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Sunday, July 1, 2018

Charactersitics of electron in one dimensional periodic potential

CRYSTAL MOMENTUM

For free electron, the quantity ℏk represents true momentum of electron as follows:

E= (ℏk)²/2m = (1/2m)*(ℏ)²*(k)² = (1/2m)*(h/2𝝅)²*(2𝝅/ƛ)² = (1/2m)*(h/ƛ)² = P²/2m

Therefore, the dynamical behavior of free electron can be represented by true momentum.

But when we consider an electron in periodic potential, ℏk doesn't represent true momentum. ℏk does not represent true momentum. The energy doesn't vary with 'k' as in previous case.

The true instantaneous momentum of an electron in presence of lattice potential is not a constant of motion and cannot be calculated by quantum mechanical method we take average value.
So in order to describe the dynamical behavior of electron in periodic potential we introduce a new type of momentum called as Crystal Momentum.

When we deal with interactions of electrons with lattice, we use conservation of crystal momentum and not of true momentum.

The crystal momentum is perfectly well defined constant for a state of given energy.


VELOCITY

The quantum mechanical part describes that the velocity of electron in a one dimensional lattice will be equal to Group Velocity of waves representing the electron.

v = (dw/dk) ...................................................................................................(1)

where 'w' is angular frequency of debroglie waves.

Eqn(1) depends on actual E-K curve.

(dE/dk) = ℏ (dw/dk) ; v=(1/ℏ)(dE/dk) ..........................................................(2)

for free electrons, substituting E= ℏk, v=p/m

giving linear variation of 'v' with 'k'.

In band theory of solids, however, 'E' is not proportional to k².

The variation of 'E' with 'k' is as sown in fig:




using this type of variation of 'E' with 'k' as shown in fig below.




We observe that at bottom (k=0) of energy band, the velocity is zero and as the value of 'k' increases ('E' increases) the velocity increases reaching its maximum at k=kₒ, where kₒ corresponds to "point of inflection" on E-K curve. Beyond this point the velocity begins to decrease and finally assumes zero value at k=𝝅/a, which is top of band. These are entirely new features which do not appear at all in behavior of free electrons.

EFFECTIVE MASS OF ELECTRON

It is known for long time that an electron has well defined mass and when accelerated by an electric field, it obeys Newtonian Mechanics. What happens when electron is to be accelerated inside a crystal? How will it react to electric field.

The mass of an electron inside the crystal appears, in general, different from free electron mass and is usually referred to as "effective mass".

The velocity of an electron in one dimensional lattice is given by

v = (2𝝅/ℏ)(dE/dk)..................................................................................................(3)

a= dv/dt = (2𝝅/h)(d²E/dk²)*(dk/dt) .......................................................................(4)

so we have to find value of dk/dt.

Let an electron is subjected to influence of an electric field of  strength 'E' for a time dt. If velocity of electron is v, the distance travelled in time dt=vdt

Therefore Work done, dE=(e𝜀)*v*dt

we know

v = (2𝝅/h)(dE/dk) ; therefore  dE= (e𝜀)*(2𝝅/h)*(dE/dk)*dt

(dk/dt) =  2𝝅e𝜀/h .....................................................................................................(5)

substituting (5) in (4), we get

a =  (2𝝅/h)² *(e𝜀)*(d²E/dk²)  ...................................................................................(6)

For free particle, a= m(dv/dt) = eE;

a=e*E/m  ..................................................................................................................(7)

comparing (6) & (7) , both forms are identical, we introduce a new mass known as effective mass given by

m* = (h/4𝝅) * (d²E/dk²)⁻¹  ......................................................................................(8)

For free electron,

m* = m

  • Effective mass can also be determined using "Cyclotron Resonance Experiment".
From experimental values of effective mass, we can conclude that
  •  Effective mass need not always be greater than 'm'. It can be smaller than 'm'.
  • It can be negative.
Variation of m* with k:  




Physically speaking near the bottom of band the effective mass m* has a constant value which is positive because the quadratic eqn [E ∝k²] is  satisfied near the bottom of band.

But as 'k' increases m* is no longer a constant, being now a function of k, because quadratic relation is no longer valid.

The degree of freedom of an electron is defined by a factor


fk = (m/m*) = (m/ℏ2)*(d²E/dk²)

fk is measure of extent to which an electron in state 'k' is free.