Why is it called Einstein equation?

Question

Why is the equation $ \frac{D_n}{\mu_n}=\frac{D_p}{\mu_p}= V_t$ called Einstein equation where $D_n$ represents diffusion coefficient of electrons, $D_p$ represents diffusion coefficient of holes, $\mu_n, \mu_p$ represents mobility of electrons and holes respectively and $V_t$ represents thermal voltage.

Answer 1

In his 1905 paper on Brownian motion, Einstein derived the equation $$D=\frac{RT}{N}\frac{1}{6\pi kP}\tag{7}$$ where $T$ is temperature, $k$ is viscosity, $P$ is the radius of a spherical molecule (the Stokes radius) and $R$ and $N$ are constants.

A more familiar form used by Wikipedia is $$D=\frac{k_BT}{6\pi\eta r}$$ where $k_B$ is Boltzmann's constant, $\eta$ is viscosity, and $r$ is the radius. The variant you have, $$\frac{D}{\mu}=V_t$$ arises from defining the thermal voltage as $$V_t\equiv\frac{k_BT}{q}$$ and writing the electrical mobility $\mu$ in terms of $q$ and $6\pi\eta r$.

Sutherland and Smoluchowski also did similar work to arrive at variations of the equation, so they, too, deserve some credit. That said, neither Smoluchowski nor Einstein (I don't have access to the full text of Sutherland's work) used thermal voltage to compactify the diffusion equation.

Answer 2

It resembles the equations Einstein wrote down when he was studying diffusion in the context of Brownian Motion.

See, e.g., https://en.wikipedia.org/wiki/Einstein_relation_(kinetic_theory) for more