Fractional Quantum Hall Effect in a Diluted Magnetic Semiconductor

We report the observation of the fractional quantum Hall effect in the lowest Landau level of a two-dimensional electron system (2DES), residing in the diluted magnetic semiconductor Cd(1-x)Mn(x)Te. The presence of magnetic impurities results in a giant Zeeman splitting leading to an unusual ordering of composite fermion Landau levels. In experiment, this results in an unconventional opening and closing of fractional gaps around filling factor v = 3/2 as a function of an in-plane magnetic field, i.e. of the Zeeman energy. By including the s-d exchange energy into the composite Landau level spectrum the opening and closing of the gap at filling factor 5/3 can be modeled quantitatively. The widely tunable spin-splitting in a diluted magnetic 2DES provides a novel means to manipulate fractional states.

The fractional quantum Hall effect (FQHE) is a collective high-magnetic field phenomenon, originating from Coulomb repulsion of electrons confined in two dimensions. At certain fractional filling ν = p/q of the Landau levels (LLs) [ν = filling factor, p, q = integers], quantized plateaus in the Hall resistance ρ xy and vanishing longitudinal resistance ρ xx herald the presence of peculiar electron correlations [1,2]. Here, the electrons condense into a liquid-like ground state that is separated by a gap ∆ from the excited states. Most experiments to date have been carried out on GaAs-based systems, being still the cleanest material system with the highest electron mobilities [3]. When the direction of the magnetic field B is tilted, the orbital LL splitting is given by the field component B ⊥ normal to the 2DES while the total field strength B determines the Zeeman splitting E Z . Early experiments on GaAs revealed that the ν = 4/3, 5/3 and 8/5 states behave differently upon tilting the sample [4,5]: While the ν = 4/3 and 8/5 states were undergoing a transition from a spin-unpolarized state to a polarized one, the ν = 5/3 state is always fully spin polarized.
Although the FQHE has been reported in quite a number of different materials [6][7][8][9][10][11][12], the FQHE was never observed in a diluted magnetic semiconductor in which atoms with magnetic moment (e.g. Mn 2+ ) are placed in the 2DES. Then, the localized spins in the magnetic impurities' d -orbitals interact with the correlated electron system via the quantum mechanical s-d exchange interaction, causing giant Zeeman splitting [13] which is tunable in magnitude, sign and field dependence [14]. The constant αN 0 ∆ specifies the sd exchange strength and is the largest energy scale in the system. It hence has remained unclear whether FQHE states survive in the presence of magnetic impurities. Below we demonstrate that (i), the FQHE indeed exists in magnetic 2DESs, and (ii), that the opening and closing of gaps in an in-plane field can be described within a modified composite fermion (CF) picture, in which the s-d exchange is taken into account.
Let us first recall the CF-model which maps the FQHE onto the integer quantum Hall effect (IQHE) by introducing new particles, composite fermions, each composed of an electron and an even number (here: 2) of flux quanta [15]. Between 1 < ν < 2 the effective magnetic field for CFs vanishes at ν = 3/2 while they encounter an effective magnetic field B CF away from this filling [16]. In the vicinity of ν = 3/2 the CF filling factor ν for composite fermions of holes is related to the one of electrons via ν = 2 − ν /(2ν ± 1) where '±' relates to CF filling factors at positive and negative effective field B CF [16]. To model the ∆ ν (B) characteristics we expand the CF model [15] to cover diluted magnetic semiconductors (DMSs). To account for fractional states in a DMS we put forward a CF-LL fan chart modified by s-d exchange interaction, The first term represents the cyclotron energy of a CF in the N th CF-LL, the second one describes Zeeman splitting in the presence of magnetic impurities [13]. It has two contributions: the first part, linear in B, is the conventional Zeeman term with g , viewed as the g-factor of CFs. The second part is due to exchange between CFs and manganese spins with  [28], for GaAs from [29,30] and for CdTe from [11]. To compare model and experiment we assume that αN 0 = 220 meV is the same for electrons and CFs. In case of ν = 5/3 the vanishing gap can be ascribed to the crossing of the 0,↑ and 0,↓ CF-LLs, occurring at vanishing spin splitting, so that E Z = g µ B B +αN 0 xS = 0.
Using that and the value at which the gap vanishes, B = 11.45 T, x = 0.24% and S = 5/2 we obtain g = −1.99. This value deviates by about 16% from the g-factor of electrons in CdTe, g = −1.67; similar matching between g-factors of electrons and CFs has been seen in experiments on GaAs heterostructures [16]. Having fixed g = −1.99 and αN 0 = 220 meV, we now use the "coincidence method" to determine ω CF c . The coincidence of the 0,↑ and 1,↓ level occurs when the gap ∆ 5/3 reaches its maximal value at B = 14.1 T (see Fig. 3a).
Then, we have that E Z (B = 14.1T) = ω CF c and obtain ω CF c = 3.54 K (m CF = 1.25m e ) at ν = 5/3. The calculated evolution of ∆ 5/3 (B) (middle panel of Fig. 3a) reproduces the experimental data well for tilt angles below θ ∼ 45 • . With increasing in-plane field (tilt angle), however, the model describes the data less perfect. In Fig. 3a (middle panel) the model predicts a constant gap above 14 T while the data deviate. This is to some extent due to the larger error in extracting the gap; however, a reduction of the gap is also expected from the coupling of the growing in-plane field to the orbital motion of the electrons in our 30 nm wide QW [21]. The gap ω CF c , obtained by the "coincidence method" is for ν = 5/3 (and also for ν = 4/3, see below) by a factor of ∼ 3 larger than the activation gaps.
Generally, gaps are overestimated by theory when disorder, LL mixing and finite thickness correction are neglected [22][23][24][25]. The difference in gap size reflects the different experimental techniques used to extract ω CF c : The fitting of the CF-LL spectrum (red dashed lines in Fig. 3a,b), obtained from the coincidence of different CF-LLs, is less affected by disorder. (c) Calculated evolution of the gaps ∆ 8/5 (B) and ∆ 7/5 (B) for the parameters given in the text.

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The cyclotron gap extracted from activated transport (top panels), in contrast, is strongly affected by disorder broadening and hence smaller [26].
Assuming that the usual CF-LL spectrum gets modified by the exchange energy αN 0 = 220 meV enables us to model the gap evolution ∆ 5/3 (B) quite reasonably. Below we check whether the modified CF-LL spectrum, extracted above is consistent with the observations made at other filling. For that we assume that αN 0 = 220 meV and g = −1.99 are independent of ν. The latter assumption is justified by previous work on GaAs where the CF g-factor was found to be independent of ν and close to the one of electrons [16,27]. In other words, we assume that the Zeeman gap closes at the same B = 11.45 T for all ν. Finally, let us now turn to the gaps at ν = 7/5 and 8/5. While we observed gaps for ν = 5/3 and 4/3 in different samples, fractional states at 7/5 and 8/5 filling were only observed in the sample discussed here. Because the extraction of gaps might be disputable since for most tilt angles no clear ρ xx minima arise we only show modeled ∆ 7/5 (B) and ∆ 8/5 (B) traces to illustrate that their evolution is, within experimental accuracy, consistent with the resistivity as function of θ. Here we assume that maxima in the resistivity, measured as a function of θ (i.e. the resistivity along the dashed lines in Fig. 2d) correspond to vanishing CF-LL gaps. Using, as before, αN 0 = 220 meV and g = −1.99 we can obtain a vanishing gap at ∼ 13 T for filling factor 8/5 in Fig. 3c (left panel). This agrees with a maximum of ρ xx at θ ∼ 38 • in Fig. 2d. For ν = 7/5 maxima in ρ xx at θ ∼ 8 • (not shown) 8 and ∼ 48 • correspond to vanishing gaps at about ∼ 11.5 T and 17 T in Fig. 3c (left panel).
This agrees reasonably well with the calculated traces.
In summary we note that without taking αN 0 in Eq. (1) into account the angular dependence of ∆ ν (B) cannot be reproduced. Especially the closing and opening of the ν = 5/3 gap, not observed in any other material, and corresponding to a change in spin polarization from 0,↓ to 0,↑ at B = 11.45 T, highlights the impact of exchange splitting on the spin-polarization of the CF ground states.