Theory of optical conductivity for dilute ${\text{Ga}}_{1-x}{\text{Mn}}_x\text{As}$

We construct a semimicroscopic theory, to describe the optical conductivity of ${\text{Ga}}_{1-x}{\text{Mn}}_x\text{As}$ in the dilute limit, $x\sim 1\%$. We construct an effective Hamiltonian that captures inside-impurity-band optical transitions as well as transitions between the valence band and the impurity band. All parameters of the Hamiltonian are computed from microscopic variational calculations. We find a metal-insulator transition within the impurity band in the concentration range, $x\sim 0.2–0.3\%$ for uncompensated and $x\sim 1–3\%$ for compensated samples, in agreement with the experiments. We find an optical mass $m_{\text{opt}}\approx m_e$, which is almost independent of the impurity concentration except in the vicinity of the metal-insulator transition, where it reaches values as large as $m_{\text{opt}}\approx 10m_e$. We also reproduce a mid-infrared peak at $\hslash \omega \approx 200\text{meV}$, which redshifts upon doping at a fixed compensation, in quantitative agreement with the experiments.

Figure 1

(Color online) Density of states for $x=0.2\%$ and 0.3%. Both the impurity-band and valence-band contributions are shown. At these concentrations the impurity band is clearly separated from the valence band. The gray area indicates the region of localized states, while delocalized states are shown as white regions. For these calculations we used a hole fraction $f=0.5$. The arrow indicates the position of the mobility edge corresponding to $f=f_c=0.8$.

Figure 2

Comparison between our theory and experimental data. The optical conductivity data presented in the inset were extracted from Ref. 11.

Figure 3

(Color online) Mid-infrared peak position extracted from the optical conductivity curves vs the effective optical spectral weight, $N_{\text{eff}}$ [defined through Eq. (28)]. Open symbols represent theoretical results while the solid dots are the experimental data extracted from Ref. 11. Calculations for different hole fractions $f$ and concentrations $x$ $\left(x=1–5\%\right)$ fall onto a single curve.

Figure 4

(Color online) Top: variationally computed hopping parameter $t$ as function of the relative distance between the Mn sites, $R$. Bottom: the energy shift $\Delta E\left(R\right)$. The inset shows the correction $\Delta \tilde{E}\left(R\right)$ to the classical Coulomb shift.

Figure 5

(Color online) Momentum matrix element (in units $\hslash =1$) as a function of the separation $R$ between the two Mn sites. For $R>8\text{Å}$ the matrix elements are very well approximated by the Peierls substitution (dashed line).

Figure 6

(Color online) Wave-vector dependence of the on-site optical matrix element computed for free and Coulomb scattering states $\left[\hslash =1\right]$. We also compare this with the result obtained for a simple Hydrogenlike variational wave function and free scattering states.

Figure 7

(Color online) Ground-state density of states for different Mn concentrations and a fixed hole fraction, $f=0.5$. The solid lines represent the DOS for the impurity band while dashed lines denote the valence-band DOS. The Monte Carlo relaxation time is fixed to $t_{MC}=1$ per Mn ion. Shaded areas indicate localized states, while unshaded regions correspond to extended states. For these concentrations we find two mobility edges in the impurity band: one separating states in the tail of the impurity band, while the other separating localized states in the impurity-band—valence-band gap.

Figure 8

(Color online) Ground-state density of states and participation ratios for different Mn concentrations but a fixed hole fraction, $f=0.5$. The Monte Carlo relaxation time is fixed to $t_{MC}=1$. The shaded regions indicate the position of the mobility edge for each concentration.

Figure 9

(Color online) Shifted local density of states for different Mn concentrations. The Monte Carlo relaxation time is fixed to $t_{MC}=1$.

Figure 10

(Color online) The real part of the optical conductivity for different Mn concentrations and hole fractions. In (a) and (c) the Mn concentrations is 0.03%, while in (b) and (d) it is $x=1\%$. In (c) and (d) the intraband and interband contributions are presented for a hole fraction $f=0.5$. The sample in panel (a) is insulating while the one in panel (b) is metallic. The peak observed at $\sim 2000–2500{\text{cm}}^{-1}$ for the metallic sample is due to the valence to impurity-band transitions while the broad Drude feature at smaller energies is generated by the intraimpurity-band contributions.

Figure 11

(Color online) Upper panels: (a) the dc limit of the intraband optical conductivity for concentrations between 0.02% and 1%. (b) The scaling exponent $\alpha$ for the conductivity in the low-energy limit as function of the carrier concentration. The MIT transition is indicated by a drop of the critical coefficient $\alpha$ down to 0.35. Bottom panel: frequency dependence of the intraband optical conductivity in the small frequency limit presented for different Mn concentrations. In all three figures the hole fraction was fixed to $f=0.8$.

Figure 12

(Color online) The optical mass as function of number of carriers. Only the impurity-band contribution was used to compute $m_{\text{opt}}$. Deep in the metallic regime, for carrier concentrations larger than $2\times {10}^{20}{\text{cm}}^{-3}$ the effective mass remains practically constant, while in the critical regime as approaching the MIT transition $\left(p\simeq {10}^{19}–{10}^{20}{\text{cm}}^3\right)$ a powerlike increase in the effective mass is observed.

Figure 13

Top: RPA series giving the leading correction to the optical conductivity. The dot represents the current vertex. Middle: (a) interband contribution to the optical conductivity and characterize excitations, (b) intraband contribution. Bottom: Bethe-Salpeter equation for the renormalized current vertex ${\tilde{\Gamma }}_{\mu \nu }$, indicated as a filled triangle.

Figure 14

Deep, strongly localized occupied levels in the tail of the valence band move together with the valence-band edge, and can give rise to impurity transitions at about 200 meV, indicated by the wavy line. Empty circles denote holes.

Figure 15

(Color online) Normalized ground-state wave function: comparison between the single parameter and the full calculations.

Figure 16

The lowest few energy levels of the ${\text{Mn}}_2$ molecule as a function of distance. The left figure presents the case where only $s$ waves were considered (18 parameters were used). The right panel shows the results when both $s$ and $p$ waves are considered (14 parameters were used).