Stacking of 2D Electron Gases in Ge Probed at the Atomic Level and Its Correlation to Low-Temperature Magnetotransport

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1 pubs.acs.org/nanolett Stacking of 2D Electron Gases in Ge Probed at the Atomic Level and Its Correlation to Low-Temperature Magnetotransport G. Scappucci,*,, W. M. Klesse,, A. R. Hamilton, G. Capellini, D. L. Jaeger, M. R. Bischof, R. F. Reidy, B. P. Gorman, and M. Y. Simmons, School of Physics, University of New South Wales, Sydney, NSW 2052, Australia Australian Research Council Centre of Excellence for Quantum Computation and Communication Technology, University of New South Wales, Sydney, NSW 2052, Australia Dipartimento di Fisica E. Amaldi, Universita di Roma Tre, Via della Vasca Navale 84, Roma, Italy Department of Materials Science and Engineering, University of North Texas, Denton, Texas 76209, United States Department of Materials Science and Engineering, Colorado School of Mines, Golden, Colorado 80401, United States *S Supporting Information ABSTRACT: Stacking of two-dimensional electron gases (2DEGs) obtained by δ-doping of Ge and patterned by scanning probe lithography is a promising approach to realize ultrascaled 3D epitaxial circuits, where multiple layers of active electronic components are integrated both vertically and horizontally. We use atom probe tomography and magnetotransport to correlate the real space 3D atomic distribution of dopants in the crystal with the quantum correction to the conductivity observed at low temperatures, probing if closely stacked δ-layers in Ge behave as independent 2DEGs. We find that at a separation of 9 nm the stacked-2degs, while interacting, still maintain their individuality in terms of electron transport and show long phase coherence lengths ( 220 nm). Strong vertical electron confinement is crucial to this finding, resulting in an interlayer scattering time much longer ( 1000 ) than the scattering time within the dopant plane. KEYWORDS: Germanium, doping, atom probe tomography, weak localization, multilayer, stacked two-dimensional gases There is a rapidly growing interest in the study of electronic transport in germanium-based low-dimensional systems for the development of innovative quantum devices. 1,2 We have recently demonstrated that phosphorus atomic layer doping in ultrahigh vacuum is a viable method to achieve a twodimensional sheet of donors (δ-doped layer) in Ge. This method provides a highly confined two-dimensional electron gas (2DEG) with electron densities in the cm 2 range. Generally δ-doping of semiconductors results in abrupt dopant profiles at the expense of the quality of the host crystal with increased surface roughness. 3 On the contrary narrow δ- doped layers with atomically flat surfaces have recently been achieved in Ge, owing to the relatively low temperatures needed to achieve high-quality growth ( C) in Ge crystals. 4 This unique property of Ge makes it possible to stack multiple δ-doped layers with near identical doping profiles in a crystal environment free from interfaces. 5 Multiple closely spaced n-type δ-layers can be used for the realization of ultrahigh-doped, low resistivity Ge thin films with possible applications ranging from the fabrication of ultrashallow junctions for ultrascaled transistors 6 to development of lasers monolithically integrated into the complementary metal oxide semiconductor (CMOS) technology. 7,8 Delta-doping can also be employed in the fabrication of planar atomic-scale devices in Ge. 9 In this approach, scanning tunneling microscopy (STM) lithography is used in ultrahigh vacuum (UHV) to pattern a 2DEG with atomic precision. This method has proven successful to embed P-doped planar nanodevices in a Ge crystal, such as 1D wires and tunnel gaps, opening an entire new playground where the quantum behavior of highly confined electrons can be studied. These achievements, together with the possibility to stack an arbitrary number of 2DEGs in a high quality crystal matrix, pave the way for STM patterning of three-dimensional epitaxial circuits, where atomic-level control over donor positioning and wave function overlap can be obtained in all three dimensions (see Figure 1). In 3D architectures two or more layers of active electronic components are integrated both vertically and horizontally to enhance performance and add functionalities, overcoming the limits of conventional CMOS technology. As an example, for new applications in the area of quantum information processing relying on single dopant-based qubit devices, 10 patterning of multiple layers makes it possible to Received: July 11, 2012 Revised: August 17, 2012 XXXX American Chemical Society A

2 Nano s electron transport? In this we present a combined atom probe tomography (APT) and low-temperature magnetotransport study of single and multiple-stacked phosphorus δ-doped layers in Ge. Using the single-atom elemental imaging capability of APT we obtain the real space 3D atomic distribution of P dopants throughout the doped regions. Low-temperature magnetotransport characterization allows us to probe how electrons in one layer are affected by electrons in the other layers by measuring all the characteristic conductivities and scattering times. The combined knowledge of structural and electrical properties ultimately provides the clues to understand if closely stacked layers still behave as independent 2DEGs. Previous transport studies on single-layer 2DEGs obtained in GaAs,12 Si,13 and recently, graphene,14 have shown that weak localization (WL) and electron electron interaction (EEI) affects the electrical conductivity at low temperatures, both giving rise to corrections with a ln(t) dependence. As a consequence, the determination of the key transport properties of the 2DEG, such as the Drude conductivity (σd), the electrically active carrier density (n0), the electrons scattering time (τe), and phase coherence time (τφ) is challenging. If we consider systems made of stacked 2DEGs, we also have to consider interlayer scattering as an additional dephasing mechanism, as recently highlighted in WL studies of multilayered 2DEGs in oxide superconductors.15,16 This effect becomes prominent when the phase coherence time τφ of electrons within a 2DEG and the interlayer scattering time τ1 are comparable and appears as a suppression of the WL feature in magnetotransport experiments, yielding an apparent saturation of τφ at low temperatures. To date, there has been no study that has correlated interlayer scattering with the atomic-scale structural properties of multilayered 2DEGs, in our case the real space 3D atomic distribution of dopants in the host Ge crystal. Two Ge:P atomic layer doped samples (S, M) were fabricated in an UHV system using PH3 as a dopant source Figure 1. 3D epitaxial circuits using P atomic-layer doping of Ge. The demonstration of highly confined 2DEGs in Ge (top left), their patterning into planar nanodevices with an atomically flat surface (bottom left), and the possibility to stack an arbitrary number of 2DEGs in a high quality crystal environment (top right) pave the way for STM patterning of three-dimensional epitaxial circuits (bottom right). increase the number of gates addressing a single dopant, a key requirement to realize qubit arrays with error correction capabilities.11 Any design of 3D circuits made of epitaxially stacked 2DEGs and/or atomic-scale Ge:P devices with confinement in more dimensions requires a detailed understanding of: (i) dopant distribution at the atomic level in all three-dimensions (ii) lowtemperature transport properties of carriers in an isolated 2DEG and how they are affected by interactions between different layers when stacked in multiple 2DEGs. In particular, one key question to be addressed is: how close can Ge:P 2DEGs be stacked while retaining their individuality in terms of Figure 2. APT of single and multilayered P δ-doped Ge. (a, b) P (solid spheres) atomic distribution along the epitaxial growth direction (scale bars = 6 nm). (c, d) P concentration profiles along the growth direction (points) and Gaussian fit (solid lines). (e) Distribution of P for sample M, 100% displayed as purple spheres, and 3D isoconcentration surfaces of P cm 3, where surface color represents the distance from the center of each respective isoconcentration surface. (f) 2D concentration maps of the in plane distribution of P in nm3 slices centered about each respective δ-layer, Di (i = 1 4). (g) In plane distribution of P in nm3 slices centered about the interlayer spacer regions, Ci (i = 1 3), between the respective doped layers (scale bars = 6 nm). B

3 Nano s and a dual-temperature, molecular beam epitaxy process to embed 2D layers of P atoms in the Ge crystal matrix. Sample S is a single δ-layer, while sample M comprises four δ-layers, each nominally identical to sample S, separated by undoped Ge spacer layers 9-nm-thick. Details of the sample deposition process are found in refs 4, 5, and 17. The three-dimensional distribution of P dopants throughout the samples was determined using pulsed laser APT. Samples were prepared using the focused ion beam in situ lift-out and annular mill method 18 (see Supporting Information for details on APT sample preparation, data acquisition, and data analysis). Selected volume reconstructions aligned with the Ge:P δ-layers and along the growth direction were used to create 3D tomographic reconstructions of samples S and M (Figure 2a and b, respectively), and 1D phosphorus atomic concentration profiles along the growth direction (Figure 2c and d, respectively). From these maps and the analysis of the atomic concentration profiles we conclude that all δ-layers are physically narrow and, importantly, nearly identical along the growth direction (see Table 1). Qualitatively, we also observe Table 1. Structural and Electrical Properties of Samples: Number N of Stacked δ-layers, Phosphorus Sheet Density along the Epitaxial Growth Direction n P, Full-Width Half- Maximum fwhm, Leading Slope λ L (nm/decade), and Trailing Slope λ L (nm/decade) of the Last Deposited δ- Layer, Preliminary Estimate of Electron Density n 4K at 4.2 K (10 13 cm 2 ), Electron Density n 0 (10 13 cm 2 ), Drude Conductivity σ D, Weak Localization Correction Δσ wl to the Conductivity, Electron-Electron Interaction Correction Δσ ee to the Conductivity, Transport Relaxation Time τ e, Mean Free Path λ (nm), Coefficient K ee of the Electron Electron Interaction Correction to the Conductivity, Pre-factor α in the Hikami Fit sample S M N 1 4 n P (10 13 cm 2 ) fwhm (nm) λ L (nm) λ T (nm) n 4K (10 13 cm 2 ) n 0 (10 13 cm 2 ) σ D (e 2 /πħ) Δσ a wl (e 2 /πħ) Δσ a ee (e 2 /πħ) τ E (10 14 s) λ (nm) l a ϕ (nm) K EE α a At T = 300 mk. the presence of few P atoms outside the plane of the δ-layers, most likely a result of dopant redistribution during the deposition process. This is evidenced by non-negligible P signal coming from the nominally undoped Ge spacer layers in panels b, d, e, and g of Figure 2. To quantify the planar dopant concentration throughout the δ-doped stack, the selected volume reconstructions were used to create discrete point plots of the position of the collected P atoms and the relative 3D iso-concentration surfaces ( Pcm 3 ) (Figure 2e and its 3D animation provided in the Supporting Information). The position of P atoms is distributed randomly within the plane of the δ-layers, as expected from the gas-phase doping process, with average nearest neighbor distances between dopants being in the range of nm (Supporting Information, Figure S1 and S2). The 3D iso-concentration surfaces suggest that the interfaces of the individual P doped δ-layers are atomically abrupt. As evident in Figure 2e, however, there is indication of the formation of filaments made of P atoms in spacers C2 and C3 connecting three adjacent δ-layers (D2 D4). These filaments appear clearly as local increase of dopant density in the 2D concentration maps of P in-plane distribution (Figure 2f and g), and we believe that they originate from fast dopant diffusion along extended epitaxial defects. 19 When probed at the atomiclevel the Ge spacers, while nominally undoped, possess a finite dopant concentration (locally up to cm 3 ). The great advantage of APT is to detect three-dimensional configurations of dopants (vertical filaments of P atoms in our case), not observable by standard secondary ion mass spectroscopy (SIMS), that can provide interlayer electron scattering sites and/or conductive pathways between the doped layers. To investigate the electrical properties of the doped layers, we performed four-probe magnetotransport Hall bars characterization to simultaneously measure the longitudinal ρ xx and transverse ρ xy components of the resistivity tensor as a function of perpendicular magnetic field B in the temperature range of K. Because carriers and dopants are vertically confined within a very narrow region, as shown by APT, our Ge:P 2DEGs represent the ultimate disordered 2DEGs with an average separation of dopants of 1 nm, leading to very short scattering times, τ e ( s). 17 As a consequence electron transport is in the diffusive regime (kt e /ħ 1) throughout the whole range of investigated temperatures. The electrical properties of the samples resulting from a careful analysis of the low-temperature magnetotransport data are listed in Table 1. At low temperatures the longitudinal conductivity σ xx of a disordered 2DEG is expressed as σxx = σd + δσwl + δσee (1) where δσ wl and δσ ee are the quantum corrections arising, respectively, from WL and EEI. 12 Overall, eq 1 reduces at B =0,to τ τ σ = σ + α e kt e xx(0) D G0ln + KeeG0ln τ ( T) ħ φ where G 0 = e 2 /πh, α is a phenomenological prefactor 1 present in the Hikami description of WL in the diffusive regime, 20 and K ee is a parameter related to the amplitude of the angle-averaged screened Coulomb interaction. 21 In disordered 2D systems small energy transfer (Nyquist) electron electron interactions are expected to be the dominant dephasing mechanism with a τ φ T 1 dependence. Thus, both WL and EEI in eq 2 produce a ln(t) correction to the longitudinal conductivity. Differently, no correction from WL is predicted for the measured Hall coefficient R H = ρ xy /B. 22 The logarithmic EEI correction to σ xx in eq 1, however, contributes via tensor inversion, such that 12,23 R H = 0 RH ( 1 + δσ 2 ee σ ) (2) D (3) C

4 Nano s where R 0 H is the classical Hall coefficient related to the classical Hall density n 0 by R 0 H = 1/n 0 e. This expression is valid in the 2 limit of σ xy σ 2 xx, verified for our samples throughout the entire magnetic field range investigated (Figure S4, Supporting Information). We separate the different quantum corrections to the conductivity by applying a self-consistent procedure to eliminate the EEI corrections in the resistivity tensor. This is done by estimating the interaction parameter K ee that yields an optimum fit of the ρ xy (B) curves to the same Hall slope at all T (see Supporting Information for details). 12,23,24 The outcome of this self-consistent procedure is shown in the upper and lower panels of Figure 3, where we plot the magnetic fields and is due to the ln(t) EEI corrections. Sample M shows an additional parabolic increase of ρ xx at high fields (B > 2 T) due to transport occurring in parallel channels. By subtraction of the EEI correction we observe a collapse of ρ xy (B) onto the same curve at high fields, where the WL correction is minimal. A more quantitiative insight into the self-consistent procedure of separating WL and EEI corrections is provided in the upper and lower panels of Figure 4 that focus, Figure 3. Effect of EEI on the measured magnetoresistivity and conductivity tensor. Two sets of graphs are shown: the ρ xy (B) curves before and after removal of EEI (upper panels); the ρ xx (B) curves before and after removal of EEI (lower panels). The EEI correction δσ ee was calculated using the prefactor values K ee listed in Table 1 and values of n 0 and σ D obtained with the self-consistent procedure described in the text. Measurement temperatures for sample S: 0.3 K (purple), 0.5 K (blue), 0.7 K (green), 0.9 K (yellow), 5 K (red); sample M: 0.2 K (purple), 0.35 K (blue), 0.5 K (cyan), 0.7 K (green), 0.9 K (yellow), 5 K (red). measured ρ xx (B) and ρ xy (B), respectively, before and after removal of the EEI correction. The fingerprints of EEI show up in the temperature dependence of the tranverse magnetoresisitivity (upper panels of Figure 3): the slope of the linear ρ xy (B) curves increases as the measurement temperature is decreased, pointing out to an apparent decrease in carrier density. This effect is more pronounced in samples with a lower density and/or conductivities as sample S. 24 Upon subracting the EEI corrections δσ ee calculated by using the values of K ee as listed in Table 1, the ρ xy (B) traces labeled EEI excluded in the upper panels of Figure 3 clearly lie onto the same slope. The value of K ee for the single layer is within 10% of theoretical expectations 21 where Coulomb repulsion is the dominant interaction between electrons. 13,25,26 This discrepancy increases to 20% for the multilayer sample, pointing to an additional interaction mechanism. The longitudinal resistivity traces (lower panels in Figure 3) show, with decreasing T, awl peak around zero magnetic field which increases in magnitude along with a displacement of the curves toward higher resistivity values. This displacement persists at the highest Figure 4. Separation of the WL and EEI corrections allowing for the correct determination of n 0 and σ D. Upper panels show carrier densities obtained by: the linear fit of the original ρ xy (B) curves (black squares); after subtraction of EEI (open squares) and temperatureaverage (n 0, solid line); including EEI interaction to n 0 (red line). Lower panels show the conductivities at zero magnetic field: as measured (black circles); including EEI corrections (red line) or WL corrections (blue line) to σ D ; including both EEI and WL corrections to σ D (dashed green line); after subtraction of EEI and WL corrections and averaged over temperature (σ D, black line). respectively, on the estimate of n 0 and σ D. The carrier densities (black squares) obtained by the simple linear fit of the original ρ xy (B) curves show a logarithmic decrease with decreasing temperature. Upon subtraction of EEI, the collapse of the ρ xy (B) curves in Figure 3 onto one slope is reflected by the carrier densities (open squares) now being scattered with a small standard deviation <1% around the temperature-averaged values n 0 (solid line in Figure 4). In other words, the theoretical values of carrier density including EEI interaction (red line) match the temperature dependence observed in the original experimental data (black squares). The conductivity plots in the lower panels of Figure 4 show the separation of WL and EEI corrections to the conductivity at zero magnetic field. As for the carrier densities, the estimated values of conductivity excluding EEI and WL lie within a small standard deviation of less than 1% around a temperature-averaged value (black line) that we assume to be the best estimate of σ D. Upon including EEI (red line) or WL (blue line) corrections, the conductivity of all samples is similarly decreased at all T. Notably, EEI and WL contributions to the conductivity are almost equal in magnitude for each sample (see Table 1). Overall, inclusion of both EEI and WL corrections to σ D (dashed green line) reproduces reasonably the conductivity measured at zero magnetic field (black circles). Upon closer scrutiny, the conductivity curves in Figure 4 reveal an interesting feature that differentiates the single layer from the multilayer. Unlike the monotonic WL correction seen D

5 Nano s for the single layer sample, sample M shows a weakening of the WL correction for T < 1 K, as evident from the σ D + δσ wl curve (black arrow in Figure 4). To explain this, we must consider the temperature dependence of the phase coherence times, which ultimately produces the σ D + δσ wl curves via eq 2. The upper panels of Figure 5 show the magnetoconductivity traces σ xx (B) of interlayer scattering on the transport properties. In the first approach additional dephasing channels are added empirically, whereas in the second approach the data is fit to the theory of WL in multilayered structures by Novokshonov. 16 The results of both methods were then used to determine the temperature dependence of the relevant relaxation times, as summarized in Figure 6. Figure 5. Analysis of the weak localization feature. Upper panels: magnetoconductivity (colored lines) and corresponding Hikami fits (black lines). Measurement temperatures for sample S: 0.3 K (purple), 0.5 K (purple), 0.7 K (purple), 0.9 K (purple), 2.3 K (purple), 4.2 K (purple), 5 K (purple), 6 K (purple); sample M: 0.2 K (purple), 0.35 K (navy), 0.5 K (blue), 0.7 K (cyan), 0.9 K (green), 5 K (orange). Lower panels: corresponding values of the fitting parameter τ φ (black circles) and fit toa T p dependence (black lines). σ xx (0) and the corresponding fits with the Hikami formula 20 using for each data set τ φ (T) and α as fitting parameters (see Supporting Information for details). In all cases the theoretical fit matched the data with α close to unity, as shown in Table 1. The phase coherence time τ φ (T) in Sample S follows a τ φ T p law throughout the whole temperature range, with p = 0.93 ± 0.01, i.e. close to unity as expected where Nyquist dephasing is the dominant mechanism in 2D. In agreement with the sharp and narrow dopant distribution observed by APT in Figure 2a and c, the detailed magnetotransport characterization of Sample S confirms that electrons are strongly confined in the plane of the δ-layer and that transport is strictly 2D. Sample M, however, shows a temperature dependence that deviates from τ φ T 1 for T < 1 K, reflected in the behavior of σ D + δσ wl observed in Figure 4. It is important to note that fitting the data to the Hikami formula, we treated Sample M as four parallel, strictly noninteracting 2DEGs. Differently from Sample S, the observed τ φ (T) dependence, instead, reveals an additional dephasing mechanism that becomes relevant at low temperature. To understand this additional dephasing mechanism we consider the realistic 3D atomic distribution of P dopants measured by APT. Due to the narrow interlayer spacing, which is of the same order of magnitude as the mean free path of carriers within one layer, the donor atoms previously shown as filaments in the interlayer regions (see Figure 2b and e) act as scattering centers, providing additional scattering paths for carriers between the layers in the δ-layer stack. Two different methods were used to calculate the effect Figure 6. Estimation of the relevant relaxation times for Sample M with the two methods described in the text: phase coherence time τ φ true obtained with the Hikami fit disregarding interlayer transitions (black circles), where the black line is a guide for the eyes; interlayer transition time τ 1 estimated with method one (white diamonds) or method 2 (gray diamonds), with shaded area as a guide for the eyes; phase coherence time including interlayer transitions with method 2 (blue squares); phase coherence time of sample S used as a reference for method one (red line). In the first method we empirically introduce the interlayer scattering time τ 1 which measures the rate at which electron are scattered between adjacent δ-layers. Given the short τ e in the strongly confined 2DEGs, the conductivity within each layer of sample M is expected to be much higher than the conductivity between layers, despite the limited presence of donors in the nominally undoped Ge spacers. As a consequence, we assume τ 1 τ e, so that quantum corrections in the multilayers are quasi-2d rather than 3D in nature. 15 If τ 1 is comparable to τ φ, however, electron scattering between adjacent layers will reduce the probability of forming closed loops available for the interference of electron waves propagating in opposite directions within the same layer. Consequently, the magnitude of the WL correction in the multilayer sample is reduced, as if there was an additional channel contributing to dephasing. Thus, the true relaxation time τ φ true obtained with the Hikami fit (black circles) comprises both the intralayer phase coherence time of electrons τ φ and the additional interlayer scattering time τ 1. We express this empirically by assuming true 1/( τφ ) = 1/ τφ + 1/ τ1 (4) We assume τ φ as obtained in sample S to be a good estimate for the intralayer phase coherence time for sample M, owing to the nearly equal electron density in both samples ( cm 2 in S and cm 2 in each of the sample M layers). 27 Following eq 4, we estimate τ 1 1 by simply subtracting the relaxation rates for Sample S from those for Sample M, obtained both from the Hikami fit shown previously in Figure 5. The interlayer transition time τ 1 (white diamonds) has a E

6 Nano s weak temperature dependence with values in the s range. These values are 1000 τ e, confirming our assumption that extended 3D states are not formed in the multiple layers, which maintain their 2D nature in transport even at thin spacer thickness used. At very low temperatures (T < 1 K), τ φ (red line) becomes increasingly long and when it is comparable with τ 1 the contribution of the interlayer scattering rate to the total relaxation rate becomes significant in eq 4. Therefore, deviations from the T 1 dependence are observed. In the second method we have fitted the magnetoconductivity curves for sample M with the WL theory with insulating boundary conditions for quasi-2d disordered multilayer 16 in which the main contribution to the probability of transitions between layers is from their elastic scattering in the random field of impurities. Unlike in the first method discussed above, the fit is now performed using both τ φ (T) and τ 1 (T) as fitting parameters. As shown in Figure 6, the obtained values of τ 1 (gray diamonds) confirm the trend observed within the previous empirical analysis, and the fitted phase coherence time τ φ (blue squares) shows the expected τ φ T p dependence throughout the whole temperature range with p = 0.85 ± The fitted τ φ values are close to those of Sample S, therefore validating the assumption in method one to use the intralayer dephasing times of Sample S as a reference for the corresponding ones in Sample M. While the previous APT analysis demonstrated from a structural point of view nearly identical δ-doped layers along the growth direction, the outcome of the magnetotransport analysis confirms the effectiveness of the atomic layer doping technique to stack 2DEGs in Ge nearly identical from an electrical point of view. Because of the ultrahigh donor P concentration, the strong variation of in-plane P-distribution from layer to layer shown by APT (Figure 2g) does not affect the electrical properties within each layer. Consequently, this allows for the use of a simple empirical method to capture, although in a qualitative way, the presence of additional dephasing channels. Ultimately, this will turn useful in future design of nanoscale 3D epitaxial circuits. To conclude our analysis, we highlight that at low temperature (300 mk) the phase coherence length l φ = (Dτ φ ) 1/2 for electrons in the δ-doped layers increases to values of 250 nm (S) and 220 nm (M), respectively. Despite the close proximity of the stacked layers the coherence length for electrons in one layer is only marginally reduced by the presence of the additional stacked layers. For studies of electron transport in multilayer systems, it implies that there is still plenty of room to fabricate two-dimensional electron systems in Ge:P with even higher carrier densities by increasing the number of stacked layers while maintaining a total thickness of the doped region t < l φ. For fabrication of atomic scale devices in Ge by STM lithography, where feature sizes as small as 5 nm have already been demonstrated, 9 phase coherence lengths in the nm range open up the possibility of creating complex 3D epitaxial circuits based on phase coherence. In summary, we have combined APT and magnetotransport measurements to demonstrate that stacked 2DEGs obtained by phosphorus atomic layer doping of germanium preserve their individuality from a structural and electrical point of view. Only at T < 1 K does interlayer scattering between carriers in adjacent δ-layers contribute to the total relaxation rate, possibly due to the occasional presence of P filaments in nominally undoped spacers. However, because of the strong vertical confinement, and large amount of disorder within the δ-layers, the interlayer scattering time is always much longer than the F scattering time within the dopant planes: while electrons are interacting due to their proximity, transport is still 2D in nature. Future transport studies with even thinner Ge spacers will help us understand the limits to which individuality and coherence can be preserved in these systems and guide us toward the design of 3D epitaxial circuits at the atomic scale. In addition, the full 3D information of dopant positions available by atom probe tomography will be crucial to understand how the random positions of P atoms within patterned 2DEGs affect the transport properties of Ge:P atomic-scale devices. ASSOCIATED CONTENT *S Supporting Information Extra information related to atom probe tomography, and magnetotransport measurement and data analysis. Four extra figures related to P distribution and correspondent statistical analysis within the plane of the δ-layers (Figure S1), isoconcentration surfaces with increasing P concentration (Figure S2), the temperature independence of transverse magnetoconductivity (Figure S3), and magnetic field dependence of the ratio between transverse and longitudinal conductivity (Figure S4). One 3D animation of dopant distribution from Figure 2e. This material is available free of charge via the Internet at AUTHOR INFORMATION Corresponding Author * giordano.scappucci@unsw.edu.au. Notes The authors declare no competing financial interest. ACKNOWLEDGMENTS This research has been supported by the University of New South Wales (UNSW) and the Australian Research Council Centre of Excellence for Quantum Computation and Communication Technology (project number CE ). G.S. acknowledges support from UNSW under the GOLDSTAR Award 2012 scheme. G.C. is thankful to UNSW for a Visiting Professor Fellowship. A.R.H. acknowledges support from the ARC DP scheme, APF, and DORA awards. M.Y.S. acknowledges an Australian Government Federation Fellowship. D.L.J., M.R.B., R.F.R., and B.P.G. acknowledge support from the Defense Advanced Research Project Agency, Space and Naval Warfare Center, San Diego, and the Emerging Technology Fund of the State of Texas under contract N C The authors are grateful to Dr. John N. Randall of Zyvex Laboratories for helpful discussions. REFERENCES (1) Pillarisetty, R. Nature 2011, 479 (7373), 324. (2) Vrijen, R.; Yablonovitch, E.; Wang, K.; Jiang, H. W.; Balandin, A.; Roychowdhury, V.; Mor, T.; DiVincenzo, D. 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