Abstract
Electronic states in disordered conductors on the verge of localization are predicted to exhibit critical spatial characteristics indicative of the proximity to a metalinsulator phase transition. We used scanning tunneling microscopy to visualize electronic states in Ga_{1}_{x}Mn_{x}As samples close to this transition. Our measurements show that dopinginduced disorder produces strong spatial variations in the local tunneling conductance across a wide range of energies. Near the Fermi energy, where spectroscopic signatures of electronelectron interaction are the most prominent, the electronic states exhibit a diverging spatial correlation length. Powerlaw decay of the spatial correlations is accompanied by lognormal 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 scaleindependent powerlaw characteristics, which are described as being multifractal (8). Given the poorly understood nature of the metalinsulator transition in the presence of disorder and electronelectron 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 Ga_{1}_{x}Mn_{x}As, over a range of Mn concentrations near the metalinsulator transition (x = 1.5 to 5%). Over the past decade, Ga_{1}_{x}Mn_{x}As 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 metalinsulator 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 Ga_{1}_{x}Mn_{x}As are Bloch states associated with either the valence bands or extended states originating from an impurity band (11–13); however, the validity of these assumptions has been questioned (14, 15). Moreover, many recent lowtemperature transport studies show evidence of electronelectron interaction and weak localization of carriers even for samples with high doping levels (16–19). We used atomicscale imaging and highresolution spectroscopy with the STM to visualize electronic states in Ga_{1}_{x}Mn_{x}As and to examine the spatial structure of electronelectron 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 Ga_{1}_{x}Mn_{x}As samples (2000 Å thick) grown by molecular beam epitaxy on a ptype Bedoped 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 ingap 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 tightbinding 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 metalinsulator transition at a relatively high level of doping (between 1 and 2%) in Ga_{1}_{x}Mn_{x}As. By increasing the Mn concentration from weakly insulating samples with variablerange hopping resistivity at 1.5% (critical temperature T_{C} = 30 K) (19, 21) to relatively conducting samples at 5% (T_{C} = 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 Ga_{1}_{x}Mn_{x}As compared with other semiconductors doped with shallow dopants (15).
Spectroscopic mapping with the STM can be used to show that Mn acceptors and other defects give rise to atomicscale 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 layerfilm interface. By contrasting the spatial dependence of electronic states in the buffer layer (with 2 × 10^{18} per cm^{3} Be acceptors) to those of Ga_{1}_{x}Mn_{x}As (with x = 0.015) in Fig. 2A, we find the Mndoped 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 ingap states (compare Fig. 2, B and C). These features of the local electronic structure of Ga_{1}_{x}Mn_{x}As 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 E_{F} lies within the range of electronic states that are spatially inhomogeneous.
In addition to strong spatial variations, electronic states of Ga_{1}_{x}Mn_{x}As are influenced by electronelectron 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 E_{F}. 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 E_{F} is observed at all doping levels. This feature is indicative of an AltshulerAronov 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 squareroot singularity in the conductance near E_{F} (Fig. 2D). Previous spectroscopic measurements of Ga_{1}_{x}Mn_{x}As 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 Ga_{1}_{x}Mn_{x}As, we examine energyresolved STM conductance maps. In Fig. 3, we show examples of such maps at different energies relative to E_{F} 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 angleaveraged autocorrelation function between two points separated by r, C(E,r) = 1/(2π) ∫ dθ ∫ d^{2}r' [g(E,r') − g_{0}(E)] × [g(E,r' + r) − g_{0}(E)], in which g(E,r) is the local value of the differential tunneling conductance that is proportional to the LDOS, g_{0}(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 longdistance correlations near E_{F}. 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 E_{F} 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 metalinsulator transition. Control experiments on Zn or Bedoped GaAs samples show no evidence of any special length scale or of a sharp peak near the E_{F} in the autocorrelation function.
Continuous phase transitions, such as the metalinsulator 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 powerlaw 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 energyresolved autocorrelation function C(E,r) for the 1.5% sample (Fig. 4C) appears to follow a powerlaw at E_{F}, 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 metalinsulator transition. However, our observation that the longestranged correlations are centered at E_{F}, as opposed to some other energy, which could be identified as a mobility edge, signifies the importance of electronelectron interactions in the observed correlations.
Given the importance of electronelectron interactions, the conductance maps are perhaps more precisely identified as probing the spatial nature of quasiparticle excitations of a manybody system rather than simply imaging singleelectron states in the noninteracting limit. Currently there are no theoretical models of the realspace structure of these excitations near the metalinsulator 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 E_{F}. Our experimental results for the least conducting sample (1.5%) indicate that the correlation length ξ is indeed suppressed away from E_{F}, roughly following (E − E_{F})^{−1} (dashed line in Fig. 4A). At E_{F}, for this sample, these correlations decay in space following a powerlaw 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 metalinsulator transition wide distributions are expected, especially in local quantities such as the LDOS, which begin to cross over from Gaussian to lognormal distributions even in the limit of weak localization (28, 29). Spectroscopic maps of the density of states at E_{F} 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 lognormal 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 lognormal 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.
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 scaleinvariant nature of critical wave functions and has been examined in great detail by numerical simulations of the singleparticle quantum states near an Anderson transition (8). Multifractal patterns, which are ubiquitous in nature, are usually described by analysis of their selfsimilarity 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 {r_{i}} that share a common exponent α, where the distribution scales locally with distance as Ψ(r_{i})^{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 twodimensional 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 metalinsulator transition and electronic correlations may play a more important role in the underlying mechanism of magnetism of Ga_{1}_{x}Mn_{x}As than previously anticipated. Beyond its application to understand the nature of states Ga_{1}_{x}Mn_{x}As, 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 metalinsulator transition should provide accurate measurements of powerlaw 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
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵ Materials and methods are available as supporting material on Science Online.
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 ↵
 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.