Hydrodynamics for pedestrians

In electron transport, different length scales are involved: Among these, there are the typical width $w$ of a sample, the mean free path $\ell_{\rm mr}$ for momentum relaxing scattering from defects, phonons, umklapp scattering, and the mean free path $\ell_{\rm mc}$ for momentum conserving electron-electron scattering processes. (With momentum we mean in all cases the total momentum of the electron «fluid», not to be confused with the momenta of the individual charged particles.) We implicitly make a relaxation-time approximation and ignore any microscopic details of the scattering processes mentioned. Three regions can be distinguished:

• $w,\ell_{\rm mc}\gg\ell_{\rm mr}$. Traditional Ohmic regime – boundary scattering as well as momentum conserving scattering are ignored. For a bar of length $\ell$, width $w$, height $h$, the resistance scales like $R\propto\ell/(wh)$. Ohm’s law is valid, it makes sense to introduce a resistivity $\rho=(A/\ell)\cdot R$ with $A$ the cross-sectional area and $\ell$ the length of the sample. This removes any geometry, in particular cross-section dependence of the resistivity of a bar, the most common setup.

• $\ell_{\rm mc},\ell_{\rm mr}\gg w$. In this case, the sample width is the smallest parameter, boundary scattering dominates, and the behavior of the resistance changes. Essentially two types can be distinguished:
  • Knudsen regime: eletrons propagate freely inside the sample and scatter diffusively from the boundary. Martin Knudsen (1871–1949) originally developed his theory in 1909 for the description of gases diffusing through an array of small filter tubes. We note that in the kinetic theory of gases collisions between the gas particles (which conserve total momentum) and with the boundaries (momentum relaxing) are essentially the only scattering mechanism considered. This is very much different to the electron gas in popular single-electron descriptions like density functional theory where in particular electron-electron scattering is ignored.
  • Ballistic regime: electrons propagate freely inside the sample and undergo specular reflections at the boundary. A description corresponding to classical chaos can give some insight into the resistance here. However momentum relaxing generating finite resistance does not happen inside the sample but must be modeled in a different way, for example by attaching semi-infinite current leads to electron reservoirs.
• $w,\ell_{\rm mr}\gg\ell_{\rm mc}$. Here, electron-electron scattering dominates – between two «hits» at the boundary, the electrons undergo multiple phase-coherent scattering events, and momentum relaxing essentially occurs only at the (rough) boundaries. For $\ell_{\rm mr}\gg w$ in addition, we can apply a classical hydrodynamic description. For a bar of length $\ell$, width $w$, height $h$, the resistance scales like $R\propto\ell/(wh)(1/w^2+1/h^2)$, see this page.

We note that in most cases collisions between electrons are strongly suppressed due to (a) the Pauli principle and (b) screening. Taking these into account, the scattering rate for electron-electron collisions roughly follows $1/\tau_{\rm mc}\approx(k_{\rm B}T)^2/(\hbar\epsilon_{\rm F})$. Rutherford scattering assuming unscreened point charges combined with Pauli’s principle would give $1/\tau_{\rm mc}\sim(k_{\rm B}T/\epsilon_{\rm F})^2$, however using a Yukawa potential with screening length $r$ instead (Thomas-Fermi-screening) gives another factor $r^2\sim\epsilon_{\rm F}$. This yields a scattering time at room temperature of $\tau_{\rm mc}\approx10^{-10}{\rm s}$ which is about four orders of magnitude larger than typical experimentally observed scattering times in Ohmic wires. Furthermore electron-electron scattering only contributes to $\rho$ if umklapp processes are involved, however they can only occur when certain Fermi surface geometry conditions are fulfilled: At low temperatures $T\ll\Theta_{\rm D}$ (and sufficiently clean samples) they are most efficient when the Fermi surface is near the Brillouin zone boundary, because then only small phonon energies are required to generate large changes in the direction of the final electron momenta.

For sufficiently pure samples however it makes sense not to suppress electron-electron scattering and instead try a semiclassical description in terms of a stationary viscous electron fluid with position dependent drift velocity $v$ in a constant electric field $E$. With a finite relaxation time $\tau_{\rm mr}=\ell_{\rm mr}/v_{\rm F}$ for momentum-relaxing scattering, the Ukrainian physicist Radii Nikolaevich Gurzhi (1930–2011) suggested a modified Navier-Stokes equation

introducing the kinematic viscosity $\nu\approx v_{\rm F}\ell_{\rm mc}/3$. From the solution for a rod with radius $w$, Gurzhi obtained a rough sketch of the temperature dependence of the rod’s resistance: At very low temperatures $T$ well below a certain characteristic temperature $T_1$ provided that $\ell_{\rm mc}\gg w$, the effective mean free path is dominated by boundary scattering, $\ell_{\rm eff}\approx w$, and the resistance is a constant independent of temperature. Around $T_1$ with $\ell_{\rm mc}\approx w$, electron-electron scattering starts to dominate, and the resistance drops accordingly reaching a minimum at a temperature $T_2$ where momentum-relaxing scattering in turn is no longer negligible with $\ell_{\rm mr}\approx w^2/\ell_{\rm mc}$. At and above this temperature, $R(T)$ remains roughly constant as long as momentum-relaxing impurity and defect scattering dominates over electron-phonon collisions up to a further temperature $T_3$ where electron-phonon processes prevail and $R(T)$ follows the usual Bloch-Grüneisen law, i. e., $R(T)\propto(T/\Theta_{\rm D})^5$ for $T\ll\Theta_{\rm D}$, and $R(T)\propto T/\Theta_{\rm D}$ for $T\gg\Theta_{\rm D}$.

From Hagen’s and Poiseuille’s law (applicable here for $\ell_{\rm mr}\to\infty$), we learn for the resistance of a viscuous fluid in a hose with a circular cross section of radius $r$ that $R\propto r^{-4}$, in contrast to Ohm’s law where $R\propto r^{-2}$. This tells us why artery narrowing by e.g. atherosclerosis or smoking is very dangerous and can become life threatening and also why it is a good idea to use at least 3/4-inch hoses instead of the standard 1/2-inch ones for watering the garden.

[1] Gotthilf Heinrich Ludwig Hagen: Ueber die Bewegung des Wassers in engen cylindrischen Röhren. Annalen der Physik 122, 423 (1839)

[2] Jean Leonard Marie Poiseuille: Recherches expérimentales sur le mouvement des liquides dans les tubes de très petits diamètres. Comptes rendus hebdomadaires des séances de l’Académie des sciences 11, 961 & 1041 (1840); 12, 112 (1841)

[3] Martin Knudsen: Die Gesetze der Molekularströmung und der inneren Reibungsströmung der Gase durch Röhren. Annalen der Physik 333, 75 (1909)

[4] Radii Nikolaevich Gurzhi: Minimum of Resistance in Impurity-free Conductors. Journal of Experimental and Theoretical Physics 17, 521 (1963)