Solid state(donors and acceptors)

[fabsuk]

A sample of silicon contains 10 to the power of 18 donors m3 and 10 to the power of 16 acceptors m3. The donor ionisation energy is 50 meV. Calculate the temperatures at which the electron concentrations are
(a) 10 times the saturated value
(b) 0.1 the saturated the saturated value.

calculate the intrinsic electron concentration, and assume for part (a) that the energy gap at the required temperature is 1.1 eV. An iterative method will be required for both parts.






Could someone please help. I don't know what the saturated value in the 1st place is and don't know what equation to use. A solution or help in the right direction would be useful.

 

[OlderDan]

A sample of silicon contains 10 to the power of 18 donors m3 and 10 to the power of 16 acceptors m3. The donor ionisation energy is 50 meV. Calculate the temperatures at which the electron concentrations are
(a) 10 times the saturated value
(b) 0.1 the saturated the saturated value.

calculate the intrinsic electron concentration, and assume for part (a) that the energy gap at the required temperature is 1.1 eV. An iterative method will be required for both parts.

Could someone please help. I don't know what the saturated value in the 1st place is and don't know what equation to use. A solution or help in the right direction would be useful.

Somewhat random thoughts on the subject that might get you thinking in the right direction: There are 100 times as many donors as acceptors, so there are potentially 100 electrons for every hole. At very low temperatures, the donor atoms hang on to their electrons and the holes remain unfilled. At higher termperatures, some of the electrons acquire enough energy to escape the donors and start migrating into the holes. I assume saturation means that there are as many of these migrating electrons as there are acceptor holes to be filled, or something along those lines, so the concentration of electrons at saturation is the acceptor concentration or 1% of the donor concentration. There is some temperature dependent energy distribution that accounts for the freeing of some fraction of the donor electrons, and that fraction increases with temperature. I assume you have some sort of distribution function to work with. Perhaps this will be helpful

http://www.ece.utep.edu/courses/ee3329/ee3329/Studyguide/ToC/Fundamentals/Carriers/explain.html

 

[fabsuk]

I am still stuck,

it would be really useful if someone can start me off with correct equations or even the solution. I have asked many people and nobody seems to know the answer.

 

[OlderDan]

I am still stuck,

it would be really useful if someone can start me off with correct equations or even the solution. I have asked many people and nobody seems to know the answer.

I don't know what you do know about this subject, but here are some things I believe to be relevant to your problem

http://hyperphysics.phy-astr.gsu.edu/hbase/solids/fermi.html

http://hyperphysics.phy-astr.gsu.edu/hbase/solids/fermi3.html#c1

http://hyperphysics.phy-astr.gsu.edu/hbase/solids/dope.html#c2
Note particularly what happens to the Fermi level with doping.

http://jas.eng.buffalo.edu/education/semicon/fermi/bandAndLevel/
Click the buttons to show everything. Read the brief Applet Tutorial and do what it suggests. Try it at different termperatures. Click on the buttons at the top to access the discussion. See also

http://jas.eng.buffalo.edu/education/semicon/fermi/heavyVSmoderate/intro.html
I think you are in the regime where MB statistics can be used, and I assume the applet uses them. The next applet compares the distributions.

http://jas.eng.buffalo.edu/education/semicon/fermi/heavyVSmoderate/index.html

http://jas.eng.buffalo.edu/education/semicon/fermi/functionAndStates/functionAndStates.html
Read the brief Applet Tutorial and do what it suggests. Try it at different termperatures.

A reference to saturation is here

http://www.mrl.ucsb.edu/~seshadri/2004_100A/100A_SemicondDevices.pdf

I have not found anything that puts it all together for a case of Donor and Acceptor doping, but I think this last reference combined with the first applet is saying that if you have Ed levels present due to donor doping, at room termperature many of those levels will be occupied. If you reset the applet at 300K and adjust the Fermi level to a donor doping of about E18, the fermi level is just below Ed and the distribution function is broad at that temperature, so many Ed states will be occupied. Ramping up the Acceptor concentration introduces some Ea states at around E12, and as the number of those states increases with Acceptor concentration, the density of conduction electrons drops a bit.

If you do the same thing at a lower temperature, say 100K, the Fermi energy is above Ed with no Acceptor doping and the distribution function is very sharp If the applet is correct, those Ed states are nearly fully occupied. Ramping up the Acceptror concentration to about E16 does almost nothing, but ramping it until the Acceptor concentration reaches Donor concentration has a dramatic effect.

At 200K it looks to me like at Donor concentration of E18 the Fermi level is just below Ed and starts to drop when the Acceptor concentration gets to be about E16.

The trend seems to be that to get the Fermi level significantly below the Ed levels, you have to go to higher temperatures, but then the distribution gets broad, so I don't see them being depleted. I don't have an answer for you about what the saturation level is, but the general behavior of the applet seems to be consistent with other sources such as the temperature dependence of the Fermi level graph in

http://photochemistry.epfl.ch/EDEY/AJMcE_2.pdf

It is of course the difference in Fermi levels in p and n type materials that accounts for the junction behavior of diodes and transistors, and this last reference talks about what is going on there.