Potential Energy and Kinetic Energy of an Electron in an Energy Band diagram

[Robotduck]

Hi,

Can anyone explain the concept of K.E. and P.E. in a semiconductor?
Why Ec (lower level of Conduction Band) is considered as P.E. ? and why the Difference between the higher energy state and Ec is Kinetic energy ?
I understand it if I look at the Bohr model but when I am not clear how to look at P.E. and K.E. in an Energy Band diagram.

Thank you.

 

[wwfeldman]

An attempt:

as you know:
consider the bohr model of hydrogen:
an electron can orbit the nucleus only in specific quantum levels
the lowest level, n = 1, is at -13.6 eV
the next level, n = 2, is at -3.4 eV
the next level, n = 3, is at -1.5 eV
reference: http://hyperphysics.phy-astr.gsu.edu/hbase/hyde.html

these hydrogen energy levels are potential energy
when the electron is at energy level 0 eV, it is no longer bound to the nucleus
when the electron is at an energy > 0 eV, it has some kinetic energy, hence some velocity and is therefore moving,
free from the nucleus

the electron can only change potential energy level by emitting or absorbing the exact energy requiared for the transition,
say 10.2 eV for the change between n = 1 and n = 2
the energy is absorbed when the electron moves from 1 to 2
the energy is emitted when the electron moves from 2 to 1.

also, as you know:
instead of discrete levels, as in hydrogen, bulk materials show the energy in bands

for materials, there is the valence band and the conduction band
with an energy gap between the top of the valence band and the bottom of the conduction band.

the spacing between the bands determines if the material is an insulator, semiconductor
or conductor

the energy diagram is a graph of the energy available to electrons in the material

the valence band represents the energy available to the outer shell of the atom of the material
this is the region where the electrons are bound to their individual atoms
as they do not have enough energy to be free of their nucleus.

the conduction band is where the electron have sufficient energy to move around essentially freely
like the electron with energy > 0 eV in hydrogen

the gap in between is the (so-called) forbidden region
what this really means is that we do not see electrons at that energy level

in insulators, the band gap is relatively large, hence electrons are NOT free to move about

in conductors, the band gap is does not really exist - the valence band and the conduction band overlap
therefore, at least some of the electrons are free to move about
good conductors (silver, gold, copper) have a larger overlap than poor conductors (lead).

in semiconductors, the band gap is relatively small
by adding dopants (contaminants) we control the band gap and the behavior of electrons between
the valence and conduction bands

http://hyperphysics.phy-astr.gsu.edu/hbase/Solids/band.html#c4


now KE and PE:
the energy levels in individual atoms (such as hydrogen) are KE levels
the lower limit of the conduction band Ec (as you called it) is like one of these atomic
energy levels - potential energy

when the electron is "higher" in the energy level diagram, i.e. higher in the conduction band,
that additional energy is kinetic energy.

i hope this helps a little
i am afraid i may have only re-stated what you already know.

 

[Robotduck]

An attempt:

as you know:
consider the bohr model of hydrogen:
an electron can orbit the nucleus only in specific quantum levels
the lowest level, n = 1, is at -13.6 eV
the next level, n = 2, is at -3.4 eV
the next level, n = 3, is at -1.5 eV
reference: http://hyperphysics.phy-astr.gsu.edu/hbase/hyde.html

these hydrogen energy levels are potential energy
when the electron is at energy level 0 eV, it is no longer bound to the nucleus
when the electron is at an energy > 0 eV, it has some kinetic energy, hence some velocity and is therefore moving,
free from the nucleus

the electron can only change potential energy level by emitting or absorbing the exact energy requiared for the transition,
say 10.2 eV for the change between n = 1 and n = 2
the energy is absorbed when the electron moves from 1 to 2
the energy is emitted when the electron moves from 2 to 1.

also, as you know:
instead of discrete levels, as in hydrogen, bulk materials show the energy in bands

for materials, there is the valence band and the conduction band
with an energy gap between the top of the valence band and the bottom of the conduction band.

the spacing between the bands determines if the material is an insulator, semiconductor
or conductor

the energy diagram is a graph of the energy available to electrons in the material

the valence band represents the energy available to the outer shell of the atom of the material
this is the region where the electrons are bound to their individual atoms
as they do not have enough energy to be free of their nucleus.

the conduction band is where the electron have sufficient energy to move around essentially freely
like the electron with energy > 0 eV in hydrogen

the gap in between is the (so-called) forbidden region
what this really means is that we do not see electrons at that energy level

in insulators, the band gap is relatively large, hence electrons are NOT free to move about

in conductors, the band gap is does not really exist - the valence band and the conduction band overlap
therefore, at least some of the electrons are free to move about
good conductors (silver, gold, copper) have a larger overlap than poor conductors (lead).

in semiconductors, the band gap is relatively small
by adding dopants (contaminants) we control the band gap and the behavior of electrons between
the valence and conduction bands

http://hyperphysics.phy-astr.gsu.edu/hbase/Solids/band.html#c4


now KE and PE:
the energy levels in individual atoms (such as hydrogen) are KE levels
the lower limit of the conduction band Ec (as you called it) is like one of these atomic
energy levels - potential energy

when the electron is "higher" in the energy level diagram, i.e. higher in the conduction band,
that additional energy is kinetic energy.

i hope this helps a little
i am afraid i may have only re-stated what you already know.

Can you please Elaborate on this ? "the lower limit of the conduction band Ec (as you called it) is like one of these atomic
energy levels - potential energy" I still am having difficulty in understanding this. How come the Ec represents the P.E of an electron ?

 

[wwfeldman]

waiting on a response from a friend

 

[wwfeldman]

response from a friend:
1) there are some errors in the discussion about hydrogen in post #2
2) Ec, the energy of the low limit of the conduction band is not specifically related to potential energy
it is just the energy o the lower boundary of the conduction band.

in a semiconductor, with the multitude of atoms and electrons, things work a little differently
and the classical concepts of kinetic and potential energy have their analogs, but it isn't as simple or
straightforward as a ball with some position, mass and velocity.

what brought on the question of Ec and potential energy vs kinetic energy?

 

[c_mitra]

Can anyone explain the concept of K.E. and P.E. in a semiconductor?

Before I try to explain, I must tell you that the two terms Kinetic and potential (energies) are basically Newtonian in origin. They are related to dynamics and you need to understand the concept of a force field.

Without a force field, you have no potential energy. The kinetic energy depends on the velocity and hence depends on the frame of observation.

In reality, they are not conserved (individually) and are therefore not constants of motion. Together they make total energy which is conserved and is therefore a constant of motion.

In reality, potential energy is an useful quantity because it is tied with the field (say gravitational or an electric field) but potential and kinetic energies are always interconvertible.

Once you go into Lagrange's equation of motion, you notice that they have the same form in terms of generalized coordinates: (1/2)*(k)*(q^2)

When we draw a band diagram, the x-axis is often unlabelled and the y axis is labelled as potential energy. But the underlying concept is Fermi level and the internal electronic structure.

Electrons in the lower filled levels cannot move; hence they do not have (in some sense) kinetic energy. All the levels in the filled bands are potential energies (except near the top).

In an energy band diagram, the KE is responsible for the transitions between the valence and conduction bands.

I hope I have made it clear.

 

[Robotduck]

response from a friend:
1) there are some errors in the discussion about hydrogen in post #2
2) Ec, the energy of the low limit of the conduction band is not specifically related to potential energy
it is just the energy o the lower boundary of the conduction band.

in a semiconductor, with the multitude of atoms and electrons, things work a little differently
and the classical concepts of kinetic and potential energy have their analogs, but it isn't as simple or
straightforward as a ball with some position, mass and velocity.

what brought on the question of Ec and potential energy vs kinetic energy?

I am asking this question in context of Energy band diagram. Topic is Band bending.
P.E.=-q * Electrostatic potential ...
In this topic, ( From Robert Pierret Book), it is said ,' If electron is excited from valence band with just Eg then it just sits at Ec, it is not free to move. But if an electron is excited from valence band with energy > Eg then it is free to move in C.B..
How would you explain this in simple form ?
Thank you

 

[wwfeldman]

all of these appeals to classical physics are a little bogus,
and there are no really good analogies, however:

consider a rocket launched at escape velocity
it is moving at escape velocity and the engine is turned off
like a baseball thrown straight up, as it rises above the earth, it slows

and, like the baseball, when it reaches maximum height, its velocity is now zero

HOWEVER, unlike the baseball that starts falling, the rocket stops and does not come back.
it had only enough energy to escape, and no more.
since it has stopped, it is far from earth and stationary

the difference is that the baseball is still caught in the gravity well of the earth,
while the rocket is free, but with no kinetic energy

in the band model of semiconductors, when an electron in the valence band acquires Eg energy,
it can jump up to the conduction band
but it has no more energy, and like the escaped rocket, isn't going anywhere

 

[c_mitra]

If electron is excited from valence band with just Eg then it just sits at Ec, it is not free to move. But if an electron is excited from valence band with energy > Eg then it is free to move in C.B..
How would you explain this in simple form ?

The y-axis is potential energy and the bands represent the potential energy of the electron.

Consider an electron at the top of the valence band is supplied with the band gap energy (anything less is not acceptable). It will now reach the bottom of the conduction band. All the externally supplied energy has now been taken up as the potential energy after the transition.

Obviously the electron is at the bottom of the conduction band and has no extra energy left as kinetic energy.

If the externally supplied energy is band_gap_erergy+delta is applied to an electron at the top of the valence band, the band_gap_energy is used to take the electron to the bottom of the conduction band and any extra left over (delta) will be available as kinetic energy.

In real life, the journey is more complex.