Charge and spin diffusion on the metallic side of the metal-insulator transition: A self-consistent approach

We develop a self-consistent theory describing the spin and spatial electron diffusion in the impurity band of doped semiconductors under the effect of a weak spin-orbit coupling. The resulting low-temperature spin-relaxation time and diffusion coefficient are calculated within different schemes of the self-consistent framework. The simplest of these schemes qualitatively reproduces previous phenomenological developments, while more elaborate calculations provide corrections that approach the values obtained in numerical simulations. The results are universal for zinc-blende semiconductors with electron conductance in the impurity band, and thus they are able to account for the measured spin-relaxation times of materials with very different physical parameters. From a general point of view, our theory opens a new perspective for describing the hopping dynamics in random quantum networks.

Figure 1

(a) Example of an irreducible diagram of second order in the hopping amplitude contributing to the average local Green function $G^{(\pm )}\left(\varepsilon \right)$. The solid lines represent the hopping amplitude matrix $\mathcal{V}$, the circles stand for $G_{00}^{(\pm )}\left(\varepsilon \right)=1/{\text{z}}_{\pm }$, and the dotted lines indicate identical sites. (b) Self-energy ${\mathrm{\Sigma }}^{(\pm )}\left(\varepsilon \right)$ corresponding to the irreducible diagram for $G^{(\pm )}\left(\varepsilon \right)$ shown in (a).

Figure 2

Diagrammatic expression of the expansion of the intensity propagator $\mathrm{\Phi }$ in terms of its irreducible component $U$ and the local average Green functions $G^{(\pm )}$ (first line), together with the resulting Bethe-Salpeter equation (35) (second line).

Figure 3

(a) Density of electronic states $\rho$, (b) spin-relaxation rate ${\tau }_{\mathrm{s}}^{-1}$, and (c) diffusion coefficient $\mathcal{D}$ in the impurity band as a function of energy for the three approaches developed in this work: the simplest self-consistent approximation (SSCA, dotted), the loop-corrected self-consistent approximation (LCSCA, dashed), and the repeated-scattering-corrected self-consistent approximation (RSCSCA, solid). Results are presented for two dimensionless impurity densities ${\mathcal{N}}_{\mathrm{i}}=0.{29}^3$ (black) and $0.{33}^3$ (red). Energies are measured using the isolated impurity level as origin and are expressed in units of $V_0$ (twice the ionization energy). The other physical constants used to define the scales of the figure are the effective Bohr radius $a$ and the Dresselhaus coupling constant $\gamma \ll a^3{V}_0$. The vertical lines mark the position of the Fermi energy corresponding to the two considered impurity densities for the RSCSCA, and are very close to those of the LCSCA. The Fermi energy for the SSCA is that of the isolated impurity level ${\varepsilon }_{00}=0$.

Figure 4

Irreducible component of the intensity propagator within the SSCA, obtained by applying the “cut and fold” procedure to the self-consistent self-energy of Fig. 1. The solid horizontal upper (lower) line stands for the hopping amplitude matrix $\mathcal{V}\left(\mathbf{r}\right) \left[{\mathcal{V}}^{*}\left(\mathbf{r}\right)\right]$, the dotted vertical lines indicate identical sites.

Figure 5

Spin-relaxation time at the Fermi energy ${\tau }_{\mathrm{s}}\left({\varepsilon }_{\mathrm{F}}\right)$, for a half-filled impurity band, as a function of the dimensionless impurity density ${\mathcal{N}}_{\mathrm{i}}=n_{\mathrm{i}}{a}^3$ for the three approaches developed in this work: the SSCA [Eq. (71), dotted], the LCSCA (dashed), and the RSCSCA (solid). The red dots are the numerical results of Ref. [20]. The theory only applies to the interval $0.017<{\mathcal{N}}_{\mathrm{i}}<0.07$ (i.e., between the critical density ${\mathcal{N}}_{\mathrm{c}}\simeq 0.017$ of the Mott transition and the hybridization density ${\mathcal{N}}_{\mathrm{h}}\simeq 0.07$); the lowest-density results are presented to show the importance of the repeated-scattering correction in this regime.

Figure 6

(a) Self-energy obtained by considering loops of arbitrary length representing processes where the electron visits different impurities before hopping back to the starting site (LCSCA). (b) Diagrammatic expression of the renormalized hopping amplitude (thick horizontal line) as an expansion of infinite order in the bare hopping amplitude (thin horizontal lines).

Figure 7

Irreducible component of the intensity propagator within the LCSCA expressed as a renormalized insertion. The thick solid horizontal upper (lower) line stands for the renormalized hopping amplitude matrix ${\mathcal{F}}^{(+)}({\varepsilon }_1,\mathbf{r}) \left[{\mathcal{F}}^{(-)}({\varepsilon }_2,\mathbf{r})\right]$, the dotted vertical lines indicate identical sites, ${\varepsilon }_{1,2}=\varepsilon \pm \hslash \omega /2$.

Figure 8

Diagrams for the self-energy including repeated scattering.

Figure 9

Diagrams for the irreducible component $U$ of the intensity operator including repeated scattering.

Figure 10

Example of a reducible diagram contributing to the average local Green function. The solid horizontal lines represent the hopping matrix $\mathcal{V}$, the circles stand for the Green functions, and dotted lines indicate identical sites.

Figure 11

(a) Example of a two-point reducible diagram. Since the central part (marked by the red rectangle) can be seen as part of the average local Green function at impurity 2 [see Fig. 1], this diagram is effectively contained in the diagram displayed in Fig. 12 below, and must therefore not be counted again when evaluating $\mathrm{\Sigma }$ in the self-consistent approach. (b) Applying the recipe for constructing $U$ diagrams from the $\mathrm{\Sigma }$ diagram of (a)—with cut at the impurity 3—yields a reducible intensity diagram, which can be expressed as the product of Green functions and two irreducible insertions (marked by the two red rectangles).

Figure 12

(a) Example of a diagram ${\mathrm{\Sigma }}_{\mathbf{i}}$ for the local self-energy, characterized by the labels $\mathbf{i}=\{1,2,0,1\}$ of the intermediate impurities (from right to left), see Eq. (B1). (b) Corresponding diagram $U_{\mathbf{i},2}$ contributing to the intensity operator, obtained according to the construction recipe described in the text by cutting the $\mathrm{\Sigma }$ diagram (a) at impurity 2.

Figure 13

Example of a reducible component of the intensity propagator expressed as the product of Green functions representing the same impurity and two irreducible insertions $U_1$ and $U_2$.