I have a conceptual question. Say at room temperature, electron has thermal energy of the order of 25 milli eV. Consider Germanium having band gap of 0.7 eV, so the corresponding temperature will be of the order of 8400 Kelvin. But as I have performed Four Probe experiment to find band gap, I went max upto 200 degree temperature. So what am I missing to understand, like to perform conduction in semiconductor do I need to go at large temperature's which is opposite to that of what we study in properties of semiconductor.
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2$\begingroup$ Hi Anshul. What are you asking? How electrons get excited across the band gap? If so then the answer is that they don't get excited across the band gap at room temp. We need to dope the germanium to make it conduct electricity. $\endgroup$– John RennieCommented Jan 22, 2023 at 7:30
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$\begingroup$ @JohnRennie Germanium, like all semiconductors, has an intrinsic carrier concentration. That is, how many thermally excited carriers you have in an undoped semiconductor. For Ge it is 2.4e13/cm3 at room temp. So yes, they most certainly do get thermally excited across the band gap at room temperature. Just not very many. $\endgroup$– MattCommented Jul 16, 2023 at 2:47
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2 Answers
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It sounds like you want to find the bandgap from your four-probe data which is a function of temperature. You need to fit this equation: $\sigma = q\ (\mu_e + \mu_p) N_c\ e^{-E_{gap}/(2kT)},$ where q is the magnitude of the electric charge, the $\mu$s are the mobility of the negative and positive charge carriers, $N_c$ is the number of carriers, and you understand the terms in the exponential. The mobility has a dependence on the band structure and scattering. The number of carriers $N_c$ also has a temperature dependence of $N_c\propto T^{3/2})$.
If you go to too high of a temperature, you will start probing parts of the band structure that violate the assumptions of this equation, and your fit won't be good. It's best to fit a small window of temperature, but still begin enough to average out scatter in your data.
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1$\begingroup$ Nc and Nv are "effective density of states", not number of carriers. $\endgroup$– MattCommented Jul 16, 2023 at 12:42
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1$\begingroup$ Can you go into more detail on which terms you fit? Mobilities and effective mass (part of Nc, Nv) seem to be things you wouldnt know in advance if trying to measure band gap energy. Are you fitting all of those? $\endgroup$– MattCommented Jul 16, 2023 at 12:44
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$\begingroup$ @Dr.Nate My question asked is very simple. By the formula we see that, temperature needed for an electron to go from CB to VB is about 8400 K, but by four probe, we go up to 500 K or so to find conductivity of a pure silicon. My question is how can we solve the conspiracy behind it even though they must gain energy of temperature equivalent about 8400 K. So that leads to my final question is that the sample placed in the four probe like silicon or germanium, if it gives conductivity at 400 K, then they must be doped Si or Ge then or is it something else? $\endgroup$ Commented Jul 21, 2023 at 7:19
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When you perform a four probe measurement you effectively voltage bias your sample.
By applying a voltage $V_0$ on it, the carriers (of charge $e$) inside it gain energy which is equal to $E=eV_0$. So you can very easily exceed the band gap with $V_0>E_{gap}$ ($0.7\,$V in your example) and produce a current you measure.
Moreover any energy value can be expressed in units of "temperature" through Boltzmann constant. This is not an actual temperature your sample is heated up at but just a way of comparing what would be the temperature of something having a thermal energy equal to $eV_0$.
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1$\begingroup$ Your voltage explanation doesn't sound right to me. By that logic, I could make diamond conduct with a 9 V battery because the bandgap is 5.5 eV. In the semiclassical picture, a particle would only gain $E = eV_0$ if it crossed the entire sample. Scattering stops particles from crossing substantial distances unimpeded. $\endgroup$– Dr. NateCommented Jul 16, 2023 at 15:15
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$\begingroup$ My question asked is very simple. By the formula we see that, temperature needed for an electron to go from CB to VB is about 8400 K, but by four probe, we go up to 500 K or so to find conductivity of a pure silicon. My question is how can we solve the conspiracy behind it even though they must gain energy of temperature equivalent about 8400 K. So that leads to my final question is that the sample placed in the four probe like silicon or germanium, if it gives conductivity at 400 K, then they must be doped Si or Ge then or is it something else? $\endgroup$ Commented Jul 23, 2023 at 5:09