Title: Ch 6 . Free Electron Fermi Gas
1Solid 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
26.1 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.? ?? ??? ?? ??.
?
36.2 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.)
46.3 Energy Levels in 1D (2)
- 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
56.4 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
66.5 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)
76.5 Free Electron Gas in 3-D (2)
- From Schrödinger eq.,
- N?? ???? (free el.)? ?? system ? ground state (?,
?? ??? - ????)??, ??? ? ?? ??? level ? 3?? ??? ? ??.
-
86.5 Free Electron Gas in 3-D (3)
- Total of orbits (? ??? ?? ?? ?)
-
-
-
-
-
- Fermi velocity at the Fermi surface
-
-
For a typical metal,
density of electrons
96.5 Free Electron Gas in 3-D (4)
106.5 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
116.6 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
126.6 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 -
??
136.7 Experimental Heat Capacity of Metals
- At low temp. (?, T lt ?D, TF)
Fig.9 ??
146.8 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
156.9 Electrical Conductivity and Ohms Law (1)
- Lorentz force (???? ??? ??? ?? ?)
- ????? (s)
- current density ( ) ?? ???, ?? ??? ???? ???
-
-
- For el.,
(??/??)
??
N/V (?? /?? ????)
? ? ???
??
collision time
166.9 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
176.10 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
186.10 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? ???, ??? ?? !
196.11 Motion in Magnetic Fields (1)
- ???? ??? ??? ?? ? Lorentz force
-
F
tt ?? Fermi surface
t0 ?? Fermi surface
206.11 Motion in Magnetic Fields (2)
Eq.
cyclotron freq.
- In the steady state in a static el. field,
E ??? ?? ??
216.12 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
Ex
v (???)
B
-e
jx
FB
FE
Ey
jx
y
x
z
B
if major ?
jx
Ey
226.12 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
-
excellent agreement of Na, K
Surface current density
236.13 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