Weak localization is a physical effect which occurs in disordered electronic systems at very low temperatures. The effect manifests itself as apositive correction to theresistivity of ametal orsemiconductor.^[1] The name emphasizes the fact that weak localization is a precursor ofAnderson localization, which occurs at strong disorder.

The effect is quantum-mechanical in nature and has the following origin: In a disordered electronic system, theelectron motion is diffusive rather than ballistic. That is, an electron does not move along a straight line, but experiences a series of random scatterings off impurities which results in arandom walk.

Theresistivity of the system is related to the probability of an electron to propagate between two given points in space.Classical physics assumes that the total probability is just the sum of the probabilities of the paths connecting the two points. Howeverquantum mechanics tells us that to find the total probability we have to sum up the quantum-mechanical amplitudes of the paths rather than the probabilities themselves. Therefore, the correct (quantum-mechanical) formula for the probability for an electron to move from a point A to a point B includes the classical part (individual probabilities of diffusive paths) and a number of interference terms (products of the amplitudes corresponding to different paths). These interference terms effectively make it more likely that a carrier will "wander around in a circle" than it would otherwise, which leads to anincrease in the net resistivity. The usual formula for the conductivity of a metal (the so-calledDrude formula) corresponds to the former classical terms, while the weak localization correction corresponds to the latterquantum interference terms averaged over disorder realizations.

The weak localization correction can be shown to come mostly from quantum interference between self-crossing paths in which an electron can propagate in the clock-wise and counter-clockwise direction around a loop. Due to the identical length of the two paths along a loop, the quantum phases cancel each other exactly and these (otherwise random in sign) quantum interference terms survive disorder averaging. Since it is much more likely to find a self-crossing trajectory in low dimensions, the weak localization effect manifests itself much more strongly in low-dimensional systems (films and wires).^[2]

In a system withspin–orbit coupling, the spin of a carrier is coupled to its momentum. The spin of the carrier rotates as it goes around a self-intersecting path, and the direction of this rotation is opposite for the two directions about the loop. Because of this, the two paths along any loop interferedestructively which leads to alower net resistivity.^[3]

In two dimensions the change in conductivity from applying amagnetic field, due to either weak localization or weak anti-localization can be described by the Hikami-Larkin-Nagaoka equation:^[3][4]

${\displaystyle \sigma (B)={\sigma }_0-\frac{e^2}{2{\pi }^2\hslash }\left[\psi (\frac{1}{2}+\frac{1}{\tau a})-\psi (\frac{1}{2}+\frac{1}{{\tau }_{1}a})+\frac{1}{2}\psi (\frac{1}{2}+\frac{1}{{\tau }_{2}a})-\frac{1}{2}\psi (\frac{1}{2}+\frac{1}{{\tau }_{3}a})\right]}$

Where${\displaystyle a=4DeH/\hslash c}$, and${\displaystyle \tau ,{\tau }_1,{\tau }_2,{\tau }_3}$ are variousrelaxation times and${\displaystyle {\sigma }_0}$ is the conductivity of the system in the absence of weak localization or weak anti-localization. This theoretically derived equation was soon restated in terms of characteristic fields, which are more directly experimentally relevant quantities:^[5]

${\displaystyle \sigma (B)={\sigma }_0-\frac{e^2}{2{\pi }^2\hslash }\left[\psi (\frac{1}{2}+\frac{H_1}{H})-\psi (\frac{1}{2}+\frac{H_2}{H})+\frac{1}{2}\psi (\frac{1}{2}+\frac{H_3}{H})-\frac{1}{2}\psi (\frac{1}{2}+\frac{H_4}{H})\right]}$

Where the characteristic fields are:

${\displaystyle H_1=H_0+H_{SO}+H_s}$

${\displaystyle H_2=\frac{4}{3}{H}_{SO}+\frac{2}{3}{H}_S+H_i}$

${\displaystyle H_3=2H_S+H_i}$

${\displaystyle H_4=\frac{2}{3}{H}_S+\frac{4}{3}{H}_{SO}+H_i}$

Where${\displaystyle H_0}$ is potential scattering,${\displaystyle H_i}$ isinelastic scattering,${\displaystyle H_S}$ is magnetic scattering, and${\displaystyle H_{SO}}$ is spin-orbit scattering. For a non-magnetic sample (${\displaystyle H_S=0}$), this can be rewritten:

${\displaystyle \sigma (B)-\sigma (0)=+\frac{e^2}{2{\pi }^2\hslash }\left[\ln \left(\frac{B_{\varphi }}{B}\right)-\psi \left(\frac{1}{2}+\frac{B_{\varphi }}{B}\right)\right]}$

${\displaystyle +\frac{e^2}{{\pi }^2\hslash }\left[\ln \left(\frac{B_{\text{SO}}+B_e}{B}\right)-\psi \left(\frac{1}{2}+\frac{B_{\text{SO}}+B_e}{B}\right)\right]}$

${\displaystyle -\frac{3e^2}{2{\pi }^2\hslash }\left[\ln \left(\frac{(4/3)B_{\text{SO}}+B_{\varphi }}{B}\right)-\psi \left(\frac{1}{2}+\frac{(4/3)B_{\text{SO}}+B_{\varphi }}{B}\right)\right]}$

${\displaystyle \psi }$ is thedigamma function.${\displaystyle B_{\varphi }}$ is the phase coherence characteristic field, which is roughly the magnetic field required to destroy phase coherence,${\displaystyle B_{\text{SO}}}$ is the spin–orbit characteristic field which can be considered a measure of the strength of the spin–orbit interaction and${\displaystyle B_e}$ is the elastic characteristic field. The characteristic fields are better understood in terms of their corresponding characteristic lengths which are deduced from${\displaystyle B_i=\hslash /4e{l}_i^2}$.${\displaystyle l_{\varphi }}$ can then be understood as the distance traveled by an electron before it loses phase coherence,${\displaystyle l_{\text{SO}}}$ can be thought of as the distance traveled before the spin of the electron undergoes the effect of the spin–orbit interaction, and finally${\displaystyle l_e}$ is themean free path.

In the limit of strong spin–orbit coupling${\displaystyle B_{\text{SO}}\gg B_{\varphi }}$, the equation above reduces to:

${\displaystyle \sigma (B)-\sigma (0)=\alpha \frac{e^2}{2{\pi }^2\hslash }\left[\ln \left(\frac{B_{\varphi }}{B}\right)-\psi \left(\frac{1}{2}+\frac{B_{\varphi }}{B}\right)\right]}$

In this equation${\displaystyle \alpha }$ is -1 for weak antilocalization and +1/2 for weak localization.

The strength of either weak localization or weak anti-localization falls off quickly in the presence of a magnetic field, which causes carriers to acquire an additional phase as they move around paths.

• Coherent backscattering

• Mesoscopic physics
• Condensed matter physics
• Electric and magnetic fields in matter

• Articles with short description
• Short description matches Wikidata