We know that high doping densities cause the band gap to shrink in semiconductor. I found the general relationship between doping density and band gap change as
delta E=(-3*q^2/16*pi*epsilon)*square root(q^2*N/epsilon*k*T)---(i)
where delta E is the change of band gap energy,N is the doping density, q is the electronic charge, epsilon is the dielectric constant of the semiconductor, k is Boltzmann's constant,T is the temperature in Kelvin and * is "multiplication sign" , / is "division sign" and ^ is "to the power" sign. For silicon,epsilon = 11.9 and T=300K.
Now how can I reduce the equation (i) to
delta E= -22.5*square root (N/10^18 cm^-3) meV
for silicon.
Hi
Are you referring to this?
http://ecee.colorado.edu/~bart/book/eband6.htm
Hi Chandra,
As an alternative to the equation you gave, consider the following two approaches, which are applied by the physicists in the silicon photovoltaic industry.
There's a simple empirical approximation that relates bandgap narrowing (BGN) to doping density, where the change in energy follows the formula ∆E = A∙ln(N/Nonset) for N > Nonset, where N is the dopant concentration, and where A and Nonset are constants that are determined empirically. They depend on the dopant type and temperature. See, for example, the papers in the publication list below led by delAlamo, Yan, and Cuevas.
There's also a more complicated approach that takes Fermi-Dirac statistics into account. E.g., see papers led by Schenk, Altermatt and McIntosh.
Finally, there's an online calculator here https://www.pvlighthouse.com.au/bandgap that calculates ∆E for silicon as a function of dopant concentration and temperature.
I hope that helps,
Keith.
Some publications on BGN in silicon:
P.P. Altermatt, J.O. Schumacher, A. Cuevas, M.J. Kerr, S.W. Glunz, R.R. King, G. Heiser, A. Schenk, "Numerical modeling of highly doped Si:P emitters based on Fermi–Dirac statistics and self-consistent material parameters," Journal of Applied Physics, 92 (6), pp. 3187–3197, 2002.
J. del Alamo, S. Swirhun and R. M. Swanson, "Simultaneous measurement of hole lifetime, hole mobility and band gap narrowing in heavily doped n-type silicon," Proceedings of the 18th IEEE PVSC, Las Vegas, pp. 290–293, 1985.
J. del Alamo, S. Swirhun and R. M. Swanson, "Measuring and modeling minority carrier transport in heavily doped silicon," Solid-State Electronics, 28, pp. 47–54, 1985.
A. Cuevas, P.A. Basore, G Giroult–Matlakowski and C. Dubois, "Surface recombination velocity of highly doped n-type silicon," Journal of Applied Physics, 80, pp. 3370–3375, 1996.
K.R. McIntosh and P.P. Altermatt, "A freeware 1D emitter model for silicon solar cells," Proceedings of the 35th IEEE PVSC, Honolulu, paper 531, pp. 2188–2193, 2010.
A. Schenk, "Finite-temperature full random-phase approximation model of band gap narrowing for silicon device simulation," Journal of Applied Physics, 84 (7), pp. 3684–3695, 1998.
D. Yan and A. Cuevas, "Empirical determination of the energy band gap narrowing in highly doped n+ silicon," Journal of Applied Physics, 114, 044508, 2013.
D. Yan and A. Cuevas, "Empirical determination of the energy band gap narrowing in p+ silicon heavily doped with boron," Journal of Applied Physics, 116, 194505, 2014.
Dear Alexander De Los Reyes ,
I am writing the equation (i) referring to that link.Actually I want to know how the calculation is done to get the answer for Si using the equation (i)
Hi Chandra
You can show the relationship if you will be careful with the units.
1.) Use the following values for the constants:
q = 1.602 x 10^-19 (Coulombs)
pi = 3.14
epsilon_s = 11.7 x 8.854 x 10^-12 (multiple relative permittivity to vacuum permittivity)
k = 1.38 x 10^-23 J/K
2.) From here, you will get -3.169 x 10^-33 in units of J*m^3/2 . Convert (-3.169x10^-33 J/m) X * (1 eV) / (1.602 x 10^-19 J) = -22.5 x 10^-15 (eV*m^3/2) or -22.5 x 10^-12 meV*m^3/2. The (10^-12)*(m^3/2) can be converted to sqrt(1/10^18 cm3).
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