# Why do electron and hole mobilities decrease with temperature?

From page 35 of "Microelectronics" by Millman Grabel

Mobility $\mu$ decreases with temperature because more carriers are present and these carriers are more energetic at higher temperatures. Each of these facts results in an increased number of collisions and $\mu$ decreases.

Limiting for now the discussion to electrons, my intuition suggests that at a higher temperature (without changing the electric field and the number of carriers), a single electron collides more frequently with the ions and so, on average, its velocity is reduced to zero more often. That causes its drift velocity to be less than it would be at a lower temperature.

I can't get an intuition of why an increased number of electrons (at a constant temperature) causes the mobility to decrease. For a given applied electric field, each electron changes its direction at each collision and, on average, it attains a drift velocity $v_d$. That happens for every electron, that are assumed to be independent from one another. The global drift velocity should still be the same $v_d$. So why the number of electrons affects the mobility (that is, the drift velocity since $v_d = \mu E$)?

What should I imagine for holes?

• Higher temperature means more phonons, so your paragraph on electrons is generally correct as one scattering process. The more free carriers then also means more ionized impurities (the donor/acceptors), so more impurity scattering. Holes will generally be the same, but with a higher effective mass so a lower mobility overall. See Sze's Physics of Semiconductor Devices, chapter 1. – Jon Custer Jan 22 '15 at 21:49
• Take a look at majority carriers, e.g. Platinum Silicide, while you're at it. – Carl Witthoft Jan 22 '15 at 21:51
• @Jon Custer. Thanks. I don't understand why more impurity scattering affects the overall drift velocity. Maybe I simply don't understand what do you mean with "impurity scattering". Do you mean more carriers, more recombination of charges, that is more chances that an electron combines with an acceptor? – the_eraser Jan 22 '15 at 22:08
• Take doped Si - the dopant atoms (B, P, etc.) aren't silicon, and electrons going by know that the potential that they see, well, just isn't quite right. So, the presence of the dopant atoms on the lattice increases carrier scattering, separately and on top of phonon scattering. So, while you may get more carriers, you also get more scattering. That is why high mobility devices need to keep the carriers away from the dopants that generated them. – Jon Custer Jan 22 '15 at 22:32
• @JonCuster. Anyway, the paragraph quoted at the beginning speaks in general, without referring to extrinsic semiconductors, so it should apply also for intrinsic ones, without dopants. The section in the book goes on saying "Mobilities are also functions of the electric field intensity and doping levels". Your explanation seems to relate more with this last sentence (the "doping levels" part). So, why in an intrinsic semiconductor, more carriers causes less mobility? – the_eraser Jan 23 '15 at 11:31

Maybe I've found an intuitive answer myself to the remaining question: "why in an intrinsic semiconductor more carriers causes less mobility?".

Probably what the book means is in terms of probability for electrons to hit the targets (ions). Thinking of one single electron moving in a lattice, it is likely that it travels all across a finite conductor without colliding with a ion. The $v_{d,1}$ for that electron is higher than $v_d$. If a second moving electron collides, its $v_{d,2}$ is less than $v_{d,1}$ and so the average is closer to $v_d$. This way of reasoning can be extended to a higher number of electrons up to the point where an increase in the number of electrons doesn't affect the average number of collisions. Maybe that point is not reached for normal temperatures.

The way mobility depends on average scattering time of the carriers is given here:

A simple model gives the approximate relation between scattering time (average time between scattering events) and mobility. It is assumed that after each scattering event, the carrier's motion is randomized, so it has zero average velocity. After that, it accelerates uniformly in the electric field, until it scatters again. The resulting average drift mobility

where q is the elementary charge, m* is the carrier effective mass, and τ is the average scattering time.

If the effective mass is anisotropic (direction-dependent), m* is the effective mass in the direction of the electric field.

The higher the temperature , i.e. the more kinetic energy the carriers have, the faster they will meet a scattering center. So in a simple model the higher the temperature the smaller the mobility.

With increasing temperature, phonon concentration increases and causes increased scattering. Thus lattice scattering lowers the carrier mobility more and more at higher temperature. Theoretical calculations reveal that the mobility in non-polar semiconductors, such as silicon and germanium, is dominated by acoustic phonon interaction.

A rough human size analogy for slow motion (long scattering time, low temperature) versus fast: A crowd in a square listening to a speaker. They are still except for random mobility of changing places for friends or access to seats.... There will be no problem with mobility, i.e a person can go through the crowd easily. If an explosion is heard , everybody will start running and hitting each other (high temperature low mobility) . People die trampled in panic situations.