I agree with @JonCuster. I would, however, like to supplement his answer by mentioning a “trick” which can be used to circumvent the mobility-dependence problem. There is a way in which the “mobility,” as per the conventional definition, does not affect the current through the device. This is possible if the device is “ballistic.” With a ballistic device, however, we would be cheating, since mobility is undefined for a ballistic device. Before elaborating on these statements, let me first describe mobility from a microscopic point of view.
The microscopic interpretation of a phenomenological parameter such as mobility is connected to the rate of carrier scattering during carrier transport. In other words, mobility is a measure of the rate at which a current-carrying particle scatters off of phonons, electrons, impurities, lattice defects, etc. Mathematically this can be expressed as$$\mu = \frac{e \tau}{m_{\rm eff}}$$where $e$ is the unit charge, $\tau$ is the overall “carrier lifetime” or “carrier relaxation time,” and $m_{\rm eff}$ is the effective mass of the carrier. The individual contributions from qualitatively different scattering processes can be included using Matthiessen’s rule:$$\frac{1}{\tau} = \frac{1}{\tau_{\rm phonon}} + \frac{1}{\tau_{\rm impurity}} + \frac{1}{\tau_{\rm defects}} + \ldots$$where (for example) $\tau_{\rm phonon}$ would be the time interval between successive collision of a carrier with a phonon.
Now, in a ballistic device, the carrier travels through the entire device without scattering even once. This is accomplished by manipulating the device geometry, material parameters, temperature, etc., which in turn affect the contribution of the different scattering sources. However, as I pointed out earlier, we are cheating. That’s because the mobility, being solely defined in the terms of scattering rate, is undefined in a ballistic device. In other words, if I am telling you that a device has mobility $\mu$, then it is implied that the device is not ballistic. Hence you cannot have a fixed well-defined mobility and still have the current be independent of mobility. This is pretty much the same thing @JonCuster pointed out.
If you are interested in how one would obtain ballistic transport, then that’s a separate question, which I would be glad to answer if necessary. In this post, I simply pointed out that mobility alone cannot completely describe semiconductor transport in all parameter regimes. If you are interested, you can also find several examples of ballistic devices in the transport literature.
