Abstract
Embedding magnetic moments into semiconductor heterostructures offers a tuneable access to various forms of magnetic ordering and phase transitions in lowdimensional electron systems. In general, the moments are introduced artificially, by either doping with ferromagnetic atoms, or electrostatically confining oddelectron quantum dots^{1,2,3,4}. Here, we report experimental evidence of an independent, and intrinsic, source of localized spins in highmobility GaAs/AlGaAs heterostructures with large setback distance (≈80 nm) in modulation doping. Measurements reveal a quasiregular distribution of the spins in the delocalized Fermi sea, and a mutual interaction via the Ruderman–Kittel–Kasuya–Yosida (RKKY) indirect exchange below 100 mK. We show that a simple model on the basis of the fluctuations in background potential on the host twodimensional electron system can explain the observed results quantitatively, which suggests a ‘disordertemplated’ microscopic origin of the localized moments.
Similar content being viewed by others
Main
Local fluctuations in the conduction band often lead to localization of singleelectron states in a dilute metallic system. These ‘impurity’ states are physically separated from the surrounding delocalized Fermi sea by a tunnel barrier, and can become magnetic for large onsite Coulomb interaction U>−ε, where ε (<0) is the energy of the localized states relative to the surrounding Fermi level. In the limit of small valency fluctuations, that is, −ε/Γ≫1, where Γ is the level broadening due to finite tunnelling, the Kondo representation of the Anderson impurity model^{5} allows the impurity spin to form a spin singlet with the local spin cloud of conduction electrons as the system is cooled below the characteristic Kondo temperature T_{K}≈E_{F}exp[πε(ε+U)/2Γ U], where E_{F} is the Fermi energy of the conduction electrons.
Because the Kondo effect contributes one additional state at each impurity site within a small bandwidth of T_{K} around E_{F}, spectroscopic techniques based on tunnelling^{6} or nonequilibrium transport^{1,2,3,4,7,8,9} reveal a resonance in the lowenergy density of states (DOS) at zero source–drain bias (V_{SD}). This is known as the zerobias anomaly (ZBA) (see Fig. 1). In the case of multiple mutually interacting impurities, the DOS is more complex and depends on a competition between T_{K} and a pairwise interimpurity exchange interaction J_{12} (refs 10, 11). For J_{12}≫k_{B}T, where T is the temperature, the Kondo resonance is suppressed at energy scales ≲J_{12} (Fig. 1), resulting in a split ZBA at zero external magnetic field, which acts as an indicator of spontaneous spin polarization and static magnetic phases within the system^{12,13,14}.
Along with the magnetic effects, the embedded localized states also act as scattering centres for conduction electrons, leading to backscattering and interference effects when the phase coherence length of the electrons exceeds the interimpurity distance (R). A completely random spatial distribution of the impurities gives rise to weak localization phenomena at low T, which are manifest as a universal aperiodic fluctuation in the linear magnetoconductance as a function of either E_{F} or a weak transverse magnetic field (ref. 15). However, when the impurities are arranged in a quasiregular manner, the transport takes place through multiple, connected Aharonov–Bohm rings encircling a discrete number of impurities. As a function of , the magnetoconductance at low T breaks into quasiperiodic phasecoherent oscillations with periodicity given by
where R_{C} is the radius of a stable ring^{16,17}.
Observation of Kondo resonance in unconfined mesoscopic onedimensional^{7} or twodimensional^{8,9} devices from highmobility semiconductor heterostructures indicates unexpected localized spin states, whose microscopic origin is not understood. Most experimental evidence so far is extracted from the ZBA in nonequilibrium transport, which remains inconclusive on the spatial layout, or the mutual interaction of the states. Here, we have exploited both spin and orbital effects of localized states by combining the investigation of nonequilibrium characteristics and linear magnetotransport in the same mesoscopic device. The results are not only consistent with the existence of local magnetic moments in highmobility heterostructures, but also, for the first time, indicate the nature of mutual interaction, and a plausible microscopic origin of these moments.
Our mesoscopic devices were fabricated from Simodulationdoped GaAs/AlGaAs heterostructures, where unintentional magnetic impurities are expected to be absent, and a high lowT mobility of ∼1–3×10^{6} cm^{2} V^{−1} s^{−1} provides a long asgrown elastic mean free path ∼6–8 μm. The lithographic dimensions of our devices were kept smaller than this length scale to ensure quasiballistic transport (Fig. 2a). Nonequilibrium transport in these systems involves measuring the differential conductance dI/dV as a function of V_{SD} at fixed voltages (V_{G}) applied on the (nonmagnetic) surface gate, which reflects the lowenergy DOS through the Landauer formalism. Experiments were carried out at electron temperatures as low as ≈30 mK.
Below ∼100 mK, the zerofield dI/dV shows rich structures in the V_{G}–V_{SD} plane (Fig. 2c). For most devices, the structures are strongest in the electrondensity range n_{2D}∼1–3×10^{10} cm^{−2} (Fig. 2b), and often visible up to G≳10–15×(e^{2}/h), where G=dI/dV at V_{SD}=0. In Fig. 2c, dI/dV consists of a repetitive sequence of two types of resonance as V_{G} is increased. We denote the singlepeak resonance as ZBAI, which splits intermittently to form a gap at E_{F}, referred to as ZBAII. We define Δ as the halfwidth at halfdepth of ZBAII. Similar nonequilibrium characteristics were observed in over 50 mesoscopic devices from five different wafers, where reducing the setback distance (d_{s}) below ∼60–80 nm was generally found to have a detrimental effect on the clarity of the resonance structures, often leading to broadening or complete suppression of both types of ZBA.
The ZBAs are logarithmically suppressed for T≳300 mK (Fig. 2f,g), and split linearly in an inplane magnetic field when Zeeman energy exceeds the corresponding Δ (Fig. 2d,e). These are characteristic features of Kondo resonance^{9}. (See Supplementary Information for arguments against alternative explanation of the ZBA.) Moreover, both linear and nonlinear transport at ZBAII indicate nonmonotonicity at the scale of Δ (Fig. 2e,g), which can be understood in a ‘twoimpurity Kondo’ model that implies the presence of multiple interacting spins^{9,13}. Because Δ∼J_{12} (Fig. 1), the reentrant splitting of ZBA in Fig. 2c thus indicates J_{12} to be oscillatory in V_{G}, where ZBAI corresponds to J_{12}≪k_{B}T.
Before exploring the microscopic origin of J_{12}, we discuss the linear lowtransversefield magnetoconductance measurements, which provide information on the spatial layout of the scattering centres. At the lowest T, mesoscopic devices that show clear resonances in dI/dV also show reproducible quasiperiodic oscillations in the linear magnetoconductance over a transverse field range . These oscillations appear over the same range of n_{2D} as the ZBA, but, apart from being slightly smaller in amplitude near a ZBAII, they are mostly insensitive to the details of the ZBA structure. In Fig. 3a, b this is illustrated with a device of configuration similar to that shown in Fig. 2a. Discrete peaks in the fastFourier power spectrum of these oscillations imply their quasiperiodicity, thereby distinguishing them from universal conductance fluctuations.
To analyse the magnetoconductance oscillations, we note that the strong peak at (Fig. 3c) is common to all traces and corresponds to the ‘unit cell’ in the scatterer array, which is an Aharonov–Bohm ring encircling a single scattering centre, with a radius of R/2 (orbit (ii)). From equation (1), we get R≈670±30 nm, which was found to be weakly device dependent, varying between 600 and 800 nm, but insensitive to the lithographic dimensions of the devices. Often peaks at higher frequency would also appear, consistent with Aharonov–Bohm rings encircling larger numbers of scattering centres (see orbit (iii) in Fig. 3c). The peak at lower frequency, which becomes stronger at (fastFourier power spectrum denoted by empty blue circles in Fig. 3c), represents orbit (i), which is inscribed by the finite crosssection of the scattering centres.
Implicitly, the phasecoherent oscillations indicate a quasiregular distribution of the scattering centres. We independently crosschecked this from commensurability of classical electron trajectories in a twodimensional array of point scatterers at finite (ref. 18). In Fig. 3d, we show the fourprobe magnetoresistance of the same device at T≃1.4 K, which consists of small peaklike structures superposed on a parabolically increasing background. On subtracting the background, the signature of commensurable orbits at 0.08, 0.05 and 0.03 T can be observed, corresponding to the cyclotron radius of the electron encircling one, two and four scattering centres respectively (inset). This gives R∼500 nm, consistent with the estimate obtained from phasecoherent oscillations.
Assuming each scattering centre hosts a localized spin state, we find that J_{12} can be naturally associated with RKKYtype indirect exchange, which is expected in a system composed of itinerant electrons and localized magnetic impurities. In this framework, the oscillatory behaviour of J_{12} in V_{G} arises from the range function Ψ(2k_{F}R) in the interaction magnitude, which reverses its sign with a periodicity of π in 2k_{F}R, being the Fermi wavevector and R being fixed for a given device. Analytically,
Figure 4a shows the direct confirmation of this, where we have plotted Δ as a function of 2k_{F}R for the device in Fig. 2c. The clear periodicity of ≈π (within ±5%) in 2k_{F}R can be immediately recognized as the socalled ‘2k_{F}R oscillations’ in the RKKY interaction, establishing the multiplelocalizedspin picture.
The absolute magnitude of J_{12}, and hence Δ, for a twodimensional distribution of spins may differ widely from the straightforward pairwise RKKY interaction, and would be affected by frustrated magnetic ordering or spinglass freezing^{19}, as well as deviation from perfect periodicity in the spin arrangements^{20}. Nevertheless, a framework for comparison of Δ in different samples can be obtained from equation (2) by normalizing Δ with E_{F}. As shown in Fig. 4b, manually adjusting for the experimental uncertainty in k_{F} and R, Δ/E_{F} for four different devices with various lithographic dimensions can be made to collapse onto the solid line proportional to the modulus of the twoimpurity RKKY range function over a wide range of 2k_{F}R (ref. 21).
A plausible mechanism for local moment formation in highmobility GaAs/AlGaAs systems would thus require us to simultaneously account for (1) the observed magnitude of R and the quasiregular impurity distribution and (2) Kondo coupling of the localized spin to the conduction electrons. We show below that background potential fluctuations arising from a strongly correlated dopant layer can address both observations quantitatively. In this scheme, which is similar to moment formation at metal–semiconductor Schottky barriers^{6}, depletion of electrons at strong potential fluctuations leads to Mott transition locally around the depletion region. Isolated localized states are formed, which become magnetic when Γ/U is reduced below ∼0.1 (the Mott criterion) with decreasing n_{2D}.
For heavily compensated donor layers of monolayerdoped heterostructures, the density correlation function φ(q) is commonly approximated as that of a nonideal plasma^{22,23}:
which in the longwavelength limit, q≪1/d_{s}, leads to an effective dopant concentration of
where ε_{0} is the vacuum permittivity, ε_{r} is the relative permittivity and N_{Δ} and T_{0} are the bare dopant density and freezingout temperature for electrons left in the dopant layer, respectively. Strong Coulomb interaction can further lead to an ordering in this effective donor distribution, with a sharp peak in the correlation function at (ref. 24). Using known system parameters T_{0}≈80 K and d_{s}=80 nm, we find that strong background potential fluctuations are expected at a length scale of 1/q_{max}∼0.4–0.5 μm, which agrees well with the observed magnitude of R, as well as results of direct experimental imaging^{25}. We note that both phenomena will be observable as long as any deviation (ΔR) from a perfect ordering satisfies ΔR≪2π/k_{F} for RKKY dynamics, or for the phasecoherent oscillations. These conditions impose ΔR/R≪0.4, which is realistic, because following ref. 24 we obtain ΔR/R∼Δq/q_{max}≲0.2.
To verify the Kondo coupling of localized moments to surrounding itinerant electrons, we have calculated the ratio ε/Γ for the localized states using the experimentally observed halfwidth of the ZBAI resonances, and equating it to T_{K}. From the trace shown in the lower inset of Fig. 2c, T_{K}≈0.035 mV∼E_{F}exp(πε/2Γ) in the limit, which gives ε/Γ≈−2.1, confirming the Kondo regime. (Here U∼e^{2}/ξ≫E_{F}, and ξ∼150 nm is the localization length obtained experimentally from the magnetoconductance oscillations corresponding to orbit (i) in Fig. 3c.) Notably, formation of magnetic quantum point contacts^{7,26,27} in a percolationtype representation of the disorder landscape at low n_{2D} (ref. 28) cannot explain the commensurability effect or phasecoherent magnetoconductance oscillations shown in Fig. 3, as they require extended and uninterrupted electron orbits.
Methods
Wafer and device characteristics
Devices were fabricated from (100) molecular beam epitaxygrown GaAs/Al_{0.33}Ga_{0.66}As heterostructures, where the twodimensional electron layer was formed 300 nm below the surface. A setback distance of d_{s}=80 nm was obtained by inserting an undoped layer of Al_{0.33}Ga_{0.66}As between the Si Δdoped layer, with dopant density N_{Δ}≈2.5×10^{12} cm^{−2}, and the heterointerface. This resulted in a heavily compensated dopant layer with a filling factor f≈0.9. The device width was determined by the width of the wetetched mesa, and fixed at ≈8 μm, and the length defined by the width of the surface gate was varied between 2 and 7 μm (Fig. 2a).
Measurements
Nonequilibrium measurements were carried out inside a dilution refrigerator (base electron temperature ≈30 mK), with an ac+dc twoprobe technique. The ac modulation (∼2–5 μV at 90 Hz) was kept much less than k_{B}T to avoid heating or other nonlinearities. Before measurement all devices were subjected to slow cooling cycles between room temperature and 4.2 K, with each cooling lasting 5–10 h. See Supplementary Information for more details on the effect of cooling and disorder.
References
GoldhaberGordon, D. et al. Kondo effect in a singleelectron transistor. Nature 391, 156–159 (1998).
Cronenwett, S. M., Oosterkamp, T. H. & Kouwenhoven, L. P. A tunable Kondo effect in quantum dots. Science 281, 540–544 (1998).
Jeong, H., Chang, A. M. & Melloch, M. R. The Kondo effect in an artificial quantum dot molecule. Science 293, 2221–2223 (2001).
Craig, N. J. et al. Tunable nonlocal spin control in a coupledquantum dot system. Science 304, 565–567 (2004).
Schrieffer, J. R. & Wolff, P. A. Relation between the Anderson and Kondo hamiltonians. Phys. Rev. 149, 491–492 (1966).
Wolf, E. L. & Losee, D. L. Spectroscopy of Kondo and spinflip scattering: Highfield tunneling studies of SchottkyBarrier junctions. Phys. Rev. B 2, 3660–3687 (1970).
Cronenwett, S. M. et al. Lowtemperature fate of the 0.7 structure in a point contact: A Kondolike correlated state in an open system. Phys. Rev. Lett. 88, 226805 (2002).
Ghosh, A., Ford, C. J. B., Pepper, M., Beere, H. E. & Ritchie, D. A. Possible evidence of a spontaneous spin polarization in mesoscopic twodimensional electron systems. Phys. Rev. Lett. 92, 116601 (2004).
Ghosh, A. et al. Zerobias anomaly and kondoassisted quasiballistic 2D transport. Phys. Rev. Lett. 95, 066603 (2005).
Jayaprakash, C., Krishnamurthy, H. R. & Wilkins, J. W. Twoimpurity Kondo problem. Phys. Rev. Lett. 47, 737–740 (1981).
Affleck, I. & Ludwig, A. W. W. Exact critical theory of the twoimpurity Kondo model. Phys. Rev. Lett. 68, 1046–1049 (1992).
Pasupathy, A. N. et al. The Kondo effect in the presence of ferromagnetism. Science 306, 86–89 (2004).
Heersche, H. B. et al. Kondo effect in the presence of magnetic impurities. Phys. Rev. Lett. 96, 017205 (2006).
Nygard, J., Cobden, D. H. & Lindelof, P. E. Kondo physics in carbon nanotubes. Nature 408, 342–346 (2000).
Thornton, T. J., Pepper, M., Ahmed, H., Davies, G. J. & Andrews, D. Universal conductance fluctuations and electron coherence lengths in a narrow twodimensional electron gas. Phys. Rev. B 36, 4514–4517 (1987).
Schuster, R., Ensslin, K., Wharam, D., Kühn, S. & Kotthaus, J. P. Phasecoherent electrons in a finite antidot lattice. Phys. Rev. B 49, 8510–8513 (1994).
Weiss, D. et al. Quantized periodic orbits in large antidot arrays. Phys. Rev. Lett. 70, 4118–4121 (1993).
Weiss, D. et al. Electron pinball and commensurate orbits in a periodic array of scatterers. Phys. Rev. Lett. 66, 2790–2793 (1991).
Hirsch, M. J., Holcomb, D. F., Bhatt, R. N. & Paalanen, M. A. ESR studies of compensated Si:P,B near the metal–insulator transition. Phys. Rev. Lett. 68, 1418–1421 (1992).
Roche, S. & Mayou, D. Formalism for the computation of the RKKY interaction in aperiodic systems. Phys. Rev. B 60, 322–328 (1999).
BéalMonod, M. T. Ruderman–Kittel–Kasuya–Yosida indirect interaction in two dimensions. Phys. Rev. B 36, 8835–8836 (1987).
Efros, A. L., Pikus, F. G. & Samsonidze, G. G. Maximum lowtemperature mobility of twodimensional electrons in heterojunctions with thick spacer layer. Phys. Rev. B 41, 8295–8301 (1990).
Buks, E., Heiblum, M. & Shtrikman, H. Correlated charged donors and strong mobility enhancement in a twodimensional electron gas. Phys. Rev. B 49, 14790–14793 (1994).
Grill, R. & Döhler, G. H. Effect of charged donor correlation and Wigner liquid formation on the transport properties of a twodimensional electron gas in modulation Δdoped heterojunctions. Phys. Rev. B 59, 10769–10777 (1999).
Finkelstein, G., Glicofridis, P. I., Ashoori, R. C. & Shayegan, M. Topographic mapping of the quantum Hall liquid using a fewelectron bubble. Science 289, 90–94 (2000).
Graham, A. C., Pepper, M., Simmons, M. Y. & Ritchie, D. A. Anomalous spindependent behavior of onedimensional subbands. Phys. Rev. B 72, 193305 (2005).
Berggren, K.F., Jaksch, P. & Yakimenko, I. Effects of electron interactions at crossings of Zeemansplit subbands in quantum wires. Phys. Rev. B 71, 115303 (2005).
Meir, Y. Percolationtype description of the metal–insulator transition in two dimensions. Phys. Rev. Lett. 83, 3506–3509 (1999).
Acknowledgements
The work was supported by an EPSRC funded project. A.G. acknowledges discussions with C. J. B. Ford, G. Gumbs, M. Stopa, P. B. Littlewood, H. R. Krishnamurthy, B. D. Simons, C. M. Marcus, D. GoldhaberGordon and K. F. Berggren. C.S. acknowledges financial support from the Gottlieb Daimler and Karl Benz Foundation.
Author information
Authors and Affiliations
Contributions
Experimental work and data analysis were carried out by C.S. and A.G., project planning by A.G. and M.P. and wafer growing by I.F. and D.A.R.
Corresponding authors
Ethics declarations
Competing interests
The authors declare no competing financial interests.
Supplementary information
Rights and permissions
About this article
Cite this article
Siegert, C., Ghosh, A., Pepper, M. et al. The possibility of an intrinsic spin lattice in highmobility semiconductor heterostructures. Nature Phys 3, 315–318 (2007). https://doi.org/10.1038/nphys559
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1038/nphys559
This article is cited by

Reversible doping polarity and ultrahigh carrier density in twodimensional van der Waals ferroelectric heterostructures
Frontiers of Physics (2023)

Magnetic lattice surprise
Nature Physics (2007)

Unexpected features of branched flow through highmobility twodimensional electron gases
Nature Physics (2007)