As atoms/cm3 . What is the equilibrium hole concentration p0 at 300K? Where is EF relative to Ei ?

• Solution:


• From no = nie(EF – Ei)/kT we see that in addition to no , ni and EF also depend on temperature

• ni dependence on temperature:

• The value of ni for a given T is a given number for a given material

Temperature Dependence of Carrier Concentrations

• Example: n-type doped Si with Nd = 1015 cm-3

• At low T, all donor electrons are bound to donor atoms

• As T increases,electrons donate to conduction band

• At 1000/T = 10, all atoms are ionized, no =

• no is constant until ni is comparable to no

• Usually want to operate within extrinsic region

Temperature Dependence of Carrier Concentrations

• We assumed semiconductor contains either Nd or Na

• Semiconductor can contain both donors and acceptors

• Example: Nd > Na

• Since Nd > Na ,material is n-type

• Compensation n0 = Nd – Na

• If we add acceptors until Nd = Na then:

• Adding more acceptors Na > Nd p-type

Compensation and Space Charge Neutrality

• Space charge neutrality: if a material is to remain electrostatically neutral, the sum of the positive charges must equal the sum of the negative charges:

• Approximation: if material is doped n-type no >> po then :

Compensation and Space Charge Neutrality

• Determine the electron and hole equilibrium concentrations in silicon at T=300K for the following doping concentrations. (a) Nd = 1016 cm-3 and Na = 0. (b) Nd = 5 ×1015 cm-3 and Na = 2 ×1015 cm-3. Recall that ni = 1.5 ×1010 cm-3 in silicon at T = 300K.

Example

• Knowledge of carrier concentration is necessary for calculating current flow in the presence of electric and magnetic fields.

• We must take into account the collisions of the carriers with the lattice and with the impurities

• Mobility: ease with which electrons and holes can flow through the crystal

• These collisions depend on:

Drift of Carriers in Electric and Magnetic Fields

• At room T, thermal motion of individual electrons may be seen as random scattering from :

• For random scattering there is no net motion for the group of electrons n / cm3

• If an electric field Ɛx is applied in the x-direction. Each electron will experience a force:

• Net motion in –x direction

• The force on the n electrons is:


• Net rate of change of momentum is zero due to collisions:

• Mean free time (ṫ) :mean time between scattering events

• Average momentum per electron:

• Average velocity per electron:

• Current density (Jx): no. of electrons crossing a unit area per unit time, multiplied by the charge on the electron:

• Current density is proportional to the electric field


• Conductivity:

• Electron mobility µn =

• mn* = conductivity effective masses. Use for charge

transport problems

• µn can be also defined as :

• Units of mobility:

• Jx in terms of µn =

• Exact same idea for holes, just change n to p, -q to +q, µn

to µp ,

• If there are both electrons and holes, then Jx =


• There are two main parameters that determine mobility: m*

and mean free time ṫ

• Lighter particles are more mobile than heavier particles

• Resistance of the bar:

• Holes move in the direction of E-field

• Electrons move in opposite direction of E-field

• Drift current is constant throughout the bar

• Space charge neutrality is maintained

Drift and Resistance

• Two types of scattering that influence electron and hole mobility: lattice scattering and impurity scattering.

• Lattice scattering: carrier is scattered by a vibration of the lattice, (from temperature)

• Lattice scattering increases as T increases

• Impurity scattering is dominant at low T

• A slow moving carrier is likely to be scattered more strongly by interaction with a charged ion than a carrier with higher mobility

• Impurity scattering decreases mobility as T decreases

• Mobilities due to different scattering mechanisms add:

Effects of Temperature and Doping on Mobility


• As concentration of impurities increases, the effect of impurity scattering are felt at higher temperatures:

•Example:

• A Si bar 0.1cm long and 100 µm2 in cross-sectional area is doped with 1017 cm-3 phosphorus. Find the current at 300K with 10V applied.

Example I

• Consider a compensated n-type silicon at T=300K, with a conductivity of 16 (ohm-cm) and acceptor doping concentration of 1017 cm-3 . Determine the donor concentration and the electron mobility.

Example II

• We assumed current density is proportional to electric field through conductivity:

• At high electric fields, J actually depends on the electric field

• At high fields, the drift velocity is saturated

• Velocity saturates at a point where any additional energy goes to the lattice and not to the carrier velocity.(Scattering limited velocity)

• Leads to constant current at high electric fields

High Field Effects

• Until now we only mentioned homogenous materials with uniform doping

• Starting next week we will start talking about non-uniform doping in a semiconductor, or junctions occurring between semiconductors.

• Main concept to follow: no discontinuity or gradient can arise in the equilibrium Fermi level EF

• Example: two materials in contact

• Each material has different Fermi function

• There is no current no net charge transport

Invariance of Fermi level at Equilibrium

• At energy E, rate of transfer of electrons is proportional to no. of filled states at E in material 1 times the no. of empty states at E in material 2.

• Conclusion: Fermi level must be constant throughout materials in intimate contact

Invariance of Fermi level at Equilibrium

chapter_3

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