Use of quantum effects as potential qualifying metrics for “quantum grade silicon”

Across solid state quantum information, materials deficiencies limit performance through enhanced relaxation, 10 charge defect motion or isotopic spin noise. While classical measurements of device performance provide cur11 sory guidance, specific qualifying metrics and measurements applicable to quantum devices are needed. For 12 quantum applications, new materials metrics, e.g., enrichment, are needed, while existing, classical metrics 13 like mobility might be relaxed compared to conventional electronics. In this work, we examine locally grown 14 silicon superior in enrichment, but inferior in chemical purity compared to commercial-silicon, as part of 15 an effort to underpin the materials standards needed for quantum grade silicon and establish a standard 16 approach for intercomparison of these materials. We use a custom, mass-selected ion beam deposition tech17 nique, which has produced isotopic enrichment levels up to 99.99998 % Si, to isotopically enrich Si, but 18 with chemical purity > 99.97% due the MBE techniques used. From this epitaxial silicon, we fabricate 19 top-gated Hall bar devices simultaneously on the Si and on the adjacent natural abundance Si substrate 20 for intercomparison. Using standard-methods, we measure maximum mobilities of ≈ (1740± 2) cm/(V · s) 21 at an electron density of (2.7× 10 ± 3× 10) cm−2 and ≈ (6040± 3) cm/(V · s) at an electron density of 22 (1.2× 10 ± 5× 10) cm−2 at T = 1.9 K for devices fabricated on Si and Si, respectively. For magnetic 23 fields B > 2 T, both devices demonstrate well developed Shubnikov-de Haas (SdH) oscillations in the longitu24 dinal magnetoresistance. This provides transport characteristics of isotopically enriched Si, and will serve 25 as a benchmark for classical transport of Si at its current state, and low temperature, epitaxially grown Si 26 for quantum devices more generally. 27

vide diagnostics that will indicate the likely performance of qubits early in a fabrication stream.This paper presents devices, methods and results for a comparative study of magnetotransport properties between 1) high isotopic enrichment, low chemical purity and 2) high chemical purity, natural abundance (low isotopic enrichement) silicon.This characterization sets the stage for determining whether coherence properties in quantum dot devices correlate with the trends in these simpler measurements, since the benefit of enrichment on coherence may outpace the liability of some additional contaminants.In a detailed theoretical study, Witzel et al., 13 illustrate that the coherence of a spin qubit can, in principle, be increased by an order of magnitude for every order of magnitude increase in the isotopic enrichment of 28 Si in the qubit's Si environment.A comprehensive experimental investigation of this prediction, however, is hindered due to the discreteness of the available isotopic enrichment levels.Among the four different enrichment levels have been reported 10,14-16 only 99.98% 28 Si 14 and 99.995% 28 Si 10 have been utilized for quantum electronic device fabrication. 11,17,18Moreover, contemporary methods for producing isotopically enriched 28 Si material are based on chemical vapor deposition (CVD) techniques and are not compatible with qubit architectures requiring low temperature processing, e.g.STM fabricated single dopant atom qubits. 19In contrast, the method used for producing 28 Si reported here is compatible with all the contemporary qubit architectures, and represents molecular beam epitaxy (MBE) grown Si more generally.While the coherence of a spin qubit is predicted to improve at higher isotopic enrichment levels, 13 how other material properties will limit the expected enhancement of qubit coherence is unclear.To the best of our knowledge, no study yet has attempted to correlate macroscopic electrical characteristics to the performance of quantum devices.Yet such a study will be an essential component for defining metrics for "quantum grade" silicon within the three main goals identified earlier.
Starting from natural abundance SiH 4 gas, we have developed a method to grow isotopically purified silicon reaching isotopic enrichments up to 99.99998 % 28 Si. 20,21 method provides the unique advantage of targeting a desired enrichment level anywhere from natural abundance to the highest possible enrichment. 22As a first step towards correlating macroscopic electrical characteristics with the performance of quantum devices, we report here on characterization of gated Hall bar devices fabricated on isotopically enriched 28 Si, and control devices on the same natural abundance Si ( nat Si) substrate but outside the isotopically enriched 28 Si spot using macroscopic manifestations of quantum effects such as Shubnikov-de Haas (SdH) effect and weak-localization effect.We compare the devices fabricated on nat Si (floatzone grown) and 28 Si (MBE grown) during the same fabrication process, eliminating possible differences due to imperfect fabrication conditions.We present results of 28 Si devices to serve as a benchmark for MBE grown iso- layer.An optical micrograph of the gated multi-terminal Hall bar device is shown in Fig. 1(c).
The isotopic enrichment of the 28 Si epilayers is measured by using Secondary Ion Mass Spectrometry (SIMS).In Fig. 1(d), the SIMS-derived isotopic ratio of 29 Si/ 28 Si is shown as a function of depth at several locations near the fabricated Hall bar device.For the device reported here, the level of isotopic enrichment measured at locations 1, 2, and 3 corresponds to ≈ 99.976 %, ≈ 99.980 %, and ≈ 99.993 % 28 Si, respectively.Figure 1(d) also reveals the thickness non-uniformity of the deposited 28 Si epilayer, i.e., the thickness of the 28 Si epilayer at location 3 is greater than that of locations 1 and 2.Moreover, separate SIMS measurements on these isotopically enriched 28 Si epilayers reveals that the films contain adventitious chemical impurities, namely C, N, O, with approximate atomic concentrations of 2 × 10 19 cm −3 , 3 × 10 17 cm −3 , and 3 × 10 18 cm −3 .However, the atomic concentrations of these chemical impurities on the handle wafer were below the SIMS detection limit (≤ 10 16 cm −3 ).We believe that these chemical im-  2(a) could be due to several reasons, e.g., magnetic impurities or inhomogeneity of the magnetic field. 28,29The SIMS of a similar 28 Si epilayer found no measurable magnetic impurities.The Hall resistance shows non-idealities particularly in the nat Si device [Fig.2(b)] where R xy is non-monotonic.These non-idealities could be due to scattering between discrete degenerate states at the tails due to level broadening. 30,31ver, a detailed discussion of the asymmetry of R xx and the flatness of the Hall plateaus is outside the scope of this article.We also see a lifting of the four-fold degeneracy at B > 5 T for nat Si, which is likely due to the spin degree of freedom, but, at this time we are unable to determine whether this is due to spin or valley degree of freedom, due to limitations in the experimental setup.
Near zero magnetic field, both devices demonstrate a peak in the sample resistance, see Fig. 2.This increase in resistance near zero magnetic field is known as weak localization (WL).Weak localization is a quantum mechanical phenomenon that can be observed in twodimensional (2D) electron systems at low temperatures where the phase coherence length (l φ ) is greater than the mean free path (l) 32,33 .Relative to the zero field resistance, the weak-localization is larger for the device fabricated on isotopically enriched 28 Si.
To further investigate the WL behavior of these de- where Ψ is the digamma function, l is the mean free path, l Φ is the phase coherence length, and α is a constant close to unity.In Fig. 3, the solid lines are the fits to experimental data (symbols) using the HLN equation.For these fits, we use the calculated values of l TABLE II.Parameters extracted from the least-squares-fits of Eq. 2 to the data in Fig. 3 inset.
Device a b c Adjusted (10 10 s −1 ) (10 10 K −1 s −1 ) (10 10 K −2 s −1 ) R-square  q ( 28 Si)  This is the author's peer reviewed, accepted manuscript.However, the online version of record will be different from this version once it has been copyedited and typeset.

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FIG. 1.(a) A schematic illustrating the device layout of a given sample.Reduced coverage of the 28 Si spot allows to fabricate devices on 28 Si and nat Si simultaneously.(b) Schematic representation of the gated Hall bar device fabricated on 28 Si is shown.(c) An optical micrograph of a gated multi-terminal Hall bar device fabricated on 28 Si is shown.(d)The isotopic ratios of 29 Si/ 28 Si at positions 1 ( ), 2 ( ), and 3 ( ) in (c) are shown.The shift in the rising edge at different positions corresponds to the thickness variation in the deposited 28 Si film.Measured 29 Si isotopic ratios at locations 1, 2, and 3 are (149 ± 18) × 10 −6 mol/mol, (128 ± 14) × 10 −6 mol/mol, and (45 ± 2) × 10 −6 mol/mol, respectively.
× 10 mm), see Fig. 1(a).Due to the re-150 duced coverage of the 28 Si spot, devices on isotopically 151 enriched and natural abundance Si can be fabricated on 152 the same Si chip [see Fig. 1(a)] at the same time.This 153 eliminates the effect of imperfections in the fabrication 154 process (e.g., oxide growth) when comparing the electri-155 cal properties of the devices.A schematic cross section 156 of a device fabricated on 28 Si spot is shown in Fig. 1(b).157 The structure of the devices fabricated on nat Si, i.e. out-158 side the 28 Si spot, is identical except without the 28 Si This is the author's peer reviewed, accepted manuscript.However, the online version of record will be different from this version once it has been copyedited and typeset.PLEASE CITE THIS ARTICLE AS DOI:10.1063/1.5128098

FIG. 2 .
FIG. 2. The magnetoresistance Rxx (right-axis) and the Hall resistance Rxy (left-axis) measured for the devices fabricated on (a) isotopically enriched 28 Si epi-layer, and (b) natural Si substrate are shown.For both devices, the corresponding filling factors (ν) are shown at the minima of Shubnikov-de Hass oscillations.In contrast to the device on isotopically enriched 28 Si epi-layer, the device on nat Si demonstrates spinsplitting for B > 3 T.Both devices are fabricated on the same Si chip, see main text for more information.The relative uncertainty associated with Rxx and Rxy is typically less than 0.1 % and is mostly due to the uncertainty of the measured current.
vices, we plot the change in conductivity ∆σ xx as a function of magnetic field B applied perpendicular to the 2D electron system [see Fig. 3].The change in conductivity due to WL ∆σ xx = σ xx (B) − σ xx (B = 0), where σ xx = ρ xx /(ρ 2 xx + ρ 2 xy ).For non-zero B, the change in conductivity due to WL in a 2D electron system can be modeled by the Hikami-Larkin-Nagaoka (HLN) equation, 34

Fig.3 inset for devices fabricated on isotopically enriched 269 28
FIG.3.The change in conductivity (∆σxx) vs external magnetic field (B) for devices fabricated on 28 Si ( ) and nat Si ( ) measured at 3 K.Solid lines are the least-square-fits to HLN equation (Eq.1).Estimated uncertainty for ∆σxx is < 0.3 %.Inset: The inelastic scattering rates (1/τ φ ) for 28 Si and nat Si vs the measurement temperature are shown.Here the solid lines are the least-squares-fit to a quadratic equation, see main text details.Error bars in the inset represent the fit uncertainty associated with the values extracted for 1/τ φ at each temperature.
FIG. 4. (a) The background subtracted (see text) Rxx, i.e., ∆Rxx, vs the inverse of the external magnetic field (1/B) for the 28 Si device is shown.(b) A "Dingle plot" of ln(A SdH /X(T )) versus 1/B.Error bars represent the uncertainty associated with extracting A SdH from ∆Rxx vs. 1/B plot.(c) The single particle lifetimes, τq, extracted from the Dingle plots and transport lifetimes, τ , at different temperatures for devices on 28 Si and nat Si.Error bars represent the uncertainty associated with calculating the values of τq (τ ) using the Dingle plots (charge carrier mobilities) at each temperature.

TABLE I .
Macroscopic materials and electrical properties of natural abundance, nat Si, and isotopically enriched, 28 Si, silicon.
Considering SIMS measured chemical impurity con-350 centrations of C, N, and O and assuming these impurities 351 acting as isolated scatters, for 28 Si where l ≈ 33 nm, we l.371tected in the28Si epilayers act as the impurity scatters in 372 the devices fabricated on 28 Si.However, higher levels of 373 adventitious chemical impurities detected in the 28 Si epi-374 layers are too high to be considered as isolated scattering 375 centers, since the nearest neighbor distance is consider-376 ably shorter than the scattering lengths extracted from 377 the transport data.Further, for these impurity levels, 378 the dipolar interactions between randomly distributed 379 electron spins associated with impurities and the central 380 spin of a potential qubit is considered to be the domi-381 nant decoherence mechanism at high enrichments. 13For 382 the worst case analysis, if all of the N and O chemical 383 impurities are considered as randomly distributed single 384 electron spins, the influence of these dipolar interactions 385 on the central spin could result in qubit coherence times 386 poorer than high purity natural abundance Si.However, 387 we are confident that the recent and planned improve-398 devices reported here will serve as a benchmark for find-399 ing the correlations between macroscopic properties and 400 the performance of future nanoscale devices, e.g.quan-401 tum dots, and lead to identifying qualifying metrics for 402 "quantum grade" silicon.