Apparent Curie temperature and nanoscale phase coexistence at the carrier-localization boundary of (Ga,Mn)As

(Ga,Mn)As, the canonical dilute ferromagnetic semiconductor, orders through exchange between localized Mn spins mediated by holes, and it lies unusually close to the Anderson-Mott metal-insulator transition (MIT). Gate-depletion experiments established that reducing the hole density monotonically lowers the Curie temperature and drives the system through a superparamagnetic-like, nanoscale phase-separated state before ferromagnetism is lost. What they could not settle is how the apparent Curie temperature $T_C^*$ behaves exactly at the localization boundary - whether it decays continuously to zero with decreasing hole density $p$, or stays finite up to a characteristic disorder threshold (with boundary values grouping near 15 K) and then jumps abruptly to zero - and whether the coexistence of ferromagnetic, superparamagnetic and paramagnetic regions is set by quantum-critical fluctuations of the local density of hole states or by a non-random distribution of the Mn ions. The problem is to determine the magnetic phase diagram of (Ga,Mn)As in the immediate vicinity of the localization boundary, including its electric-field tunability and its dependence on film thickness, effective Mn content and host semiconductor, and to disentangle the two candidate mechanisms for the magnitude and spatial distribution of the apparent $T_C^*$.

Background

(Ga,Mn)As is the most thoroughly studied dilute ferromagnetic semiconductor: gallium arsenide in which a few percent of Ga sites are replaced by Mn during low-temperature molecular-beam epitaxy. A substitutional Mn atom ($\mathrm{Mn}_{\mathrm{Ga}}$) carries a localized spin $S=5/2$ and acts as an acceptor, so each magnetic impurity also supplies a hole - a positive charge carrier residing either in the host valence band or in a Mn-derived impurity band (a narrow band of states split off from the valence band by the magnetic impurities). Ferromagnetic order arises because the holes mediate an effective exchange between the Mn spins - the $p$-$d$ Zener mechanism, in which the valence $p$ orbitals of the host couple to the $d$ shell of Mn - and the Curie temperature $T_C$, below which a spontaneous magnetization appears, is in the simplest mean-field picture controlled jointly by the Mn content and the hole density $p$. What makes (Ga,Mn)As exceptional among ferromagnets is that it sits very close to an Anderson-Mott metal-insulator transition (MIT): the disorder from random Mn substitution and from compensating defects (Mn atoms in interstitial positions, arsenic antisites) localizes the holes once their density is low enough. The Anderson-Mott transition is the disorder-driven quantum transition between extended and localized carrier states in a doped semiconductor; near it the local density of states (the number of available carrier states at a given point and energy) fluctuates strongly on nanometer scales. Whether localization enhances or destroys carrier-mediated magnetic correlations - and whether any ferromagnetism survives on the insulating side - is a long-standing question on which theory has offered diverging scenarios: mean-field valence-band Zener ferromagnetism, impurity-band models in which the holes remain partially localized, bound-magnetic-polaron percolation (a bound magnetic polaron is a carrier locally trapped by the magnetic ions it polarizes, forming a small magnetized cluster; long-range order then appears only when such clusters percolate), and various nanoscale phase-separation pictures. Experimentally, Sawicki and co-workers (2010) attacked this question with metal-insulator-semiconductor (MIS) field-effect structures, in which a gate voltage across an insulator depletes or accumulates holes in a thin (Ga,Mn)As channel, measuring the spontaneous magnetization directly with a superconducting quantum interference device (SQUID), the standard ultrasensitive magnetometer. Gate depletion lowered $T_C$ monotonically, with no maximum at intermediate hole density of the kind some impurity-band models predict, and the ferromagnetic-to-paramagnetic transformation proceeded through a superparamagnetic-like arrangement - a system of small magnetized regions whose orientations fluctuate thermally and are pinned only below a blocking temperature (superparamagnetism) - implying nanoscale coexistence of ferromagnetic, superparamagnetic and paramagnetic regions, which the authors assigned to quantum-critical fluctuations of the local hole density of states at the Anderson-Mott transition. Earlier work had shown that ferromagnetism in (Ga,Mn)As appears already deep in the insulating regime, with the hopping energy of localized transport a good predictor of $T_C$ (Sheu and co-workers, 2007), and polaron-percolation theory predicted ordering that terminates at a percolation threshold (Kaminski and Das Sarma, 2002). Three things were nevertheless left open. First, the apparent Curie temperatures - the ordering temperatures actually registered by magnetization or transport in these inhomogeneous systems, denoted $T_C^*$, which can be fixed by the blocking of superparamagnetic clusters rather than by true long-range order - reported for (Ga,Mn)As at the ferromagnetism boundary were noted to group around a common value of roughly 15 K before jumping to zero, but the data did not exclude a continuous decay of $T_C^*$ with $p$ all the way to the boundary; distinguishing the two requires magnetic measurements finer in, and much closer to, the localization boundary than existed. Second, because the magnitude and spatial pattern of $T_C^*$ could equally be governed by non-random aggregation of the Mn ions themselves rather than by carrier-density fluctuations, the two mechanisms must be disentangled experimentally. Third, the depth-dependent hole density $p(z)$ in gated structures is set partly by uncharacterized interface states and compensation, which limits how reproducibly the boundary can be reached by an electric field; and the phase coexistence had been established only for particular thicknesses, Mn contents and host semiconductors.

Problem Statement

Determine how carrier-mediated ferromagnetism in (Ga,Mn)As ends at the carrier-localization (Anderson-Mott) boundary of the metal-insulator transition, and which of the competing behaviours the existing experimental record supports, which it leaves open, and which measurement would discriminate them. (1) The functional form of the apparent Curie temperature $T_C^*$ near the boundary - the central dichotomy being whether, as the hole density $p$ is reduced toward the transition, $T_C^*$ decays continuously to zero or remains finite up to a characteristic disorder threshold, with reported boundary values in (Ga,Mn)As grouping near 15 K, and then jumps abruptly to zero - together with, as far as the record permits, the rest of the magnetic phase diagram on both sides of the boundary: its dependence on effective Mn content $x_{\mathrm{eff}}$, disorder, film thickness and host semiconductor, and its electric-field tunability. (2) The mechanism setting the magnitude and spatial distribution of the apparent $T_C^*$ and the observed coexistence of ferromagnetic, superparamagnetic and paramagnetic regions - quantum-critical fluctuations of the local density of hole states at the Anderson-Mott transition, or a non-random distribution and aggregation of the Mn magnetic ions - and how general that coexistence and its gate dependence are across the III-V dilute ferromagnetic semiconductor family. Reproducible quantitative gate control of the depth-dependent hole density $p(z)$, and hence of the magnetic state, depends on characterizing and controlling interface-state densities, surface oxides and compensation in the gated (Ga,Mn)As structures.

References

1. M. Sawicki, D. Chiba, A. Korbecka, Y. Nishitani, J. A. Majewski, F. Matsukura, T. Dietl, H. Ohno, 'Experimental probing of the interplay between ferromagnetism and localization in (Ga, Mn)As', Nature Physics 6, 22-25 (2010). DOI: 10.1038/nphys1455, https://doi.org/10.1038/nphys1455
2. B. L. Sheu, R. C. Myers, J.-M. Tang, N. Samarth, D. D. Awschalom, P. Schiffer, M. E. Flatté, 'Onset of Ferromagnetism in Low-Doped Ga1-xMnxAs', Physical Review Letters 99, 227205 (2007). DOI: 10.1103/PhysRevLett.99.227205, https://doi.org/10.1103/PhysRevLett.99.227205
3. A. Kaminski, S. Das Sarma, 'Polaron percolation in diluted magnetic semiconductors', Physical Review Letters 88, 247202 (2002). DOI: 10.1103/PhysRevLett.88.247202, https://doi.org/10.1103/PhysRevLett.88.247202
4. S. R. Dunsiger, J. P. Carlo, T. Goko, G. Nieuwenhuys, T. Prokscha, A. Suter, E. Morenzoni, D. Chiba, Y. Nishitani, T. Tanikawa, F. Matsukura, H. Ohno, J. Ohe, S. Maekawa, Y. J. Uemura, 'Spatially homogeneous ferromagnetism of (Ga, Mn)As', Nature Materials 9, 299-303 (2010). DOI: 10.1038/nmat2715, https://doi.org/10.1038/nmat2715
5. M. Dobrowolska, K. Tivakornsasithorn, X. Liu, J. K. Furdyna, M. Berciu, K. M. Yu, W. Walukiewicz, 'Controlling the Curie temperature in (Ga,Mn)As through location of the Fermi level within the impurity band', Nature Materials 11, 444-449 (2012). DOI: 10.1038/nmat3250, https://doi.org/10.1038/nmat3250
6. S. Ohya, K. Takata, M. Tanaka, 'Nearly non-magnetic valence band of the ferromagnetic semiconductor GaMnAs', Nature Physics 7, 342-347 (2011). DOI: 10.1038/nphys1905, https://doi.org/10.1038/nphys1905
7. Y. Yuan, C. Xu, R. Hübner, R. Jakiela, R. Böttger, M. Helm, M. Sawicki, T. Dietl, S. Zhou, 'Interplay between localization and magnetism in (Ga,Mn)As and (In,Mn)As', Physical Review Materials 1, 054401 (2017). DOI: 10.1103/PhysRevMaterials.1.054401, https://doi.org/10.1103/PhysRevMaterials.1.054401
8. Y. Yuan, M. Wang, C. Xu, R. Hübner, R. Böttger, R. Jakiela, M. Helm, M. Sawicki, S. Zhou, 'Electronic phase separation in insulating (Ga, Mn) As with low compensation: super-paramagnetism and hopping conduction', Journal of Physics: Condensed Matter 30, 095801 (2018). DOI: 10.1088/1361-648X/aaa9a7, https://doi.org/10.1088/1361-648X/aaa9a7
9. L. Gluba, O. Yastrubchak, J. Z. Domagala, R. Jakiela, T. Andrearczyk, J. Żuk, T. Wosinski, J. Sadowski, M. Sawicki, 'Band structure evolution and the origin of magnetism in (Ga,Mn)As: From paramagnetic through superparamagnetic to ferromagnetic phase', Physical Review B 97, 115201 (2018). DOI: 10.1103/PhysRevB.97.115201, https://doi.org/10.1103/PhysRevB.97.115201
10. M. Lonsky, J. Teschabai-Oglu, K. Pierz, S. Sievers, H. W. Schumacher, Y. Yuan, R. Böttger, S. Zhou, J. Müller, 'Investigation of a possible electronic phase separation in the magnetic semiconductors Ga1-xMnxAs and Ga1-xMnxP by means of fluctuation spectroscopy', Physical Review B 97, 054413 (2018). DOI: 10.1103/PhysRevB.97.054413, https://doi.org/10.1103/PhysRevB.97.054413
11. Y. Takeda, S. Ohya, N. H. Pham, M. Kobayashi, Y. Saitoh, H. Yamagami, M. Tanaka, A. Fujimori, 'Direct observation of the magnetic ordering process in the ferromagnetic semiconductor Ga1-xMnxAs via soft x-ray magnetic circular dichroism', Journal of Applied Physics 128, 213902 (2020). DOI: 10.1063/5.0031605, https://doi.org/10.1063/5.0031605
12. N. A. Bogoslovskiy, P. V. Petrov, N. S. Averkiev, 'Phase diagram of a ferromagnetic semiconductor: Origin of superparamagnetism', Physical Review B 109, 024436 (2024). DOI: 10.1103/PhysRevB.109.024436, https://doi.org/10.1103/PhysRevB.109.024436
13. H. Wang, J. Ma, J. Zhao, 'Giant modulation of magnetism in (Ga,Mn)As ultrathin films via electric field', Journal of Semiconductors 40, 092501 (2019). DOI: 10.1088/1674-4926/40/9/092501, https://doi.org/10.1088/1674-4926/40/9/092501
14. X. Wang, C. Zhang, H. Wang, Y. Yuan, Z. Shang, B. Tan, T. Liu, D. Wei, H.-X. Deng, J. Zhao, 'Exploring the Metal-Insulator Transition in (Ga,Mn)As by Molecular Absorption', Nano Letters 22, 9190-9197 (2022). DOI: 10.1021/acs.nanolett.2c03203, https://doi.org/10.1021/acs.nanolett.2c03203
15. Y. K. Edathumkandy, K. Das, K. Gas, D. Sztenkiel, D. Hommel, H. Przybylińska, M. Sawicki, 'Ferromagnetic resonance in a magnetically dilute percolating ferromagnet: An experimental and theoretical study', Physical Review B 112, 024424 (2025). DOI: 10.1103/5rzr-478b, https://doi.org/10.1103/5rzr-478b
16. X. Liu, S. Dong, L. Riney, J. Wang, Y.-L. Wang, R.-K. Zheng, S.-K. Bac, J. Kossut, M. Dobrowolska, B. A. Assaf, J. K. Furdyna, 'Crossover behavior of the anomalous Hall effect in Ga1-xMnxAs1-yPy across the metal-insulator transition', Physical Review B 103, 214437 (2021). DOI: 10.1103/PhysRevB.103.214437, https://doi.org/10.1103/PhysRevB.103.214437
17. X. Liu, L. Riney, J. Guerra, W. Powers, J. Wang, J. K. Furdyna, B. A. Assaf, 'Colossal negative magnetoresistance from hopping in insulating ferromagnetic semiconductors', Journal of Semiconductors 43, 112502 (2022). DOI: 10.1088/1674-4926/43/11/112502, https://doi.org/10.1088/1674-4926/43/11/112502
18. C. Xu, M. Wang, Y. Yuan, G. Larkin, M. Helm, S. Zhou, 'Hole compensation effect in III-Mn-V dilute ferromagnetic semiconductors', Journal of Physics D: Applied Physics 52, 355301 (2019). DOI: 10.1088/1361-6463/ab25dd, https://doi.org/10.1088/1361-6463/ab25dd
19. F. Matsukura, Y. Tokura, H. Ohno, 'Control of magnetism by electric fields', Nature Nanotechnology 10, 209-220 (2015). DOI: 10.1038/nnano.2015.22, https://doi.org/10.1038/nnano.2015.22
20. T. Jungwirth, J. Wunderlich, V. Novák, K. Olejník, B. L. Gallagher, R. P. Campion, K. W. Edmonds, A. W. Rushforth, A. J. Ferguson, P. Němec, 'Spin-dependent phenomena and device concepts explored in (Ga,Mn)As', Reviews of Modern Physics 86, 855-896 (2014). DOI: 10.1103/RevModPhys.86.855, https://doi.org/10.1103/RevModPhys.86.855
21. T. Dietl, H. Ohno, 'Dilute ferromagnetic semiconductors: Physics and spintronic structures', Reviews of Modern Physics 86, 187-251 (2014). DOI: 10.1103/RevModPhys.86.187, https://doi.org/10.1103/RevModPhys.86.187
22. T. Dietl, A. Bonanni, H. Ohno, 'Families of magnetic semiconductors - an overview', Journal of Semiconductors 40, 080301 (2019). DOI: 10.1088/1674-4926/40/8/080301, https://doi.org/10.1088/1674-4926/40/8/080301 (preprint arXiv:1909.02999, https://arxiv.org/abs/1909.02999)
23. G. Bouzerar, R. Bouzerar, 'Unraveling the nature of carrier-mediated ferromagnetism in diluted magnetic semiconductors', Comptes Rendus Physique 16, 731-738 (2015). DOI: 10.1016/j.crhy.2015.09.003, https://doi.org/10.1016/j.crhy.2015.09.003