Theory of weak localization in ferromagnetic (Ga,Mn)As

We study quantum interference corrections to the conductivity in (Ga,Mn)As ferromagnetic semiconductors using a model with disordered valence-band holes coupled to localized Mn moments through a $p\text{−}d$ kinetic-exchange interaction. We find that at Mn concentrations above 1% quantum interference corrections lead to negative magnetoresistance, i.e., to weak localization (WL) rather than weak antilocalization (WAL). Our work highlights key qualitative differences between (Ga,Mn)As and previously studied toy-model systems and pinpoints the mechanism by which exchange splitting in the ferromagnetic state converts valence-band WAL into WL. We comment on recent experimental studies and theoretical analyses of low-temperature magnetoresistance in (Ga,Mn)As which have been variously interpreted as implying both WL and WAL and as requiring an impurity-band interpretation of transport in metallic (Ga,Mn)As.

Figure 1

(Color online) Diagrammatic expressions for the quantum interference correction to the conductivity and the Cooperon. Crosses stand for scattering events off impurities.

Figure 2

(Color online) M2DEG: Quantum correction to conductivity in the absence of an external magnetic field for (Ref. 32) ${ϵ}_F\tau =20$, ${\tau }_{\varphi }\simeq 42\tau$, and $u^z=0$. There is a WAL-to-WL crossover when the exchange field becomes comparable to the SO interaction strength. (Note that the band splitting due to the exchange field is $2h_z$). In limiting cases there is good agreement with the expressions of Table . The fact that $\delta \sigma \left(\lambda =0\right)$ is weakly dependent on $h_z$ indicates that an exchange splitting per se does not suppress weak localization unless it becomes comparable to ${ϵ}_F$. We have separately evaluated the $h_z=0$ curve for a different values of $l_{\varphi }$ (not shown) and found a good agreement with Fig. 1 of Ref. 3.

Figure 3

(Color online) M2DEG: Exchange field dependence of the conductivity correction for $\lambda k_F=0.5{ϵ}_F$, ${ϵ}_F{\tau }_{\varphi }\simeq 840$, and $u^z=0$. Note that the value of the exchange field at which the transition from WAL to WL occurs ($h_z\simeq 0.25{ϵ}_F$ in this figure) is independent of the scattering rate. This is true because the WAL-to-WL transition in a M2DEG is largely determined by changes in intraband correlations.

Figure 4

M2DEG: effect of exchange scatterers in a unmagnetized $\left(h_z=0\right)$ two-dimensional electron gas with Rashba SO interactions and ${ϵ}_F{\tau }_0=20$, where ${\tau }_0$ is the scattering time due to nonmagnetic impurities. Exchange impurities dephase the singlet Cooperon responsible for WAL and leave the triplet Cooperons unchanged. Hence there is a WAL-to-WL transition. The magnitude of WL keeps increasing at larger $u^z$ because the total scattering rate too is increasing.

Figure 5

(Color online) (Ga,Mn)As: Interband vs intraband conductance correction vs Mn fraction $x$ for $l_{\varphi }\simeq 100\text{nm}$ and $u^z=0$ in absence of an external field. $p$ stands for the hole density. In this figure we have set interband matrix elements of the velocity operator to zero in order to make the distinction between interband and intraband contributions to quantum interference clearer. The interband correlations responsible for WAL in GaAs decay rapidly as the exchange field due to Mn ions lifts band degeneracies. In contrast, intraband correlations are relatively indifferent to exchange splitting and are responsible for the eventual crossover to WL. For $p=0.4{\text{nm}}^{-3}$ and ${ϵ}_F\tau =2.5$ (these values are within the experimentally relevant range), $h_z=1/\tau$ is satisfied at $x=0.05$—note that interband correlations are nearly vanished at that point. Assuming a thin-film geometry with square cross section, the conductance in this figure is defined as $G=\sigma t$, where $\sigma$ is the calculated conductivity and $t$ is the thickness of the film (100 nm).

Figure 6

(Color online) (Ga,Mn)As: Total conductance correction for two different values of disorder. Since the interband contribution which favors positive conductance contributions vanishes when $h_z\sim 1/\tau$, the value of the Mn concentration at which the WAL-to-WL crossover occurs depends on the scattering rate. This behavior is different from that of the M2DEG model.

Figure 7

(Color online) (Ga,Mn)As: Magnetoresistance for $u^z=0$ for several different Mn concentrations. The short mean-free paths of (Ga,Mn)As imply that the magnetoresistive signal persists up to relatively large fields. The conductance is defined as $G=\sigma t$, where $\sigma$ is the calculated conductivity and $t$ is the thickness of a square film (100 nm). At very small $x$, we find the positive MR characteristic of WAL; the sign changes as a function of Mn concentration.

Figure 8

(Color online) (Ga,Mn)As: Magnetoresistance for $u^z=0,p=0.4{\text{nm}}^{-3},{ϵ}_F\tau =2.5$ at several different Mn concentrations near the crossover from negative to positive MR.

Figure 9

(Color online) (Ga,Mn)As: Exchange impurities reverse the sign of magnetoresistance in GaAs by decreasing the WAL correlations while leaving the WL correlations unaffected. ${\tau }_0$ is the scattering time of nonmagnetic impurities.

Figure 10

(Color online) Dependence of magnetoresistance on the relative orientation between the electric current and the magnetic easy axis $\left(\widehat{z}\right)$ in (Ga,Mn)As. ${\sigma }_0=\left({\sigma }_{xx}+{\sigma }_{zz}\right)/2$ is the average value of the Boltzmann conductivity.