Ch 6 . Free Electron Fermi Gas

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Solid State Physics
Ch 6 . Free Electron Fermi Gas
Prof. J. Joo (jjoo_at_korea.ac.kr) Department of
Physics, Korea University http//smartpolymer.kore
a.ac.kr
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6.1 Introduction

• (Fig.1) Na (sodium) metal
• sea of conduction electrons
• ??? ??
• (Cu, Au, Ag)
• Free el. model ? ?? ??? ??? ??? ??? ? ??
• ??? ??(valence el.)?? ??? ??? ? ?? conduction
  el. ?? ??,
• ??? ?? ???? ???? ??? ? ??.
• ltNotegt conduction el. ? ion ?? ???? ? Ch.7
• Simple metals (ex Na, K, Rb, Cs ) ?? ???
  15 ??
• Alkali metals 3s 4s 5s 6s
  ion core ? ????.
• ?? ?? simple, classical(????) free el. model
  ?
• ???? ? ? Ohms law ? Electrical and
  thermal conductivity
• ???? ? ? Heat capacity ? Magnetic
  susceptibility due to cond. el.
• (Maxwell ????) ? ???? ??
• ? free el. (or conduction el.) ?? ?? ???? ????
  ???? ??? ? ???? ? ion core ? ???
• ? Pauli exclusion ??? ?? ?? el.? ?? ??? ?? ??.

?
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6.2 Energy Levels in 1D (1)

• Consider a free el. gas in 1D, taking account of
  quantum theory and of the Pauli exclusion
  principle
• An el. (mass m) is confined to a length L
  (infinite barrier)
• energy levels
• wavefunctions
• Let wavefunction of el. (solution
  of Schrödinger eq.)

3
???(n)
2
1
0
free el.
the energy of el. in the orbital
to denote a solution of the wave eq. for a system
of only one el. (no interaction with other els.)
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6.3 Energy Levels in 1D (2)

• Apply b.c.

• lt ???? gt
• Degeneracy (???) of orbits with the same
  energy (?, ?? ???? ?? ?? ???)
• nF the topmost filled (energy) level, where we
  start filling the levels from the bottom (n1)
  and continue filling higher levels with electrons
  until all N electrons are accomodated (p.145)
• Fermi energy (eF) the energy of the topmost
  filled level

boundary condition ? ????? ??? 2pL/?nnp
ms ½
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6.4 Effect of Temperature on the Fermi-Dirac
Distribution

• El. gas ? ?? ???? ??? ???? ?? ????.
• ? ? ? ???? el. ? ??? Fermi-Dirac ????? ???.
• ???? µ(T) chemical potential
• at T0 , µeF

gives the probability that an orbital at
energy e will be occupied in an ideal el. gas in
thermal equilibrium
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6.5 Free Electron Gas in 3-D (1)

• 3?? free-particle ? Schrödinger ???
• ltNotegt 1??
• ??, el. ? ?? L? cube? ?? ??, wavefunction?
  standing wave ??? ???
• ??? ? ?(xL, y, z)?(x, y, z)

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6.5 Free Electron Gas in 3-D (2)

• From Schrödinger eq.,
• N?? ???? (free el.)? ?? system ? ground state (?,
  ?? ???
• ????)??, ??? ? ?? ??? level ? 3?? ??? ? ??.

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6.5 Free Electron Gas in 3-D (3)

• Total of orbits (? ??? ?? ?? ?)
• Fermi velocity at the Fermi surface

For a typical metal,
density of electrons
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6.5 Free Electron Gas in 3-D (4)

• D.O.S D(e)

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6.5 Free Electron Gas in 3-D (5)
D(e)
? e1/2
f(e)D(e) finite temp.(??)
f(e)1
T0 (??)
e
eF
At finite temp., electrons are thermally excited
from to
Important for the physical property of matter
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6.6 Heat Capacity of the Electron Gas (1)

• Classical model ??? N?? ???? N?? ?????
• ?? Cv ?
• ???, ??? ? Cv (at room temp.)
• ??? ?????
• ???? ????? heat capacity? ???? ?? ????
• ?? ? Pauli exclusion principle ? ????.
• ? Fermi-Dirac ????
• ?? ??? T0 K ??? heating ?? ?, ???? ?? ???? kBT
  ???
• ???? ?? ?? ???. ?? Fermi level ??? ???? kBT ?
  ????
• ?? thermally excited ??.
• the el. in this region are responsible
• for the heat capacity

TF
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6.6 Heat Capacity of the Electron Gas (2)
agree with the experimental results

• More detail derivation of Cv (due to electrons)
• Sommerfeld Expansion (at low temp. T lt TF)
• (???) Ashcroft / Mermin ? SSP
  Ch.2

??
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6.7 Experimental Heat Capacity of Metals

• At low temp. (?, T lt ?D, TF)

Fig.9 ??
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6.8 Thermal Conductivity

• ?? ???? thermal conductivity (p.166)
• el. mean free path
• average velocity of el.
• heat capacity / vol. due to el.
• ???,

? Pure metals el. contribution is dominant,
compared with that due to phonon ? In disordered
system of impure metal, phonon term can not be
ignored. (?el. ? mean free path ? impurity ???
collision ??? ???)
collision time
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6.9 Electrical Conductivity and Ohms Law (1)

• Lorentz force (???? ??? ??? ?? ?)
• ????? (s)
• current density ( ) ?? ???, ?? ??? ???? ???
• For el.,

(??/??)
??
N/V (?? /?? ????)
? ? ???
??
collision time
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6.9 Electrical Conductivity and Ohms Law (2)

• The definition of electrical conductivity is
• And, the electrical resistivity ?
• At room temp.
• s(Ag)6.21?105 S/cm (?-1cm-1)
• s(Cu)5.88?105 S/cm
• s(Nb)0.69?105 S/cm
• s ? n ( density of charge)
• t (collision time) ?? l (mean free path)
• Almost pure Cu
• sCu (T4K) 105? sCu (T295K) 1011 S/cm
• ? t 2?10-9 (s) at T4K
• and
• ? lvFt 1.57?108?2?10-9 0.3 cm
• the velocity at the
  very long mean

? ??gt VIR (Ohms Law)
? 3 ??
SI unit
CGS unit for s sec-1
l (T4K,Cu) 0.3 cm l (T295K,Cu) 3?10-6 cm
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6.10 Experimental Electrical Resistivity of
Metals (1)

• Electrical resistivity due to collision
• ? Collision with (lattice) phonons at high
  temp.
• ? Collision with impurities and imperfection
  (??? p.150, Fig.11??)
• important at low temp. (4.2K)
• (?no lattice vibration for collision)
• Net relaxation time (or net collision time)
• ? ??? ??
• due to impurities
• due to phonons
• Net resistivity is given by
• due to phonon Open, independent of
  temperature
• ? defects ??? ??? ?? ??? ??

due to scattering of el. by defects or impurities
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6.10 Experimental Electrical Resistivity of
Metals (2)

• Electrical resistivity ? ??? ?????
• ? Matthiessens rule (???? ??? ??)

lt Fig.12 gt
Potassium(K)
2 ?? impurity ??? ?? sample (??? ??)
Phonon contribution is dominant
Resistivity ratio residual
resistivity pure metal ??? ? (RT)? ??
? (T?0)? lower ? ? ? higher
lt??gt
??? ????? phonon contribution? ?? impurity
scattering? dominant ! Impurities? ??
scattering? ???, ??? ?? !
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6.11 Motion in Magnetic Fields (1)

• ???? ??? ??? ?? ? Lorentz force

F
tt ?? Fermi surface
t0 ?? Fermi surface
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6.11 Motion in Magnetic Fields (2)

• ?? ????

Eq.
cyclotron freq.

• In the steady state in a static el. field,

E ??? ?? ??
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6.12 Hall Effect (1)

• ??? n (charge density)? ?? ??? ?? ??? ???? ?? ???
  ??
• Hall field? ?? (?????? ???? ?? Lorentz force ? ??
  
• 
  ?? ??? ??? ??)
• external mag. field Bz
• external el. field jx
• ?? ??? y ???? ????? ??
• Eq. ?? let vy 0

y
z
x

• z ?? ??? ?? (??? ? ??)

Ex
v (???)
B
-e
jx
FB
FE
Ey
jx
y
x
z
B
if major ?
jx
Ey
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6.12 Hall Effect (2)

• Define Hall coefficient (RH)
• ? negative for free el.
• ? positive for hole (positive el.)
• ? RH ? greater when n is lower
• ? RH ? ?????? n (carrier concentration) ? ???
  ? ??
• Notegt 2D Hall resistivity

• Table 4 Experimental RH

excellent agreement of Na, K
Surface current density
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6.13 Ratio of Thermal to Electrical Conductivity

• Wiedemann-Franz law (not at low temp.)

due to free electron
?T
independent of particular metals
lt H.W. gt Ch.6, 1 and 6