299
Lab Experiments
Experiment-133
S
FERMI ENERGY
Dr .D Sudhakar Rao and Ms Chaithra G M
Dept of Physics, St. Aloysius College, Mangalore-575 003. INDIA.
Email:[email protected]
Abstract
Fermi energy and Fermi temperature of copper, iron, gold and silver is
determined by studying resistance variations at different temperatures. The values
obtained are compared with the standard values.
Introduction
“Fermi level” is the term used to describe the top of the collection of electronic energy levels
at absolute zero temperature. This concept comes from Fermi – Dirac statistics. Electrons are
fermions and by the Paulis Exclusion Principle cannot exist in identical energy states. So at
absolute zero they pack into the lowest available energy states and build up a “Fermi sea” of
electron energy states. The Fermi level is the surface of that sea at absolute zero where no
electrons will have enough energy to rise above the surface. The concept of the Fermi energy
is important for the understanding of the electrical and thermal properties of the solids. Both
ordinary electrical and thermal processes involve energies of a small fraction of an electron
volt. But the Fermi energies of metals are of the order of few electron volts. This implies that
the vast majority of the electrons cannot receive energy for these processes because there are
no available energy states for them to go to within a fraction of an electron volt of their
present energy. At higher temperatures a certain fraction, characterized by the Fermi
function, will exist above the Fermi level. For a metal, the density of conduction electrons
can be implied from the Fermi energy. The Fermi energy also plays an important role in
understanding the mystery of why electrons do not contribute significantly to the specific
heat of solids at ordinary temperatures.
Further, in metals, Fermi energy gives us information about the velocities of the electrons
which participate in ordinary electrical conduction. The Fermi velocity VF of these
conduction electrons can be calculated from the Fermi energy EF using the relation,
VF =
Where
2E F
m
…1
m = 9.1 x 10-31 kg is the mass of electron.
EF is Fermi Energy
VF is Fermi Velocity
This speed is a part of the microscopic Ohm’s Law for electrical conduction. A Fermi gas is a
collection of non-interacting fermions. It is quantum mechanical version of ideal gas.
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Electrons in metals and semiconductors can be approximately considered as Fermi gases. The
energy distribution of the fermions in a Fermi gas in thermal equilibrium is determined by
their density, the temperature and the set of available energy states using Fermi-Dirac
statistics. It is possible to define a Fermi temperature below which the gas can be considered
degenerate. This temperature depends on the mass of the fermions and the energy. For
metals, the electron gas’s Fermi temperature is generally many thousands of Kelvin, so they
can be considered degenerate. Fermi temperature TF can be obtained by the relation
EF = kTF
…2
Where k = 1.38 x 10-23 J K-1 is Boltzmann constant.
Theory
The number of free electrons in metal per unit volume is given by,
n=
Nρ
M
Where
…3
N = 6.023 x 1026 per m3 is Avogadro number
ρ = density of the metal
M = Mass number of the metal
The electrical conductivity of the metal,
σ=
L
Ra
Where
…4
L is the length of the metal wire
R is its resistance at a reference temperature
a is the area of cross-section of the wire.
The relaxation time is given by,
τ=
σm
ne 2
…5
Where e = 1.602 x 10-19 C is electron charge.
If VF is Fermi velocity, then mean free path of electrons,
λF =VFτ
…6
Now Fermi energy,
2
 ne 2 πAr 2   ∆R  2
 x
EF = 

 L (2m)   ∆T 
…7
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301
Where the constant A = λF x T
T is the reference temperature of the wire in Kelvin,
r is the radius of the wire
∆R
is the slope of the straight line obtained by plotting resistance of the
∆T
metal against absolute temperature of the metal.
Once the Fermi energy is found, Fermi temperature can be calculated using the equation-2.
Apparatus Used
DC regulated power supply, digital milli ammeter, Digital milli voltmeter (DMM), Heating
arrangements, Thermometer 0-160 degree, and Wires of copper, gold, silver and iron. The
complete experimental setup is shown in Figure-1
Figure-1, Fermi Energy Experimental Setup
Experimental Procedure
1. About 2 meter length copper wire is taken and its radius is determined and cross sectional
area is calculated. Its mass number and density are noted from Clark’s table.
LCopper = 2.2m, Radius r =0.165x10-3m
Cross sectional area =πr2 = 85.52x10-9m2
Density ρ = 8930 Kg m-3
Mass number M = 63.54 gm
2. The wire is wound over an insulating tube (10-15mm dia) to form a coil. The coil is
immersed in pre heated liquid paraffin as shown in the experimental setup. The two end
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Lab Experiments
of the coiled wire is connected to a power supply through a milli ammeter. And milli
voltmeter is connected across the coil.
3. A thermometer is immersed in the beaker containing liquid paraffin and coil. When the
thermometer attains steady temperature the temperature is noted.
4. The power supply is switched on and voltage and currents are noted In Table-1. The
liquid is allowed to cool and power supply is switched off until another steady
temperature is reached.
5. Trial is repeated taking reading in the interval of 5 degree and until the temperature reach
45 degree. At each temperature the voltages and currents measured are noted in Table-1.
6. A graph is drawn taking temperature in degree K along X-axis and resistance on Y axis as
shown in Figure-2. The slope of straight line is calculated.
Resistance (Ohms)
Table-1
Temperature
Voltage
Current
ºC
ºK
(mV)
(mA)
98
371
24.6
42.9
90
363
24.2
43.1
85
358
23.8
43.2
80
353
23.5
43.2
75
348
23.1
43.1
70
343
22.8
43.1
65
338
22.5
43.2
60
333
22.3
43.3
55
328
21.8
43.5
45
318
21.5
43.6
Resistance variation with temperature
Resistance
(Ω)
0.5730
0.5614
0.5509
0.5439
0.5359
0.5290
0.5208
0.5150
0.5057
0.4650
0.595
0.575
0.555
0.535
0.515
0.495
0.475
300
320
340
360
380
Temperature (K)
Figure-2, Variation of resistance with temperature for copper wire
Slope of the straight line in Figure-2
∆R
= 1.4414 x10 − 3 ΩK-1
∆T
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Lab Experiments
7. Experiment is repeated for Iron, Silver and Gold wires
Calculations
At 318 degree K
Electron density n =
Nρ 6.023 x10 26 x8930
=
= 8.464 x10 28 /Kgmol
M
63.546
Electrical conductivity σ =
L
2.2
=
= 5.531x107 / Ωm
−9
Ra 0.465 x85.52 x10
The Fermi velocity for copper VF = 1.57x106m/sec
The relaxation time τ =
σm
5.531x107 x9.1x10 −31
=
= 2.317 x10 −14 sec
ne 2 8.464 x10 28 x(1.602 x10 −19 ) 2
Mean free path λF = VFτ = 1.57x1062.317x10-14=3.638x10-8m
Constant A = λF T= 3.638x10-8 x318 = 11.57x10-6
Fermi Energy
2
 ne 2 πAr 2   ∆R  2
 x
EF = 

 L (2m)   ∆T 
2
 8.464 x10 28 (1.602 x10 −19 ) 2 3.14 x11.57 x10 −6 (0.165 x10 −3 ) 2 
−3 2
=
 x (1.44 x10 )
−31
2.2 2 x9.1x10


−18
= 1.09 x10 J
= 6.8 eV
Fermi Temperature
TF =
E F 1.09x10−18
=
= 78.98x103 K
k 1.38x10− 23
The various parameters connected with this experiment is tabulated in Table-2 for
comparison
Table-2
28
7
Metal
Z
nx10
σx10
τx10-14
λFx10-8
Ax10-6
Slope X 10-3
Iron
26
17
0.60
0.62
0.25
0.79
1.375
Copper
29
8.46
5.5
2.3
3.6
11.5
1.441
Silver
47
5.86
5.7
3.4
4.8
14.5
0.545
Gold
79
5.90
4.1
2.4
3.4
10.34
3.209
Comparison of various parameters of metals used in this experiment
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Results
The results obtained are tabulated in Table-3
Metal
Iron
Copper
Silver
Gold
Mass No
Z
26
29
47
79
Fermi Energy (eV)
Fermi Temperature (T) x103K
Expt
Thet
Expt
Thet
11.8
11.1
13.69
12.89
6.8
7.0
7.9
8.13
5.23
5.49
6.07
6.37
5.57
5.53
6.47
6.42
Experimental Results
References
1. Eisburg, R and Resnik, R., Quantum physics of atoms, molecules, solids, Nuclei and
particles. 2nd Ed. Newyork: Willy,1985
2. Neil W. Asheroft and N. David Mermin, Solid State Physics
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