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Visualizing Critical Correlations Near the Metal-Insulator Transition in Ga1-xMnxAs

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Science  05 Feb 2010:
Vol. 327, Issue 5966, pp. 665-669
DOI: 10.1126/science.1183640

Abstract

Electronic states in disordered conductors on the verge of localization are predicted to exhibit critical spatial characteristics indicative of the proximity to a metal-insulator phase transition. We used scanning tunneling microscopy to visualize electronic states in Ga1-xMnxAs samples close to this transition. Our measurements show that doping-induced disorder produces strong spatial variations in the local tunneling conductance across a wide range of energies. Near the Fermi energy, where spectroscopic signatures of electron-electron interaction are the most prominent, the electronic states exhibit a diverging spatial correlation length. Power-law decay of the spatial correlations is accompanied by log-normal distributions of the local density of states and multifractal spatial characteristics.

Since Anderson first proposed 50 years ago that disorder could localize electrons in solids (1), studies of the transition between extended and localized quantum states have been at the forefront of physics (2). Realizations of Anderson localization occur in a wide range of physical systems from seismic waves to ultracold atomic gases, in which localization has recently been achieved with random optical lattices (3). In electronic systems, the signatures of localization have long been examined through electrical transport measurements (4, 5), and more recently by local scanning probe techniques that have imaged localized electronic states (6, 7). For noninteracting systems, the electronic states at the mobility edge are predicted to have a diverging localization length with scale-independent power-law characteristics, which are described as being multifractal (8). Given the poorly understood nature of the metal-insulator transition in the presence of disorder and electron-electron interactions, direct imaging of electronic states can provide insights into the interplay between localization and interactions.

We report on scanning tunneling microscopy (STM) and spectroscopy studies of electronic states in the dilute magnetic semiconductor Ga1-xMnxAs, over a range of Mn concentrations near the metal-insulator transition (x = 1.5 to 5%). Over the past decade, Ga1-xMnxAs has emerged as a promising material for spintronic applications with a high ferromagnetic transition temperature (9, 10). Mn atoms substituted at Ga sites act both as acceptors that drive the metal-insulator transition and as localized spins that align at low temperatures to give rise to magnetism. The nature of the electronic states underlying magnetism in these heavily doped semiconductors is still debated. It is often assumed that the carriers that mediate magnetism in Ga1-xMnxAs are Bloch states associated with either the valence bands or extended states originating from an impurity band (1113); however, the validity of these assumptions has been questioned (14, 15). Moreover, many recent low-temperature transport studies show evidence of electron-electron interaction and weak localization of carriers even for samples with high doping levels (1619). We used atomic-scale imaging and high-resolution spectroscopy with the STM to visualize electronic states in Ga1-xMnxAs and to examine the spatial structure of electron-electron correlations in this system. Our results indicate that spatial heterogeneity and electronic correlations must be considered in understanding the mechanism of magnetism in highly doped semiconductors.

Figure 1 shows STM topographs of cleaved Ga1-xMnxAs samples (2000 Å thick) grown by molecular beam epitaxy on a p-type Be-doped GaAs buffer layer (20, 21). Before measurements at a temperature of 4.2 K, the degenerately doped substrates are cleaved in situ to expose a (110) or equivalent surface of the heterostructure in cross section (Fig. 1A, inset). The topographs show in-gap states, dominated by individual Mn acceptor wave functions, although other defects such as As antisites are observed as well. Using previous STM studies of samples with more dilute Mn concentrations (22, 23) and the results of tight-binding model calculations (21, 24, 25), we can identify the topographic signatures of individual Mn acceptors in layers from the surface to the third subsurface layer (Fig. 1B). The size of an individual Mn acceptor state wave function is about 20 Å, due to its deep binding energy, which results in a metal-insulator transition at a relatively high level of doping (between 1 and 2%) in Ga1-xMnxAs. By increasing the Mn concentration from weakly insulating samples with variable-range hopping resistivity at 1.5% (critical temperature TC = 30 K) (19, 21) to relatively conducting samples at 5% (TC = 86 K, annealed), we find that higher concentrations of dopants appear in STM topographs on top of the atomically ordered GaAs lattice (Fig. 1C). All characteristic lengths, such as dopant separation or mean free path (~10 Å), are much shorter for Ga1-xMnxAs compared with other semiconductors doped with shallow dopants (15).

Fig. 1

STM topography of the in-gap states of GaMnAs. (A) STM topograph (+1.5 V, 20 pA) of Ga0.985Mn0.015As over a 1000 Å by 500 Å area. The inset shows a topograph (+1.8 V, 20 pA) of the heterostructure junction between the Mn- and Be-doped layers of GaAs. The size of this area is 80 Å by 125 Å. (B) A topograph (+1.5 V, 30 pA) over a smaller area of size 150 Å by 150 Å of Ga0.985Mn0.015As. Several substitutional Mn’s in various layers are identified in the lower part of the panel and marked by the layer in which they appear, with the surface labeled as zero. An As antisite is also shown. (C) STM topograph (+1.5 V, 20 pA) of the Ga0.95Mn0.05As sample after annealing for 6 hours at 200°C. The size of the area is 150 Å by 195 Å. (B) and (C) have the same scale.

Spectroscopic mapping with the STM can be used to show that Mn acceptors and other defects give rise to atomic-scale fluctuations in the local electronic density of states (LDOS) over a wide range of energies. Figure 2A shows tunneling spectra of states from within the valence band (with the voltage V < 0) to the conduction band edge (V > 1.5), measured along a line perpendicular to the buffer layer-film interface. By contrasting the spatial dependence of electronic states in the buffer layer (with 2 × 1018 per cm3 Be acceptors) to those of Ga1-xMnxAs (with x = 0.015) in Fig. 2A, we find the Mn-doped region to have strong spatial variations in the electronic states at the valence and conduction band edges and a broad distribution of states within the GaAs band gap. Increasing the Mn concentration gives rise to a larger number of in-gap states (compare Fig. 2, B and C). These features of the local electronic structure of Ga1-xMnxAs are difficult to reconcile with a weakly disordered valence or impurity band picture and show the importance of compensation and disorder in this compound. The Fermi energy EF lies within the range of electronic states that are spatially inhomogeneous.

Fig. 2

Local dI/dV spectroscopy for various dopings. (A) The differential conductance (dI/dV) (ΔV = 10 mV) across a line normal to the growth surface of length 1000 Å. The first 200 Å is the Be-doped GaAs buffer layer, and 800 Å of 1.5% Mn-doped GaAs follows. The junction between the two is marked by a dashed line. (B and C) The dI/dV spectra (ΔV = 5 mV) for energies close to the Fermi level, across a line of length 450 Å over the 1.5% sample (B) and the 5% sample (C). Top of the valence band and the in-gap states are shown. (D) The spatially averaged differential conductance for several samples. The inset shows the same data as the main panel, with the square root of the voltage on the horizontal axis.

In addition to strong spatial variations, electronic states of Ga1-xMnxAs are influenced by electron-electron interactions (Fig. 2D). STM spectra, spatially averaged across large areas for several samples with increasing doping levels, illustrate a strong suppression of the tunneling density of states near EF. The evolution from weakly insulating (1.5%) to relatively conducting samples (5%) is well correlated with the increase in the density of states at the Fermi level, yet a suppression centered at EF is observed at all doping levels. This feature is indicative of an Altshuler-Aronov correlation gap that is expected to occur in the tunneling density of states of a disordered material on the metallic side of the phase transition (26), appearing as a square-root singularity in the conductance near EF (Fig. 2D). Previous spectroscopic measurements of Ga1-xMnxAs with macroscopic tunneling junctions have also reported similar correlation gaps (16).

To determine whether there are any specific length scales associated with the spatial variation of the LDOS in Ga1-xMnxAs, we examine energy-resolved STM conductance maps. In Fig. 3, we show examples of such maps at different energies relative to EF for the 1.5% doped sample. These maps show that, in addition to modulations on the length scales of individual acceptors, there are spatial structures in the LDOS with longer length scales. To characterize these variations, for each conductance map, we compute the angle-averaged autocorrelation function between two points separated by r, C(E,r) = 1/(2π) ∫ dθ ∫ d2r' [g(E,r') − g0(E)] × [g(E,r' + r) − g0(E)], in which g(E,r) is the local value of the differential tunneling conductance that is proportional to the LDOS, g0(E) is the average value of the conductance at each energy E, and θ is the angle characterizing the direction of the separation vector r. Displaying C(E,r) in Fig. 4A, we find a marked increase of the long-distance correlations near EF. At this energy, the correlations remain measurable to length scales well beyond that of single Mn acceptor states, which dominate the behavior on short length scales at all energies. The increased correlation length can also be seen directly in the size of the patches of high and low conductance (Fig. 3C). We have observed the enhanced spatial correlations near EF for all doping levels examined in this study (up to 5%, Fig. 4B); however, this effect is most pronounced for the least doped samples (1.5%) closest to the metal-insulator transition. Control experiments on Zn- or Be-doped GaAs samples show no evidence of any special length scale or of a sharp peak near the EF in the autocorrelation function.

Fig. 3

Spectroscopic mapping in Ga0.985Mn0.015As. The differential conductance (dI/dV) measurements over an area of 500 Å by 500 Å show the spatial variations in the LDOS for the states in the valence band as well as states deep inside the semiconductor gap: (A) −100 mV, (B) −50 mV, (C) 0 mV, (D) +50 mV, (E) +100 mV, and (F) +150 mV.

Fig. 4

Correlation length for Mn dopings close to the metal-insulator transition. The autocorrelation was calculated from LDOS maps and is presented on a logarithmic scale for (A) Ga0.985Mn0.015As and (B) as-grown Ga0.95Mn0.05As. The dashed line in (A) is ~ (EEF)−1 and is a guide to the eye. (C) The autocorrelation for the dI/dV map at the Fermi level of Ga0.985Mn0.015As, as well as one valence band (−50 mV) and one in-gap energy (+50 mV), are plotted. The inset shows the autocorrelation for the same energies on a semi-logarithmic scale.

Continuous phase transitions, such as the metal-insulator transition, are typically characterized by a correlation length, which describes the exponential decay of spatial fluctuations when a system is tuned near the phase transition. At the critical point, this correlation length diverges and spatial fluctuations and other physical properties display power-law spatial characteristics. In the noninteracting limit, the transition between a metal and an insulator occurs by tuning the chemical potential relative to the mobility edge. Mapping the spatial structure of the electronic states as a function of energy can be used to determine the correlation length and to probe the critical properties for such a transition between extended and localized states (4, 5, 8). In our experiments, the distance dependence of the energy-resolved autocorrelation function C(E,r) for the 1.5% sample (Fig. 4C) appears to follow a power-law at EF, while at nearby energies it falls off exponentially (see inset). These observations, together with the apparent divergence of the correlation at a specific energy, are indeed signatures of the critical phenomena associated with a metal-insulator transition. However, our observation that the longest-ranged correlations are centered at EF, as opposed to some other energy, which could be identified as a mobility edge, signifies the importance of electron-electron interactions in the observed correlations.

Given the importance of electron-electron interactions, the conductance maps are perhaps more precisely identified as probing the spatial nature of quasiparticle excitations of a many-body system rather than simply imaging single-electron states in the noninteracting limit. Currently there are no theoretical models of the real-space structure of these excitations near the metal-insulator transition in a strongly interacting and disordered system, although there is continued effort to understand the nature of such transitions in the presence of interactions (5, 27). Nevertheless, we suspect that the correlation length associated with these excitations will become shorter due to multiparticle processes and inelastic effects at energies away from EF. Our experimental results for the least conducting sample (1.5%) indicate that the correlation length ξ is indeed suppressed away from EF, roughly following (EEF)−1 (dashed line in Fig. 4A). At EF, for this sample, these correlations decay in space following a power-law r−η, with η = 1.2 ± 0.3.

Despite the importance of strong interactions, many of the predictions for the noninteracting limit still appear to apply. Weakly disordered extended states are expected to show Gaussian distributions of the LDOS, indicating that these states have a finite probability to be present over the entire system. In contrast, near the metal-insulator transition wide distributions are expected, especially in local quantities such as the LDOS, which begin to cross over from Gaussian to log-normal distributions even in the limit of weak localization (28, 29). Spectroscopic maps of the density of states at EF for three different dopings (Fig. 5, A to C) show different degrees of spatial variations; however, their histograms (Fig. 5D) are similar in being skewed log-normal distributions where the mean is not representative of the distribution due to rare large values. Decreasing the doping skews the distribution further in a systematic fashion away from Gaussian and toward a log-normal distribution. For comparison, a histogram of the LDOS for states deep in the valence band for the least doped sample (gray circles in Fig. 5D) shows a Gaussian distribution.

Fig. 5

The spatial variations of the LDOS at the Fermi level, with their histogram and multifractal spectrum. The LDOS mapping of a 700 Å by 700 Å area of (A) Ga0.985Mn0.015As, (B) Ga0.97Mn0.03As, and (C) Ga0.95Mn0.05As. (D) The normalized histogram of the maps presented in (A) to (C). The local values of the dI/dV are normalized by the average value of each map. The inset shows the multifractal spectrum, f(α), near the value α0 where the maximum value occurs. For comparison, the results of a similar analysis over a LDOS map at −100 mV (valence band states) for the 1.5% doped sample are also shown.

Based on the predictions for the noninteracting limit, we expect critical states to have a spatial structure that is multifractal in nature. This property is directly related to the scale-invariant nature of critical wave functions and has been examined in great detail by numerical simulations of the single-particle quantum states near an Anderson transition (8). Multifractal patterns, which are ubiquitous in nature, are usually described by analysis of their self-similarity at different length scales through their singularity spectrum f(α). Physically, f(α) describes all the fractal dimensions embedded in a spatial pattern, such as those associated with a quantum wave function and its probability distribution. It is calculated by splitting the probability distribution into sets of locations {ri} that share a common exponent α, where the distribution scales locally with distance as |Ψ(ri)|2 ~ L−α, and measuring the fractal dimension of each set (8, 21). A variety of techniques have been developed to compute f(α), which has been used to distinguish between various models of the Anderson transition (21, 30). Application of such an analysis to our conductance maps (Fig. 5D, inset) shows an f(α) spectrum that is peaked at a value away from 2, which is indicative of anomalous scaling in a two-dimensional map. The f(α) spectrum also shows a systematic shift with decreasing doping, indicating a trend from weak toward strong multifractality with decreasing doping. In contrast, these signatures of multifractal behavior are absent for states deep in the valence band (gray curve) that, despite the strong disorder, show scaling consistent with those expected for extended states.

Our findings suggest that proximity to the metal-insulator transition and electronic correlations may play a more important role in the underlying mechanism of magnetism of Ga1-xMnxAs than previously anticipated. Beyond its application to understand the nature of states Ga1-xMnxAs, our experimental approach provides a direct method to examine critical correlations for other material systems near a quantum phase transition. In principle, experiments at the lowest temperatures for samples closest to the metal-insulator transition should provide accurate measurements of power-law characteristics that can be directly compared to theoretically predicted critical exponents.

Supporting Online Material

www.sciencemag.org/cgi/content/full/327/5966/665/DC1

Materials and Methods

Figs. S1 to S4

References

  • * These authors contributed equally to this work.

References and Notes

  1. Materials and methods are available as supporting material on Science Online.
  2. This work was supported by grants from Office of Naval Research, Army Research Office, the Keck Foundation, NSF, and the NSF–Materials Research Science and Engineering Center program through the Princeton Center for Complex material. P.R. acknowledges a NSF graduate fellowship.
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