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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

=

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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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