Intrinsic transmission magnetic circular dichroism spectra of GaMnAs

Ferromagnetic semiconductors (FMSs) have attracted much attention because of their intriguing features originating from the interplay between ferromagnetism and semiconducting properties. In order to investigate the ferromagnetism and the spin-polarized band structures of FMSs, magneto-optical (MO) measurements have been extensively carried out.^1–11 Magnetic circular dichroism (MCD) is one of the MO effects, in which right and left circularly polarized lights (RCP, LCP) are absorbed or reflected differently. Due to the spin dependent optical selection rules, the spin polarization of the band can be observed in MCD. In particular, the band structure of GaMnAs, which is one of the most prototypical FMSs, has been investigated by MCD spectroscopy.^{1–7,12–14} In transmission MCD spectra of GaMnAs, a large broad peak around photon energy E ∼ 1.7 eV and additional anomalous peak or kinks at E ∼ 1.5 – 2.5 eV are observed;^1–7 however, the origin of these additional structures has not been clarified because their peak positions do not correspond to the photon energies of the optical transitions at the critical points of GaMnAs, whose energy positions are similar to those of GaAs. In previous studies, some band pictures were proposed by interpreting the spectral shapes of MCD, and these additional MCD peaks were attributed to some origins such as the contribution from light hole bands^3,4 or that from the impurity bands.^1,2 However, in these studies, optical interference, which is caused by the layered structure of the measured sample, was not appropriately considered. Recently, we have systematically performed reflection MCD spectroscopy measurements on GaMnAs and have shown that the spectral shape of the reflection MCD is strongly influenced by the optical interference effect.¹⁵ Also, we derived the off-diagonal element of the dielectric tensor of GaMnAs, which is intrinsic to the material, and the intrinsic reflection MCD spectra.¹⁵ For further understanding of the MCD characteristics and band structure of GaMnAs, it is important to understand the transmission MCD of GaMnAs. In this study, we have calculated the transmission MCD spectra by taking into account optical interference and also calculated the intrinsic transmission MCD spectra of GaMnAs. We have revealed that the anomalous peak appears at ∼2 eV even in the intrinsic MCD spectra of GaMnAs.

Transmission MCD is defined by the difference of the transmittance between RCP and LCP, and is expressed by

where d is the thickness of the magnetic film, and T₊ and T_- are the transmittance of a sample for RCP and LCP, respectively. We calculated the transmission MCD spectra of two kinds of GaMnAs-based thin film samples: Sample I (inset of Fig. 2(a)) composed of only Ga_1-xMn_xAs (d₁ nm) and Sample II (inset of Fig. 3(a)) composed of Ga_1-xMn_xAs (d₁ nm)/GaAs (d₂ nm). For the calculation, we used the transfer matrix method.¹⁶ Figure 1(a) shows the schematic structure of Sample II and the propagating light in each medium. The light (RCP or LCP) is perpendicularly incident to the surface of the sample, in the positive direction of the z-axis. The light is multiply reflected at the interface between GaMnAs and GaAs, and at the surfaces of the sample. In Fig. 1(a), mediums 0, 1, 2, and 3, whose refractive indices are n_m (m = 0 – 3), are vacuum, Ga_1-xMn_xAs, GaAs, and vacuum, respectively. We assumed n₀ = n₃ = 1, and n₁ is dependent on the polarization of the light. For n₁ and n₂, we used the values reported in previous studies.^15,17, We note that n₁ is estimated from the reflection MCD spectroscopy measured at 5 K with an external magnetic field of 1 T applied perpendicular to the sample plane.¹⁵ Figure 1(b) shows the photon energy E dependence of the real $n1±′$ and imaginary part $n1±″$ of $n1±(=n1±′+i⋅n1±″)$ in Ga_0.92Mn_0.08As, where + (–) denotes RCP (LCP), derived in our previous study.¹⁵ The difference between $n1+′$ and $n1−′$⁠, and that between $n1+″$ and $n1−″$ give rise to MCD. The amplitudes of electric fields in each medium are E₀, E₁, E₂, and E₃, as shown in Fig. 1(a), where $Em|→$ and $Em→|$ represent the amplitude of the electric field of the light which propagates in the positive direction of the z-axis at the interface between mediums m and m–1 (m = 1, 2, and 3), and at the interface between mediums m and m+1 (m = 0, 1, and 2), respectively. Similarly, $Em|←$ and $Em←|$ represent the amplitude of the electric field of the light which propagates in the negative direction of the z-axis at the interface between mediums m and m–1 (m = 1, 2, and 3), and at the interface between mediums m and m+1 (m = 0, 1, and 2), respectively. Here, electric fields E_m-1 and E_m (m = 1, 2, and 3) satisfy

In Eq. (2), I_m-1,m is expressed by

where r_m-1,m and t_m-1,m are the Fresnel coefficients at the interface between mediums m–1 and m; $rm−1,m=nm−nm−1nm+nm−1$⁠, $tm−1,m=2nm−1nm+nm−1$⁠. In medium m (m = 1 and 2), electric fields $Em|→,Em→|,Em|←$⁠, and $Em←|$ satisfy

where $δm=2πλnmdm$⁠, and λ is the wavelength of the light. By using Eqs. (2) and (4), we obtain

By substituting $E3|←=0$⁠, the transmittance is given by

Thus, when the refractive index of each layer is given, T_± and transmission MCD of Sample II can be calculated. We calculated the T_± and transmission MCD of Sample I in the same way.

Figure 2(a) shows the calculated transmission MCD of Sample I with various d₁ (= 100 – 150 nm) as a function of E. The transmission MCD spectra are strongly dependent on d₁; the peak energy and the sign of the MCD are varied with d₁. In particular, we found that the positive peak around E ∼ 2.2 eV is emphasized when d₁ = 110 and 120 nm, which were often observed in previous studies.^3–5 Figures 2(b) and 2(c) show the calculated transmission spectra and transmission MCD spectra of Sample I as a function of d₁, respectively. As shown in Fig. 2(c), the transmission MCD spectrum is periodically varied with d₁ due to the optical interference shown in the transmittance (Fig. 2(b)). The constructive and destructive interference conditions are represented by black and red broken curves in Figs 2(b) and 2(c). At E > 2.5 eV, three (positive and negative) peaks, which originate from the optical transitions at critical points (E₁ ∼ 3.0 eV, E₁+Δ₁ ∼ 3.2 eV, and E₀’ ∼ 4.5 eV) are observed as well. The influence of interference is little when E > ∼2.75 eV because of the large extinction coefficient of GaMnAs (> ∼2).¹⁵ 

We also found that the spectral shape of transmission MCD depends on the thickness of a non-magnetic layer (e.g. medium 2 in Fig. 1(a)), which usually remains due to the difficulty of the exact depth control of the etching of samples from the backside. Figure 3(a) shows the calculated transmission MCD of Sample II with various d₂ (= 100 - 150 nm) at d₁ = 100 nm (fixed) as a function of E. The transmission MCD spectra are strongly dependent on d₂. The transmittance and transmission MCD spectra of Sample II with various d₂ are summarized in Figs. 3(b) and 3(c), respectively. The red and black broken curves represent the constructive and destructive interference conditions. The transmission MCD spectrum is varied with d₂, reflecting the d₂ dependence of the transmittance. These results mean that the MCD spectra are apparently influenced by the optical interference effect that is determined by the thickness of the non-magnetic GaAs layer.

We fit the calculated MCD spectra to the experimental results. In Fig. 4, solid black and red curves show the experimental transmission MCD spectra of Ga_1-xMn_xAs (x = 6.4% and 6.8%, respectively) in ref. 3. These samples were composed of Ga_1-xMn_xAs (∼100 nm)/Al_0.70Ga_0.30As (0.2 μm). In that study,³ the difference in the measured MCD spectra between the two GaMnAs samples was explained by the difference in the effective Mn concentration of the samples. Figure 4 also shows the calculated MCD spectra with the broken black and red curves. In the calculation, we assumed the sample structure composed of Ga_0.92Mn_0.08As (d₁ = 100 nm)/Al_0.27Ga_0.73As (d₂ = 216 nm and 228 nm), whose thicknesses were determined to obtain the better fitting. For the refractive index of Al_0.27Ga_0.73As, we used the value obtained in ref. 18. In Fig. 4, the values of MCD were multiplied by a factor 0.65 (or 0.8) for better fitting. The calculated MCD intensity is larger than the experimental signals, because x of the Ga_1-xMn_xAs sample used in our previous study (x = 8%) to determine the dielectric tensor of Ga_1-xMn_xAs is higher than those of the GaMnAs samples used in this study (x = 6.4% and 6.8%). The shapes of the experimental MCD spectra are similar to those of the calculated ones. The deviations of the peak positions and the peak heights may originate from the difference between the compositions of each element in the samples (for example, Mn content and the composition of Al and Ga in AlGaAs). The peaks of the experimental spectra are broader than the calculated ones, which may originate from the surface and interface roughness of the samples. Nevertheless, the calculated result apparently indicates that the difference in the strength of the second peak (∼ 2 eV) can be explained by the difference in d₂. This result indicates that we cannot avoid the optical interference in the transmission MCD measurements of GaMnAs especially in the low photon energy region (E < ∼2.5 eV).

In order to understand the transmission MCD spectra correctly, we derived intrinsic transmission MCD spectra of GaMnAs, which are not influenced by optical interference. We define the intrinsic transmission MCD as the one which occurs when the light propagates the infinitesimal distance d in GaMnAs. Because the value of the transmission MCD depends on the light propagation length, we defined the intrinsic one when d → 0. By using Eqs. (1), (4), and (6), the intrinsic transmission MCD can be written as follows:

Figure 5 shows the intrinsic transmission MCD spectra at x = 1%, 2%, and 8%, which were calculated with Eq. (7). The spectrum at x = 8% is also shown in Figs. 2(a) and 3(a) with thick black solid curves, and in Fig. 4 with the blue broken curve. In the intrinsic transmission MCD spectra of Ga_1-xMn_xAs, a broad positive peak rises at the bandgap energy of GaAs (∼1.5 – 2 eV) regardless of x, as were observed in previous studies.^1–4 These results mean that the bandgap energy of Ga_1-xMn_xAs is nearly the same as that of GaAs even when Ga_1-xMn_xAs shows metallic behavior (x > 2%), which is consistent with the recent understanding of the valence band structure of GaMnAs.^{1–4,15,19–23} Interestingly, as shown in Fig. 5, another peak or shoulder peak is observed around E ∼ 2.2 – 2.4 eV even in the intrinsic spectra of Ga_1-xMn_xAs. Our results indicate that the peak or shoulder is intrinsic to the MCD spectra of GaMnAs. In GaMnAs, the spin polarized impurity band is formed above the valence band top, and the spin imbalance in the valence band states is caused by the hybridization of the impurity band and the valence band. Both of the impurity band and the valence band contribute to the MCD spectra. The peak may originate from the imbalance in the valence band states of Ga_1-xMn_xAs which is caused by the hybridization of the impurity band and the light hole band of Ga_1-xMn_xAs, as is proposed in ref. 3. This band picture is consistent with the result of the angle resolved photo emission spectroscopy (ARPES).²² For further quantitative discussions, band structure calculations will be required because we must consider the spin polarization and the density of states of GaMnAs simultaneously.

In summary, we have calculated transmission MCD spectra of GaMnAs-based thin film samples with the transfer matrix method. We have revealed that the spectral shape of transmission MCD strongly depends on the sample profile due to the optical interference effect. The anomalous peak, which was observed at around E ∼ 2 eV in previous studies, is enhanced in certain sample profiles. This means that transmission MCD spectroscopy requires the analysis taking into account the optical interference of the layered structure. We must consider not only the thickness of ferromagnetic films, but also the existence of additional non-magnetic layers in samples. To understand the transmission MCD correctly, we also calculated the intrinsic spectra that are not influenced by optical interference. The anomalous peak exists at 2.2 – 2.4 eV even in the intrinsic transmission MCD spectra of Ga_1-xMn_xAs. The possible origin of this peak is the spin imbalance in the valence band states of Ga_1-xMn_xAs which originates from the hybridization of the impurity band and the light hole band of Ga_1-xMn_xAs.

This work was partially supported by Grants-in-Aid for Scientific Research, CREST of JST, and the Spintronics Research Network of Japan.