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FREE ELECTRON THEORY
ARC
TOPICS TO BE COVERED
Classical free electron theory and its limitations
Deduction of Ohm’s law, Statement of Weidemann-Franz law
Quantum theory of free electrons, Fermi level
Density of states, Fermi-Dirac distribution
Thermionic emission, Richardson equation
Chief Characteristics of Metals
• Metal possesses high electrical and thermal conductivity
• Metals obey Ohm’s law
• Conductivity of metals decreases with rise of temperature
• Metals obey Wiedemann-Franz law
• Near absolute zero, certain metals exhibit superconductivity
Classical Free Electron Theory
• Initially stated by Drude in 1900
 He was working
prior to the
development of
quantum
mechanics, so he
began with a
classical model.
(1863-1906)
CONCEPT
• In Drude model, the valence electrons from each atom become detached
and wander freely through the metal, while the metallic ions remain intact
and play the role of the immobile positive particles.
• Each electron behaves as a perfect gas molecule
• Each electron is free to move through the entire volume of the metal
• System of free electrons in a metal = free electron gas
Free Electron Model
Schematic model of metallic crystal,
such as Na, Li, K, etc.
Equilibrium positions of the atomic cores are
positioned on the crystal lattice
+
+
+
+
+
and surrounded by a sea of conduction
electrons.
+
+
+
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+
+
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Drude’s Assumptions
1. Matter consists of light negatively charged electrons which are mobile & heavy, positively
charged ions.
2. The only interactions are electron-ion collisions, which take place in a very short time t.
• The neglect of the electron-electron interactions is THE INDEPENDENT ELECTRON
APPROXIMATION.
• The neglect of the electron-ion interactions is THE FREE ELECTRON APPROXIMATION
3. Electron-ion collisions are assumed to dominate. These will abruptly alter the electron
velocity & maintain thermal equilibrium.
4. The mean time between collisions is 𝜏
5. Electrons emerge from each collision with their velocity changed.
•Till the application of an external
electric field, the electrons move
about in a random manner
Drift Velocity
In presence of applied electric field, electrons move in a specific direction.
This directional motion of the free electrons is called DRIFT.
Average velocity gained during this drift motion is called DRIFT VELOCITY.
Steady state drift velocity produced for unit electric field is called MOBILITY
(μ)
Fig. Ref. Google
Relaxation Time (𝜏)
When the applied electric field is switched off, the electrons again
undergo collision. The electron gas resumes its equilibrium condition.
Such a process which leads to the establishment of equilibrium in a
system from which it was previously disturbed is called the relaxation
process. The time taken for this process is RELAXATION TIME.
< 𝑣𝑥 > = < 𝑣𝑥 >0 𝑒 −𝑡
τ
Mean free path (λ)
It is the average distance travelled by the conduction electron between
successive collisions with the lattice ions.
Mean collision time (𝜏𝑪 )
The average time taken by an electron between two successive collisions of an
electron with lattice points during its motion. (averaging is done over a large
number of collisions)
Drift Velocity Expression
An electric field is applied. The equation of motion of free electron of mass 𝑚𝑒 is
Integrating, we get
If
is the average time between collisions then the average
drift velocity is
𝒆𝑬𝝉
𝒗𝒅 = −
𝒎
10
Ohm’s Law
• Basic law concerning the flow of electricity.
• Ohm's law states that the current through a conductor between two
points is directly proportional to the potential difference across the
two points.
• Constant of proportionality, resistance, is introduced
• In mathematical terms, V = I x R where V is voltage, I is current, and R
is resistance
• When an electric field, E is applied to a conductor, an electric current
begins to flow, and the current density by Ohm’s law is J  E
• Materials that obey Ohm’s law are said to be ohmic
E
Ohm’s Law
Experimental observation:
V I
V  RI
L
 I
A
I
 L  LJ
A
V
 J
L
E  J
If J = current density for electric field E, then
J  E , where σ = conductivity
Amount of charge passing per unit time = -𝑛𝑒𝑣𝑑 𝐴 = I
So, current density
𝐼 −𝑛𝑒𝑣𝑑 𝐴
J= =
= −𝑛𝑒𝑣𝑑 ……… (1)
𝐴
Distance covered =𝑙 = 𝑣𝑑 𝑡
𝐴
We know,
𝒆𝑬𝝉
𝒗𝒅 = −
𝒎
………… (2)
S𝑜, 𝑣𝑜𝑙𝑢𝑚𝑒 𝑐𝑜𝑣𝑒𝑟𝑒𝑑 𝑖𝑛 𝑢𝑛𝑖𝑡 𝑡𝑖𝑚𝑒 = 𝑣𝑑 𝐴
From (1) and (2),
𝑛𝑒 2 𝞽 𝐸
𝐽=
𝑚
σ
This is the form of Ohm’s Law in terms of free electrons.
Wiedemann-Franz Law
Wiedemann and Franz law states that the ratio of thermal and electrical
conductivity of all metals is constant at a given temperature
(for room temperature and above).
Thermal conductivity
Electrical Conductivity
𝐾
σ
= constant
Later it was found by L. Lorenz that this constant is proportional to the absolute temperature
𝐾
3 𝑘𝐵2
=
𝑇
2
σ
2 𝑒
L = Lorentz Number
Drawbacks of Classical Free Electron Theory
• Specific Heat:
Classical free electron theory
all valence electrons in a metal can absorb
thermal energy. So, molar electronic specific heat
1.5 times R, where R =
universal gas constant.
This is about 100 times greater than experimentally predicted values.
• Mean free path:
Experimental value of λ is much greater than the theoretical value
• Temperature (T) Dependence of Electrical Conductivity(σ)
Classical free electron theory
σ is inversely proportional to 𝑇
experiments
σ is inversely proportional to 𝑇
Drawbacks (continued..)
• Wiedemann- Franz law:
At low temperatures, K/σT is not a constant. But in classical free electron theory,
it is a constant at all temperatures.
• Paramagnetism of Metals:
Theoretical value of paramagnetic susceptibility is greater than the experimental
value. Experimental fact that paramagnetism of metals is nearly independent of
temperature could not be explained
Salient features of Quantum Free Electron Theory
Proposed by Sommerfeld in 1928
•
Electrons obey the laws of quantum mechanics
•
Energy levels of electrons are quantized
•
Electrons occupy energy orbitals according to Pauli’s exclusion principle
•
Distribution of electrons in different energy levels are according to FermiDirac statistics
•
Retained concept of free electrons moving in a uniform potential but
prevented them from escaping the crystal by very high potential barriers at
the surfaces
FERMI LEVEL : Highest energy level occupied by electrons at Absolute zero. All the energy
states upto Fermi level are OCCUPIED and all energy levels above Fermi level are VACANT.
FERMI ENERGY: Energy corresponding to Fermi Level.
Constant for a particular system
Probability of an electron occupying a particular energy level ‘E’ is given by
Fermi-Dirac Distribution
f (E) 
1
e
( E  E F ) / k BT
1
On increasing the temperature, electrons get excited to higher energy
level.
Distribution of electrons in different energy levels gets determined by
Fermi-Dirac Distribution function.
At T = 0 K and for E < E F , f(E) = 1
for E > EF , f(E) = 0
For lower energies,
f tends to 1.
For higher energies,
f tends to 0.
Fermi Distribution Function at Different Temperatures
For temperatures greater than zero, Fermi function plot begins to fall close to E F
𝟏
and at E = EF , f(E) =
𝟐
FERMI VELOCITY = velocity associated with Fermi Energy
i.e. velocity of electrons occupying Fermi Level
FOR SODIUM,
𝐸𝐹 = 3.2 𝑒𝑉
1
𝑚𝑣𝐹2
2
𝑣𝐹 =
= 𝐸𝐹 = 3.2 X 1.6 X 10−19 J
2𝐸𝐹
𝑀
So, 𝑣𝐹 = 1.1 𝑋 106 𝑚/𝑠
FERMI VELOCITY
FERMI TEMPERATURE = Temperature associated with Fermi energy
𝑘𝐵 𝑇𝐹 = 𝐸𝐹
So, 𝑇𝐹 = 37100 𝐾
Ref: Eisberg, R. and Resnick, R. Quantum Physics of Atoms, Molecules, Solids, Nuclei, and Particles, 2nd ed. New York: Wiley, 1985.
FERMI TEMPERATURE
DENSITY OF ENERGY STATES
• In a macroscopically small energy interval, there are many discrete energy levels.
• Difference between neighbouring energy levels is as small as 10−6 eV
CONCEPT OF DENSITY OF ENERGY STATES simplifies OUR CALCULATION. It helps in finding
the number of energy states with a specific energy.
Density of states (DOS) of a system describes the number of available states in a unit volume
per unit energy range.
In a system, if N(E) = number of electrons with energy E,
g(E) = number of energy states with energy E,
f(E) = probability of an electron to occupy energy state E,
then
N(E) dE = g(E)dE f(E)
We consider a free electron of mass ‘m’ trapped inside a cubical metal block of side length, ‘a’.
According to quantum mechanics, energy of the free electron,
𝐸=
𝑛2 ℎ 2
----- (1) where h = Planck’s constant
2
8𝑚𝑎
where 𝑛2 = 𝑛𝑥2 + 𝑛𝑦2 + 𝑛𝑧2
𝑛𝑥 , 𝑛𝑦 , 𝑛𝑧 𝑎𝑟𝑒 𝑝𝑜𝑠𝑖𝑡𝑖𝑣𝑒 𝑖𝑛𝑡𝑒𝑔𝑒𝑟𝑠 𝑔𝑟𝑒𝑎𝑡𝑒𝑟 𝑡ℎ𝑎𝑛 𝑧𝑒𝑟𝑜
Let us consider a space of points represented by coordinate system 𝑛𝑥 , 𝑛𝑦, 𝑛𝑧 along the three
mutually perpendicular directions.
Let each point with integer values of the coordinates represent an energy state.
Let n be the radius vector from origin (0,0,0) to a point represented by (𝑛𝑥 , 𝑛𝑦, 𝑛𝑧 ).
So, 𝑛2 = 𝑛𝑥2 + 𝑛𝑦2 + 𝑛𝑧2 -------- (2)
All points on the surface of the sphere of radius ‘n’ will have the same energy.
As per quantum condition, values of 𝑛𝑥 , 𝑛𝑦, 𝑛𝑧 are restricted to be positive.
Only in one octant of the sphere, each point corresponds to only positive values of 𝑛𝑥 , 𝑛𝑦, 𝑛𝑧 .
SPACE OF POINTS
If n = radius of sphere whose octant encloses all the points upto an energy ‘E’, then
Number of allowed energy values upto an energy E
= number of points in the octant of sphere of radius ‘n’
We consider another sphere of radius n+dn whose octant encloses all points upto an energy ‘E+dE’, then
Number of allowed energy values upto an energy E + dE
= number of points in the octant of sphere of radius ‘n + dn’
So, number of allowed energy states in energy range dE
= number of points in the space between two octant shells of radii
n and n+dn
= (volume of space between two octant shells of radii n and n+dn)
X (number of points / unit volume)
1
=
4 π 𝑛2 𝑑𝑛 𝑋 1
8
1
2
= π 𝑛2 𝑑𝑛
----------------
(3)
Since 𝑛𝑥 , 𝑛𝑦, 𝑛𝑧 are all integers,
A unit volume of plot consists of
Just one point
If g(E) = number of energy states per unit energy range, then number of energy states in the energy interval dE
= g(E)dE
π
So, g(E) dE =
2𝐸
8𝑚𝑎
From (1), 𝑛2 =
ℎ2
𝑛=
2
𝑛 𝑛 𝑑𝑛
------------ (4)
---------- (5)
8𝑚𝑎2 𝐸
ℎ2
1
2
---------- (6)
Differentiating (5) , we get
2 𝑛 𝑑𝑛 =
So,
𝑛 𝑑𝑛 =
8𝑚𝑎2
𝑑𝐸
ℎ2 2
1 8𝑚𝑎
2
ℎ2
dE
----------- (7)
Using (4), (6) and (7), we get
π
𝑔 𝐸 𝑑𝐸 =
4
3
8𝑚𝑎2 2
ℎ2
1
𝐸2
𝑑𝐸
---------- (8)
Each energy value is applicable to two energy states, one for an electron with spin-up, and the
other for an electron with spin down (Pauli’s exclusion principle).
So, the number of allowed energy states in the energy interval dE
𝑔 𝐸 𝑑𝐸 =
=
π
2𝑋
4
3
8𝑚𝑎2 2
ℎ2
3
8𝑚𝑎2 2
π
2
ℎ2
1
2
𝐸 𝑑𝐸
1
2
𝐸 𝑑𝐸
Hence, the number of energy states present in unit volume having energy values lying between E and E + dE (DOS) is given by
=
π 8𝑚
2 ℎ2
3
2
1
2
𝐸 𝑑𝐸
[𝑎3 = 𝑣𝑜𝑙𝑢𝑚𝑒 𝑜𝑓 𝑚𝑒𝑡𝑎𝑙 𝑏𝑙𝑜𝑐𝑘 𝑖𝑛 𝑤ℎ𝑖𝑐ℎ 𝑡ℎ𝑒
𝑒𝑙𝑒𝑐𝑡𝑟𝑜𝑛𝑠 𝑎𝑟𝑒 𝑝𝑟𝑒𝑠𝑒𝑛𝑡]
Density of energy states for a free electron gas
General Expression for
𝑁 𝐸 𝑑𝐸 = =
π
2
3
8𝑚𝑎2 2
ℎ2
1
2
𝐸 𝑑𝐸 𝑓(𝐸)
TASK: Find out the number of electrons present per unit volume of a cubical
metal block at absolute zero temperature
Thermionic Emission
The emission of electrons from a metal under the effect of thermal energy is called
THERMIONIC EMISSION.
Emitted electrons are called THERMIONS.
Electrons are free to move inside the metal
Electrons cannot come out of the metal surface on its own as high potential barrier is present at
the surface
but when the temperature of the metal is sufficiently high, electrons gain sufficient energy to
overcome the barrier and ESCAPE from the metal surface
Free electron theory assumes that the potential within the metal is constant.
The minimum energy to be supplied to the electron for its emission from the metal is termed as
WORK FUNCTION (Ф) of the metal
Richardson’s Equation
If W = minimum energy of the electron for its emission from the surface, E F= Fermi energy of the metal,
then, Ф = W – EF = work function of the metal.
No. of energy states / unit volume in energy range E to E + dE,
π 8𝑚
D(E)dE=
2 ℎ2
3
2
1
2
𝐸 𝑑𝐸
We know, 𝐸 = 𝑝2 2𝑚,
So, density of energy states per unit volume
in momentum range p to p+dp,
8π𝑝2 𝑑𝑝
2
2
𝐷 𝑝 𝑑𝑝 =
=
4π𝑝
𝑑𝑝
ℎ3
ℎ3
----------- (1)
We construct a plot in ‘momentum space’ such that each point represents a particular combination of
momenta components 𝑝𝑥 , 𝑝𝑦, 𝑝𝑧 of an electron along x-, y- and z-directions. So,
𝑝2 = 𝑝𝑥2 + 𝑝𝑦2 + 𝑝𝑧2
Volume element in momentum space,
𝑑𝑝𝑥 𝑑𝑝𝑦 𝑑𝑝𝑍 = 4π𝑝2 𝑑𝑝
-------- (2)
From (1) and (2), the density of states in momentum space,
𝐷(𝑝𝑥 𝑝𝑦 𝑝𝑧 )𝑑𝑝𝑥 𝑑𝑝𝑦 𝑑𝑝𝑧 =
2
𝑑𝑝𝑥 𝑑𝑝𝑦 𝑑𝑝𝑧
ℎ3
---------
(3)
Hence, no. of electrons/unit volume having momenta in the range 𝑝𝑥 and 𝑝𝑥 + 𝑑𝑝𝑥 , 𝑝𝑦 and 𝑝𝑦 + 𝑑𝑝𝑦 , 𝑝𝑧 and 𝑝𝑧 + 𝑑𝑝𝑍 ,
𝑛(𝑝𝑥 𝑝𝑦 𝑝𝑧 )𝑑𝑝𝑥 𝑑𝑝𝑦 𝑑𝑝𝑧 = ℎ23
𝑑𝑝𝑥 𝑑𝑝𝑦 𝑑𝑝𝑧
−−−−− −
𝐸 − 𝐸𝐹
𝑒 𝑘𝑇 +1
(4)
Now, we consider the metal plate to be in Y-Z plane. Electrons will be emitted in a direction perpendicular to Y-Z plane
i.e. along x-axis. Only those electrons will be emitted whose energy, E > W.
𝐸 − 𝐸𝐹 ≫ kT. So, 1 in denominator can be neglected. We know, 𝐸 = 𝑝2 2𝑚
2 𝐸𝐹
𝑛 𝑥 = 3 𝑒 𝑘𝑇
ℎ
2 𝐸𝐹
𝑛 𝑥 = 3 𝑒 𝑘𝑇
ℎ
∞
∞
∞
𝑒 −(𝑝
𝑝𝑥0
∞
2
2𝑚𝑘𝑇
2
2
)𝑑𝑝𝑥 𝑑𝑝𝑦 𝑑𝑝𝑧
−∞ −∞
∞
∞
2
𝑒 −(𝑝𝑥 + 𝑝𝑦 + 𝑝𝑧 )
𝑝𝑥0
−∞ −∞
2𝑚𝑘𝑇
)𝑑𝑝𝑥 𝑑𝑝𝑦 𝑑𝑝𝑧
2 𝐸𝐹
𝑛 𝑥 = 3 𝑒 𝑘𝑇
ℎ
∞
∞
∞
2
𝑒 −𝑝𝑥 2𝑚𝑘𝑇 𝑑𝑝𝑥
𝑝𝑥0
𝑒
2 + 𝑝2 ) 2𝑚𝑘𝑇
−(𝑝𝑦
𝑧
𝑑𝑝𝑦 𝑑𝑝𝑧
−∞ −∞
Standard Integral Form
∞
2
𝑒 −𝛼𝑥 𝑑𝑥 =
−∞
4𝜋𝑚𝑘𝑇 𝐸𝐹
𝑛 𝑥 =
𝑒 𝑘𝑇
3
ℎ
∞
𝑒
−𝑝𝑥2 2𝑚𝑘𝑇
𝑑𝑝𝑥
𝑝𝑥0
So, current density,
𝑝𝑥
𝐽 = 𝑛 𝑥 𝑒𝑣𝑥 = 𝑛 𝑥 𝑒
𝑚
=
=
𝑒 4𝜋𝑚𝑘𝑇 𝐸 /𝑘𝑇 ∞ −𝑝2 2𝑚𝑘𝑇
𝑥
𝑒 𝐹
𝑒
𝑝𝑥 𝑑𝑝𝑥
3
𝑝
𝑚 ℎ
𝑥0
𝑒 4𝜋𝑚𝑘𝑇 𝐸 /𝑘𝑇 −𝑝2 2𝑚𝑘𝑇
𝐹
𝑥
𝑒
[𝑒
(2𝑝𝑥
𝑚 ℎ3
2𝑚𝑘𝑇) 𝑝𝑥 ]
∞
𝑝𝑥0
𝜋
𝛼
J =
𝑒 4𝜋𝑚𝑘𝑇 𝐸 /𝑘𝑇 −𝑝2 2𝑚𝑘𝑇
𝐹
𝑒
. 𝑒 𝑥0
(𝑚𝑘𝑇)
3
𝑚 ℎ
J=
4𝜋𝑚𝑘 2 𝑇 2 𝐸 /𝑘𝑇 −𝑊 𝑘𝑇
𝑒
𝑒 𝐹 .𝑒
3
ℎ
2
[since 𝑊 = 𝑝𝑥0
2𝑚]
J=
4𝜋𝑚𝑘 2 𝑇 2 𝐸 /𝑘𝑇 −(𝐸 + Ф)/𝑘𝑇
𝐹
𝐹
𝑒
𝑒
.
𝑒
ℎ3
[since 𝑊 = 𝐸𝐹 + Ф]
J=
4𝜋𝑚𝑒𝑘 2
ℎ3
𝑇 2 𝑒 − Ф/𝑘𝑇
𝐽 = 𝐴 𝑇 2 𝑒 −Ф
𝑘𝑇