DC to GHz measurements of a near-ideal 2D material: P + monolayers

P + monolayers in Si are of great scientific and technological interest, both intrinsically as a material in the “ideal vacuum” of crystalline Si and because they are showing great promise as qubits of electron and nuclear spin. The GHz complex conductivity σ ( ω ) can allow one to elucidate basic physical properties and is also important for fast devices, but measuring σ ( ω ) in 2D materials has not been easy. We report on such measurements, including showing (i) qualitatively a lack of any resonances up to 5 GHz (indicating no energy splittings below about 0.02 meV); and (ii) the quantitatively ideal Drude behavior of this novel material up to 5 GHz, showing a lower bound on the scattering rate of about 2 × 10 10 s − 1 . We also discuss deconvolving the confounding effect of the distributed resistance and capacitance of the monolayer.


I. INTRODUCTION
In recent years, workers have developed the ability to generate a subsurface monolayer of P + dopants (typically about 1 4 of a full monolayer) surrounded by lattice-matched crystalline Si. 1,2 In addition, the ability to pattern the monolayer with atomic precision using an STM has resulted in an exciting variety of device possibilities under the rubric of "digital manufacturing," 3 in which nominally the fabrication can be atomically perfect; this variety has resulted in rapid progress in quantum coherent manipulation in both electron and nuclear spin. 4,5owever, some of the basic properties of this novel 2D material are not yet fully understood, including the basic conduction mechanism and energy spectrum.For instance, such a system's conductivity spectrum depends strongly on its energy level spectrum and, therefore, is a direct reflection of the ground state and excited state physics.The high-frequency conductivity is, in addition, of great importance for the high-speed operation of conventional and quantum devices in this STM-patterned P + monolayer architecture.
In our previous work, 6 we proposed a quantitative, non-contact method of sensitively characterizing the GHz response of 2D flakes and showed simulated results on a candidate material.In the present work, we have experimentally demonstrated the applicability of this technique to the important case of a blanket P + monolayer in Si.
The closest previous result to the characterization of a blanket P + monolayer embedded in Si in this high-frequency regime is the demonstration of exchange qubits, 4 as it includes the transmission of signals in "wide" P + lines (10 nm), from which we can surmise qualitatively that some level of GHz transmission was occurring.Our observation of a smooth frequency dependence of the transmission (i.e., no resonances) and ideal Drude behavior in the complex conductivity adds to our understanding of this 2D material.In the remainder of this publication, we give experimental details of the fabrication, measurement, and analysis, show the raw data for transmission and DC resistance and the deduced complex conductivity, and then conclude.

A. Fabrication
We prepared blanket P + layers using our standard process; 7 briefly (samples W18-F3 and W18-F4), we cleaned atomically flat Si (001) chips in UHV, exposed the sample to a saturation dose of PH3 gas that resulted in about 1 4 of a monolayer of P + , incorporated P atoms substitutionally by heating to 350 ○ C for 2 min, then overgrew epitaxial Si (33.0 ± 2.7 nm for W18-F4 and 30.8 ± 1.6 nm for W18-F3) with a 2 nm room-temperature locking layer.After this, we etched mesas of size (7 ×50 μm 2 ) to a depth of 52 nm, deposited about 185 nm of Al, and performed standard photolithography and chemical etching to produce coplanar waveguides (CPWs) (both continuous and with gaps in the center conductor).The continuous CPWs were used to normalize the transmission data to derive the conductivity function.
For series-gapped CPWs, we placed the gap centered over the mesa; see Fig. 1 for a sketch of the final device.We note that, as a result of depositing Al over the side of the mesa-etched monolayer, we provided a weak resistive connection R Contact between the CPW signal line and the monolayer-in the future work, we plan on providing an insulating barrier to avoid this complication.Please see Appendix A for a theoretical analysis of the effect of this resistance.
Finally, we placed the 4 × 10 mm 2 dies into a sample box assembly, wirebonded them to Al pads, and mounted the assembly onto the mixing chamber plate of a dilution refrigerator (DR).In this publication, we report measurements taken at the base temperature (the thermometer read 10 mK).
We also produced control devices (W22-62 and W22-85) nominally identical with the previous two, with the exception that we did not expose the samples to PH3 and, therefore, they had no P + .We note that the control samples had a nominally identical overgrown Si layer.

B. Measurement details
For the high-frequency measurements, we used standard Vector Network Analyzer (VNA) techniques yielding magnitude |t| = |S 21 | and phase ϕ = arg(S 21 ) of the transmission function, with SMA cable assemblies going to the sample box assembly in the DR.The cable assemblies were heat-sunk at all stages using 0 dB attenuators.
To deduce the complex conductivity from |t| and ϕ, we used techniques derived in Ref. 6, modified to account for the effects of the distributed impedance in the P + monolayer on the capacitive coupling to the signal lines (see Appendix A and Ref. 8 for more details).
For the DC resistance measurement of the sample, we etched Hall bars on the same chips with metalized contact pads on the samples.We made four-point measurements using a closed cycle cryostat at a temperature of 4 K (to suppress parallel conduction in the Si substrate).We then extracted the sheet resistance of the phosphorous monolayer using the geometry of five squares between each of the voltage leads.

III. DATA AND ANALYSIS
Figure 2 shows representative data for the magnitude and phase of transmission t in several structures.
First, the two top curves in the upper panel show continuous CPWs (no gap), which demonstrate both the expected low pass filter frequency dependence, good (low) insertion loss below about 1 GHz, and excellent reproducibility (the two measurements were taken on different devices in the same cooldown using different wiring).
Second, we note the dynamic range (about a factor of 10 2 or 40 dBm) separating the monolayer transmission data from both the continuous CPW and the CPW with no monolayer.Additionally,

FIG. 2.
Raw data for the magnitude and phase of the transmission coefficient vs frequency.Note the excellent signal dependence on the μm-sized conducting monolayer: (i) above the monolayer, the gapped CPW has a much larger transmission than above the undoped Si; (ii) the gapped CPW has a much smaller transmission than a continuous CPW.Note also the good reproducibility between the two "continuous CPW" spectra (same cooldown, different wiring) as well as between the two "gapped CPW over P + monolayer" (same cooldown, different samples, different wiring).Above about 5 GHz, coupling through the wirebonds removes the difference between monolayer and bare Si (see text).The typical power supplied by the Vector Network Analyzer was −10 to −15 dBm.

ARTICLE
pubs.aip.org/aip/adv the reproducibility shown in sets of data taken on similar configurations (e.g., gapped CPWs) demonstrates the P monolayer is the dominant material being probed.In particular: (1) The transmission in the gapped CPW over undoped Si is about 10 2.5 smaller than in the continuous CPW (power is 10 5 smaller); this shows that the transmission in 5 m of the transmission line is totally dominated by the gap of size a few μm.(2) For the dynamic range, we note that the transmission magnitude for the devices with a conducting P + monolayer is far larger than the "control" device (no conducting P + monolayer) up to about 5 GHz.Above this frequency, the transmission intensity across a series-gapped CPW begins to suffer from crosstalk and loses reliability.This qualitative change in the frequency-dependence of the transmission spectrum can be seen in t of Fig. 2, which shows that the transmission across the series gapped CPW over undoped Si sharply increases with frequency above ∼1 GHz, eventually converging to the transmission spectrum across the series gapped CPWs over P + mesas, indicating a loss of sensitivity to the P + layers.Shortening the wirebond length increased the maximum frequency at which transmission was dominated by transmission through the series gap between the signal lines.Comparing different samples of the same type, please note the excellent agreement between the two curves with the P + monolayer, given that the two measurements were taken on different devices in the same cooldown using different wiring.
Our technique (AC transmission in μm-scale 2D materials using non-contact gapped CPWs) has two main methods of analysis.The first is the qualitative one of looking for resonant features in the transmission.We can see from Fig. 2 that, up until the 5 GHz extrinsic limit, we see no such resonant features.Therefore, we can conclude that there are no excited states within 0.02 meV of the ground state of the conduction electrons in our P + monolayer (see Appendix B for a discussion of the prediction of the Drude model as extended to include possible resonances).
The second method is the quantitative one of deducing the frequency-dependent conductivity σ(ω) from the complex transmission spectrum mathematically.In our previous publication, 6 we derived the complex transmission spectrum expected to result from the circuit diagram shown in Fig. 4(c), where Z Couple , the coupling impedance between each signal line and the P + monolayer, replaces C Couple .C Series is the capacitive coupling between the signal lines, σ(ω) is the conductivity of the P + monolayer, Z CPW is the CPW impedance (≈50 Ω), and α is the ratio of the series gap to the signal linewidth.Simply put, when the gap in the CPW looks like an open ), the signal is transmitted through a series coupling capacitance between the one signal line and the sample of interest, the conductivity of the sample of interest, and capacitive coupling to the second signal line.Since the frequency dependence of the capacitors is smooth and monotonic, features in the sample conductivity vs frequency become detectable.
However, we realized in the course of this work that the simple formula was not sufficient.As previously discussed in Refs.9 and 10 for the limit of infinitely large R Contact , and for the general case in Appendix A, the finite conductivity in the monolayer that forms the capacitor plate leads to a substantial modification in the transmission and, therefore, we need the result in Appendix A to accurately deduce σ(ω), resulting in Here, u = iωC Couple l wσ(ω) , where l and w are the length and width, respectively, of the mesa-etched monolayer underneath both signal lines.The formula for the limit of infinitely large R Contact can be found in Refs.9 and 10.
We have thus taken the complex transmission in Fig. 2 and deduced σ(ω) using Eqs.( 1) and ( 2), shown in Fig. 3.This deduction requires accurate normalization for the transmission; as discussed in Appendix A, we used the data from the continuous CPW to normalize.We also note that the significant dispersion (dependence on f in Fig. 2) arises from the complex impedance of the RC delay embodied in Z Couple ; thus, the simple Drude results in Fig. 3 are consistent with the dispersion observed in Fig. 2.
The values of R Contact used to deduce the conductivity spectra were 138 and 495 kΩ for the devices with a 2 and 8 μm gap, respectively.These values were taken from electronic transport measurements.The values of C Couple were obtained by fitting the transmission spectra between 10 and 100 MHz, where the signal is dominated by the capacitive coupling.It is also worth noting that the values of C Couple used to fit the spectra are close to theoretical expectations.
We note the good reproducibility between the two samples for both real and imaginary parts; the real part of the complex conductivity is about 20% above the DC value measured in Hall bars.We further note that (i) Re[σ(ω)] is frequency-independent and that (ii) Im[σ(ω)] ≈ 0 (within the statistical uncertainty).From the discussion in Appendix B, we thus conclude that an upper bound on the scattering time τ is ∼ 0.2 2π(4 GHz) ≈ 10 ps based on the negligible frequency-dependence of the real and imaginary components of σ(ω).
The discrepancy of about 20% could be due to either (i) lateral inhomogeneity in the P + monolayer 2D resistance (in some FIG. 3. Complex conductivity deduced from data in Fig. 2, using Eqs.( 1) and ( 2) and the parameters in Table I.The two different devices (2 and 8 μm gap) were on different dies and were measured in the same cooldown with different wiring.Note that 1.1 ×10 −3 sq/Ω is the value of resistance per square measured in the Hall bars.
cases, the resistance range across chips is larger than 20%); (ii) a true frequency dependence resulting in a change below about 100 MHz (the lowest frequency we measured with this technique); or (iii) the uncertainty of our measurement of t and of the deduction of σ(ω).

IV. CONCLUSION
From the qualitative lack of resonances in the raw data in Fig. 2, we can conclude that there are no excited states within about 0.02 meV of the ground state (as expected) (see Appendix B for details).From the deduced conductivity in Fig. 3, we can see that this nearly ideal conductor in the "ideal vacuum" of crystalline Si shows behavior identical to the simple Drude model within the uncertainty.We note the upper bound on the scattering time of τ < 10 ps.
We again note the discrepancy between the real part of the deduced complex conductivity and the DC value measured in Hall bars.As described earlier, this is likely due to inhomogeneity in monolayer resistivity across a chip.While very unexpected, frequency dependence below 100 MHz cannot be ruled out entirely; we hope to extend one technique or the other to determine the existence of such a dependence.
We can also comment on the strength of our measurement technique generally: While there have been a number of previous techniques to measure GHz conduction in 2D materials (see Ref. 6), Fig. 3 clearly demonstrates that our technique provides this new capability, with an estimated relative uncertainty δσ (ω) σ(σ) of at most about 20%, and provides a wide bandwidth while avoiding the need for Ohmic contact.
In addition to gaining additional scientific information about the P + monolayer conducting behavior, our results also bear on the burgeoning use of P + qubits and ancillary devices in quantum dots. 4he natural frequency of the electron spin in a typical magnetic field is of p order 10-40 GHz and, therefore, to produce one-qubit rotations and two-qubit coupling requires that the P + leads can transmit signals at this frequency range and speed (for pulses).The excellent results obtained recently provide an inference about the ability of the P + leads to transmit high-speed signals, and our results provide quantitative confirmation of this inference.

APPENDIX A: RC COUPLING DERIVATION
In our previous publication 6 on the theoretical framework for this technique, we assumed a circuit diagram as in Fig. 4, with capacitive coupling to the flake Z(ω).However, in the present work, we realized that much (essentially all) of the frequency dependence of the t(ω) in Fig. 2 is not reflective of intrinsic dispersion in the 2D material P + but rather arises due to the dispersion of the distributed resistive/capacitive network in the P + .Therefore, in this section, we derive a framework for the two modifications mentioned in Fig. 1: (i) R Contact as mentioned in Sec.II A and (ii) a distributed resistance R Electrode in the plate of C Couple arising from the non-zero resistivity of the monolayer; both of these modifications can be seen in Figs.4(c) and 4(d).We note that we have chosen to put capacitors at both ends of the distributed RC network; 8 in the limit of large N (see Fig. 4 caption), this boundary condition becomes numerically insignificant.We note that the authors of Refs. 9 and 10 previously analyzed the case of infinitely large R Contact .
In the following sections, we will (i) recursively derive an expression for ZN and then (ii) approximate Z Couple = limN→∞ ZN.

Deriving a general expression for Z N
It is clear that R and C decrease in value as N increases.For most of this section, we will suppress this dependence and simply treat R and C as values that are constant for all elements in the circuit for a given value of N and then substitute for the decreasing values at the end.

a. Deriving Z 2
It is straightforward to derive and, therefore, In analogy with Eq. (A2), in this subsection, we will first show that the trial solution in Eq. (A2) satisfies the recursion relation and the boundary conditions set by with the constraints that for all coefficients x ∈ [a, b, c, d], x j j = 0 whenever i < 0 or i > j − 1.We then solve for the coefficients a, b, c, and d.We note, as shown later, that the decreasing nature of the coefficients ensures that the series converges.
We start by observing that Now, we will use Eq.(A2) as a trial solution and show that it gives a consistent result for Z N+1 .Substituting the trial solution from Eq. (A2) into the recursion relation Eq. (A3), we obtain  In the main text, we combine Eqs. ( 1) and (2) to deduce σ(ω).An experimental complication arises from the need for normalization of the transmission data, as otherwise, non-idealities in the wiring will substantially degrade this analysis. 6As shown in Fig. 2, by comparing the "continuous CPW" to the "gapped CPW" data, we can clearly see that the transmission magnitude is dominated by the P + monolayer; from this observation, we concluded that the best normalization is the continuous CPW data.Therefore, when deducing σ(ω) in Fig. 3, for each device, we replaced t(ω) in Eq. (1) with t gapped (ω) t continuous (ω) .
Finally, we wish to compare our technique to the closest previous one. 8In the previous work, the authors used a similar circuit as in Fig. 4 in order to deduce σ [not σ(ω)] for a MOSCAP.In order to achieve quantitative results, they limited their technique to frequencies such that 1 l σw ≪ 1 ωC Couple ; they also did not solve for the equivalent impedance and, therefore, could not deduce the frequency dependence.In contrast, in this work, we solve for the impedance [Eq.(A32)], do not limit ourselves in frequency, and deduce the frequency dependence.

APPENDIX B: DRUDE MODEL AND RESONANCES
The Drude model (no resonances) result for the complex conductivity is where σ(0) = ne 2 τ m for simple metals, i = √ −1, and where τ is the scattering time.
Note that σ(ω) will vary from the low-frequency limit (flat real part and zero imaginary part) by about 20% when ωτ ≈ 0.2 (see Fig. 7); we can thus put an upper bound on the scattering time of τ < 10 ps.
An extension of this model in the case of resonances can be derived as follows: The dielectric response for multiple resonances is 11 where λj, ω 0,j , and τj are, respectively, the weights, center frequencies, and lifetimes of the various resonances.
In addition, we can convert σ(ω) = ω 4πi ε(ω), 12 using the second convention, and with the understanding that in our experiment, the measured transmission corresponds to the conductivity σ(ω) arising from all (bound and free) electrons.Importantly, we note that Eq. (B3) collapses to Eq. (B2) in the limit of no resonances (only one term in the sum) and with ω 0,j = 0 and τj = τ.
We can now use Eq.(B3) to achieve an estimate of the absence of any excited states in the P + monolayer, given the frequencyindependent Re[σ(ω)] within ∼±10% (Fig. 3).Very simply, this leads to | ω 2 0, j τ j ω − ωτ j | < 0.1; assuming ωmaxτj < 0.1 where ωmax ≈ 2π(4π GHz) is the maximum experimental frequency from Fig. 3, we obtain ω 0, j < √ 0.1ωmax/τ j .If we then approximate τj as being due to thermal broadening (not intrinsic linewidth of the resonance) at 1 K, we finally obtain a bound on the minimum energy of any resonance of ̵ hω 0, j < √ (0.1 ≈ 0.1 meV.Finally, we note that this analysis requires that the possible observation of conductivity peaks from Eq. (B3) depends on λ j τ j λτ being not very small compared to unity; understanding "oscillator strengths" is a complicated topic of its own and, therefore, a detailed discussion of this assumption is beyond the scope of this paper.

FIG. 1 .
FIG. 1.Above: Sketch of the fine area of the gapped CPW device: the purple region is the P + monolayer; gray is Si and the teal color is the metal CPW.The gap in the CPW allows us to focus the dominant transmission effect on the μm-sized mesaetched conducting monolayer.Below: Sketch with the lumped-parameter circuit elements added.

FIG. 4 .
FIG. 4. (a) Magnified portion of Fig. 1, with lumped-parameter electrical elements.Indicated [corresponding to panel (b)]; (b) Circuit diagram equivalent to our previous publication, 6 with simple capacitors C Couple .(c) Modified diagram, where we have replaced C Couple with the more general Z Couple .(d) The details of Z Couple , in particular the R Contact and the distributed RC combination.There are N capacitors and (N − 1) resistors; in the end, we will take the limit as N → ∞.

FIG. 6 .
FIG. 6. Theoretical calculation of the frequency dependence of Z Couple , from Eq. (A9), and comparison to the simple parallel combination of R Contact and C Couple .Note the agreement at low frequencies where l wσ < 1 ωC Couple.All parameters are given in TableIfor the 8 μm gap.

TABLE I .
6arameters used in deduction of σ(ω).α and l come from the nominal geometry, R Contact from Hall bar measurements, and C Couple from low-frequency measurements where C Couple dominates the transmission.6Thevalues of C Couple are within 10% of independent estimates from the nominal geometry.