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Transcript of Intro to PN Junctions: II - nanoHUB PNJunctionsII... · PDF file Lundstrom ECE 305 S16...

  • Lundstrom ECE 305 S16

    ECE-305: Spring 2016

    Intro to PN Junctions: II

    Professor Mark Lundstrom Electrical and Computer Engineering

    Purdue University, West Lafayette, IN USA [email protected]

    2/18/15

    Pierret, Semiconductor Device Fundamentals (SDF) pp. 195-209

  • NP junction (equilibrium)

    2

    N P

    p0 ! NA

    ρ ! 0 n0 ! ND

    ρ ! 0

    xp−xn 0

    “transition region”

    Lundstrom ECE 305 S16

    p0 < NAn0 < ND

  • energy band diagram

    3

    EF

    EC

    EV

    x

    E

    Ei

    x = xpx = 0x = −xn

    qVbi

    p0 < NAn0 < ND

    Lundstrom ECE 305 S16

  • electrostatics: V(x)

    4

    V

    x

    N P

    xp−xn

    qVbi = kBT ln NAND ni 2

    Lundstrom ECE 305 S16

  • electrostatics: E (x)

    5

    E

    x N P xp−xn

    Lundstrom ECE 305 S16

  • carrier densities vs. x

    6

    log10 n x( ), log10 p x( )

    x N P xp−xn

    p0P = NA

    p0N = ni 2 ND

    n0N = ND

    n0 p = ni 2 NA

    n0N

  • electrostatics: rho(x)

    7

    ρ

    x

    N P

    ρ = q p0 x( ) − n0 x( ) + ND+ x( ) − NA− x( )⎡⎣ ⎤⎦

    xp −xn

    qND

    −qNA

    Lundstrom ECE 305 S16

    n0N

  • NP junction electrostatics

    8

    How do we calculate rho(x), E(x), and V(x)?

    Lundstrom ECE 305 S16

  • Gauss’s Law

    9

    +Q n̂

    “Gaussian surface”

    ! D = ε0

    ! E

    ! D = KSε0

    ! E

    ! D i d

    ! S"∫ = Q

    Lundstrom ECE 305 S16

  • Gauss’s Law in 1D

    10

    ! D i d

    ! S"∫ = Q

    xx x + dx

    ρ x( )C/cm3 D x + dx( )D x( )

    n̂n̂

    Area = A

    −D x( )A+ D x + dx( )A = Q

    Q = ρ x( )Adx

    D x + dx( )− D x( ) dx

    = ρ x( )

    dD dx

    = ρ x( ) Lundstrom ECE 305 S16

  • the Poisson equation

    11

    dE dx

    = ρ x( ) KSε0

    dD dx

    = ρ x( )

    ∇ i ! D = ρ x( )

    ! D i d

    ! S"∫ = Q

    D = KSε0E

    Lundstrom ECE 305 S16

  • electrostatics: rho(x)

    12

    ρ

    x

    N P

    ρ = q p0 x( ) − n0 x( ) + ND+ x( ) − NA− x( )⎡⎣ ⎤⎦

    xp −xn

    qND

    −qNA

    “depletion approximation”

    Lundstrom ECE 305 S16

    n < ND

    p < NA

  • the “depletion approximation”

    13

    dE dx

    = ρ x( ) KSε0

    ρ

    x

    N P

    −xn

    ρ = +qND

    xp

    ρ = −qNA

    qNDxn = qNAxp

    NDxn = NAxp Lundstrom ECE 305 S16

  • but first

    14

    E

    x

    V = 0V > 0

    d

    E = - dV

    dx

    V = − E

    x1

    x2

    ∫ dx

    E = V

    d

    dE dx

    = ρ x( ) KSε0

    = 0 →E is constant

    Lundstrom ECE 305 S16

  • electric field between the plates

    15

    E

    x

    + d 2

    − d 2

    E = V

    d V = − E

    −d /2

    +d /2

    ∫ dx

    Lundstrom ECE 305 S16

  • the NP junction

    16

    dE dx

    = ρ x( ) KSε0

    is not constant!

    ρ

    x

    +qND

    ρ = −qNA

    xn + xp

    P

    V = 0

    N

    V =Vbi > 0

  • 1)  Make depletion approximation

    2)  Solve

    3)  Find

    NP junction electrostatics

    17

    dE dx

    = ρ x( ) KSε0

    ρ

    x

    N P

    −xn

    ρ = +qND

    xp

    ρ = −qNA

    E x( ),V x( ), xn , xp