Dear hadi,

You can solve this problem as follows:

You are given that the intrinsic  concentration ni as

ni^2 = Nc Nv exp - Eg/kT if Eg is measured in electron volt  eV, kT must be also in eV,

It is known that the Nc= Ac N^3/2, Nv=Av T^3/2, Ac and Av are known constants since Nc and Nv  are known at room temperature T= 300 K.

To get the requited temperature T you need only to:

ni = Na or Nd. If you have curve for ni as a  function of T you can read the temperature corresponding to certain ni = Nd or Na.

wish you success

Dear hadi,

You can solve this problem as follows:

You are given that the intrinsic  concentration ni as

ni^2 = Nc Nv exp - Eg/kT if Eg is measured in electron volt  eV, kT must be also in eV,

It is known that the Nc= Ac N^3/2, Nv=Av T^3/2, Ac and Av are known constants since Nc and Nv  are known at room temperature T= 300 K.

To get the requited temperature T you need only to:

ni = Na or Nd. If you have curve for ni as a  function of T you can read the temperature corresponding to certain ni = Nd or Na.

wish you success

This is tricky.  Using np=ni^2 and electrical neutrality n+Na=p+Nd, one gets a quadratic equation for n or p which is easily solved.  What that shows is if ni is 10 (20) times the majority impurity concentraion then the majority carrier concentration is within 5% (2.5%) of ni.  You can then use standard expressions for the temperature dependence of ni to determine the temperature for these conditions.  I don't know if this is what Plummer had in mind but that is how you solve the problem with first principals correcting, if you wish, for the temperature dependence of the band gap etc.

Put  Nc = Nv,         n+Na = p+Nd,     ni = Na+n or Nd+p   and   ni^2 = Nc Nv exp - Eg/kT  calculate  T.

Dear research gate colleagues,

please take care of the question. The condition of the material becoming intrinsic  by heating is given in the question by ni= Nd or Na, this condition will be applied one on the n-side and on the p-side separately. Then the most straight forward solution will be as i described above. It is so that the material becomes intrinsic by heating if ni gets=> than the majority carrier concentration which is p0=Na for ptype and n0= Nd for ntype since at such elevated temperature mostly all impurities will be ionized.

Best regards

In my opinion the quadratic expression is better suitable for calculating n or p at a given temperature. Maybe in this case, in which the temperature is actually the unknown, and as the conditions for both semiconductors to become intrinsic are well established in the problem, the procedure detailed by Abdelhalim is appropriate. You can use experimental measurements or simply the expression of ni given above, but as Thomas claims, be careful because the dependence of EG with temperature should therefore be taken into account, mainly for the highly doped n-type side.

Unfortunately the homework problem is badly written.  Intrinsic implies n=p.  What the problem should of ask was the process temperature for which ni equaled the impurity concentration on each side and this has nothing to do with the intrinsic condition.

Usually, a semiconductor is assumed to be extrinsic, if Nd, or Na (more generally, the absolute value of Nd-Na) is >ni. So to it is necessary to find the temperature when dopant concentration equals to intrinsic concentration. Typically, The temperature can be determined from expr.:

ni^2 = Nc Nv exp - Eg/kT

In some semiconductor materials and conditions (e.g. degenerate doping, large ionization energy of dopant states) the above expr. is not accurate (e.g. in some narrow gap semiconductors) and more accurate expression, specific to the material should be used.

I agree with the answer of Abdelhalim. However, if we do not have the plot of ni as a function of T the problem can not be solved analytically. Is a transcendental equation as Nc= Ac T^3/2 and Nv=Av T^3/2 and even more the Gap also is temperature dependent. In addition I also agree with Thomas about the question is not properly written as intrinsic implies n=p. The fact ni=Nd or Na has nothing to be with the intrinsic nature of the semiconductor. 

The answer of Professor Abdelhalim Zekry is right. This is the way to calculate the intrinsic value of silicon for different temperatures, or vice versa. I recommend seeing the graph from the book "Physics of Semiconductor Devices", Sze_3rd.Ed_2007, page 20, fig. 9. The intrinsic temperature is reached at ~260°C for NA=1x1015 cm-3 and ~1325°C for ND=1x1019 cm-3. Just a comment, the melting point of silicon is 1414°C, so the intrinsic temperature for ND=1x1019 cm-3 is very close to the melting point.

Hi

If you know the Na or Nd density, you can calculate the temperature T to the that the semiconductor has its intrinsic behavior, that is, when ni>=Na or Nd. Then, Na or Nd=(NcNv)^0.5*exp(-Eg/2kT) and you solve the equiation for T. Easy!!