# Temperature dependence of the relaxation time in Boltzmann equation for impurity scattering in metals

Is there any temperature dependence of relaxation time in impurity scattering of conducting electrons?

It seems to me that there is none. But, some people claim that there is. So if you could explain, how temperature dependence comes into play if it does at all?

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Might want to emphasise 'impurity scattering' in the title, since this is the key part to this question ... – Brendan Dec 2 '12 at 0:07
I don't know the answer :-) Pretty much all I know about the impurity term is that it has a zero temperature contribution to resistivity. Actual impurities in a material should not increase with T but the effects could potentially increase with T. As far as I am aware, most 'textbook' models leave the impurity contribution as a constant but these are simplified and only reflect current established theory. The likely answer is that temperature dependence of the impurity contribution in most metals etc. is slight but in some cases it may be important ... Sorry I can't give a definitive answer! – Brendan Dec 2 '12 at 15:25
Yes, that is what I know, too. I am actually wondering, if you exclude lattice and electron-electron collisions, what of temperature dependence is left? Just the average energy of carriers coming from temperature dependence of Fermi-Dirac distribution, because, after all, impurity scattering is very much affected by kinetic energy of carriers, meaning, it is small for high energies..so, one way of increasing energy is by increasing temperature...anyone? – Zarko Dec 2 '12 at 22:07

Is there any temperature dependence of relaxation time in impurity scattering of conducting electrons?

Since in the original post, a precise reference (to Ziman’s book) is missing, I presume the referenced material is

Ziman, J. M. “Principles of the Theory of Solids” (1999), p. 215,

and the question is referring to the “relaxation time approximation” in Boltzmann’s semi-classical approach to transport in electronic systems in presence of static impurities.

In the Boltzmann’s approach, the change in the occupation number of momentum states $| \mathbf{k} \rangle$ due to collisions is obtained by using Fermi's golden rule which yields an integral equation, $$\left( \frac{\partial}{\partial t} n_{\mathbf{k}} \right)_{collisions} = - \frac{n_{imp}}{V} \sum_{\mathbf{k}'} [ n_{\mathbf{k}} (1 - n_{\mathbf{k}'}) W_{\mathbf{k}', \mathbf{k}} - n_{\mathbf{k}'} (1 - n_{\mathbf{k}}) W_{\mathbf{k}, \mathbf{k}'} ] ~,$$ where $\frac{n_{imp}}{V}$ is the density of impurities, $n_{\mathbf{k}}$ denotes the occupation probability of the state $\mathbf{k}$ (not necessarily the equilibrium Fermi-Dirac distribution), and $W_{\mathbf{k}, \mathbf{k}'}$ is the transition rate from state $\mathbf{k}$ to $\mathbf{k}'$. The first term in the sum represents the rate of scattering out of state $\mathbf{k}$ and the second term, represents the rate for scattering into the state $\mathbf{k}$.

Often, one uses a simpler approximation for the collision term, the “relaxation time approximation”, in which $$\left( \frac{\partial}{\partial t} n_{\mathbf{k}} \right)_{collisions} \approx - \frac{n_{\mathbf{k}} - n_{\mathbf{k}}^{eq}}{\tau}$$ where $n_{\mathbf{k}}^{eq}$ is the equilibrium distribution function, and $\tau$ is the “relaxation time”. This time scale is roughly equal to lifetime of the electrons due to the impurity scattering [1]. The $\frac{1}{\tau}$ factor is essentially an approximation to the transition rate $W$, through which only forward scattering (no change in the momentum) is kept.

Notice that Fermi's golden rule is only valid at zero temperature. So, strictly speaking, the result above is valid only for zero temperature. Furthermore, Boltzmann transport equation describes a non-equilibrium situation where the temperature is ill-defined. So, one concludes that, in the strict sense, the relaxation time cannot depend on temperature. It is related to the ground-state (hence, zero-temperature) properties of the electronic system and the impurities.

However*, it is possible to extend the Boltzmann’s semi-classical equations to include quantum effects; for this, one uses a quantum many-body theory and obtains the quantum Boltzmann equations for a non-equilibrium setting in presence of an external field (e.g., an electromagnetic field). Then, the temperature dependence in the parametres appearing in the Boltzmann equation (like the relaxation time) appears due to the initial equilibrium state – i.e., the state of the system before applying the external field. In the current case (where $\tau$ is directly related to the lifetime of conduction electrons), one concludes the microscopic theory of Fermi liquids that $\tau \sim T^2$ at low temperatures; so, $\tau$ would have only a weak temperature dependence.

For details, consult, e.g., Rammer, J. “Quantum Transport Theory”, (2004), chp. 10.8.

[1] This part is based on Bruus, H., and K. Flensberg. “Many-body quantum theory in condensed matter physics” (2004), section 15.3.
$^\ast$ This note is added thanks to a comment by garyp.

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If the analysis is valid only at zero temperature, how does one conclude that the relaxation time cannot be different at a non-zero temperature? – garyp Dec 4 '15 at 19:46
The Boltzmann equation is, sensu stricto, valid at zero temperature or in a non-equilibrium setting; a temperature-dependence can only added via a phenomenological justification. @garyp [as far as I understand your question] – Philosophiæ Naturalis Dec 4 '15 at 19:54
Ok, so the simple version of the Boltzmann eq. that you present has nothing to say about non-zero temperature, so it really can't be used to address this question. No? – garyp Dec 4 '15 at 20:08

Ok, so, since nobody is answering, let me try. If we neglect everything but electrons we can write for conductivity an integral with derivative of fermi dirac function under it. So, there it is. This derivation is a delta f at T=0, but it gets fuzzier while temperature increases. So in that change lies T dependence, I guess. But, why then Ziman in his book on transport says that there is no impurity scattering temperature dependence?

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Zarko, please add a proper reference to the Ziman's book mentioned in your answer; i.e., the bibliographic data of the book and the page or section you are referring to. – Philosophiæ Naturalis Dec 4 '15 at 17:17
Zarko, is this the precise reference you are referring to: Ziman, J. M. “Principles of the Theory of Solids” (1999), p. 215. – Philosophiæ Naturalis Dec 4 '15 at 22:26
I think the argument proposed in Zarko's answer can be extended to finite temperature using the Sommerfeld expansion. This answer by @RobJeffries, and the references therein, could be helpful. However, I do not think that it provides a rigorous analysis. – Philosophiæ Naturalis Dec 4 '15 at 22:30