Question 26 on Electronic Devices from GATE (Graduate Aptitude Test in Engineering) 2012 Electronics and Communication Engineering paper.
Q26. The source of a silicon (
), n-channel MOS transistor has an area of 
and a depth of 
. If the dopant density in the source is 
, the number of holes in the source region with above volume is approximately
(A) 
(B) 
(C) 
(D) 
Solution
To answer this question, had to dig up the copy of Solid State Electronic Devices, Ben G Streetman, Sanjay Kumar Banarjee (buy from Amazon.com
Electrons in solids obey Fermi-Dirac statistics. The function Fermi-Dirac distribution function, gives the probability that an available energy state at
will be occupied by an electron at absolute temperature
,
.
The quantity is the Fermi level and
is the Boltzmann’s constant.
The concentration of electrons in the conduction band is,
, where
is the density of states
in the energy range
.
The integral is equivalently stated as,
, where
is the effective density of states at conduction band edge
.
The Fermi function at is approximately,
.
So in this condition the concentration of electrons in the conduction band is,
.
Similarly, the concentration of holes in the valence band is,
, where
is the effective density of states at valence band edge
.
The term,
.
So in this condition the concentration of holes in the valence band is,
.
The product of and
at equilibrium is a constant for a particular material and temperature even if doping is varied.
, where
is the gap between the conduction band and valence band.
Similarly, for an intrinsic material, the product of and
is,
For an intrinsic material, the electron and hole concentration are equal i.e. .
The constant product of electron and hole concentration can be written as
With the above understanding, let us try to find the solution to the problem. In our question,
and
.
The hole concentration is,
.
The volume of the source region is
.
The number of holes is,
Based on the above, the right choice is (D) 0.
References
[1] GATE Examination Question Papers [Previous Years] from Indian Institute of Technology, Madras http://gate.iitm.ac.in/gateqps/2012/ec.pdf
[2] Solid State Electronic Devices, Ben G Streetman, Sanjay Kumar Banarjee (buy from Amazon.com
[3] Valence band http://en.wikipedia.org/wiki/Valence_band
[4] Conduction band http://en.wikipedia.org/wiki/Conduction_band