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Homework Statement
An intrinsic semiconductor with a direct gap have valance band ##\epsilon_k = E_v-b|k|^2## and conduction band ##\epsilon_k = =E_c+a|k|^2##, with ##E_v=6.0##eV, ##E_c = 5.5eV##, ##a=5.0eV\cdot Å^2##, and ##b=3.0eV\cdot Å^2##.
Calculate the chemical potential ##\mu## and the number of charge carriers ##n+p##.
Homework Equations
Effective mass:
##m^*_e = \frac{\hbar^2}{2a}##
Intrinsic semiconductor ##n=p##:
##p=n = \sqrt{pn} = \frac{1}{\sqrt{2}}\left( \frac{k_BT}{\pi \hbar^2}\right)^{3/2}(m_e^* m_n^*)^{3/4}e^{-E_g/(2k_BT)}##
##e^{(2\mu-E_g)/k_BT}= \left( \frac{m_h^*}{m_e^*}\right)^{3/2}##
##\mu = \frac{E_g}{2}+\frac{3}{4}k_BT\ln \frac{m_h^*}{m_e^*}##
At ##T\approx 300##, ##k_BT \approx 25.7meV##.
The Attempt at a Solution
For the first part I get the correct answer just plugging in the numbers, with ##E_g = E_c-E_v = 0.5##eV and then shifting the energy by ##E_V## and noting that ##m_h/m_e=a/b##
##\mu = 5.5+0.25+25.7/1000 \cdot ln(5/3) \approx 5.76eV##.
For the second part we can rewrite
##n=\frac{1}{\sqrt{2}}\left( \frac{k_BT}{2\pi \sqrt{ab}}\right)^{3/2}e^{-E_g/(2k_BT)}##
plugging in the numbers here we get ##n \approx 1.45\cdot 10^{-9} Å^{-3}= 1.45\cdot 10^{21} m^{-3}##. However the answer claims I should get ##3.9\cdot 10^{23}m^{-3}##.
Any obvious error I'm doing or is the answer incorrect here? The factor in front can change a bit from simply changing ##k_BT## and the answer may be by a factor 2 as well if they mean ##n+p## but the factor of 100 is troubling.
Here is the numerical calculation in matlab
Code:
eg=0.5;
kbt = 25.7e-3;
sqab=sqrt(15);
c=1e30*(kbt/(2*pi*sqab))^(3/2)*exp(-eg/kbt/2)/sqrt(2)