4. Free Electron Models - Dipartimento di Fisica | Università …bracco/serpchem/4a_Free_Electro… · · 2014-10-134. Free Electron Models ... to the electrical conductivity was

4. Free Electron Models4. Free Electron Models

G. Bracco - Material Science - SERP CHEM 1

� Introduction

� Drude Model� Ohm’s law

� Newton’s law

� Hall effect

� AC conductivity

� Plasma oscillations

� Thermal conductivity

� Thermoelectric effect

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 <v

T2>≠0

therefore the average thermal energy <½mvT

2 > ≠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.

Introduction


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.

Kinetic theory

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

→ ρρρρe=Z N

Aρρρρm

/A


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


• 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.

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/ττττ, ττττ=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.


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

Ohm’s 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 τthe average (drift) velocity is

<v> = vD

= [(-e)E/m] τ

� =��

�E and the conductivity σ

= 1/ρ = ��

�

Ohm Law

Around room temperature ρ is linear in T but falls

off more steeply at lower temperatures G. Bracco - Material Science - SERP CHEM 8

This allows an estimation of

τ= �

��

at room temperature 10-1410-15 s

An estimation of the mean free path

between collisions is

� = v0τ

by the equipartition theorem the

electron velocity is

½mv0

2=(3/2) kB

T→ v0=105 m/s

and � = 1 − 10Å

But at low T, � increases

�>>atomic spacing !


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)

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

Newton’s Law



Magnetic field: Hall effectTypical 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

�� =(-e)�� ×B=(-e)p×B�

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

��

��= −

�

�+(-e) E+

�×B

� =

−��

�+ −� � + −�

��

�

−��

�+ −� �� − −�

��

�

−��

�+ −� ��


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)

Positive charge?

negative


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) = −�(ω)

�+(-e)E(ω) and introducing j=

�

�p

j(ω)=

��

�

�� ω → j(ω)=

��

�� 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


AC electrical conductivityConsidering the Maxwell equations (International System of Units SI)

� ∙ � = 0� ∙ � = 0� × � = −��

�� × � = �

0(+

0

��

��) taking the curl

� × � × � = −�2�=−� ×��

��= iω� × � = iω�

0(σ�− �ω

0�)

−�2� = iω�0σ + ω

2�00� = iω�

0σ+

ω2

�2� =

ω2

�2(1 + i

σ

0ω)�

Wave equation −�2� =ω

�

��(ω)� in a medium with dielectric constant

ω = (1 + iσ

��)= )=(1+i

�

��(�� )

)

for frequencies � >>1 � ω =(1-��

��

)=(1-�

�

�

��)

�� =��

��

is the plasma (angular) frequencyThe frequency is

��

��= 11.4

��

��

��/�

10�� Hz


AC electrical conductivityFor �< �� 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


Plasma oscillations

Another consequence of � ω =(1-�

�

�

��) can be found considering the equation

of continuity and Gauss law

−��

��= � ∙ � → i�� = � ∙ � � ∙ � =

�

��

� = ��

→ i�� = ��

��

→ � 1 + ��

��

= 0

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/ε

0,

Eq.of motion Nm�� = �� = −��σe/ε

0=−��2�� /ε

0,

��=−�2�/(mε0)d → ��=−��

2 d


Thermal conductivityEmpirical 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)


Thermal conductivityIn 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.

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.

Electrons in a hotter

place have a

higher energy.

Electron arriving from

other places, after a

collision, are

thermalized at this

temperature

Electrons in a colder

place have a lower

energy.

Electron arriving from

other places, after a

collision, are

thermalized at this

temperature


Thermal conductivity

Let’s consider a very simple model in which the heat flow is along the x direction

jTx= −κ

�

�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


Thermal conductivityDue 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]) −

��

��

��

��v�

2�)

And the total current density jTx

= jcTx

+ jhTx

=−n ��

�

�

�v�

2�

x x+�x- �

vD


Thermal conductivityBut n=N/V (N number of electron in the volume V)

→n ��

� =N

��

� /V=

��

� /V=c

v

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�)

< v�2 > = < v�

2 > = < v�2 > = �

�⁄ vD2 mean square electronic speed

jT

= − ��⁄ v

D2�c

v� → κ= � �⁄ v

D2�c

v= �

�⁄ vD

2�cv

Dividing by the electrical conductivity σ

κ

σ=��

2��

��


Thermal conductivity

Drude calculated this ratio by using results of the classical ideal gas

E=½�vD2 =

�

�� and c

v=�

�� where ��

is the Boltzmann’s constant

κ

σ=�

�

��

�

T trend in agreement with the empirical Wiedemann and Franz law

κ

σ�= 1.11 10-8 watt-ohm/K2.

→ 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.


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



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 isv�= (vh

� + vh�)/2=[v(x-v��)+v(x+v��)] ≈ -

��

��v��=

-��

�

��/2

Extension to 3D, assuming an isotropic velocity distribution �=(v�, v�, v�)

< v�2 > = < v�

2 > = < v�2 > = �

�⁄ vD2 mean square electronic speed

And considering the velocity function of T(x)

v�= �

�⁄ (−��

�

��) ∇T

the drift velocity due to the electric field is v�= −�

�

For the steady condition v +v�= 0 → Q=−(� �⁄ )��

��

�

��=−

!

��

x x+�x- �

vD



Q=−!

��

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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4. Free Electron Models - Dipartimento di Fisica | Università …bracco/serpchem/4a_Free_Electro… · · 2014-10-134. Free Electron Models ... to the electrical conductivity was