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Physics Capsule

Drude–Lorentz DC electrical conductivity

Solid-state physics

Drude–Lorentz DC electrical conductivity

We have previously seen the Drude model describing the microscopic structure of metals. We will now see how this explains electrical conductivity of metals.

We have previously seen the Drude model describing the microscopic structure of metals. We will now see how this explains electrical conductivity of metals. We begin with a small step by understanding DC conductivity and, over the next couple of articles, we will attempt to understand AC conductivity and certain electromagnetic phenomena.

Ohm’s law describes current through a conductor as being proportional to the potential drop across, specifically as being proportional by a factor known as the resistance of the material of the conductor. Mathematically, this means R = \frac{V}{I}.

It has been found that this resistance varies with the dimensions of the conductor — the bigger the cross-section of the wire is perpendicular to the direction of current flow, the smaller its resistance to the current flowing through it. It is helpful to think of a water pipe, since the larger the cross-section of the pipe, the more water flows through it, i.e. the less it resists the flow of water. While this idea is not entirely accurate, it is a good analogy to gain an initial understanding of the phenomenon.

Electric field and current density

Our approach begins with an understanding of electric fields and yet another concept somewhat closely related to resistance, called resistivity. The reason we introduce resistivity is precisely because resistance depends on the conductor dimensions, which is something we do not want hindering our calculations. An alternate way to define resistance, independent of Ohm’s law, is as R = \frac{\rho \vec{L}}{\vec{A}}, for a conductor of length L, cross-sectional area, A, and a new, intrinsic property of the material of the conductor, \rho, which depends only on the nature of the material — the resistivity.

\rho describes the nature of the conducting material — specifically a metal in our example — by comparing an applied electric field and the current density it induces. Current density is the amount of charge flowing in unit time through a unit cross-sectional area of a conductor, perpendicular to the direction of current flow. In fact, \rho is just a proportionality constant between the two: \vec{E} \propto \vec{j} \Longrightarrow \rho = \frac{\vec{E}}{\vec{j}}.

A moment of investigation will reveal that the two formulae involving \rho above are one and the same thing. For an electric field \vec{E} applied on a material, the potential difference across two points on it, V, is given by \vec{E} = \frac{\VEC{V}}{\vec{L}}, where L is the separation between the two points.

In other words, \vec{E} = \rho \vec{j} \Rightarrow \vec{V} = \frac{\rho \vec{I} \vec{L}}{\vec{A}}, or, in other words, R = \rho \vec{L} \vec{A}^{-1}.

Electrical conductivity

The current density \vec{j} = - n e \vec{v}, where the velocity \vec{v} incorporates the various directions of electron motion that arise as a result of their thermal energies. Suppose an electron (according to the Drude model) is traveling with some velocity \vec{v}, on collision, its velocity changes by some amount, \Delta \vec{v}. Over several collisions separated by an averaged out time, called the relaxation time, \tau, we can use the following simple mathematical calculations to arrive at a new property of the material called its conductivity.

We know that \vec{a} = \frac{\vec{F}}{m}, or \frac{\Delta \vec{v}}{t} = \frac{-e\vec{E}}{m}. But we have redefined our time interval as \tau, which means our velocity is now averaged out as well to, say, \vec{v_{avg}}, which in turn gives us, \vec{v}_{avg} = -\frac{e\vec{E}\tau}{m}.

The current density, \vec{j}, therefore becomes, \vec{j} = \left( \frac{ne^2 \tau}{m} \right) \vec{E}. We define the bracketed quantity as the conductivity (\sigma) of the conducting material.

    \begin{align*} \therefore \sigma = \frac{ne^2 \tau}{m} \end{align*}

Relaxation time and mean free path

As a result of our use of \tau in defining the electrical conductivity of a material, we have before us two inter-related ideas: when charge carriers move about, whether randomly or, in the case we are interested in, under the influence of an applied current, they collide. The average time between successive collisions, as stated above, is the relaxation time. Similarly, the average length covered between successive collisions is known as the mean free path.

Working backwards from the equation for \sigma we get a considerably accurate measure of the relaxation time:

    \begin{align*} \sigma = \frac{ne^2 \tau}{m}\\ \Longrightarrow \tau = \frac{m}{ne^2\rho} \end{align*}

where \sigma = \rho^{-1} as usual: the resistivity and conductivity of a material are inverses of each other. The relaxation time is generally of the order of one to ten femto seconds \approx 10^{-15}10^{-15}.

The mean free path (\lambda) is simply the product of the velocity and relaxation time: \lambda=v\tau. The value of the average velocity is about 10^5ms^{-1} (this can be determined by solving \frac{1}{2}mv^2=\frac{3}{2}k_BT for k_B being Boltzmann’s constant). And this gives an average value of \lambda as 1 to 10 \AA. All of this is, of course, at room temperature of about 293K.

Unlike relaxation time, the mean free path obtained this way, while accurate, is not representative of real world observations, where \lambda can sometimes go as high as a few centimetres.

An important observation to make at this point, one where the fallacy of the Drude model comes to light, is in three particular formulae. First of all, using \frac{1}{2}mv^2=\frac{3}{2}k_BT to calculate v, the magnitude of the average velocity of charge carriers in a conductor, means v \propto \sqrt{T} and, secondly, \tau \propto \frac{1}{\sqrt{T}}.

Finally, the value of \sigma being calculable from the above formula, we can see that it must be related to T as \sigma \propto \frac{1}{\sqrt{T}}.

This, however, is not entirely true, and the correction for this will come from more sensitive quantum mechanical calculations which will be the subject of another article. The general idea is that it tell us the variation of \tau is not simply as an inverse proportionality agains \sqrt{T} and, by extension, neither is the variation of \sigma. As a starting idea, and for everyday “classical” use — and as far as conductivity itself is concerned — the Drude model is an excellent approximation.

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V.H. Belvadi

V.H. Belvadi is a physics postgraduate specialising in solid-state physics and also interested in astroparticle physics. When he is free he makes photographs and short films, writes on his personal website, makes music, reads voraciously, or plays his violin.

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