Dephasing rates for weak localization and universal conductance fluctuations in two dimensional Si:P and Ge:P δ-layers

We report quantum transport measurements on two dimensional (2D) Si:P and Ge:P δ-layers and compare the inelastic scattering rates relevant for weak localization (WL) and universal conductance fluctuations (UCF) for devices of various doping densities (0.3–2.5 × 1018 m−2) at low temperatures (0.3–4.2 K). The phase breaking rate extracted experimentally from measurements of WL correction to conductivity and UCF agree well with each other within the entire temperature range. This establishes that WL and UCF, being the outcome of quantum interference phenomena, are governed by the same dephasing rate.

Delta doped Si:P and Ge:P devices offer an excellent platform for devices in quantum circuits due to their exceptionally low electrical noise 1 . Experimental and theoretical investigations of δ-doped devices over the past decade have led to realization of high conductivity interconnects with extremely low noise 2 and single/few donor based devices of phosphorous in silicon 3 . These delta-doped devices have been made possible through scanning tunneling microscope (STM) lithography combined with molecular beam epitaxy (MBE). This technological advancement in STM-MBE fabrication takes us forward to address the solid state quantum bits and perform the required quantum operation. These systems, though aimed to be the building blocks of a solid state quantum computation scheme can have exotic and novel physical properties as they are a natural realization of a half-filled impurity band in 2D. The P δ-layers in Si and Ge provide a pure semiconductor based two dimensional (2D) electron system to investigate quantum coherent phenomena such as weak localization (WL) and universal conductance fluctuations (UCF). Here, we compare the phase breaking rate (which is the inverse of phase coherence time) relevant to the phenomena of WL, τ φ − ( ) WL 1 , and UCF, τ φ − ( ) UCF 1 , in δ-doped 2D electron systems in silicon and germanium. Using measurements of magnetoconductance and time-dependent conductance fluctuations in a perpendicular magnetic field, we show that τ τ = φ φ WL UCF which implies that WL and UCF are governed by the same scattering rates.
Quantum transport in low dimensional disordered systems at low temperatures is highly sensitive to the phase coherence of interfering wave functions of electrons. Investigation of quantum transport in disordered systems have often been carried out by measuring the temperature dependence of conductivity, the magnetoconductivity in a perpendicular magnetic field ⊥ B or through conductivity fluctuations. The coherent interference of wave functions of electrons as they are multiply backscattered by random impurities lead to the ubiquitous phenomena of WL and UCF which are fundamental concepts in disordered systems and have been extensively studied both experimentally and theoretically for a variety of systems over the past few decades [4][5][6][7][8][9][10][11][12][13][14][15][16] . WL is characterized by a reduction in conductivity as the temperature is reduced 6,17,18  fluctuations are most prominent when the sample dimensions are of the order of the phase coherence length φ l of the sample. Both WL and UCF are characteristic to the diffusive regime of electron transport where , where τ is the elastic scattering time.
It is known that the time scale determining the WL correction and its temperature dependence is the phase breaking time τ φ WL , which can be experimentally evaluated by fitting the WL theory of disordered 2D conductors to the magnetoconductivity in ⊥ B 12,[19][20][21][22] . The relevant rate for UCF τ φ UCF however, has been a subject of controversy with different theoretical and experimental reports over the last two decades [23][24][25][26][27][28] . The scattering time relevant to the expression for UCF was thought to be the out-scattering time τ out which is related to the frequency of inelastic collisions and differs from τ φ WL by a logarithmic factor for 2D systems 24 . This scenario was supported by experiments on quasi-2D thin silver films by Hoadley et al. who carried out WL and time-dependent UCF measurements to extract τ φ WL and τ φ UCF and found that τ for low temperatures where the electron-electron interactions cause dephasing 25 . This was contradicted by later theoretical work of Aleiner and Blanter who showed that the time scales for WL and UCF are exactly identical for dephasing by electron-electron scattering 26 . Specifically, Aleiner and Blanter showed that UCF is governed by the time-scale defined by a pole in the UCF diffuson which is exactly identical to the time scale for WL. Subsequent experiments by Trionfi et al. have supported this claim by demonstrating excellent agreement between τ φ WL and τ φ UCF extracted from WL and UCF measurements respectively, for quasi-one dimensional (1D) and quasi-2D AuPd wires 27 . Thus there exists contradicting experimental and theoretical reports regarding the equivalence of τ φ WL and τ φ UCF . Trionfi et al. have suggested that the discrepancy between τ φ WL and τ φ UCF observed in some experiments, could result from an inaccurate expression for the magnetic field dependence of UCF (due to incorrect assumption that the UCFs are unsaturated) 28 .
In this report, we have studied the quantum transport in atomically thin P δ-layers in elemental semiconductors Si and Ge through the measurements of WL and UCF. We show that the phase breaking time relevant for WL and UCF are identical (i.e. τ φ WL = τ φ UCF ) for the Si:P and Ge:P δ-layers at low temperatures (0.3-4.2 K) where the dephasing is due to electron-electron scattering as confirmed by the temperature dependence of τ φ WL . Our results are significant from the material aspect since δ-doped systems are the building blocks of solid state quantum computation architecture. From the fundamental physics aspect, we have studied the existing problem of the equivalence of τ φ WL and τ φ UCF in atomically thin 2D systems as opposed to quasi-2D systems and thin films investigated earlier.

Results
All the Si:P and Ge:P δ-layers were fabricated in an ultra-high vacuum (UHV) variable-temperature STM system with a base pressure of 5 × 10 −11 mbar. The STM-chamber has PH 3 dosing system for P doping and a sublimation source (Si or Ge) for homoepitaxial growth. For electrical transport measurements, Hall bars were formed using electron beam lithography (EBL) and reactive ion etching. Ohmic contacts were made to the Hall bars by EBL and subsequent metallization. The details of the growth and fabrication process can be found in refs 29-31. A false color scanning electron microscope image of a Ge:P δ-doped Hall bar is shown in Fig. 1a.
Quantum transport in Si:P and Ge:P δ-layers. The effect of quantum coherent transport in these devices is manifested in the logarithmic decrease of conductivity with temperature as shown in Fig. 1b for Si-1 and Ge-1. This logarithmic decrease in conductivity arises due to corrections from WL and electron-electron interaction effects as has been observed previously 32 . The characteristic feature of WL is a dip in the magnetoconductivity in a perpendicular magnetic field, ⊥ B , at = ⊥ B 0 as shown in Fig. 1c and d for device Ge-1 and Si-1 respectively at different temperatures. To extract τ φ WL , the magnetoconductivity σ ∆ has been fitted with the Hikami formula for a disordered 2D system 12 , is the phase breaking field with D being the electron diffusivity.
2 , has been estimated using the experimentally calculated Drude conductivity σ D and number density n obtained from Hall measurements. The phenomenological prefactor α has a value in the range 1-1.2 for all devices at all temperatures. The solid black lines in Fig. 1c and d are the fits using Eq. 1 for the devices Ge-1 and Si-1 respectively with τ φ WL as the fitting parameter. Thus the phase breaking rate relevant to the phenomena of WL has been extracted for all devices at all temperatures and it agrees well with earlier reports on Si:P and Ge:P δ-layers 29,32 . Magneto-fingerprinting and noise due to UCF. The UCF is manifested as aperiodic reproducible fluctuations (~e h / 2 ) in conductance as a function of ⊥ B , called magneto-fingerprinting, when the sample dimensions are of the order of phase coherence length τ = φ φ l D . The magneto-fingerprinting has been observed at 0.2 K for device Si-1 of width 2 μ m when the length of the δ-layer between the voltage probes is 1 μ m as shown by the orange trace in Fig. 2a. As the length increases, the fluctuations decrease due to ensemble averaging and for the largest dimension, µ φ  m l 30 the conductance fluctuations are e h / 2 (olive green trace in Fig. 2a). To study the dependence of UCF on external parameters like magnetic field and temperature, we have measured the low frequency conductance fluctuations, which has been previously used to probe UCF and metal-to-insulator transition in bulk doped silicon [33][34][35] . The slow time-dependent fluctuations in conductance are recorded, which because ergodicity is analogous to ensemble fluctuations 23,33,[36][37][38] . We employ an AC four probe balanced bridge technique to measure the time-dependent conductance fluctuations which is then digitally processed to obtain the power spectral density, PSD (S V ), by using a fast fourier transform technique. The PSD of the out-of-phase component of voltage fluctuations is the background Johnson's noise along with amplifier noise (S V Y ) which is subtracted from the total noise (PSD of the in-phase component, S V X ) to obtain the noise from the sample (Fig. 2c). The details of the process can be found elsewhere [39][40][41] . From the measured S V , and known current I and voltage V, 2 is computed. The normalized variance of conductance fluctuations can be calculated as, Hz and ≈ f 7 2 Hz define the experimental bandwidth.
Using this procedure we have estimated the magnitude of UCF at different temperatures and ⊥ B . As a function of temperature, δG G / 2 2 increases as the temperature reduces (for  T 15 K) as shown in Fig. 2d for Si-1 ( ≈ . × − n 2 5 10 m 18 2 ) and Si-2 ( ≈ . × − n 1 1 10 m 18 2 ) and is associated with quantum interference effects. The noise in this region is due to UCF which increases at low temperatures as 1/T as shown by the dotted black lines in Fig. 2d. The 1/T dependence can be understood from the Feng-Lee-Stone mechanism of conductivity fluctuations due to the movement of a single impurity atom. In this mechanism, the conductance fluctuations within a phase coherent box δ φ G is 37,38  where k F , l and n s are the Fermi wavevector, mean free path and the density of active scatterers respectively. The function α(x) represents the phase change of the electron wavefunction due to scattering off the moving impurity at a distance δr. It is important to mention that for these devices for the entire range of temperature and hence Eq. 2 is applicable. As δ φ G ( ) 2 from different boxes are statistically independent of each other, the conductance fluctuations of the entire sample, δG ( ) 2 , is related to δ φ G ( ) 2 as, is the conductance of a single phase coherent box, Ω = φ φ l 2 is the volume of one phase coherent box (or area for 2D systems) and V is the volume of the sample (or area L L x y in the 2D case with L x and L y being the sample dimensions in the x and y direction respectively). Substituting for δ ϕ G ( ) 2 using Eq. 2, we get T 0 5 (as obtained from WL fits to the magnetoconductivity shown in Fig. 1  B , where φ B is the phase breaking field which introduces one flux quantum within a phase coherent box, as shown in Fig. 3a for the the highly doped Ge-1 δ-layer at 0.7 K (open squares) and 4.2 K (filled circles). A reduction in noise magnitude by a factor of two at the scale of φ B is expected for low-dimensional systems at low temperatures in the quantum coherent regime where the low frequency noise arises due to UCF 36 . A perpendicular magnetic field removes the Cooperons' (self-intersecting trajectories, schematic in Fig. 2b) contribution to noise thus suppressing its magnitude by a factor of two as has been observed for various systems like metal films 37 , mesoscopic wires 43 , bulk Si 33 and graphene 44 . Similarly for Si-1 δ-layer, the N G reduces as ⊥ B increases beyond φ B   where, τ min,max denotes the relaxation time of noise causing processes. From the magneto-fingerprinting data (Fig. 2a) . This suggests that the time-dependent noise has to be integrated over 1000 decades of frequency to saturate the UCF. Hence we can safely assume that the Si:P and Ge:P δ-layers are in the unsaturated regime.
The ⊥ B -dependence of N G (Fig. 3) was fitted with the crossover function ν ⊥ B T ( , ), appropriate for unsaturated UCF, given by 36.
where c is the fractional UCF noise. The solid olive lines in Fig. 3a and b show fits to Eq. 6 and Eq. 7 respectively with φ l UCF as the fitting parameter. The right inset of Fig. 3a and b shows the same plot and fits in the logarithmic scale. Using this procedure we have calculated φ l UCF for all devices at different temperatures. The corresponding phase breaking time τ φ UCF can then be calculated using the relation τ =  Fig. 3c and d respectively). Since the characteristic field scales φ B and B Z decreases with decreasing T, the reductions in N G occur at a lower value of ⊥ B for lower T as shown in Fig. 3a,b,c and 3d. The observation that N G reduces by factors of two at φ B and B Z further confirms UCF as a major source of noise in these systems at low temperatures.
Comparison of phase breaking rates relevant to WL and UCF. The comparison between phase breaking rate τ φ −1 for WL and UCF, as extracted from theoretical fits to magnetoconductivity and N G vs ⊥ B respectively, is illustrated in Fig. 4 for different temperatures and density. The magnitude of τ φ WL and τ φ UCF agree well with each other at all temperatures for Ge-1 and Si-1 as shown in Fig. 4a and b Fig. 4a and b) indicating that the dephasing is due to electron-electron scattering (or Nyquist dephasing) 11 .
The equivalence of τ φ WL and τ φ UCF is illustrated for devices of varying density in Fig. 4c. For Ge:P δ-layers the comparison between the two scattering times have been made in the density range 0.32-1.35 × 10 18 m −2 . However for the Si:P δ-layer, we were not able to extract τ φ UCF for lower density devices since the UCF noise was suppressed at = ⊥ B 0 due to a spontaneous breakdown of time reversal symmetry in the strongly interacting regime as has been explained in Ref. 42. The inset of Fig. 4c shows that φ l UCF and φ l WL (phase breaking length corresponding to   Table 1. The phase coherence lengths φ l WL and φ l UCF extracted from weak localization fits to magnetoconductivity in a perpendicular magnetic field (B ⊥ ) and theoretical crossover function fits to B ⊥ dependence of conductance fluctuations, respectively for various Si:P and Ge:P δ-layers at different temperatures T. n is the carrier density as determined from Hall measurements.
Scientific RepoRts | 7:46670 | DOI: 10.1038/srep46670 UCF and WL respectively) agree with each other for all densities investigated in our experiment. The dotted black line shows that φ l WL (or φ l UCF ) ∝ n 0.9 . Table 1 shows the values of φ l WL and φ l UCF for devices of varying density at different temperatures. The ratio φ φ l l / WL UCF ranges from ~0.95-1.2 implying a maximum difference of ~20% between the two values which is within the limits of statistical uncertainty.

Conclusion
In conclusion, we have studied the quantum interference phenomena in 2D Si:P and Ge:P δ-layers. Weak localization corrections to conductivity and universal conductance fluctuations have been measured as a function of temperature and perpendicular magnetic field. The phase breaking rate τ ∝ φ − T 1 , which indicates that electron-electron scattering is the dominant dephasing mechanism in these systems. The relevant dephasing rate (and hence the phase coherence length) corresponding to WL and UCF are in quantitative agreement with each other for devices of different doping densities indicating that the same scattering times govern WL and UCF for dephasing due to electron-electron interaction. The result is significant as WL and UCF are fundamental concepts in solid state physics and an equivalence of the corresponding scattering rates reiterates that the two phenomena are analogous even when the system is purely two dimensional in nature.