Across solid state quantum information, material deficiencies limit performance through enhanced relaxation, charge defect motion, or isotopic spin noise. While classical measurements of device performance provide cursory guidance, specific qualifying metrics and measurements applicable to quantum devices are needed. For quantum applications, new material metrics, e.g., enrichment, are needed, while existing classical metrics such as mobility might be relaxed compared to conventional electronics. In this work, we examine locally grown silicon that is superior in enrichment, but inferior in chemical purity compared to commercial-silicon, as part of an effort to underpin the material standards needed for quantum grade silicon and establish a standard approach for the intercomparison of these materials. We use a custom, mass-selected ion beam deposition technique, which has produced isotopic enrichment levels up to 99.999 98% ^{28}Si, to isotopically enrich ^{28}Si, but with chemical purity >99.97% due to the molecular beam epitaxy techniques used. From this epitaxial silicon, we fabricate top-gated Hall bar devices simultaneously on ^{28}Si and on the adjacent natural abundance Si substrate for intercomparison. Using standard-methods, we measure maximum mobilities of ≈(1740 ± 2) cm^{2}/(V s) at an electron density of (2.7 × 10^{12} ± 3 × 10^{8}) cm^{−2} and ≈(6040 ± 3) cm^{2}/(V s) at an electron density of (1.2 × 10^{12} ± 5 × 10^{8}) cm^{−2} at *T* = 1.9 K for devices fabricated on ^{28}Si and ^{nat}Si, respectively. For magnetic fields *B* > 2 T, both devices demonstrate well developed Shubnikov-de Haas oscillations in the longitudinal magnetoresistance. This provides the transport characteristics of isotopically enriched ^{28}Si and will serve as a benchmark for the classical transport of ^{28}Si at its current state and low temperature, epitaxially grown Si for quantum devices more generally.

Conventional electronics have been industrialized for decades; consequently, precise metrics based on macroscopic properties, such as chemical purity, charge carrier mobility, and defect density, are established for qualifying a material, e.g., silicon, for conventional electronics. While silicon has been the workhorse of conventional electronics, it is also becoming a promising host for spin based quantum information processing devices.^{1,2} Specifically, spin qubits have already shown promising advancements with regard to long coherence times,^{3,4} manipulation with high gate fidelity,^{3,4} and scalability.^{5,6}

Even though silicon has improved tremendously over the decades to meet demands of today’s state-of-the-art transistors, this excellent material is still not sufficient to support quantum information. For example, in spin-based quantum information systems, the presence of the ^{29}Si isotope in natural abundance silicon reduces coherence times due to the nonzero nuclear spin of *I* = 1/2. Nuclei with nonzero spin in the host lattice act as a source of decoherence for spin based qubits,^{7} as they interact with the electron spin through hyperfine interactions.^{8,9} However, by placing a spin qubit in an isotopically enriched 99.995% ^{28}Si environment,^{10} the development of silicon based quantum devices has gained considerable momentum, with reports of exceptionally long quantum coherence times.^{11,12}

The need for some level of enrichment provides an example of how “semiconductor grade” silicon quality may be necessary but is not sufficient to meet the needs of quantum. Furthermore, the metrics for conventional silicon may not always be relevant for quantum, e.g., the ease of carrier motion as quantified by mobility may not be directly relevant to quantum device performance where confinement and coherence in the *absence* of motion are critical. Additionally, as we establish properties and their numerical thresholds that are sufficient for quantum, relatively simple qualifying metrics that act as general proxies for properties more challenging to measure are invaluable. However, it may be noted that mobility in and of itself is not important, but it could be a good proxy for estimating spin-qubit relaxation or coherence.

As part of a larger program to identify and quantify “quantum grade” silicon, we are identifying (1) properties beyond those considered for semiconductor grade silicon critical to quantum; (2) the relevance and priority of properties currently considered critical for semiconductors; and (3) standard methods that may be used for new properties or provide a general indicator for challenging properties, e.g., coherence time, as three main goals that are paramount for the development of metrics for “quantum grade” silicon. This work is part of a broader effort to find ways besides making and measuring qubits to provide 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 enrichment) 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} illustrated 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 that 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.^{11,17,18} Moreover, contemporary methods for producing isotopically enriched ^{28}Si materials 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.^{19} In 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 with 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 enrichment up to 99.999 98% ^{28}Si.^{20,21} This method provides the unique advantage of targeting a desired enrichment level anywhere from natural abundance to the highest possible enrichment.^{22} As a first step toward correlating macroscopic electrical characteristics with the performance of quantum devices, we report here on the 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 the Shubnikov-de Haas (SdH) effect and weak-localization effect. We compare the devices fabricated on ^{nat}Si (float-zone grown) and ^{28}Si (MBE grown) during the same fabrication process, eliminating possible differences due to imperfect fabrication conditions. We present the results of ^{28}Si devices to serve as a benchmark for MBE grown isotopically enriched ^{28}Si and a basis for comparing macroscopic electrical characteristics within silicon quantum electronics.

Starting with 99.999% pure, commercially available, natural isotopic abundance SiH_{4} gas, isotopically enriched ^{28}Si is grown using a hyperthermal energy ion beam deposition system.^{20} Gated Hall bar devices are fabricated on isotopically enriched ^{28}Si epilayers in order to electrically characterize the material. Typically, the isotopically purified ^{28}Si spot is ≈2 mm^{2}–3 mm^{2} in area and covers only a small fraction of the starting float-zone grown, natural abundance, intrinsic Si substrate (4 × 10) mm^{2}; see Fig. 1(a). Due to the reduced coverage of the ^{28}Si spot, devices on isotopically enriched and natural abundance Si can be fabricated on the same Si chip [see Fig. 1(a)] at the same time. This eliminates the effect of imperfections on the fabrication process (e.g., oxide growth) when comparing the electrical properties of the devices. A schematic cross section of a device fabricated on a ^{28}Si spot is shown in Fig. 1(b). The structure of the devices fabricated on ^{nat}Si, i.e., outside the ^{28}Si spot, is identical except without the ^{28}Si layer. An optical micrograph of the gated multiterminal Hall bar device is shown in Fig. 1(c).

The isotopic enrichment of the ^{28}Si epilayers is measured by 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 nonuniformity of the deposited ^{28}Si epilayer, i.e., the thickness of the ^{28}Si epilayer at location 3 is greater than those of locations 1 and 2. Moreover, separate SIMS measurements on these isotopically enriched ^{28}Si epilayers reveal that the films contain adventitious chemical impurities, namely, C, N, and 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 impurities were being introduced by the ion beam as a result of the non-UHV compatible ionization source that was used to create the ion beams during the ^{28}Si deposition and has since been upgraded.

The magnetoresistance (*R*_{xx}) and the Hall resistance (*R*_{xy}) at 1.9 K for isotopically enriched ^{28}Si and natural abundance Si are shown in Fig. 2(a) and Fig. 2(b), respectively. Using low field magnetotransport data, we find that maximum mobilities at *T* = 1.9 K for ^{28}Si and ^{nat}Si are, respectively, $\mu \u2009\u200928Si\u2009=\u2009(1740\xb12)\u2009cm2/(V\u2009s)$ at an electron density *n* of (2.7 × 10^{12} ± 3 × 10^{8}) cm^{−2} and *μ*_{nat}Si = (6040 ± 3) cm^{2}/(V s) at an electron density of (1.2 × 10^{12} ± 5 × 10^{8}) cm^{−2}. Charge carrier mobilities for these devices are within the typical range of mobilities for Si-MOS (Metal Oxide Semiconductor) devices fabricated using non-MBE (e.g., CVD) growth techniques,^{23,24} the maximum mobility for a Si-MOS device to date being >4 × 10^{4} cm^{2}/(V s).^{25} In contrast, mobilities reported for Si-MOS devices fabricated on MBE grown Si range from 900 cm^{2}/(V s) to 1250 cm^{2}/(V s).^{26,27}

In order to estimate the percolation electron density *n*_{p}, we extrapolate the electron density as a function of gate voltage (as determined from Hall measurements) back to the threshold voltage (as determined from the channel current *I*_{sd} vs *V*_{g}), i.e., *n*_{p} = *n*_{e}(*V*_{th}). Using this method, we find percolation densities of (2.3 ± 2) × 10^{11} cm^{−2} for ^{nat}Si and (4.2 ± 2) × 10^{11} cm^{−2} for ^{28}Si. While the relative uncertainties are large due to the extrapolation, we think that the ≈2× larger value for ^{28}Si is significant. A summary of these macroscopic materials and electrical properties for the on-chip ^{nat}Si and ^{28}Si is provided in Table I.

. | . | Material . | |
---|---|---|---|

Property . | ^{nat}Si
. | ^{28}Si
. | |

Avg. ^{28}Si concentration | 92.23% | 99.983% | |

Impurities (cm^{−3}) | C | ≤ 10^{16} | 2 × 10^{19} |

N | 3 × 10^{17} | ||

O | 3 × 10^{18} | ||

Max. mobility μ (cm^{2}/(V s)) | (6040 ± 3) | (1740 ± 2) | |

Percolation density n_{p} (10^{11} cm^{−2}) | (2.3 ± 2) | (4.2 ± 2) |

. | . | Material . | |
---|---|---|---|

Property . | ^{nat}Si
. | ^{28}Si
. | |

Avg. ^{28}Si concentration | 92.23% | 99.983% | |

Impurities (cm^{−3}) | C | ≤ 10^{16} | 2 × 10^{19} |

N | 3 × 10^{17} | ||

O | 3 × 10^{18} | ||

Max. mobility μ (cm^{2}/(V s)) | (6040 ± 3) | (1740 ± 2) | |

Percolation density n_{p} (10^{11} cm^{−2}) | (2.3 ± 2) | (4.2 ± 2) |

Both devices show well developed SdH oscillations in *R*_{xx} with accompanying plateaus in *R*_{xy}. The slight asymmetry in *R*_{xx} in Fig. 2(a) could be due to several reasons, e.g., magnetic impurities or inhomogeneity of the magnetic field.^{28,29} SIMS of a similar ^{28}Si epilayer found no measurable magnetic impurities. The Hall resistance shows nonidealities particularly in the ^{nat}Si device [Fig. 2(b)] where *R*_{xy} is nonmonotonic. These nonidealities could be due to scattering between discrete degenerate states at the tails due to level broadening.^{30,31} However, 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 the spin or valley degree of freedom, due to the 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 two-dimensional (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 devices, we plot the change in conductivity Δ*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 Δ*σ*_{xx} = *σ*_{xx}(*B*) − *σ*_{xx}(*B* = 0), where $\sigma xx=\rho xx/(\rho xx2+\rho xy2)$. For nonzero *B*, the change in conductivity due to WL in a 2D electron system can be modeled by the Hikami-Larkin-Nagaoka (HLN) equation,^{34}

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* using the relation $l=2D\tau $. Here, *D* is the diffusion coefficient defined as $D=vF2\tau /2$, where the Fermi velocity $vF$ = *ℏk*_{F}/*m*^{*} and *τ* is the elastic scattering time, also known as the transport lifetime, defined as *τ* = *μm*^{*}/*e*. The effective mass *m*^{*} is defined as *m*^{*}/*m*_{0} = 0.19, where *m*_{0} is the rest mass of an electron.^{35,36} The Fermi wavelength *k*_{F} can be calculated for a 2D electron system in Si as $kF=(4\pi n2D/gsgv)1/2$, where *n*_{2D}, *g*_{s}, and $gv$ are the charge carrier density, spin degeneracy, and valley degeneracy, respectively. We leave *α* and *l*_{ϕ} as the free fitting parameters, constraining the value of *α* to be close to unity. From the fit-extracted values of *l*_{ϕ}, we calculate 1/*τ*_{ϕ}, where inelastic scattering time $\tau \varphi =l\varphi 2/D$. The fit derived values of 1/*τ*_{ϕ} as a function of *T* are plotted in the inset of Fig. 3 for devices fabricated on isotopically enriched ^{28}Si and natural abundance Si, respectively. The solid lines in the inset of Fig. 3 are the least-squares-fit to the data using the equation

The linear in the *T* term captures the scattering from impurities, and the quadratic in the *T* term is related to the electron-electron scattering.^{37} Table II shows the parameters extracted from the least-squares-fit to the data, the fit uncertainties for both devices, and the adjusted R-square. For the natural abundance Si, the best fit is achieved when the linear term is set to zero, i.e., *b* = 0. Consequently, for natural abundance Si, the dominant scattering mechanism appears to be the electron-electron (long-range) scattering. In contrast, for isotopically enriched ^{28}Si, the best fit is achieved with a significant linear in the *T* term. This large linear term implies that impurity (short-range) scattering is a significant contribution in ^{28}Si. The temperature independent parameter *a* is similar (within the uncertainties) for both the devices indicating that the processes (e.g., interface roughness) contributing to *a* are likely the same.

. | . | . | . | Adjusted . |
---|---|---|---|---|

Device . | a (10^{10} s^{−1})
. | b (10^{10} K^{−1} s^{−1})
. | c (10^{10} K^{−2} s^{−1})
. | R-square . |

^{28}Si | 6.6 ± 2 | 3.1 ± 0.7 | 0.60 ± 0.08 | 0.997 |

^{nat}Si | 5.1 ± 0.4 | … | 1.5 ± 0.1 | 0.989 |

. | . | . | . | Adjusted . |
---|---|---|---|---|

Device . | a (10^{10} s^{−1})
. | b (10^{10} K^{−1} s^{−1})
. | c (10^{10} K^{−2} s^{−1})
. | R-square . |

^{28}Si | 6.6 ± 2 | 3.1 ± 0.7 | 0.60 ± 0.08 | 0.997 |

^{nat}Si | 5.1 ± 0.4 | … | 1.5 ± 0.1 | 0.989 |

Line shape analysis of the SdH oscillations as a function of temperature is also used to investigate the underlying scattering mechanisms in 2D electron systems. The amplitude of the SdH oscillations can be written as *A*_{SdH} = *X*(*T*)*R*_{0} exp(−*π*/*ω*_{c}*τ*_{q}),^{38,39} where *R*_{0} is the zero field resistance, *X*(*T*) = (2*π*^{2}*k*_{B}*T*/*ℏω*_{c})/sinh(2*π*^{2}*k*_{B}*T*/*ℏω*_{c}) is the temperature damping factor, and *ω*_{c} = *eB*/*m*^{*} is the cyclotron frequency. Here, *k*_{B} is Boltzmann’s constant and *τ*_{q} is the single particle (quantum) lifetime.^{38–40} To extract the amplitude of SdH oscillations, we first subtract a slow varying background from *R*_{xx}^{41} to isolate the oscillatory part of *R*_{xx}. The *R*_{xx} after background subtraction (Δ*R*_{xx}) is plotted against 1/*B* in Fig. 4(a). Then, we extract the amplitude *A*_{SdH} as schematically defined in Fig. 4(a) at each minimum of Δ*R*_{xx} and calculate ln(*A*_{SdH}/*X*(*T*)). Figure 4(b) is a plot of ln(*A*_{SdH}/*X*(*T*)) vs 1/*B*, also known as the “Dingle plot”^{38,39} for the device fabricated on ^{28}Si measured at *T* = 3 K. The approximately linear dependence of ln(*A*_{SdH}/*X*(*T*)) on 1/*B* [see Fig. 4(b)] indicates a magnetic field independent quantum lifetime, *τ*_{q}. In Fig. 4(c), we plot the quantum lifetimes, *τ*_{q}, for devices fabricated on ^{28}Si and ^{nat}Si extracted from a linear least-squares-fit to Dingle plots at each temperature. The calculated values of the transport lifetimes, where *τ* = *μm*^{*}/*e*, using the magnetotransport measurement at low magnetic fields for both devices, are also plotted in Fig. 4(c).

For the device fabricated on ^{28}Si, the ratio of *τ*/*τ*_{q} ≈ 1, and for the device on ^{nat}Si, the ratio of *τ*/*τ*_{q} ≈ 1.4. The transport lifetime *τ* is primarily affected by the large angle scattering events that cause a large momentum change, whereas *τ*_{q} is affected by all the scattering events.^{42} When the background impurities dominate, the scattering ratio *τ*/*τ*_{q} is less than or equal to 10, whereas it is ≈1 when the scattering is dominated by short-range isotropic scattering,^{42} e.g., surface roughness scattering.^{43} The thickness of the gate oxide for the devices reported here is ≈60 nm. We therefore neglect the scattering due to remote interface roughness (i.e., the interface between the gate oxide and the gate metal) as a dominant scattering mechanism for these devices.^{44} Therefore, the ratio *τ*/*τ*_{q} implies that the charge carrier mobility is limited by the background impurity scattering. Furthermore, the charge carrier mobility of the device on isotopically enriched ^{28}Si may also be limited by the interface roughness scattering.

The analysis of the weak-localization, SdH oscillations, and low-field magnetotransport data indicates the shortest scattering length scale to be the elastic (transport) scattering length *l* calculated as ≈33 nm and ≈71 nm for ^{28}Si and ^{nat}Si, respectively. Capacitance voltage (CV) measurements of MOS capacitors fabricated on natural abundance silicon (data not shown) with gate oxides grown using similar conditions to the devices reported here reveal a fixed charge density of approximately 3 × 10^{10} cm^{−2} corresponding to the nearest neighbor distance of ≈58 nm. This nearest neighbor distance is in close agreement with the transport scattering length *l*. Considering SIMS measured chemical impurity concentrations of C, N, and O and assuming that these impurities act as isolated scatters, for ^{28}Si where *l* ≈ 33 nm, we estimate the fraction of C, N, and O impurities contributing to scattering to be ≈0.2%, ≈9.3%, and ≈1.0%, respectively.

In conclusion, we have reported on the first low temperature electrical measurements of MBE grown isotopically enriched ^{28}Si. For this report, we fabricated and characterized the low temperature magnetotransport of gated Hall bar devices fabricated on highly enriched ^{28}Si. In comparison to control devices fabricated on float-zone grown, intrinsic, natural abundance Si on the same substrate, the charge carrier mobility on isotopically enriched ^{28}Si is approximately a factor of 3 lower. Nevertheless, the magnetotransport measurements of devices fabricated on isotopically enriched ^{28} Si demonstrate strong manifestations of quantum effects. Based on the analysis of temperature dependence of the weak localization and SdH oscillations, we believe that the dominant scattering mechanism is short-range scattering (impurity scattering). We believe that adventitious chemical impurities detected in the ^{28}Si epilayers act as the impurity scatters in the devices fabricated on ^{28}Si. However, higher levels of adventitious chemical impurities detected in the ^{28}Si epilayers are too high to be considered as isolated scattering centers since the nearest neighbor distance is considerably shorter than the scattering lengths extracted from the transport data. Furthermore, for these impurity levels, the dipolar interactions between randomly distributed electron spins associated with impurities and the central spin of a potential qubit are considered to be the dominant decoherence mechanism at high enrichment.^{13} For the worst case analysis, if all the N and O chemical impurities are considered as randomly distributed single electron spins, the influence of these dipolar interactions on the central spin could result in qubit coherence times poorer than high purity natural abundance Si. However, we are confident that the recent and planned improvements, as well as techniques for depleting impurities near the surfaces, will allow us to move forward and study the tension between chemical impurities and enrichment on quantum coherence.

Next, we plan to fabricate quantum dot devices on control (natural abundance) and isotopically enriched ^{28}Si to more rigorously assess the impact of purity and enrichment, e.g., charge offset drift, as the chemical purity of these MBE grown ^{28}Si films is improved. Therefore, macroscopic transport and material characteristics of the devices reported here will serve as a benchmark for finding the correlations between macroscopic properties and the performance of future nanoscale devices, e.g., quantum dots, and lead to identifying qualifying metrics for “quantum grade” silicon.

The authors acknowledge stimulating discussions with Michael Stewart, Neil Zimmerman, Roy Murray, Ryan Stein, Binhui Hu, and Peihao Huang.

## REFERENCES

^{28}Si “semiconductor vacuum

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^{30}Si single crystals: Isotope separation, purification, and growth

^{28}Si CVD grown epilayer on 300 mm substrates for large scale integration of silicon spin qubits

^{28}Si beyond 99.999 8% for semiconductor quantum computing

Highest isotopic enrichment measured to date is 99.999 98% ^{28}Si and is planned for future publication. Data can be made available upon a reasonable request.

Targeted enrichment results will be published soon.