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Fermi Surface: Notes By: Shahzaib Shahid

The document discusses the Fermi surface model of electrons in metals. It explains that: 1) At zero temperature, electron velocities in a metal fill a sphere in velocity space called the Fermi sphere, with all points inside occupied and outside unoccupied. 2) The radius of the Fermi sphere is called the Fermi velocity, which is independent of temperature. Raising the temperature slightly excites a few electrons from inside to outside the sphere. 3) Applying an electric field displaces the Fermi sphere, producing a small number of "uncompensated" electrons that generate a net current proportional to their drift velocity.

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Shazaib Mirza
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392 views7 pages

Fermi Surface: Notes By: Shahzaib Shahid

The document discusses the Fermi surface model of electrons in metals. It explains that: 1) At zero temperature, electron velocities in a metal fill a sphere in velocity space called the Fermi sphere, with all points inside occupied and outside unoccupied. 2) The radius of the Fermi sphere is called the Fermi velocity, which is independent of temperature. Raising the temperature slightly excites a few electrons from inside to outside the sphere. 3) Applying an electric field displaces the Fermi sphere, producing a small number of "uncompensated" electrons that generate a net current proportional to their drift velocity.

Uploaded by

Shazaib Mirza
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
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Fermi Surface

Notes By:
SHAHZAIB SHAHID
The electrons in metal are continuous in random motion and their energy is
purely Kinetic

𝟏
𝑲. 𝑬 = 𝒎𝒗𝟐
𝟐
Let us introduce velocity space with axes 𝑣𝑥 𝑣𝑦 𝑣𝑧

Each point in space represents different velocity


and magnitude.
Consider electrons in this velocity space----
Velocity of each electron is random so the
points representing them fill up the sphere
uniformly. At Temperature T= 0
There Exists a sphere outside which all i) The points inside the sphere
points are empty , this sphere is called b are al occupied
Fermi sphere ,its surface is called Fermi ii) The points outside the
surface . sphere are all unoccupied
The radius of this sphere is called Fermi because there energy is
greater then Fermi energy
speed 𝑣𝑓
𝐸𝑓
There is no appreciable affects of temperature on Fermi surface . The Fermi velocity is
Independent of Temperature .
When temperature is raised only few electrons from inside of Fermi surface are
excited to outer surface of sphere .

If we substitute Energy as 𝐸𝑓 for energy of electrons we have

1
𝐸𝑓 = 𝑚𝑣𝑓 2
2
So Fermi velocity is
2𝐸𝑓
𝑣𝑓 =
𝑚
Electrical conductivity : Effects of the Fermi surface
In absence of Electric field:
i) Fermi sphere is centered at origin
ii) Electrons are moving randomly each carrying individual current

The total current of the system is zero because for each electron with v there exist
electron with -v .

Fermi
Sphere in
absence of
Electric field

FS at
Equilibrium
The Situation changes when field is applied

If the Field is applied in Each Electron 𝒆𝝉


+ve x direction acquire
𝒗𝒅 = −( )
𝒎
Drift velocity

The whole Fermi sphere


is displaced to left ,
although the
displacement is very
small.

The electrons in the shaded area are called


uncompensated electrons which produce currents
The calculation of Current density is as follows:
𝑣𝑑
The fraction of uncompensated electrons =
𝑣𝑓

𝑣𝑑
Concentration of these electrons = N( )
𝑣𝑓
Each electron has velocity = -𝑣𝑓
So The current density is :
𝒗𝒅
𝑱 = −𝒆(𝑵 ( ) ) (−𝒗𝒇 )
𝒗𝒇

𝑱 = eN𝒗𝒅

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