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Milad Dagher

For a length within the ballistic regime, why the current value changing with the length ?

Hi !

Ok, I considered the relaxation time to find the relaxation length ( L = tau * Vf = 36.8 um  ; taking Vf = 8.10^5m/s )

And so I chose 2 values for the length which were much less than this value, this should mean ballistic regime right ? (If my logic is flawed, this is where the mistake would probably be.)

The 2 values I chose were 1000 nm and 500 nm, and they gave 2 different values for the current. Shouldn’t the quantum conductance be in effect in this regime and thus show no difference for a change in a geometry?

Thanks in advance, milz

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    Zlatan Aksamija

    Interesting question—although the Boltzmann transport equation formalism is semi-classical, ie. it treats electrons/holes as classical particles, but introduces scattering quantum-mechanically by using perturbation theory, it should, in principle, reproduce the quantum limit simply by correctly counting the quantum states (for a discussion, see chapter 8, section 5 in “Advanced Theory of Semiconductor Devices” by Karl Hess). Therefore, I don’t think your logic is flawed when you assume that having very long relaxation times would lead to the ballistic regime. So let’s consider what should happen to the distribution function in this regime of very low scattering/large relaxation time. In such a case, the distribution function will simply advance in time under the influence of the field and should end up being displaced in energy by the amount of the applied voltage. Then the current should end up being proportional to the applied voltage without depending on the field (or length of the tube, which sets the electric field along the tube through F=V/L). A few caveats: the velocity is not necessarily always equal to the Fermi velocity—this only holds near the Dirac point, while the velocity will typically go down as we go away from the Dirac point; also, this is a transient simulation and it takes time to reach a steady state. How fast the simulation progresses toward steady state depends on the field which is set by the ratio of voltage and length; it also depends on the relaxation time since this parameter sets how fast the electron distribution changes. The total simulation time should be much longer than the relaxation time in order to make sure the system has reached steady state at the end of the simulation. Finally, the applied voltage at the ends of the tube sets the strength of the electric field inside the tube, which in turn determines how much the electron distribution changes at each step. If the voltage is set to be high (a few hundred milivolts or more) and the tube is short (less than a micron), this could lead to very high fields along the tube and result in nonlinear effects that reduce the current, leading to a departure from the expected ballistic behavior.

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