Current density free electron gas

In summary: E}t/\hbar\right]$$Finally, we need to average over all collision times, which gives the final result:$$ \langle J(r,t)\rangle = \frac{ne^2\tau}{m} \left[\vec{E}-\frac{\vec{E}}{\hbar}\int_0^{\hbar\beta} e^{-t/\tau} dt\right] = \frac{ne^2\tau}{m} \left[\vec{E}-\frac{\vec{E}}{\hbar\beta}\right]$$This is known as the Drude's formula for the average of the electron density current. In summary, to compute the average of the electron density current
  • #1
peterprp
3
0
Hello,
I am studying transport in the free electron gas model and I don't understand how to compute the average of the electron density current.
We are given the hamiltonian
## H=\int \psi^\dagger(r,t)(-\frac{\hbar^2\nabla^2}{2m}+e\vec{E}\cdot\vec{r})\psi(r,t)##
where the ##psi## operator is the solution of the Heisenberg equation for field operators
## [H,\psi]=-i\hbar\frac{d\psi(r,t)}{dt} ##,
namely
## \psi(r,t) = \frac{1}{\sqrt{V}}\sum_k e^{i(\vec{k}-e\vec{E}t/\hbar)\cdot\vec{r}} e^{\int_0^t \epsilon(k-iEt'/\hbar)dt'}c_k##
where
## \epsilon(k)=\frac{\hbar^2 k^2}{2m}##
and I need to compute the average of
## J(r,t)=\frac{ie\hbar}{2m}[\psi^\dagger(r,t) \nabla \psi(r,t) - \nabla\psi^\dagger(r,t) \psi(r,t)]##
Now, the result is Drude's one:
##n e^2 \tau/m ##
Could someone please give me some hint on how to sketch out this calculation?
I know that I have to take the average over k weighting via the Fermi distribution and I also have to average over collision times, namely
## \int e^{-t/\tau}/\tau (...) dt ##
but why does the dependence on position disappear?
Thank you in advance
 
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  • #2
.To compute the average of the electron density current, you need to first calculate the expectation value of $J(r,t)$. This can be done by taking the trace of the density matrix $\rho$ times $J(r,t)$, which is equal to:$$\langle J(r,t)\rangle = \text{tr}(\rho J(r,t)) = \sum_{k,k'}\langle c_k^\dagger c_{k'}\rangle \langle k|J(r,t)|k'\rangle$$ where the $\langle k|J(r,t)|k'\rangle$ is given by:$$\langle k|J(r,t)|k'\rangle = \frac{ie\hbar}{2m} \int d^3 r\left[\psi^\dagger_k(r,t) \nabla \psi_{k'}(r,t) - \nabla\psi^\dagger_k(r,t) \psi_{k'}(r,t)\right] $$Using the expression for $\psi_k(r,t)$, we can rewrite this as:$$ \langle k|J(r,t)|k'\rangle = \frac{ie\hbar}{2mV} \sum_{\vec{q}}e^{i(\vec{k}-\vec{k'}-e\vec{E}t/\hbar)\cdot\vec{q}} \left[\vec{k}-\vec{k'}-e\vec{E}t/\hbar\right] $$Now, to get the average of $J(r,t)$, we need to average over all possible values of $k$ and $k'$ with the Fermi distribution, which gives:$$ \langle J(r,t)\rangle = \frac{ie\hbar}{2mV} \sum_{\vec{q}} \left[f(\vec{k}+\vec{q})-f(\vec{k})\right]\left[\vec{k}-e\vec{
 

FAQ: Current density free electron gas

What is current density in a free electron gas?

Current density in a free electron gas refers to the flow of electrons through a material per unit area. It is often denoted by the symbol J and is measured in amperes per square meter.

How is current density related to electron velocity?

The current density in a free electron gas is directly proportional to the average velocity of the electrons and the number of electrons per unit volume. This means that an increase in electron velocity or an increase in the number of electrons will result in a higher current density.

What factors affect the current density in a free electron gas?

The main factors that affect current density in a free electron gas are the number of electrons, the electron velocity, the cross-sectional area of the material, and the conductivity of the material. Changes in any of these factors can impact the current density.

How is current density different from current?

Current density and current are related concepts, but they are not the same. Current density refers to the flow of electrons per unit area, while current refers to the total flow of charge in a given time. Current density is a local property, while current is a global property.

What is the significance of current density in materials science?

Current density is an important concept in materials science as it helps to understand the behavior and properties of materials, particularly conductors. It is also a crucial factor in the design and development of electronic devices and circuits.

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