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Chapter 5 The Drude Theory of Metals
• Basic assumption of Drude model• DC electrical conductivity of a metal• Hall effect• Thermal conductivity in a metal
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Basic assumptions of Drude model
* A “ gas of conduction electrons of mass m, which move against a background of heavy immobile ions
Electron density
A
Zn mρ24
106022.0 ×=
24106022.0 × Avogadro’s number
mρMass density in g/cm3
Atomic mass in g/mole
Z
A
Number of electron each atom contribute
sr Radius of a sphere whose volume is equal to the volume per conduction electron
3
3
41sr
nN
Vπ==
3/1
4
3
=
nrs
π
32~0
−a
rsin typical metal
Bohr radius
The density is typically 103 times greater than those of a classical gas at normal T and P.
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* Between collisions the interaction of a given electron, both with others and with the ions, is neglected.
* Coliisons in the Drude model are instantaneous events that abruptly alter the velocity of an electron.
Drude attributed them to the electrons bouncing off the impenetrable ion cores.
* We shall assume that an electron experiences a collision with a probability per unit time τ1
Probability during time interval dtτ
dt
τ : relaxation time
* Electrons are assumed to achieve thermal equilibrium with their surroundings only through collisons
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DC Electrical Conductivity of a Metal
n electrons per unit volume all move with velocity . electrons will cross an area A perpendicular to
the direction of flow.vr
( )Avdtn
Charge crossing A in time dt: nevAdt−
nevAdt
nevAdtj −=
−= vnej
rr−=
vr
the average electronic velocity
When , 0=Er
0=vr
In a electric field Er
t: time elapse since last collision
0vr
randomly oriented, and does not contribute to average vr
Acquired velocity: tm
Eer
− tm
Eev
rr
−= scmv /1.0~ at 1A/mm2
Em
tnevnej
rrr
=−=
2
The average t is relaxation time τ
Em
nej
rr
=
τ2
Ejrr
σ=m
ne τσ
2
=
5
2ne
m
ρτ = τ ~ 10-14 to 10-15 sec at RT
Mean free path τ0vl = Tkmv B2
3
2
1 2
0 =
v0 ~ 107 cm/sec ~ 1 – 10 A at RTlEstimate of v0 is an order of magnitude too small
Actual l ~ 103 A at low temperature, a thousand times the spacing between ions
• Use Drude model without any precise understanding of the cause of collisions.
• -independent quantities yield more reliable informationτ
• calculated using is accurateτ 2ne
m
ρτ =
• Be cautious about quantities such as average electron velocity v, and electron specific heat cv
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At any time t, average electronic velocity m
tpv
)(r
r=
m
tpnej
)(r
r−=
Momentum at time t)(tpr
)( dttp +r at time t+dt
Fraction of electrons without suffering a collision from t to (t+dt)τ
dt−1
Each of these electrons acquire an additional momentum under the influence of an external force :)(tfr
dttf )(r
After a collision, the electronic velocity is randomly directed, and the average velocity is 0.
Fraction of electrons that undergo a collision: τ
dt
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The acquired velocity for each of these electrons after dt: dttf )(~r
The contribution to momentum per electron: ( )2)(~ dt
tf
τ
r
neglected
[ ] ( )22)()()()()()(1)( dtOdttftp
dttpdtOdttftp
dtdttp ++
−=++
−=+
rrrrrr
ττ
( )2)()()()( dtOdttftp
dttpdttp ++
−=−+
rrrr
τ)(
)()(tf
tp
dt
tpd rrr
+−=τ
equivalent to a damping term
Hall Effect and Magnetoresistance
H
xE
xj
+ + + + + + + + + + + + +
- - - - - - - - - - - - - - -
yE
x
y
z
xv
Hverr
×−
-e
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Lorentz force Hvc
e rr×− deflects electrons in –y direction
Electric field build up in y direction that oppose electron motion in y direction. In equilibrium the traverse field (Hall
field) Ey balance the Lorentz force
Two important parameters: resistivity
x
x
j
EH =)(ρ Hall coefficient
Hj
ER
x
y
H =
negative value for electrons, and positive
value for positive charge
To calculate and , consider the current density and in the presence of an electric field with
arbitrary components , and in the presence of magnetic field
)(Hρ HRxj yj
xE yEzH
×+−=
c
HvEef
rrrr
The momentum per electronτ
pH
mc
pEe
dt
pdr
rr
rr
−
×+−=
In steady state the current is independent of time
τω
τω
xxcy
xycx
ppeE
ppeE
−+−=
−−−=
0
0
wheremc
eHc =ω
Multiply by and using m
neτ− vnej
rr=
yxcy
xycx
jjE
jjE
+−=
+=
τωσ
τωσ
0
0 Where m
ne τσ
2
0=
In steady state 0=yjxx
cy j
nec
HjE
−=
−=
0σ
τωSonec
RH
1−=
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