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Transcript of Chapter 6. Free Electron Theory of Metals - YUyu.ac.kr/~hyulee/ssp_lecture/Chap6.pdf¢ ...

  • Electronic Properties of Materials Hee Young Lee

    Chapter 6.

    Free Electron Theory of Metals

  • Electronic Properties of Materials Hee Young Lee

    Electron Energy Levels in a Rigid Well

    (Source: D.A. Neamen, Semiconductor Physics and Devices, Irwin, 1997)

    textof (6.1) and (3.42) (eV) 103.75

    82 2

    2

    -19

    1 2

    2

    2222

    n L

    En mL

    nh m k

    En ×

    ≅=== h

    kxAA sin=ψ

    kxAS cos=ψ

    2 0

    2

    LL x −=

  • Electronic Properties of Materials Hee Young Lee

    Calculated Energy levels for 20 Å wide 20 eV deep potential well

    Eel = 0.314, 1.80, 2.83, 7.26, 7.79, 14.95 and 16.97 eV � Quantized Levels

    L = 20 Å

    �electrons confined

    around an atom or a molecule

  • Electronic Properties of Materials Hee Young Lee

    What Happens If L Increases to 20 mm?

    Eel : many energy levels � Nearly Continuous Distribution � “Free”

    L = 20 mm

    �electrons confined

    within a metal piece

    (mm)

    (X 1 0

    - 1 9

    e V

    )

  • Electronic Properties of Materials Hee Young Lee

    Assumption: consider valence electrons in a metal to be confined or bound within three dimensional potential well

    Allowed energy levels for an electron are quantized, i.e. discontinuous. For an infinite potential well,

    ( ) ( )

    integers. positive are and ,, where

    eV 1075.3

    828 222

    2

    19 222

    2

    2

    2

    222

    2

    22

    zyx

    zyxzyxn

    nnn

    nnn L

    nnn mL h

    mL n

    mL nh

    EE ++ ×

    ≅++===≡ −πh

    Free Electron Theory of Metals

  • Electronic Properties of Materials Hee Young Lee

    Notes :

    a) Several electrons with different wave functions can have the same

    energy (=degeneracy)

    b) For “normal” sizes (L>1 mm), these levels are very closely spaced (~continuous).

    kTEeEF /)( −=

    The probability that a M-B gas particle has energy E, from kinetic theory

    It is treated by Maxwell-Boltzmann (M-B) statistics. Its velocity distribution is

    )]2/(exp[)( 222/3 kTmvvTvN −∝ −

  • Electronic Properties of Materials Hee Young Lee

    In fact, there is a certain “density of energy states” function, Z(E) which determines how many particles can have certain energies.

    eunit volumper particles of # )( )(

    and

    )( )()(

    0∫ ∞

    ==

    =

    dEEZEFN

    EZEFEN

    These concepts also apply to the electron gas in the metal, I.e.

  • Electronic Properties of Materials Hee Young Lee

    dE Z(E)F(E) g(E) 1/N g(E)

    :determined becan g(E)energy of

    functionany of n valueexpectatioor average theLikewise, 5.

    gas) B-M clasicalfor (3/2)kT(

    Z(E)dEF(E) E 1/N E 4.

    eunit volumper particles of #dE Z(E)F(E)N 3.

    derived.) be also will(This

    F(E) states thoseof occupation ofy Probabilit 2.

    shortly.) derived be will(This

    Z(E) states available ofDensity 1.

    0

    0

    0

    ∫ ∫

    =><

    =

    >=<

    ==

  • Electronic Properties of Materials Hee Young Lee

    Density of States Function

    ( ) EECEm h

    ENEZ ∝=== 2/3 3

    2 4

    )]( )[( π

    Using infinite square well approximation, the distribution of allowed energy states in a metal can be derived, and the result is given below.

    where C is the proportionality constant. The above equation can also be applied to semiconductors and insulators at the bottom of the conduction band and the top of the valence band, simply replacing m with m*, i.e. the effective mass.

  • Electronic Properties of Materials Hee Young Lee

    Density of States Function (cont’d)

    ( )

    ( ) 9)-(1 24)(

    8)-(1 2 4

    )(

    2/3* 3

    2/3* 3

    EEm h

    EN

    EEm h

    EN

    Vhh

    Cee

    −=

    −=

    π

    π

    where me* and mh* are the effective mass values of an electron near the bottom of the conduction band and a hole near the top of the valence band, respectively.

  • Electronic Properties of Materials Hee Young Lee

    Usually drawn to represent surface, as follows

  • Electronic Properties of Materials Hee Young Lee

    Fermi-Dirac statistics

    (a) The Fermi-Dirac distribution function for a Fermi energy of 2eV and for

    temperatures of 0K, 600K and 6000K

    (b) The classical Maxwell-Boltzmann distribution function of energies for the

    same temperatures.

  • Electronic Properties of Materials Hee Young Lee

    .E level of "degeneracy"the called is Eenergy at states SThe (

    energy) of ion(conservat ENE

    N)constantelectrons(# NN :Conditions

    iii

    i ii

    i i

    ∑ =

    ===

    1)kT/)EEexp(( 1

    )E(F F +−

    =

    Definig EF =-α/β, and requiring that the above function approach the classical Maxwell-Boltzmann distribution at high temperatures, we can write this as

    Which is the “Fermi-Dirac distribution function”

  • Electronic Properties of Materials Hee Young Lee

    0K => all states < Ef occupied all states > Ef empty

    (Source:http://jas.eng.buffalo.edu/applets/education/semicon/fermi/functionAndStates/functionAndStates.html)

  • Electronic Properties of Materials Hee Young Lee

    0>K => some e- have E>Ef some holes exist E

  • Electronic Properties of Materials Hee Young Lee

    (Source:http://jas.eng.buffalo.edu/applets/education/semicon/fermi/functionAndStates/functionAndStates.html)

  • Electronic Properties of Materials Hee Young Leehttp://jas.eng.buffalo.edu/applets/

    (Source:http://jas.eng.buffalo.edu/applets/education/semicon/fermi/functionAndStates/functionAndStates.html)

  • Electronic Properties of Materials Hee Young Lee

    (Source: B.G. Streetman, Solid State Electronic Devices, Prentice-Hall, 1980, p.71.)

    Fermi Function

  • Electronic Properties of Materials Hee Young Lee

    The free electron theory of metal can largely account for the following phenomena:

    • Specific heat (and thermal conductivity)

    •Thermoelectric effect

    •Thermionic emission

    •Schottky effect and field emission

    •Photoemission

    •Contact potential

  • Electronic Properties of Materials Hee Young Lee

    The specific heat is the amount of energy required to raise temperature by 1 degree Kelvin (or Celsius).

    )text of 22.6 equ( E

    3(kT) kT)

    2

    3 (

    E

    2kT E

    F

    2

    F

    =≅><

    The specific heat of a metal (electron contribution) is some 1/100 times less than expected, all because of the Pauli exclusion principle.

    text) of 6.23 (equ E

    k6

    dT

    Ed C

    F

    2

    V = ><

    =

    Electronic Specific Heat

  • Electronic Properties of Materials Hee Young Lee

    Let’s now consider several phenomena where electrons are actually emitted from metals ;

    Thermionic, Schottky, Field Emission and Photoemission

  • Electronic Properties of Materials Hee Young Lee

    The work function is defined as the energy required to take an electron out of metal, form EF (I.e. the minimum energy remove an electron). Φ is more critical, therefore, in emission processes, than is the value of EF

    3/22

    2/3 3

    2/3

    0

    2/1

    0 3

    2/3

    8 3

    2

    3 2)2(4

    )2(4 )( )(

      

      =

    =

    == ∫∫ ∞

    π

    π

    π

    N m

    h E

    E h m

    dEE h m

    dEEFEZN

    F

    F

    EF

  • Electronic Properties of Materials Hee Young Lee

    Estimate of φ

    The potential energy developed between these two charges, when the electron is removed, equals the work function

    x+ -

    Image Charge

    surface

    vacuum

    metal

  • Electronic Properties of Materials Hee Young Lee

    eV1 )104)(1085.8(16

    106.1 U

    distance)ic (Interatom 4xsay

    eV in x16

    q U

    x16

    q U

    1012

    19

    0

    00

    00

    2

    ≅ ××

    × =≡

    =

    − =

    −−

    π

    πε

    πε

    φ

    This is a little on the small side, but is the right order of magnitude.

  • Electronic Properties of Materials Hee Young Lee

    used. wasdNeVdJ where

    dE}kT/)EE(exp{EV )m2(e4)h/A(

    dNeVAI