Lecture 1 - Review

Kishore Acharya

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Agenda

• Transport Equation (Conduction through Metal)

• Material Classification based upon Conductivity

• Properties of Semi Conductor

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Electrical Conduction through a Conductor/Metal

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Simple transport equation Drude’s Method for conduction in Metals Mq (dv/dt) = q E Mq is the mass of the charge q, E is the electric field v = (q E t)/Mq + c At the instant of collision t=0 and v =0 Velocity change in between collision v = (q E t)/Mq The average velocity between collision is called drift velocity (Vd) Vd = (q E )/Mq m/sec where is the mean time between collision Vd = E Where mobility = (q ) / Mq m2/volt sec

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Electric current I is the amount of charge crossing area A = WT in a secondTotal Charge Q = nqAVd contained within the Parallalopiped P will cross the marked area A in 1 sec. Where n is the number of particle with vharge q per unit volume. Therefore,I = nqAVd or J = I/A = nqVd = nq (q / Mq) E  Noting that J = E from the theory of Electro magnetic field theory it is seen that the conductivity = nq and the resistivity = 1/(nq

Conductivity and Resistivity

T

L

WVd

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Modern interpretation Mq must be replaced the effective mass Mq* and is calculated based upon Quantum Mechanics Mq* is based upon the curvature of the energy band and is therefore depends upon the crystal structure of the material. Since Mq* is smaller for Ge, GaAs than Si the drift velocity Vd is higher for Ge and GaAs than Si The collision of the conductive particle is with the lattice and therefore depends upon lattice vibration and the concentration of defects and impurities that distorts the shape of the lattice. Thus 1/T and 1/Nimp where T is temperature and Nimp is the concentration of the impurities

Effective Mass Concept

Mq* = ħ2/(2E/k2)

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Variation of carrier density with Temparature

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Variation of electron mobility with temperature and impurity concentration

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A Note on Device Performance Specification

• Since mobility decreases with increased temperature

• Since drift velocity decreases as the electric field decreases

• Device speed is stipulated at a temperature of 120 C and 4.5 Volts

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Conductivity based Material classification

Type 

Conductivity Range

Carrier Concentration

Band Gap (ev)

Conductivity Variation

Carrier Type

Conductor/ Metal

106 mho 1022 /cc 0 Decreases as temperature is increased

Electron

Semi Conductor

10-2 to 102 mho 1010 /cc 0.7 to 1.6 Increases with temperature.

Electron and Holes

Insulator 10-6 mho 104 /cc 2.0 NA NA 

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Properties of Semi Conductor 

Group II Group III Group IV Group V Group VI

  B* C N O

  Al* Si P* S

Zn Ga*Ga* Ge As*As* Se

Cd In Sn Sb* Te

 

Elemental Semi Conductor: Si, Ge (elements from Group IV) Alloy Semi Conductor: InP, GaAs, ZnS, CdS, CdTe* Elements used for Doping intrinsic semi conductor

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Energy Bands and Interatomic Spacing

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Derivation of Effective Mass

• E = p2/2Mq* p = momentum, E= Energy E = (ħk)2/2M*q since p = ħk where ħ=h/(2)or E/k = ħ2k/M*qor 2E/k2 = ħ2/M*qor M*q = ħ2/ 2E/k2

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A note regarding Effective mass

Effective mass depends on curvature of the E-k curve a fundamental property of the material based upon atomicspacing and crystal structure. In 0.09 m Silicon technology the curvature is manipulatedby stressing the silicon. The stressed silicon has flatter curvature than standard silicon. The flatter curvature lowers the effective mass in stressed silicon resulting in higher speeddevice.

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Si150A

SiGe400A

Strained Si

Typically 70% Si and 30% Ge

Formation of Strained Si

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Mobility Enhancement

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Building Devices

• Insulator cannot be used for building devices– Not enough charge carriers

• Conductors/ Metals cannot be used for building devices– No internal field for carrier flow control

• Semiconductors can be used for building devices– Supports internal field for carrier flow control– Carrier concentration must be increased to 1014/cc

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Intrinsic Semiconductor

ni = n = p = 1010/cc

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Impurities used for Semiconductor Doping

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N type Semiconductor P type Semiconductor

Semiconductor Doping

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Increasing Carrier concentration by Doping

• np = ni2

– n- elrctron concentration, p- hole concentration– ni = Intrinsic carrier concentration=1010/cc for silicon

• For N type semi conductor– n = Nd donor concentration [1014 to 1018 /cc] – majority carrier– p = ni

2 /Nd – minority carrier

• For P type semi conductor– p = Na acceptor concentration [1014 to 1018 /cc] – majority carrier– n = ni

2 /Na – minority carrier

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Practice Problem

• Given Nd = 1016/cc determine n and p for a N type semiconductor– n = Nd = 1016/cc – majority carrier– p = ni

2 /Nd = (1010)2/1016 = 104/cc – minority carrier

• Given Na = 1014/cc determine n and p for a P type semiconductor– p = Na = 1014/cc – majority carrier– p = ni

2 /Nd = (1010)2/1014 = 106/cc – minority carrier

Note: As the concentration of majority carrier increases Concentration of minority carrier decreases