4. Free Electron Models
Introduction
Drude Model
Ohm’s law
Newton’s law
Hall effect
AC conductivity
Plasma oscillations
Thermal conductivity
Thermoelectric effect
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Introduction
The most important characteristic of a metal is its high electrical conductivity.
1897: J. J. Thompson's discovery of the electron
1900: Drude proposed a simple model based on Kinetic Theory and Classical Mechanics
that explained a well-known empirical law, the Wiedermann-Franz law (1853)
This law stated that at a given temperature the ratio of the thermal conductivity
to the electrical conductivity was the same for all metals.
The assumptions of the Drude model are:
(i) a metal contains free electrons which form an
electron gas
(ii) the electrons have a random motions through
the metal with <vT>= 0 even though <vT2>≠0
therefore the average thermal energy <½mvT2 > ≠0
The random motions result from collisions with the
ions. This is the mechanism to to achieve thermal
equilibrium.
(iii) because the ions have a very large mass, they
are essentially at rest.
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Kinetic theory
Kinetic theory was successfully applied to explain the properties of gas.
The number density of a gas at STP (Standard Temperature and Pressure:
273.15 K, and 105 pascals) is (PV=NkBT →) N/V=P/(kBT) ρgas=2.65 1025
particles/m3= 2.65 1019 particles/cm3
In a metal, the free electrons are those weakly bound to the nuclei and can be
named as conduction electrons.
There are NA=6.02 1023 atoms per
mole
and the number of moles for a metal
of mass density ρm is n= ρm/A
(A=atomic weight), if Z is the number
of conduction electrons per atom
→ ρe=Z NA ρm/A
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Free Electron Density is 103
times higher than gas density
in similar condition.
We expect a strong e-e and
e-ion interactions that are
neglected in the Drude model
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• Free electron approximation: neglect any e-ion interaction.
This means that is neglected the periodical arrangement of
ion cores
• Independent electron approximation: neglect any e-e
interaction.
This means that is neglected the correlation between
electrons
Drude model is both a free and
independent electron model. The
e-ion interaction is not completely
neglected since electrons are
confined inside the metal.
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In a gas, the collisions are instantaneous events between similar gas particles,
in the Drude model the collisions are instantaneous events between an
electron and ionic cores, e-e collisions (analogous of particle-particle
collisions in gases) are neglected.
While the latter assumption is generally true, the former is wrong: the
collisions are between electrons and imperfections of the lattice, but
assuming an unspecified source of collisions the conclusions of the Drude
model give often a qualitative picture of phenomena.
After a collision, an electron undergoes an abrupt change of velocity with a
probability per unit time 1/ττ, τ=relaxation time or collision time or mean
free time which is assumed to be independent of the electron position and is
independent of time.
Therefore in a time dt the probability to undergo a collision is dt/τ, instead
(1- dt/τ) is the probability to move freely in dt without collisions.
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Ohm’s Law
At a macroscopic level, applying an electric field E to a metal a current I is
established, if V is the electrical potential we know
V=RI, R=resistance of the metal which depends on the metal type, length
L and section A of the wire.
R=ρL/A introducing the resistivity ρ.
In any point of the wire (V/L)= ρ I/A,
for a uniformly distributed current on a wire of section A → |j|=I/A
The microscopic Ohm’s law is
E= ρ j, with j the current density, vector parallel to the flow of charge
The number of electrons of density n crossing the area A in a interval dt
→ n A (v dt) and the transported charge in unit time → j =(-e) n vD
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Ohm Law
j =(-e) n vD
Where vD is the drift velocity
The acceleration of an electron is
a=(-e)E/m
Integrating in time v=[(-e)E/m] t
with the assumption that in average
there is a collision after an interval τ
the average (drift) velocity is
<v> = vD = [(-e)E/m] τ
=
௡௘ మ ఛ
E and the conductivity σ
௠
మ = 1/ρ =
Around room temperature ρ is linear in T but falls
off more steeply at lower temperatures
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This allows an estimation of
τ=
మ
at room temperature 10-1410-15 s
An estimation of the mean free path
between collisions is
= v0τ
by the equipartition theorem the
electron velocity is
½mv02=(3/2) kB T→ v0=105 m/s
and = 1 − 10Å
But at low T, increases
>>atomic spacing !
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Newton’s Law
Applying an electric field E a free electron of charge (-e) undergoes a force
f=(-e)E.
This force accelerate electrons but in average only for a time τ. The
collisions stop their motion and afterward electrons have no memory of
the previous motions.
The effect of the E is superimposed on the random thermal motions.
The momentum p=mv of an electron at time t+dt (p(t+dt)) changes due to
the force but only if there are no collisions in the time interval t(t+dt)
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Newton’s Law
The effect of the collisions is equivalent to a viscous damping force
(proportional to velocity) with viscosity coefficient (m/τ)
= − If the drift velocity is out of equilibrium (vD≠0) without any external force,
it decays to exponentially to zero with a relaxation time τ
This equation can be applied to treat the motion of electron under the
effect of electrical or magnetic fields
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Magnetic field: Hall effect
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Typical experimental setup to
measure Hall effect: an electric
field E is applied between red and
black leads therefore a current I is
flowing in the specimen and a
magnetic field B is applied in a
perpendicular direction (z)
Electrons undergo a Lorentz force
p×B
=(-e) ×B=(-e)
Therefore they tend to accumulate on a side of the specimen, on the other side
the lack of electron determines a positive charge
→ a transverse electric field is generated
The Newton’s law reads
−
ௗ‫ܘ‬
ௗ௧
=
‫ܘ‬
−
ఛ
+(-e)
‫×ܘ‬B
E+
௠
= −
௣ೣ
ఛ
௣೤
ఛ
+ − ௫ + −
+ − ௬ − −
−
௣೥
ఛ
௣೤ ஻೥
௠
௣ೣ ஻೥
௠
+ − ௭
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Magnetic field: Hall effect
=0
in a stationary state, multiplying by
and introducing the current density
j, imposing that there is not a transverse current
=
= → Hall coefficient =− which depends only on the electron
density and the electron charge (including the sign)
negative
Positive charge?
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AC electrical conductivity
The Drude model can be applied also in case of an alternating electric field
E(t)=Re(E(ω) )
= − +(-e)E
seeking a steady-state solution p(t)=Re(p(ω) )
ω(ω) = −
‫(ܘ‬ω )
ఛ
డ
[డ௧ = −]
+(-e)E(ω) and introducing j= p
మ
j(ω)=
೙೐
೘
ω → j(ω)=
బ
ω =σ() ω
1) No B since the Lorentz force due to magnetic field is negligible
2) equation is valid for mean free path ≪ λ of the electric field
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AC electrical conductivity
Considering the Maxwell equations (International System of Units SI)
∙ = 0 ∙ = 0 × = −
× × = −2=− ×
డ۰
డ௧
డ۰
డ௧
× = 0( + 0
డ۳
)
డ௧
taking the curl
= iω × = iω0(σ − ω
0)
−2 = iω0σ + ω 20
0 = iω0σ +
ω2
ω2
σ
=
(1 + i
)
2
2
0ω
ωଶ
Wave equation −2 = ଶ (ω) in a medium with dielectric constant
௖
σ
ఙబ
)
ω = (1 + i ఌ ன)= )=(1+iఌ ன(ଵି௜ఠఛ)
଴
଴
for frequencies >>1 ω =(1-
=
ଶ
଴
బ
௣ଶ
)=(1)
଴ଶ
ଶ
is the plasma (angular) frequency
The frequency is
ఠ௣
ଶగ
= 11.4
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௥ೞ ିଷ/ଶ
10ଵହ
௔బ
Hz
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AC electrical conductivity
For < ω is real and negative → the solution decay exponentially in space
(no wave propagation)
For > ω is real and positive → the solution is oscillatory and the wave
can propagate in the metal
Alkali metals become transparent in the ultraviolet region
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Plasma oscillations
Another consequence of ω =(1of continuity and Gauss law
−
= ∙ → i = ∙ → i =
଴
→ 1+
௣ଶ
)
ଶ
∙=
଴ can be found considering the equation
଴
= =0
௣ଶ
)
ଶ
for frequencies >>1 (1−
=0
hence a non trivial solution for ρ requires = Which implies the propagation of a charge density wave: Plasma oscillation
A simple model: the electron density is displaced by a small
amount d, at the sides two surface charge density σe=±nde
appear which produce an electric field E= σe/ε0,
Eq.of motion Nm = = −σe/ε0=−2 /ε0,
=−2/(mε0)d → =−‫݌‬2 d
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Thermal conductivity
Empirical observation: Metal conducts heat better than insulators therefore the
lattice contribution is less important and can be neglected
→ thermal current is carried only by electrons
Defining the thermal current density jT as
the thermal energy per unit time crossing a
unit area perpendicular to the flow with
direction parallel to the heat flow, for small
temperature gradient
(Fourier’s law) jT = −κ with κ the
thermal conductivity
• An empirical law (Wiedemann and
Franz) states that κ/σ ∝ T
(Lorenz number κ/(σ T) is constant)
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Thermal conductivity
In Drude model after a collision an electron has the speed appropriate to the
local temperature and the average speed is zero in any point is zero because
there is no net force on the electrons.
Electrons in a hotter
Electrons in a colder
place have a
place have a lower
higher energy.
energy.
Electron arriving from
Electron arriving from
other places, after a
other places, after a
collision, are
collision, are
thermalized at this
thermalized at this
temperature
temperature
On the other hand, before a collision, electrons arriving in the central part
from a hotter place have a higher energy with respect to electrons arriving
from a colder place → there is a net energy (heat) flow from hotter places to
colder ones, i.e. in the direction of heat flow.
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Thermal conductivity
Let’s consider a very simple model in which the heat flow is along the x direction
jTx = −κ
ௗ்
ௗ௫
and all the electron can only move along x and half of them are
arriving from the hotter region and half from the colder one.
Thermal energy per electron E=E(T) but T depends on the position x
the electron arriving in x on average have suffered the last collision in x+, with
=v , if they are arriving from the colder region on the right → E=E(T[x+v ])
and x- if they are arriving from the hotter region on the left → E=E(T[x-v ])
x- ݈
x
x+݈
vD
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Thermal conductivity
Due to the small mean free path we keep first order contributions
Cold region → E=E(T[x+v ]) ≈ E(T[x])+
v = Ec
Hot region → E=E(T[x-v ]) ≈ E(T[x])+
(−v )= Eh
The thermal current density from the colder region is jcTx = (n/2) (−v ) Ec =
(n/2) (−v ) E(T[x])+
v = ( )(−v E(T[x]) −
v 2)
from the hotter region is jhTx = (n/2) (+v ) Eh= ( )(v E(T[x]) −
And the total current density jTx = jcTx + jhTx =−n
x- ݈
x
v 2)
ௗா ௗ்
v 2
ௗ் ௗ௫ ஽
x+݈
vD
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Thermal conductivity
But n=N/V (N number of electron in the volume V)
→n
ௗா
ௗா
ௗா
=N /V= ்
ௗ்
ௗ்
ௗ்
/V=cv
where is the total energy of N electrons and cv is the electronic specific heat
To extend the result in the 3D case, we observe that v inthepresentcase is
the x-component of the speed.
Assuming an isotropic velocity distribution =(v , v, v)
< v2 > = < v2 > = < v2 > = ⁄ vD2 mean square electronic speed
jT = − ଵ⁄ଷ vD2cv → κ= ଵ⁄ଷ vD2cv = ଵ⁄ଷ vD2cv
Dividing by the electrical conductivity σ
κ ୈ2
=
σ ଶ
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Thermal conductivity
Drude calculated this ratio by using results of the classical ideal gas
E=½vD2 = and cv= where is the Boltzmann’s constant
κ =
σ ஻ T
trend in agreement with the empirical Wiedemann and Franz law
κ
= 1.11 10-8 watt-ohm/K2.
σ୘
→ the estimated value is about half of the experimental one.
On the other hand the electronic contribution cv is too large by a factor 100 but
this compensates the speed which is a 100 fold too small. Hence it was a
fortuitous cancellations of error that gave almost the correct value.
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Thermoelectrical effect
We have approximated the velocity with a single value v in x but its value
should be different from the two sides
vh > vc → mean velocity different from zero.
This fact add another contribution to the thermal current and, in turn, a different
velocity means that there is a flow of charges from the hotter region to the
colder one → electrical current
On the other hand thermal conductivity measurements are performed in open
circuit condition therefore current accumulates electrons on the colder region
producing a retarding field that allows the system to reach steady conditions
where no current can flow → mean velocity is zero.
x- ݈
x
x+݈
vD
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Thermoelectrical effect
As observed a thermal gradient is accompanied by an electric field (Seebeck
effect) E=Q ∇T with Q the thermo power.
The mean velocity in x isvொ = (vh + vh )/2=[v(x-v )+v(x+v )] ≈ ଶ
ವ
v =
- ವ /2
Extension to 3D, assuming an isotropic velocity distribution =(v , v, v)
< v2 > = < v2 > = < v2 > = ⁄ vD2 mean square electronic speed
And considering the velocity function of T(x)
ଶ
vொ = ⁄ (− ವ ) ∇T
the drift velocity due to the electric field is v = −
For the steady condition v +v = 0 → Q=−(⁄)
x- ݈
x
భ
ವ ଶ
మ
=−
!౬
x+݈
vD
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Thermoelectrical effect
Q=−
!౬
Drude used the classical relationship cv= → Q=− ஻ = −0.4310" /
Experimental values of Q are 100 fold smaller than the Drude value.
For the Wiedemann-Franz law there is compensation of errors which is not
present here
→ Classical mechanics cannot be applied to treat the dynamics of electrons in a
solid although in some cases can provide a semi-quantitative model.
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