From Donald Neamen's book on " Semiconductor Physics and Devices (4th edition)", page 113 quotes Nc and Nv values to be 2.8 x10^19/cm^3 and 1.04 x10^19/cm^3 for electron and hole effective density of states in silicon at T = 300 K. The band gap of silicon is 1.12 eV ( T = 300 K). Using k = 8.625 x10^-5 eV/K and T = 300 K and using ni^2 = Nc Nv exp(-Eg/kT), the value of ni comes to be 6.806 x 10^9/cm^3 at T = 300 K where as for all non degenerate free carriers calculation for example calculating minority carriers from known majority carriers, a value of ni = 1.5 x 10^10/cm^3 at T = 300 K is quoted throughout the exercises and solved-out problems in this book. Even though the book is widely referenced for semiconductor device related courses in top ranked US universities at least for undergraduate program, this erroneous result (much too low value) of ni coming out of listed Nc and Nv values for silicon in this textbook through computation by using above equation, does not even come close to used value 1.5 x10^10/cm^3. In the other highly popular textbook " Advanced Semiconductor Fundamentals (2nd edition)" by Robert F. Pierret, page 113 quotes Nc and Nv values for silicon at T = 300 K to be 3.23 x10^19/cm^3 and 1.83 x 10^19/cm^3. Using Eg = 1.12 eV at T = 300 K and k = 8.625 x10^-5 eV/K and ni^2 = Nc Nv exp(-Eg/kT), value of ni comes out to be 9.696 x10^9/cm^3 which is much closer to be ni value 1 x10^10/cm^3 quoted by Pierret in his two textbooks for solved-out problems and exercises. R.F. Pierret's textbooks along with Donald Neamen's textbook are reference textbooks in many university teaching on semiconductor device physics and principles world-wide. The two textbooks written by R.F. Pierret provide more accurate analysis than book by Donald Neamen although I have recollection that Donald's textbook was included as reference textbook in undergraduate course instruction in Arizona State University Tempe, Arizona, USA. The books must be standardized with respect to precisional quoting of semiconductor device parameters from which other device parameters are extracted and accuracy of these values are important for both solved-out examples and exercise problems.
Sincerely,
Dr. Nabil Shovon Ashraf
Associate Professor
Department of ECE
North South University
Dhaka, Bangladesh.
Dear Dr. Nabil,
welcome
I think the different values of the calculated ni come from the effective masses used to calculate the effective density of states. This issue was investigated in an old paper and it was found that the dependence of the effective masses on temperature is the main cause of the discrepancies found in the calculated effective masses. So, in order to eliminate this discrepancy one has to take care of the temperature of the effective masses. For more information please refer to the link:Article Effective mass and intrinsic concentration in silicon
Best wishes
Dr. Zekry: Regards. I am fully cognizant of your explanation and I knew that the 3D density of effective mass for electron and hole and also as a function of temperature are key here to determine Nc and Nv at T = 300 K. The point is that different textbooks are using these different or erroneous Nc and Nv values which give below par ni values which goes into calculation of many parameters like bulk potential, Ec-Ef or Ef-Ev difference and also calculation of minority carriers. For high injection cases, minority carriers gets highly multiplied and hence ni needs proper evaluation . The books give examples and exercises where for example when the author refers a ni = 1.5 x 10^10/cm^3 but which does not come out of relevant equation as I have described in this posed discussion, it creates illustration related problem for teaching students as I am an instructor in a university and I do follow Donald Neamen's book. My point of this discussion is authors should pay more attention when composing a book for graduate and undergraduate level instruction to precisely quote the relevant device parameters for calculation of other precise device parameters, in case for semiconductor device related courses.
Sincerely,
Dr. Nabil Shovon Ashraf
Associate Professor
Department of ECE
North South University
Dhaka, Bangladesh.
This has been a recurring problem in the literature. The most commonly used value in the past for the silicon intrinsic concentration was 1.45 x 10^10 cm^-3. Hence, the slightly simpler value in textbooks with a pre-factor of 1.5. Actually, scanning the literature it seems impossible to find out where 1.45 came from. Various paper have been published over the years, showing that the value should be closer to 1.0 (in fact, I know at least one textbook from the 1980's that used n_i = 10^10 cm^-3). The work by Sproul and Green (1990) did support this value. More recent work by Altermatt (2003) and even some prior work by others have suggested that the value might be even lower, perhaps 9.65 x 10^9 cm^-3. So you are right, most authors of textbooks have neglected to keep up with the literature and this keeps propagating inconsistencies. This said, textbook models for silicon do not describe silicon very well in general. This is the most widely used semiconductor but it has a very complicated band structure which does not make it a good prototype for simplified treatments. So, what we actually have in textbooks is a very simplified material model which on average behaves like silicon well enough for elementary purposes and which is fine for semi-quantitative or qualitative analysis. For sure one should never use textbook tables of parameters for advanced research purposes. There are other germane cases in different fields. In electromagnetism, textbooks use 3x10^8 m/s for the speed of light in vacuum, a value slightly higher than the official one but certainly easier to remember, although unphysical if one thinks about it. All works quite fine in most cases without detracting from pedagogy since dimensions are referred usually to units of wavelength. But just develop applications for lossy transmission lines (usually avoided in elementary courses but unavoidable in real life) and the minute error in speed of light may lead to spectacular differences for specific designs, with results which are not even close.
Thank you Professor Ravaioli for your highly illuminating response. I am very elated that you shared your views and perspectives with regard to this discussion. In fact the problem of defining ni = 1 x 10^10/cm^3 ( as per Professor Pierret's textbook) was not deducible even from the listed Nc and Nv values in the textbooks by Professor S. M. Sze, another renowned author whose books are highly followed in leading universities in USA and around the world. I believe Professor Ravaioli you agree that even though with research work in semiconductor devices, a researcher quotes the relevant device parameters as accurately as possible from investigation of a myriad of reference materials, it is time that these research data inputs regarding device parameters get also introduced in textbooks for undergraduate and graduate level study, for instance, semiconductor device theory in this case. This is also due to the reasons that the solved-out problems and exercise problems in different chapters of these textbooks cannot be deemed erroneous due to error prone device parameters used to solve these problems.
Sincerely,
Dr. Nabil Shovon Ashraf
Associate Professor
Department of ECE
North South University
Dhaka, Bangladesh.
As this discussion is centered around how important is to precisely calculate ni in silicon, significant error in calculation of intrinsic carrier concentration ni is associated with proper assignment of Nc and Nv values. Now incomplete ionization which needs to be accurately determined for ionized dopants at lower than room temperature, i.e., 300 K, take a look at the equation (4.55) page 133 of Donald Neamen's book "Semiconductor Physics and Devices" (4th edition). He uses Nc value as 2.8 x 10^19/cm^3 as is used to determine room temperature ni in silicon. From ni calculation as I have shown in this discussion, if Nc value quoted in this book is not precise, all the incomplete ionization related problems covered in this book will be also imprecise. On the other hand R.F. Pierret in his book "Advanced Semiconductor Fundamentals" (2nd edition) uses equations (4.72 a to 4.76 b) (page 122-123)to arrive at donor doping related incomplete ionization equations. From equation (4.73) in Pierret's book a parameter needs to be calculated which depends on Nc and if the quoted value as is illustrated in this discussion 3.23 x 10^19/cm^3 is right, more accuracy can be ensured as incorporation of this Nc and Nv as listed in Pierret's book gives ni very close to 1x10^10/cm^3 (verified from journal reference citation in his book) which he uses as reference ni in all the problems of this book. Then using this Nc, the incompletely ionized dopant concentration coming out of equation 4.76 b of this book will be also more accurate and the two books I composed with Morgan & Claypool publisher (2016 and 2018) on lower substrate temperature related device modeling, I have stuck to the derivations of R.F. Pierret's two graduate level textbooks just to be precise in parametric value related substitutions. These anomalies in assigned Nc and Nv values, calculated ni, etc., may also result in calculation errors of many device parameters related to drive current (on and subthreshold) and threshold voltage of MOSFET in important device simulation TCAD softwares like Silvaco, Sentaurus and ISE-TCAD (specially important for nanoscale MOSFET where the drive voltage is of the order of 0.4 V or below).
Sincerely,
Dr. Nabil Shovon Ashraf
Associate Professor
Department of ECE
North South University
Dhaka, Bangladesh.
One of the exercise an educator needs to demonstrate to class students by solution is the Fermi level positioning in an extrinsic semiconductor as a function of substrate temperature in a course on semiconductor device theory. Partly solution of the full domain of the temperature from 100 K or 200 K to 600 K should be ideal for this calculation. Textbooks by both Donald A Neamen and Robert F Pierret show plot of Fermi level positioning with respect to intrinsic Fermi level as a function of substrate temperature for different extrinsic doping concentrations for both n-type and p-type semiconductor although neither book discusses illustrative problems solution in relation to this figure by value substituted equation based computations. Textbook by Donald Neamen has additional accuracy problem in addition to its reference of Nc and Nv values at T = 300 K in silicon that I discussed above in a questionnaire form. The book does not provide the effective mass variation in silicon for electron and hole with substrate temperature which are required for calculating Nc and Nv values at other temperatures using room temperature reference value in silicon. Fortunately R.F. Pierret in his book "Semiconductor Device Fundamentals" provided these effective mass variation equations of electron and hole for silicon with substrate temperature. Also it is found out that the band gap of a semiconductor is also function of temperature as a result of which at lower substrate temperature than 300 K, since Eg shifts up the corresponding Ec value, Ec-Ed (where Ed is dopant energy level for donor) also widens (if we assume fixed donor states for shallow level doping where dopants generally will be less mobile keeping Ed approximately non-variant with lowering of substrate temperature but Ec varies as band gap shifts up) as a result of which its effect on incomplete ionization is slightly more pronounced. I did not see any of the solved out problems book by Donald A Neamen utilized the concept of band gap widening at lower substrate temperature to recalculate Ec-Ed from its known room temperature value for the calculation of incomplete ionization. R. F. Pierret book also does not discuss this feature explicitly with an illustrative problem solution.
Sincerely,
Dr. Nabil Shovon Ashraf
Associate Professor
Department of ECE
North South University
Dhaka, Bangladesh.
I am attaching an illustration here with regard to this discussion where I have shown how an error prone reference of intrinsic carrier concentration in silicon at T = 300 K generates gross error in n and p product of excess minority carrier generation case with error in quasi-Fermi level Efn and Efp.
Sincerely,
Dr. Nabil Shovon Ashraf
Associate Professor
Department of ECE
North South University
Dhaka, Bangladesh.
With regard to the attached illustration above, we have to further note that case 1 has a Efn-Efp = 0.7274 eV and case 3 has a Efn-Efp = 0.7091 eV. Depending on actual excitation voltage or photon flux applied to generate the excess carriers used in this illustration, case 1 will require higher excitation voltage or photon flux to generate the same np product as in case 3 and this is another clue which of the two cases 1 and 3 will need to be appropriately selected due to the reason that both cases satisfy the equation 4 with negligible error as shown in the above attached illustration.
Sincerely,
Dr. Nabil Shovon Ashraf
Associate Professor
Department of ECE
North South University
Dhaka, Bangladesh.
With regard to the illustration composed by me as an attachment, equation (4) can be satisfied by ni = 1.5 x 10^10/cm^3 (which does not anyway comes from band structure defined ni) but the difference for all cases is the value for Efn and Efp. Since Efn-Efp = Vapp or applied voltage or phi (photon flux), we need to know experimental determination of the exact value of excess carriers for electron and hole for a given applied voltage or photon flux in silicon at T = 300 K. Then from equation (4) we can determine the required ni by knowing np product, Efn and Efp. In this way we will be sure about the material specific ni which then can be related to band structure defined parameters Nc, Nv and Eg for silicon at room temperature. This ni should match required Vapp = Efn-Efp and the outcome of excess carrier generation or np product.
Sincerely,
Dr. Nabil Shovon Ashraf
Associate Professor
Department of ECE
North South University
Dhaka, Bangladesh.
Please check a technical paper I recently composed on this questionnaire and discussion session initiated by me. The paper was published in Chronicons magazine by IEEE North South University Students Branch in Issue 1 of this magazine last year. The correct reference of ni in device modeling applications is therefore imperatively insinuated here. Sincerely, Dr. Nabil Shovon Ashraf.
As illustrated in this question-answer-suggestion-observation based post posed by myself, proper computation of intrinsic carrier concentration for silicon (ni) at T = 300K is more important at highly elevated temperatures where the modified values of effective density of states for electrons (Nc) and holes (Nv) at the elevated temperatures are extracted based on reference or default values of these two parameters at T = 300 K. For instance, many of the educators while instructing solid state devices based courses discuss the figure of conductivity of a silicon semiconductor slab as a function of temperature. As we know the conductivity is dominated by intrinsic generation at highly elevated temperatures superseding the usual substrate doping values. Since intrinsic generation at higher temperatures rises exponentially with temperature, the correct value of conductivity is of utmost importance at these elevated temperatures for devices like thermistors and memristors where conductivity-induced device resistance change as a function of temperature stress is the key technology parameter. Incorrect reference of Nc and Nv at T = 300 K for silicon, as some widely-followed textbooks list for silicon, can induce substantial error in the precise value of ni at the elevated temperatures impacting the thermistor or memristor data collection or modeling being highly prone to error.
Sincerely,
Dr. Nabil Shovon Ashraf
Associate Professor
SAC 932
Department of ECE
North South University
Dhaka, Bangladesh.
Hi Dr. Ashraf,
I just came across your discussion here... A few years ago, I made just the same experience when trying to prepare a simulation of semiconductor behaviour using Excel as a tool. I had the Neamen book, also the booklets of Pierret, and found some semiconductor data in the internet, but with different figures (depending on the year of publication). I spent a lot of work in solving my problem, but finally I succeeded with a rather good (imho) calculation using figures from the site http://www.semiconductors.co.uk/.
Maybe this will help you as well.
Best regards
Axel
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