Unit 5
Solid State Physics
Band Theory of Solids
, Free electron models: CFE & QMFE
The free electron models of metals (preceding chapter) gives us a good deal of insight
into several properties of metals.
Yet several properties that could not be explained by these models are: (i) why when
chemical elements crystallize to become solids, some are good conductors, some are
insulators, and yet others are semiconductors with electrical properties that vary
greatly with temperature.
These differences are not minor, but rather remarkable. The resistivity may vary
_
from 𝝆~𝟏𝟎 𝟖 𝒐𝒉𝒎 − 𝑚for a good conductor to 𝝆~𝟏𝟎𝟐𝟐 𝒐𝒉𝒎 − 𝑚 for a good
insulator.
understand the differences between insulators and conductors by extending the free
electron model to take into account the interaction of the electrons with the positive
ion lattice.
In the QMFE model, we assumed that the potential energy inside the solid was
uniform.
It would be more realistic to assume that it is a periodic (alternating uniformly)
function of x, y, z. This is reasonable because of the periodic distribution of the lattice
ions in a crystalline solid.
Our purpose here is simply to show the existence of bands and the general
characteristics. This can be achieved with idealized models and by using qualitative
arguments.
, BLOCH'S THEOREM
To study the motion of an electron in a periodic potential, we should mention a
general property of the wave functions in such a periodic potential.
The difference in electrical conductivities of conductor, semiconductor and
insulators can be understood by extending the free electron model to take into
account the interaction of the electrons with the positive ion lattice. It would
be more realistic to assume that the potential energy inside the lattice is
periodic. This is responsible because of the periodic distribution of the lattice in
a crystalline solid.
For a free particle (electron) with 𝑽 = 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕, the space part of the wave function
𝝋 𝒙, 𝒕 = 𝑨𝒆𝒊(𝒌𝒙−𝝎𝒕) , called the eigenfunction, and is written as,
𝝋 𝒙 = 𝑨𝒆±𝒊𝒌𝒙 (𝟏)
For a periodic potential with period 𝑳, 𝑽 𝒙 = 𝑽 𝒙 + 𝑳 = 𝑽 𝒙 + 𝟐𝑳 . 𝑳 is spacing
between ions. Bloch theorem states that for a particle moving in a periodic potential,
the eigenfunction 𝝋 𝒙 are of the form
𝝋 𝒙 = 𝒖𝒌 (𝒙)𝒆±𝒊𝒌𝒙 (𝟐)
, BLOCH'S THEOREM
These eigenfunctions are plane waves modulated by a function 𝒖𝒌 𝒙 , where 𝒖𝒌 (𝒙) has
the same periodicity as the potential energy. The specific form of the function 𝒖𝒌 (𝒙)
will depend on the form of the function 𝑽(𝒙).
Because the potential is periodic, one expects that the probability of finding a particle at
a given 𝒙 is the same as that of finding it at 𝒙 + 𝑳.
Here,
𝝋∗ 𝒙 𝝋 𝒙 = 𝒖∗𝒌 𝒙 𝒆−𝒊𝒌𝒙 𝒖𝒌 𝒙 𝒆+𝒊𝒌𝒙 = 𝒖∗𝒌 𝒙 𝒖𝒌 𝒙
Therefore, when 𝒖𝒌 𝒙 = 𝒖𝒌 𝒙 + 𝑳
𝝋∗ 𝒙 𝝋 𝒙 = 𝝋∗ 𝒙 + 𝑳 𝝋 𝒙 + 𝑳 (𝟑)
Solid State Physics
Band Theory of Solids
, Free electron models: CFE & QMFE
The free electron models of metals (preceding chapter) gives us a good deal of insight
into several properties of metals.
Yet several properties that could not be explained by these models are: (i) why when
chemical elements crystallize to become solids, some are good conductors, some are
insulators, and yet others are semiconductors with electrical properties that vary
greatly with temperature.
These differences are not minor, but rather remarkable. The resistivity may vary
_
from 𝝆~𝟏𝟎 𝟖 𝒐𝒉𝒎 − 𝑚for a good conductor to 𝝆~𝟏𝟎𝟐𝟐 𝒐𝒉𝒎 − 𝑚 for a good
insulator.
understand the differences between insulators and conductors by extending the free
electron model to take into account the interaction of the electrons with the positive
ion lattice.
In the QMFE model, we assumed that the potential energy inside the solid was
uniform.
It would be more realistic to assume that it is a periodic (alternating uniformly)
function of x, y, z. This is reasonable because of the periodic distribution of the lattice
ions in a crystalline solid.
Our purpose here is simply to show the existence of bands and the general
characteristics. This can be achieved with idealized models and by using qualitative
arguments.
, BLOCH'S THEOREM
To study the motion of an electron in a periodic potential, we should mention a
general property of the wave functions in such a periodic potential.
The difference in electrical conductivities of conductor, semiconductor and
insulators can be understood by extending the free electron model to take into
account the interaction of the electrons with the positive ion lattice. It would
be more realistic to assume that the potential energy inside the lattice is
periodic. This is responsible because of the periodic distribution of the lattice in
a crystalline solid.
For a free particle (electron) with 𝑽 = 𝒄𝒐𝒏𝒔𝒕𝒂𝒏𝒕, the space part of the wave function
𝝋 𝒙, 𝒕 = 𝑨𝒆𝒊(𝒌𝒙−𝝎𝒕) , called the eigenfunction, and is written as,
𝝋 𝒙 = 𝑨𝒆±𝒊𝒌𝒙 (𝟏)
For a periodic potential with period 𝑳, 𝑽 𝒙 = 𝑽 𝒙 + 𝑳 = 𝑽 𝒙 + 𝟐𝑳 . 𝑳 is spacing
between ions. Bloch theorem states that for a particle moving in a periodic potential,
the eigenfunction 𝝋 𝒙 are of the form
𝝋 𝒙 = 𝒖𝒌 (𝒙)𝒆±𝒊𝒌𝒙 (𝟐)
, BLOCH'S THEOREM
These eigenfunctions are plane waves modulated by a function 𝒖𝒌 𝒙 , where 𝒖𝒌 (𝒙) has
the same periodicity as the potential energy. The specific form of the function 𝒖𝒌 (𝒙)
will depend on the form of the function 𝑽(𝒙).
Because the potential is periodic, one expects that the probability of finding a particle at
a given 𝒙 is the same as that of finding it at 𝒙 + 𝑳.
Here,
𝝋∗ 𝒙 𝝋 𝒙 = 𝒖∗𝒌 𝒙 𝒆−𝒊𝒌𝒙 𝒖𝒌 𝒙 𝒆+𝒊𝒌𝒙 = 𝒖∗𝒌 𝒙 𝒖𝒌 𝒙
Therefore, when 𝒖𝒌 𝒙 = 𝒖𝒌 𝒙 + 𝑳
𝝋∗ 𝒙 𝝋 𝒙 = 𝝋∗ 𝒙 + 𝑳 𝝋 𝒙 + 𝑳 (𝟑)
