Hello all,
Recently I was going through some semiconductor basics to brush up some of my concepts where I stumbled across this nice problem. I am a bit confused over the solution though. The problem goes like this:
The substrate concentration of an N-type semiconductor is Nd = 10^15 (cm-3). The wafer is doped with Na = 1.1 X 10^15 (cm-3), so that a very lightly doped P-type region is created at the surface. What is the concentration of electrons in the P-type region if the semi-conductor is Ge (Germanium)?
Intrinsic concentration of Ge (ni) = 2.4 X 10^13 (cm-3)
Answer : 5.9 X 10^12 (cm-3)
My approach for the answer:
Since for any such case, the electroneutrality equation always holds. then we have,
p0 - n0 +Nd - Na = 0 ....................(1)
Since we also know that
n0.p0 = ni^2..............(2)
n0 = ni^2/p0...........(3)
Putting this into eqn(1) and substituting values of Nd and Na, we get,
p0^2 - 10^14p0 - ni^2 = 0 ....(4)
Solving this quadratic equation,I get p0 = 1.05 X 10^14 (cm-3) and
n0 = 5.46 X 10^12 (cm-3) but the answer is 5.9 X 10^12 (cm-3)
Can anyone please help me rectify where I am making the mistake? Please help me on this!
Thanks!
Dear Sourangsu ,
Your calculations of p0 and n0 are correct. One has to solve the neutrality equation and the mass action law p0 n0= ni^2. This will give the most accurate answer.
However one has to assume complete ionization of the doping atoms which is valid for germanium at room temperature. I would assume there is some approximation in the given answer and also may have a slightly different ni.
Your procedure and answer are okay.
Best wishes
Dear Sourangsu ,
Your calculations of p0 and n0 are correct. One has to solve the neutrality equation and the mass action law p0 n0= ni^2. This will give the most accurate answer.
However one has to assume complete ionization of the doping atoms which is valid for germanium at room temperature. I would assume there is some approximation in the given answer and also may have a slightly different ni.
Your procedure and answer are okay.
Best wishes
I would like to suggest to read this paper:
http://www-inst.eecs.berkeley.edu/~ee143/fa10/lectures/Lec_21.pdf
Dear Sourangsu Banerji,
I am afraid that the problem is in fact ill-posed. Without knowing where are the energy positions of your Nd(shallow donor ?) and Na(deep acceptor ?) within Ge band gap, you can not solve the problem, particularly if you want a precision to within two decimals (5.46 as compared with 5.9).
What has to be found is the position of the chemical potential (Fermi level) for your system, when the system is modelled with 4 energies with the appropriate densities of states. Ec(Nc), Ev(Nv), Ed(Nd) and ENa(NNa). Deep in the bulk (far away from surfaces and/or interfaces), it can be assumed that there is indeed electrical charge neutrality and the equation can be solved numerically (or approximately analytically). Once you know the position of the chemical potential, you know everything, including the density of electrons at Ec.
However, you have got another problem in your system, namely an interface between your Na doped thin region and the rest of your wafer (Nd only). The difference in the chemical potential across this interface is quite large (larger than half the gap) and there will be inflow of electrons from this region into your thin p-type region. Depending on the thickness of your p-type region, the p-type region can even revert to be n-type again and at thermal equilibrium, there will not be charge neutrality.
I have not checked the numerical result, but it might be that the approximate formula you are using is good enough. Also, I would definitely consider the agreement between your result and given answer as quite good.
With best regards
Petr
Dear Sourangsu Banerji,
I am happy that we agree. And you are right, the problem could be formulated so, that it could interesting for the graduate students. In that way, one is forced to analyse the problem in its complexity before going for analytic approximations.
With best regards
Petr
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