# Mobility and resistivity as a function of temperature

## Summary:

How is it that in semiconductors when T increases, both resistivity and mobility decrease?

## Main Question or Discussion Point

Referencing: http://www.vlsiinterviewquestions.org/2012/07/21/inverted-temperature-dependence/

Mobility decreases in a MOSFET with increasing temperature

However, referencing: https://www.quora.com/Why-does-resistivity-of-semiconductors-decrease-with-increase-in-temperature

Resistivity decreases with increasing temperature.

As mobility is inversely proportional to resistivity (referencing: http://ecee.colorado.edu/~bart/book/mobility.htm)

How is it that when T increases, both resistivity and mobility decrease?

## Answers and Replies

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TeethWhitener
Gold Member
For conductivity (and its reciprocal, resistivity), I use the mnemonic "How many? How much? How fast?" In other words:
$$\sigma = ne\mu$$
where $\sigma$ is the conductivity, $n$ is the number of charge carriers (how many), $e$ is the charge on each carrier (how much), and $\mu$ is the mobility (how fast). The point here is that resistivity is proportional to mobility, but not just mobility.

As the temperature increases, the number of electrons that are thermally excited to the conduction band in a semiconductor (how many) increases, and it can increase faster than the electron-phonon interaction can decrease the mobility of the carriers (how fast).

Is this trend true for the full range of T? Is there a point where Coulomb and phonon scattering dominates over the number of thermal carriers such that conductivity actually decreases?

Lord Jestocost
Gold Member
Because of the typical temperature dependence of the mobility and the charge carier concentration, the temperature dependence of the conductivity looks for extrinsic semiconductors like that depicted in Figure 5 in:
[PDF]Lecture 8: Extrinsic semiconductors - mobility - Nptel

Because of the typical temperature dependence of the mobility and the charge carier concentration, the temperature dependence of the conductivity looks for extrinsic semiconductors like that depicted in Figure 5 in:
[PDF]Lecture 8: Extrinsic semiconductors - mobility - Nptel
If a doped semiconductor has increasing conductivity as a function of T, why is it that high performance electronics need heat sinks? Wouldn't they just conduct better at higher and higher T according to Fig 5?

ZapperZ
Staff Emeritus
If a doped semiconductor has increasing conductivity as a function of T, why is it that high performance electronics need heat sinks? Wouldn't they just conduct better at higher and higher T according to Fig 5?
Just because something has a high electrical conductivity does not mean that it also has a high thermal conductivity. Those two are not the same thing.

Electrical conductivity depends on the properties of the charge carriers. Thermal conductivity depends not just on charge carriers (electrons gas has a very low specific heat, by the way), but also on the properties of the lattice vibrations and how such vibrations are transported.

Zz.

Lord Jestocost
Gold Member
If a doped semiconductor has increasing conductivity as a function of T, why is it that high performance electronics need heat sinks? Wouldn't they just conduct better at higher and higher T according to Fig 5?
All electronic devices and circuitry generate excess heat and thus require thermal management to improve reliability and prevent premature failure. It makes no sense to focus on the performance of one component as electronic devices and circuitry consist of many material components.
https://en.wikipedia.org/wiki/Thermal_management_(electronics)
https://en.wikipedia.org/wiki/Reliability_engineering

I bring this up because in a separate post (https://www.physicsforums.com/threads/threshold-voltage-shift-vs-temperature.982727/#post-6282457), I showed the same figure of conduction vs temperature for an extrinsic semiconductor and was told that I was confusing it with an intrinsic semiconductor.

I said:
As I understand it, conductivity increases with temp in a semiconductor rather than decrease.
Fig 2.2.39 here: https://archive.cnx.org/contents/64d67245-4b32-4106-b5cf-b5bca68c2a6b@1/sspd-chapter-2-2-6-drift-velocity-in-semiconductor-metal-and-its-conductivity
Which is the same plot as:
Because of the typical temperature dependence of the mobility and the charge carier concentration, the temperature dependence of the conductivity looks for extrinsic semiconductors like that depicted in Figure 5 in:
[PDF]Lecture 8: Extrinsic semiconductors - mobility - Nptel
And the reply was:
You are mixing here the behavior of intrinsic conduction in high-purity semiconductor (actually intrinsic conduction was experimentally observed for germanium, but required materials purity is too good for modern silicon technology) and behavior of actual electronic devices which nearly never use intrinsic conduction, instead relying on fully ionized doping to provide charge carriers. Devices based on doped semiconductor have approximately fixed number of carriers and carriers mobility roughly proportional to T^(-2.5).
I'm a bit confused now. Does conductivity increase or decrease for a doped semiconductor/device as a function of increasing temperature?

ZapperZ
Staff Emeritus
I'm a bit confused now. Does conductivity increase or decrease for a doped semiconductor/device as a function of increasing temperature?
You are confused because you refuse to distinguish (or can't seem to comprehend) between ELECTRICAL conductivity and THERMAL conductivity. The former is the transport of charge. The latter is the transport of thermal vibration!

For semiconductors, as temperature increases, the ELECTRICAL conductivity increases.

For semiconductors, as temperature increases, the THERMAL conductivity does not necessarily follow the same trend. Semiconductors are typically thermal insulators. In fact, they are "band insulators" with smaller band gap than typical insulators. So they do not conduct heat very well!

So what is confusing here?

Zz.

I understand why we need a heat sink, to alleviate thermals.

The confusion is that you say electrical conductivity increases in an extrinsic semiconductor as T increases, which is in contrast to trurle and the referenced post and quote, stating that this is only true for intrinsic semiconductors and not devices. For doped semiconductors/devices (in contrast to the intrinsic case), the conductivity decreases with increasing T because dopant density >> intrinsic carrier density and thus conductivity is primarily dictated by the mobility/lattice scattering, so the conductivity degrades with T despite the continued increase in thermal carriers as those are negligible due to dopants >> n_i

ZapperZ
Staff Emeritus
I understand why we need a heat sink, to alleviate thermals.

The confusion is that you say electrical conductivity increases in an extrinsic semiconductor as T increases, which is in contrast to trurle and the referenced post and quote, stating that this is only true for intrinsic semiconductors and not devices. For doped semiconductors/devices (in contrast to the intrinsic case), the conductivity decreases with increasing T because dopant density >> intrinsic carrier density and thus conductivity is primarily dictated by the mobility/lattice scattering, so the conductivity degrades with T despite the continued increase in thermal carriers as those are negligible due to dopants >> n_i
"Devices" is not simple. Devices are often made up of not only more than one semiconductors (such as pn junctions), but also other conductors. There are surface layers to consider, what type of bias is being applied, how the connectors are made, etc... etc. Devices is a "black box" consisting of numerous combination of STUFF! It is not just a simple semiconductor, intrinsic or not.

Zz.

I'm a bit confused now. Does conductivity increase or decrease for a doped semiconductor/device as a function of increasing temperature?
For doped semiconductor, electrical conductivity decrease at high temperature.
You can see for example
http://www.vishay.com/docs/91237/91237.pdf
Fig. 4 - Normalized On-Resistance vs. Temperature

Resistance of NMOS channel doubles at 110C compared to room temperature.

Lord Jestocost
Gold Member
Does conductivity increase or decrease for a doped semiconductor/device as a function of increasing temperature?
As you can read from Figure 5 in: [PDF]Lecture 8: Extrinsic semiconductors - mobility - Nptel

In the so-called "ionization region", the conductivity increases with increasing temperature. In the so-called "extrinsic (or saturation) region", the conductivity is - so to speak - constant to a certain degree. In the so-called "intrinsic region", the conductivity increases with increasing temperature. The temperature dependence is thus related to the temperature range where you semiconductor operates.

fluidistic
Gold Member
All electronic devices and circuitry generate excess heat (...)
To be extremely pedantic, I do not think this is necessarily true. The Joule effect is always accompanied with the Thomson effect, usually at least 2 orders of magnitude lower due to the currents used. But nothing forbids one to pick a very small current and a material such that its Thomson coefficient temperature dependence is such that there would be an overall cooling when a current is passing through such a material.

fluidistic
Gold Member
I do not really understand the claim that $\kappa$ has almost nothing to do with $\sigma$ in a semiconductor. The more you dope a semi conductor, the more it becomes metallic and the more Wiedemann-Franz law will resemble that of a metal. For good thermoelectric semiconductors (usually highly, but not too highly doped semiconductors), it is common to think of $\kappa = \kappa_\text{electrons/holes} + \kappa_\text{lattice}$ where $\kappa_\text{electrons/holes}$ does satisfy W-F law.
I also do not understand why ZZ mentions that the electronic specific heat due to electrons is low (which of course it is, unless at very cold temperatures), but who cares how much energy one can store in the electrons when one is talking about the thermal conductivity, I mean $\kappa$ and $c_V$ are different concept, right? Or am I missing something?

I do not really understand the claim that $\kappa$ has almost nothing to do with $\sigma$ in a semiconductor. The more you dope a semi conductor, the more it becomes metallic and the more Wiedemann-Franz law will resemble that of a metal. For good thermoelectric semiconductors (usually highly, but not too highly doped semiconductors), it is common to think of $\kappa = \kappa_\text{electrons/holes} + \kappa_\text{lattice}$ where $\kappa_\text{electrons/holes}$ does satisfy W-F law.
I also do not understand why ZZ mentions that the electronic specific heat due to electrons is low (which of course it is, unless at very cold temperatures), but who cares how much energy one can store in the electrons when one is talking about the thermal conductivity, I mean $\kappa$ and $c_V$ are different concept, right? Or am I missing something?
I agree with you. I think ZZ might have it backwards. I would instead say: just because something has a high thermal conductivity does not mean that it also has a high electrical conductivity. Crystals will always have phonons, but not always free electrons. Note though there is some wiggle room. He might be able to mention special cases where we can disagree on which values are considered high or not.

Regarding the specific heat and conductivity, when you are dealing with the DOSs of electrons near the Fermi surface, one should expect to come across similar combinations of constants. In many metals (not in semiconductors) the electronic thermal conductivity is $\vec K=\frac{\pi^2}{3} (\frac{k_b}{e})^2 T\vec \sigma$, which doesn't require the specific heat in order to be derived. This has some similarity to the specific heat of an electron gas: $c_v=\frac{\pi^2}{3}k^2_bTg(E_F)$.

Equations are Eq. 2.81 and Eq. 13.58 from Ashcroft and Mermin.