3 Electrons and Holes in Semiconductors 3 Electrons and Holes in Semiconductors 3.1. Introduction Semiconductors

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  • 1

    3 Electrons and Holes in Semiconductors

    3.1. Introduction

    Semiconductors : optimum bandgap

    Excited carriers → thermalisation

    Chap. 3 : Density of states, electron distribution function, doping, quasi thermal equilibrium

    electron and hole currents

    Chap. 4 : Charge carrier generation, recombination, transport equation

    3.2. Basic Concepts

    3.2.1. Bonds and bands in crystals

    Free electron approximation to band

    Ref. CLASSIC “Bonds and Bands in Semiconductors” J. C. Phillips, Academic 1973

    Atomic orbitals → molecular orbitals → bands

    Valence band : HOMO

    Conduction band : LUMO

  • 2

    Semiconductors :  eVEg 35.0 

    Semi-metals :  eVEg 5.00 

    Si : sp3 orbital, diamond-like structure

    3.2.2. Electrons, holes and conductivity

    Semiconductors

    At  KT 0 , all electrons are in valence band – no conductivity At elevated temperature,

    some electrons in conduction band, and some holes in valence band

  • 3

    3.3. Electron States in Semiconductors

    3.3.1. Band structure

    Electronic states in crystalline

    Solving Schrödinger equation in periodic potential (infinite)

    Bloch wavefunction

        rkk rrk  ii eu, (3.1)

    for crystal band i and a wavevector k, which is a good quantum number.

    The energy E against wavevector k is called by crystal band structure.

    Typical band diagram plotted along the 3 major directions in the reciprocal space.

           111,011,001,000 LMX

    a)

    Conduction Band

    Valence Band

    Electrons

    a)’

    Electron in CB

    b) b)’

    Hole in VB

    Fig. 3.4. Electron in CB and Hole in VB

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    Point a

    k 

     is called the Brillouin zone boundary.

    For zinc blende structure,    

      

      

      

      

      

     

    aaa L

    aa K

    a X

     ,0

    2

    3

    2

    3 ,00

    2 ,000

    3.3.2. Conduction band

    At the vicinity of the minimum  kE in the conduction band 0ckk  , approximation

      *

    2

    0

    2

    0 2 c

    c

    c m

    EE kk

    k 

     

    (3.2)

    where  00 cc EE k and *

    cm is a parameter with dimensions of mass, and defined by

      2

    2

    2*

    11

    k

    kE

    m

    c

    c 

      

    (3.3)

    is called the parabolic band approximation.

    eg. *

    cm for GaAs 0 * 067.0 mmc  , for Si 0

    *

    // 19.0 mmc  0 * 92.0 mmc  ,

    where 0m is the static electron mass.

    *

    cm is determined by the atomic potentials, should actually be a tensor,

    but is usually treated as a scalar, and is called the effective mass of electron.

    Momentum of an electron and the applied force are related as

    dt

    d

    dt

    d kp F  (3.4)

    Following eq. (3.2), the velocity and momentum can be obtained as

       

    *

    01

    c

    c

    m E

    kk kv k

      

     (3.5)

     0 *

    ccm kkvp   (3.6)

    3.3.3. Valence band

    At the vicinity of the maximum  kE in the valence band 0vkk 

      *

    2

    0

    2

    0 2 v

    v

    v m

    EE kk

    k 

     

    (3.7)

  • 5

    where  00 vv EE k and *

    vm is the effective mass of hole.

       

    *

    01

    v

    v

    m E

    kk kv k

      

     (3.8)

     0 *

    vvm kkvp   (3.9)

      2

    2

    2*

    11

    k

    kE

    m

    v

    v 

      

    (3.10)

    eg. *

    vm for GaAs 0 * 5.0 mmv  , for Si 0

    * 54.0 mmv  ,

    *

    cm is the curvature of the band.

    3.3.4. Direct and indirect band gaps

    Whether or not minimum of conduction band and maximum of valence band occur at the same

    wavevector 00 vc kkk 

    Reciprocal lattices for (a) fcc and (b) bcc crystals

  • 6

    When the band gap is indirect, transition requires involvement of phonons for conservation of

    momentum.

    (Sze)

    3.3.5. Density of states

    According to Pauli’s exclusion principle, each quantum state only supports one electron (two,

    considering spins). Also, from Heisenberg’s uncertainty principle,

    hxp ~ ∴ hxk ~ ∴ 2~xk 

    Therefore, for a crystal of volume LLL 

    Lx

    k  22

    ~  

    3

    2  

      

    L states in this unit volume.

    Hence, the density of electron states per unit crystal volume considering spin is

       

    kkk 3

    3

    3

    2

    2 ddg

      (3.11)

    If the band structure  kE is spherically symmetric and isotropic around 0k

        dkkkgdg 23 4kk (3.12)

    Therefore,

        dE dE

    dk kkgdEEg  24 (3.13)

    ∴     dE

    dk kEg  2

    3 4

    2

    2 

     (3.14)

  • 7

    From eq. (3.2)   *

    2

    0

    2

    0 2 c

    c

    c m

    EE kk

    k 

     

    ∴  02 *

    2 2 c

    c EE m

    k  

    (3.15)

    Similarly, for the valence band (hole)

    ∴  EE m

    k v v  02

    *

    2 2

     (3.15’)

    Differentiating (3.15)

      210

    21

    2

    *2 c

    c EE m

    k  

     

     

     ∴   210

    21

    2

    *2

    2

    1  

     

      c

    c EE m

    dE

    dk

    Therefore, from (3.14)

         

     

         

          210

    23

    2

    *

    2

    21

    0

    23

    2

    *

    3

    21

    0

    21

    2

    *

    02

    *

    3

    21

    0

    21

    2

    *

    2

    3

    2

    3

    2

    2

    12

    2

    4

    2

    2

    12 4

    2

    2

    2

    2

    1 4

    2

    2 4

    2

    2

    c

    c

    c

    c

    c c

    c c

    c c

    c

    EE m

    EE m

    EE m

    EE m

    EE m

    k dE

    dk kEg

     

     

     

     

     

     

     

     

     

     

     

    

    

    

     

     

     

    (3.16)

    Similarly,

        210

    23

    2

    *

    2

    2

    2

    1 EE

    m Eg v

    v v 

     

     

     (3.17)

    Question : Derive the DOS for 2D and 1D materials. (Box 3.3)

    Excitons

    A bound state of a pair of electron and hole of same k .

    The wavefunction can be described as

       rkrk  ,, heex 

    Satisfying a hydrogenic effective mass equation

    exexex

    s

    ex E q

     

     rr

    42

    2 2

    *

    2

    * : reduced effective mass of e-h pair

    exE : binding energy of the exciton

  • 8

    22

    2

    0

    0

    *

    l

    Ryd

    m E

    s

    ex 

     

    3.3.6 Electron distribution function

    Fermi function in semiconductors

     

    1exp

    1

     

      

      

    Tk

    EE Ef

    B

    F

    This can be expressed in terms of  rk,f although the energy is only dependent on the wavevector.

          krkkkr 33 , dfgdn 

    The total electron density is given by

          k krkkr CB c dfgn 3, (3.20)

            k krkkr VB v dfgp 3,1 (3.21)

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