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SEMICONDUCTOR
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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
Electron and Hole Distribution at Equilibrium
• 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
Electron and Hole Distribution at Equilibrium
• 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 =
Electron and Hole Distribution at Equilibrium
• 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:
Electron and Hole Distribution at Equilibrium
• 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:
Drift of Carriers in Electric and Magnetic Fields
• 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
Drift of Carriers in Electric and Magnetic Fields
• 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 =
Drift of Carriers in Electric and Magnetic Fields
• 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
Effects of Temperature and Doping on Mobility
• As concentration of impurities increases, the effect of impurity scattering are felt at higher temperatures:
•Example:
Effects of Temperature and Doping on Mobility
• 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
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