# Ohm's Law derivation from Drude Model

• iLIKEstuff
In summary, the derivation of Ohm's law from the Drude model involves using a simple sum of forces and modeling collisions with a first order approximation. The relaxation time is a free parameter that needs to be determined through experiments and the Drude model is not an exact representation of reality. More detailed models are needed for accurate predictions in solid state devices.

#### iLIKEstuff

I'm having trouble understanding the derivation of Ohm's law from the drude model.

$$\Sigma F_x = - e \: E + F_{collision} = 0$$ (my understanding is that there are only two forces in Drude's model: those from electron-ion collisions and applied external electric field forces)

Now I'm confused about how $$F_{collision}$$ is defined. The text I'm reading states

$$\bar F_{collision} = \frac{\Delta p_x}{\Delta t} \approx \frac{-mv}{\tau}$$ (there is a bar over the F, which I'm assuming means averaged)

where tau is the relaxation time

It seems like they did something like this:

$$F=\frac{dp}{dt}=\frac{\Delta p}{\Delta t} =\frac{p}{t} =\frac{mv}{t}$$

This is the part i don't understand. how does $$\Delta p = p$$ ?

Sure you can do that for say position:

$$\frac{\Delta x}{\Delta t} = \frac{x}{t}$$

but you must assume that the initial position and time were at x=0 and t=0 respectively. For momentum, how can you assume that the initial momentum is 0?

The derivation then continues with simple algebraic rearrangement.

$$- e \: E + \frac{-mv}{\tau} = 0 \: \: \Rightarrow \: \: eE = \frac{-mv}{\tau} \: \: \Rightarrow \: \: v=\frac{-eE \tau}{m}$$ , where v is the drift velocity

Thanks guys.

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The Drude model is not at all "exact" in any way; the whole idea is to model what happens "on average" for a single particle in the electron gas and use that to model properties of the bulk. Hence, the details of every single collision is irrelevant.

$$\bar F_{collision} = \Delta p_x/\Delta t \approx \frac{-mv}{\tau}$$ is just a common way of modelling collisions in classical mechanics. In this case it is not meant to be an exact description of what is happening, it is merely a good approximation where $$\tau$$ is just essentially a free parameter with the dimension of time that has to be determined using experiments (i.e. $$\tau$$ does not neccesarily have anything to do with the time it takes for a single, real, collsion, and the latter would obviously need to be modeled using QM and can't be described using classical mechanics).

$$\Delta p_x/\Delta t$$ is just a first order approximation to $$dp/dt$$ which would be exact if the momentum was changing in a linear fashion during the collision (this type of model is often introduced by considering a rubber ball hitting the floor). The reason why this works is because $$\Delta p$$ is NOT the total momentum but the CHANGE in momentum that takes place during the time $$\Delta t=\tau$$.

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The equality $$\Delta p = p$$ is, IIRC, made assuming that during a collision, the electron stops, or that any given electron might come out of a collision event in a random direction, so that the average p after a collision is 0.

In reality, the scattering time is just a phenomenological fitting parameter. You know that the collision force is going to be some function of the velocity, and so you expand it in a Taylor series, F = a_0 + a_1*v + higher order terms. However, you know that F=0 when v=0, and so the zeroth-order term must vanish. If you then assume that the velocity is small, then you can neglect the higher order terms and just get a term that is linear in v.

And, as others have noted, the Drude model is nowhere near exact. I can tell you from personal experience that more detailed models for carrier transport and scattering in solids are extremely important for a lot of solid state devices.

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Manchot said:
And, as others have noted, the Drude model is nowhere near exact. I can tell you from personal experience that more detailed models for carrier transport and scattering in solids are extremely important for a lot of solid state devices.

Exactly, see this post...

My advisor, a couple of his collaborators and I at one time may have been the experts in transport theory in crystalline systems, I believe I did the only work in exact transport in anisotropic (tetragonal) lattices.

Drude is a reasonable 1st order estimate in cubic materials, you have to get into solving the Boltzmann Equation for really good estimates.

## 1. What is Ohm's Law?

Ohm's Law is a fundamental principle in physics that relates the current flowing through a material to the voltage applied across it. It states that the current (I) is directly proportional to the voltage (V) and inversely proportional to the resistance (R), expressed mathematically as I = V/R.

## 2. What is the Drude Model?

The Drude Model is a classical model that describes the behavior of electrons in a material. It assumes that electrons are free to move in a material and are scattered by collisions with atoms or other electrons. This model is often used to explain the electrical conductivity of metals.

## 3. How is Ohm's Law derived from the Drude Model?

Ohm's Law can be derived from the Drude Model by considering the motion of electrons in a material. The model assumes that electrons move with a constant average velocity and experience collisions with a characteristic frequency. By equating the average force on an electron to the electric force, the expression for Ohm's Law can be derived.

## 4. What are the assumptions of the Drude Model?

The Drude Model makes several simplifying assumptions, including that electrons are free to move and experience collisions, but do not interact with each other. It also assumes that the average velocity of electrons is constant and that the collisions are elastic, meaning there is no loss of energy.

## 5. What are the limitations of the Drude Model in explaining Ohm's Law?

The Drude Model is a simplified model and does not fully capture the complex behavior of electrons in a material. It does not take into account quantum mechanical effects or the interactions between electrons. As a result, it is not accurate for all materials and may not fully explain the observed behavior of electrical conductivity in some cases.