# Why is there no conservation principle for n and p (electrons/cm^3 and electron hole density)?

I'm taking analog circuits next quarter and I'm watching Ali Hajimiri's course at Cal Tech to prepare while doing problems from my own textbook (we cover the same material roughly). He poses this question and sort of gives an answer to it but I'm unsatisfied. I've opened up my copy of Griffith's electrodynamics and have not found anything either.

To properly state his question: "why is it that the product of n & p are constant but not the sum?"

I actually thought about the question before he posed it. We have many conservation laws in physics. Conservation of mass, conservation of charge, KVL, KCL, etc.

It's really non-trivial to me why in a qualitative way it is the product not the sum that is constant.

I worked through the

$$n + p= n_ie^{\frac{E_f - E_i}{kT}}+n_ie^{\frac{E_i-E_f}{kT}}$$

which works out to

$$n + p = 2n_i\cosh (\frac{E_f - E_i}{kT})$$

that doesn't tell me much about this relationship other than the sum is catenary in nature, in other words it's describing an inverted trajectory which makes some sort of intuitive sense but I'm still unsatisfied. I think knowing that equation is obfuscating what's really important about this question.

Ali talks about probabilities and modeling of women and men interacting in a system to explain this phenomenon but I'm unsatisfied with that.

Is this a counting problem?

To rephrase my question: Is the hole density being reused by electrons making the sum of n and p non-constant?

Intuitively that's what would seem to make sense from the physical behavior but I'm sure if that is wrong you can tell me. The electrons are bouncing in and out of holes. The geometric region they exist within is constant but they're constantly moving making their sum fluctuate as a function of each other.

Thank you for your time.

• Like the law of mass action in chemistry, where for example the products of the concentrations of H$^+$ and of OH$^-$ are constant (about $10^{-14}$ at room temperature) when one adds some acid or some base. If the sum were constant, what would happen if you added more than $10^{-7}$ of acid? – Pieter Dec 17 '17 at 2:54

## 1 Answer

The constancy of the $np$-product holds for a semiconductor as long as the Fermi-level is located in the band gap not too close to the conduction and valence band edges $E_C$ and $E_V$, respectively. Then the electron distribution in the conduction band and the hole distribution in the valence band are approximated by Boltzmann distributions so that the electron concentration in the conduction band can be written $$n=N_C\exp{-\frac{E_C-E_F}{kT}}$$ and the hole concentration in the valence band can be written $$p=N_V\exp{\frac{E_V-E_F}{kT}}$$ $N_C$ and $N_V$ are the effective densities of state of the conduction and valence band, respectively, $E_C$ and $E_V$ the conduction and valence band edge energies , and $E_F$ is the Fermi level. Thus the product is independent of $E_F$: $$np=N_CN_V\exp{-\frac{E_C-E_V}{kT}}=N_CN_Vexp{-\frac{E_G}{kT}}=n_i^2$$ where $n_i$ is the so-called intrinsic concentration and $E_G$ is the semiconductor band gap.That the $np$ product is largely independent of the position of the Fermi-level in the band gap and thus n- or p-doping simplifies calculations considerably. It has nothing to do with conservation laws of electrons or holes.

On the other hand, in a semiconductor with electrons and holes, there is a conservation law for the conservation of charge related to the conservation of the number of electrons (holes are just missing electrons). Thus, in a semiconductor also the general charge conservation law holds: $$div \vec j=-\frac {\partial \rho}{\partial t}$$ were the total current density is the sum of the electron and the hole current densities $$\vec j= \vec j_n+\vec j_p$$ and the total mobile charge density is the sum of the electron charge and the hole charge densities $$\rho=\rho_n+\rho_p=-qn+qp$$ ($q$ is the elementary charge). It has also to be taken into account that in a semiconductor electron and hole pairs can be created and annihilated by generation and recombination processes.

• So Ali in the video I shared explained the product relationship of n & p, which you do too, but my question is why is it that n and p are not conserved in a semiconductor? My question is much more qualitative. Equations are just what we have to describe relationships we observe. What is it that we observe that is not conserved with respect to n and p? – jake mckenzie Dec 16 '17 at 21:12
• I have added a paragraph addressing the conservation of charge and its relation to electrons and holes in a semiconductor. – freecharly Dec 16 '17 at 22:56
• I had to think about your response for a while but I think the bit you added helped a lot. I read some more about the generation and recombination process which helps. – jake mckenzie Dec 19 '17 at 21:46
• @jake mckenzie - In the end it means that the total number of of electrons minus holes are conserved in a semiconductor. Conservation laws for electrons and holes similar to the total charge conservation law can be formulated separately but then generation and recombination terms have to be added to these equations. – freecharly Dec 19 '17 at 22:09