Electrical conductivity of the materials according to quantum mechanical consideration

We know that conduction current density,  J = NVe = σE

Where, N= number of electron

            V= electron velocity

           σ = conductivity

According to quantum mechanical consideration, when an electric field E is applied, only the number of electrons (N^/) participate in conduction those have approximately fermi velocity (V_F) 

So, we can write  J = N^/ V_F e --------(i)

Now number of displaced electrons can be found from the relation of population density which reflects the following figure-

Figure:. Population density N(E) versus energy for free electrons and displacement ∆E by an electric field. N^/ is the number of displaced electrons per unit volume in the energy interval ∆E. N(E) is defined per unit energy and in the present case, also per unit volume.

We get-      N^/  =  N(E_F) ∆E

Where, N(E_F) = population density with Fermi energy

     And  ∆E = (dE/dK) ∆K

Therefore, N^/  =  N(E_F) (dE/dK) ∆K

The factor dE/dk is calculated by using the E versus |k| relationship known for free electrons as

E = ( ђ² k )/ 2m

Or, dE/dK  = ( ђ² K) / m

                   = ( ђ²/ m)(P/ђ)      [ since, K = P/ђ  ]

                   = ( ђ²/ m)(mV_F/ђ)    [ since, P = mV_F  ]

                   = ђV_F

Where, P = mass velocity     

            V_F = electron velocity at fermi level

            Ђ = plank constant

Now we also know (Newton’s Law)  that  

         F = m (dV/dt)

Or, eE = m (dV/dt)  [ applied force]

Or, d(mV)/dt    = eE

Or, dP / dt  =  eE

Or, ђ ( dK/dt ) = eE

Or, dK = (eE / ђ )  dt

Or, ∆K = (eE / ђ )  ∆t

So,  ∆K = (eE / ђ )  τ

Where, τ = relaxation time = ∆t

Putting all the values in (i) we get-

J = V_F² e² N(E_F) E τ    

One more consideration needs to be made because all the displaced electrons mayn’t move towards the applied electric field. If the electric field vector points in the negative V_KX direction, then only the components of those velocities that are parallel to the positive V_KX direction contribute to the electric current. The V_KY components cancel each other pairwise.

In other words only the projections of the velocities V_F  on the positive V_KX  axis ( V_FX = V_F cosϴ ) contribute to the current.    

Therefore,

This is a two dimensional view consideration. But real world is three dimensional. A similar calculation for spherical Fermi surface yields as

J = (1/3) e² N(E_F) E τ V_F²

Then the conductivity, σ = J/E

                                    = (1/3) e² N(E_F)  τ V_F²

Therefore according to quantum mechanical consideration, conductivity depends on population density ( other parameters may be fixed ) . The higher the population densities greater the conductivity.

Monovalent metals have partially filled valance bands. Their electron population densities near the Fermi energy are high which results in a large conductivity. Bivalent metals electron population densities near Fermi energy is small which leads to a comparatively low conductivity.

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What is data, information and signal ( any confusion about their difference ! )

[What is Data?]

Data are simply values or sets of values of qualitative or quantitative variables, belonging to a set of items. It may be in the form of numbers, letters, or a set of characters. It is often collected via measurements. In data computing or data processing, data is represented by in a structure, such as tabular data, data tree, a data graph, etc.

[What is Information?]

Once the data is analyzed, it is considered as information. In other words, meaningful or processed data is known as information. Example- Suppose we have a collection of numbers like 23, 25, 87 etc. These can be called as data. Now if we specified these numbers with an attribute like ID of some people in a system then they call information. Because this time each number represents a user identity.

[What is signal?]

In communication or electronic system we need to transfer information. And that is happened electrically.  In order for data or information to be transferred electronically, it must first be converted into electromagnetic signals. The signal can then be used to transfer data from one device to another device. The signal can be either analog or digital in nature. The analog signal refers to a continuous stream of data, whereas digital signal converts the data into discrete states as discus in a post before. But remember that all the part of the signal don't contain information. That is why we can say that some portion of signal convey or is data.

Hence, data and signal are two completely different things. Data is a collection of values in any form . Signal, on the other hand, is used to transfer this data from one point to another.

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Different types of noises & signals in communication sustem

What is Noise?  

The noise is the unwanted signals that tend to disturb the operation of a system and over which we have incomplete control.

Two broadly used categories of noise are given below:

External noises: The noise whose sources are external to the communication system is known as external noise. Example- atmospheric noises, extraterrestrial noises, man-made noises etc.

Internal noises: The noise which get generated within the communication system is known as internal noise. Example- shot noise, thermal noise, transit-time noise, miscellaneous internal noises etc.       

Signal: A signal is formally defined as a function of one or more variables that conveys information on the nature of a physical phenomenon.

Several types of signal’s are given below-

Continuous-time signal:- A signal that is specified for every value of time is a continuous time signal. Example- telephone and video camera output.

Discrete-time signal:- A signal that is specified for only discrete values of time, is a discrete-time signal. Example- monthly sales of a corporation.

Analog signal:- A signal whose amplitude can take on any value in a continuous range is an analog signal.

Digital signal:- A signal whose amplitude can take on only a finite number of values is known as digital signal.

Periodic signal and aperiodic signal:- A signal g(t) is said to be periodic if for some positive constant T_{0 ,}

                                              g(t) = g(t + T₀ )

The smallest value of T₀ is the period of g(t). A signal is aperiodic if it is not periodic.

Deterministic signal:- A signal whose physical description is known in either a mathematical form or a graphical form is a deterministic signal.

Random signal:-A signal which is known in terms of probabilistic description such as mean value, mean squared value and so on rather than its complete mathematical or graphical description is a random signal.

Energy signal:- A signal with finite energy is an energy signal. In other words, a signal g(t) is an energy signal if 

Power signal:- A signal with finite power is a power signal. In other words, a signal g(t) is a power signal if

Causal signal:- A signal that does not start before  t=0 is called a causal signal.

                                           g(t) = 0            , t<0 span=""> 

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Electrical conductivity of the materials according to classical electron theory

According to classical theory (described by drude ) generally free electrons move randomly in all possible directions and no net velocity results. If an electric field E is applied, the electrons are then accelerated with a force eE towards the anode.

So according to Newton’s Law, 

            m (dV/dt)  =  eE      ------(i)  [ as F = ma ]

Where, m = mass of electron        e = charge of electron

            E = applied E-field              V = drift velocity of electron

This electron motion will be counteracted by a frictional force ϒV  due to collisions.

Under this consideration (i) be modified as

           m (dV/dt)  + ϒV =  eE  ------(ii)  

where, ϒ = a constant

For the steady state case(immediate situation just before collisions) we obtain 

V = V_f

 

dV/ dt = 0

Then (ii) reduces to           ϒV_f  =  eE

                                     Or,    ϒ = eE / V_f

Where, V_f = final drift velocity

Putting this value into (ii) we get-

           m (dV/dt)  + (eE / V_f ) V =  eE  

or,     m dV     = [  eE - (eE / V_f ) V  ] dt

or    m V_f    =   [  eE - (eE / V_f ) V  ] t     

[ Here we use the integration range minimum to maximum . That is

for V ;     0 to V_f            for t ;  0 to t   ]

or,    (m V_f² )/ t  =  eEV_f -  eEV

or,   V_f  - V =  (m V_f² )/ eEt

or,       V    =  V_f  -   (m V_f² )/ eEt

                  =   V_f  [ 1 - (m V_f )/ eEt ] ----------(v)

In (v) the factor   (mV_f) / eE has the unit of a time 

which is defined by   τ  = (mV_f) / eE             where, τ = relaxation time

or,  V_f =  (τ e E) / m  ---------------(vi)

Now we know that conduction current density J = N_f V_f e = σE 

Where, σ = conductivity   and N_f = number of free electron

So,      σ = ( N_f V_f e ) / E

               = ( N_f e² τ ) / m     [ from (vi) ]

               = ( N_f e² l ) / Vm     

Where, l = Vτ = mean free path

This is the required relation. So the conductivity is large for a large number of free electrons and for a large relaxation time.
Relaxation time is defined as the average time between two consecutive  collisions. The distance passing by electron during relaxation time is  known as mean free path.

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