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
1
Basic assumptions of Drude model
* A “ gas of conduction electrons of mass m, which move against a background of heavy
immobile ions
Electron density
0.6022 ×10 24
ρm
n = 0.6022 × 10 24
Avogadro’s number
Mass density in g/cm3
A
Atomic mass in g/mole
Z
Number of electron each atom contribute
rs
Radius of a sphere whose volume is equal to the volume per conduction electron
V 1 4 3
= = πrs
N n 3
rs
Zρ m
A
a0
~ 2−3
1/ 3
 3 
rs = 

 4πn 
in typical metal
Bohr radius
The density is typically 103 times greater than those of a classical gas at normal T and P.
2
* 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.
1
* We shall assume that an electron experiences a collision with a probability per unit time
Probability dt
τ
τ
τ
during time interval dt
: relaxation time
* Electrons are assumed to achieve thermal equilibrium with their surroundings only through
collisons
3
DC Electrical Conductivity of a Metal
n electrons per unit volume all move with velocity
the direction of flow.
Charge crossing A in time dt:
j=
r
v
When
r
v
.
n(vdt )A
electrons will cross an area A perpendicular to
− nevAdt
r
r
j = −nev
− nevAdt
= −nev
Adt
the average electronic velocity
r
E=0 ,
r
v =0
r
In a electric field E
t: time elapse since last collision
r
v0
r
randomly oriented, and does not contribute to average v
r
r
eE
r
e
E
t
v ~ 0.1cm / s
Acquired velocity: −
v =− t
m
m
r
r  ne 2t  r
The average t is relaxation time
 E
j = −nev = 
 m 
2
r
r
r  ne 2τ  r
ne
τ
 E
j = 
j
=
σ
E
=
σ
 m 
at 1A/mm2
τ
m
4
τ=
m
ρne 2
Mean free path
τ
l = v0τ
v0 ~ 107 cm/sec
~ 10-14 to 10-15 sec at RT
1
3
2
mv0 = k BT
2
2
l ~ 1 – 10 A at RT
Estimate 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.
τ
•
•
τ
-independent quantities yield more reliable information
calculated using
τ=
m
is accurate
ρne 2
• Be cautious about quantities such as average electron velocity v, and electron specific heat cv
5
At any time t, average electronic velocity
Momentum
r
p (t )
r
p (t + dt )
r
r p (t )
v=
m
r
r
nep(t )
j =−
m
at time t
at time t+dt
Fraction of electrons without suffering a collision from t to (t+dt)
1−
dt
τ
Each of these electrons acquire an additional momentum under the influence of an external force
r
f (t ) :
r
f (t )dt
dt
Fraction of electrons that undergo a collision: τ
After a collision, the electronic velocity is randomly directed, and the average velocity is 0.
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r
~ f (t )dt
The acquired velocity for each of these electrons after dt:
r
f (t )
neglected
The contribution to momentum per electron: ~
(dt )2
τ
r
r
r
r
 dt  r
 dt  r
2
p (t + dt ) = 1 −  p (t ) + f (t )dt + O(dt ) 2 = p (t ) −   p (t ) + f (t )dt + O(dt )
 τ 
τ 
[
]
r
r
dp (t )
p(t ) r
=−
+ f (t )
dt
τ
r
r
r
 dt  r
2
p(t + dt ) − p (t ) = −  p (t ) + f (t )dt + O(dt )
τ 
equivalent to a damping term
Hall Effect and Magnetoresistance
z
y
H
Ex
x
+ + + + + + + + + + + + +
Ey
jx
- - - - - - - - - - - - - - -
vx
-e
r r
− ev × H
7
er r
− v×H
c
Lorentz force
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:
To calculate ρ ( H ) and
arbitrary components
RH
Ex E y
resistivity
E
ρ (H ) = x
jx
, consider the current density
Hall coefficient RH =
jx
and
jy
In steady state the current is independent of time
0 = −eE y + ωc p x −
In steady state
and using
jy = 0
Hz
r
r
r
dp
 r p r p
= −e E +
×H −
dt
mc

 τ
0 = −eE x − ωc p y −
Multiply by
in the presence of an electric field with




The momentum per electron
neτ
−
m
jx H
negative value for electrons, and positive
value for positive charge
, and in the presence of magnetic field
r
r
r r H
f = −e E + v ×
c

Ey
So
r
r
j = nev
px
τ
px
ωc =
eH
mc
τ
σ 0 E x = ωcτ j y + j x
σ 0 E y = −ωcτ j x + j y
ω τ 
 H 
E y = − c  j x = −
 jx
 nec 
 σ0 
where
Where
ne 2τ
σ0 =
m
RH = −
1
nec
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