Download Real and effective mass of the electrons inside crystals

Real and effective mass of the electrons inside crystals
(1) Eng. Marius FETOIU, (2) Eng. Alexandru CRACIUN, (3) Eng. Cornelia POPA
(1), (2), (3) SC IPA SA CIFATT Craiova, e_mail: [email protected]
Abstract. The analysis and the design of the devices
and of the electronic circuits depend on the profound
understanding of the notions of Physics equivalent to
the solid conductor materials, semiconductors and
insulators. Because of the technical and economical
necessities of the actual society, integrated circuits
must be made at a nano level, where the quantum
mechanic laws govern.
In the present paper one presents the difference
between the real mass and the effective mass of the
electrons inside crystals and a simplified model of the
energetic band for a solid body in which one would
show that the electron mass is not the real one but the
effective one.
Introduction
The study of the function of the devices and of the
electronic circuits involves the study of the behavior of
the solid materials both in the presence of some external
disturbance factors (electric fields, magnetic fields,
electro-magnetic or nuclear radiations, etc.) and also in
their absence. For this, two concepts are introduced:
thermo-dynamic equilibrium and thermo-dynamic nonequilibrium.
Thermo-dynamic equilibrium represents a global
condition that consists in the simultaneous fulfillment of
four equilibrium conditions:
- Thermal equilibrium
- Electric equilibrium
- Chemical equilibrium
- Mechanical equilibrium
The non-fulfillment of one of the four equilibrium
conditions will place the material in a thermo-dynamic
non- equilibrium situation.
Because a solid material contains around 1023 particles in
a square centimeter, particles that present small distances
between them. For this reason their behavior is studied on
the basis of two extensions of the classical Physics:
probable quantum that solves the problem of small
distances between particles and static mechanics that
solves the problem of the great number of particles.
These extensions provide main study instruments of the
materials involved in the making of the devices and of the
electronic circuits:
-the model of the energetic bands developed on the basis
of two fundamental postulates regarding the dual
character of the material particles and of the electromagnetic radiation:
- Planck’s Condition (E=h*f)
- Broglie Hypothesis (p=h/λ)
- Fermi- Dirac statistics is the one that presents the
distribution of the electrons according to their momentum
in the thermo-equilibrium.
For the introduction of the effective mass notion one
should use the model of the energetic bands. For this one

should take into account the energy bands W1 (k ) ,


W2 ( k ) ,…, Wn (k ) for the Brilluoin zone. The energy

functions Wi (k ) have the following properties.
1. In the domains of the reduced zone where the function

Wl (k ) does not cut across or touches another, this
function is continuous and admits derives from any order.
In the contingence points or of cutting points of these
functions, one should obtain situations named nongenerated for the value (level) of the energy. At the cut
across or at the contingence of only two functions one
should obtain an energetic level double non-generated.
2. The function

Wi (k ) is even


Wi (k )  Wi (k )
W
3. On the reduced zone
frontiera  0
n
(1)
(2)
In daily practice importance is given by the extremes of
the energy functions

Wi (k ) -maximum and minimum
extremes.
In the point of the reduced zone

k0 where there is an
extreme point (maximum or minimum) of energy the
following condition “(2)” is fulfilled:
 W

 k

  0
 k0
(3)
Where: α = x, y, z and represent the Cartesian coordinates
of the three-dimensional space.
If a Taylor series is developed the function energy
 
Wi (k ) around the point k  k0 and if we keep the first
terms of the development, we obtain:


1   2Wi
Wi (k )  Wi (k0 )   
2  ,   k k 

 |  (k  k )( k   k )
0
0
 k0

(4)


1
Wi  Wi (k )  Wi (k0 )     ,  s s
2  ,
(7)
Where:
Where α = x, y, z and β = x, y, z are the coordinates of
the three-dimensional space
  Wi
 2
  kx
2
  Wi    2Wi


 k k    k y k x
  2Wi

 k z k y
 Wi
k x k y
 2Wi
 2k y
 2Wi
k z k y
2
 Wi 

k x k z 
 2Wi 

k y k z 
 2Wi 

 2k z 
(8)
s  k  k 
(9)
s  k  k 
(10)
  ,
The sum from the expression (4) can be put in matrix,
2
 2W

k k 
2
(5)
With these notions a simplified model of the energetic
bands will be written for a solid body. In this model one
will show that the mass of the electron from the crystal
structure is not the real mass but the effective mass.
Due to the symmetrical property (1), the matrix (5)
follows some transforming rules.
 2Wi
 2Wi

k k  k  k
Relation (2) can be written in the following way:
(6)
References
[1] Draganescu, M. “Electronica corpului solid”, Editura Tehnica,
Bucuresti, 1972
[2] Doicaru, Elena “Dispozitive electronice”, Editura Universitaria,
Craiova, 2002
[3] Stanciu, Margareta “Fizica atomului si moleculei”, Repografia
Universitatii din Craiova, 1998