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PHYSICAL REVIEW LETTERS
PRL 112, 017001 (2014)
Identification of the Local Sources of Paramagnetic Noise in Superconducting Qubit Devices Fabricated on -Al2 O3 Substrates Using Density-Functional Calculations Donghwa Lee,* Jonathan L. DuBois, and Vincenzo Lordi† Condensed Matter and Materials Division, Lawrence Livermore National Laboratory, Livermore, California 94550, USA (Received 20 May 2013; published 7 January 2014) Effective methods for decoupling superconducting qubits (SQs) from parasitic environmental noise sources are critical for increasing their lifetime and phase fidelity. While considerable progress has been made in this area, the microscopic origin of noise remains largely unknown. In this work, first principles density functional theory calculations are employed to identify the microscopic origins of magnetic noise sources in SQs on an -Al2 O3 substrate. The results indicate that it is unlikely that the existence of intrinsic point defects and defect complexes in the substrate are responsible for low frequency noise in these systems. Rather, a comprehensive analysis of extrinsic defects shows that surface aluminum ions interacting with ambient molecules will form a bath of magnetic moments that can couple to the SQ paramagnetically. The microscopic origin of this magnetic noise source is discussed and strategies for ameliorating the effects of these magnetic defects are proposed. DOI: 10.1103/PhysRevLett.112.017001
PACS numbers: 74.78.Na, 03.65.Yz, 71.15.Mb, 74.50.+r
Superconducting qubits (SQs) represent a promising route to achieving a scalable quantum computer [1,2]. However, the coupling between SQs and (as yet largely unidentified) parasitic noise sources has so far limited the functionality of current SQs by reducing the coherence time of the quantum states below a practical threshold for measurement and manipulation [3]. These noise sources also more generally limit the performance of a variety of devices based on superconducting circuits, including high energy-resolution cryogenic radiation detectors, sensitive magnetic detectors (superconducting quantum interference devices, or SQUIDs), nanomechanical motion sensors, and quantum-limited parametric amplifiers [4–8]. Magnetic flux noise with an approximate 1=f spectral density is known to be one of the dominant contributors to loss of fidelity in both SQUIDs [9] and SQs [10,11]. A number of experimental studies [10,12] clearly implicate surface spins as a significant contributing factor to this noise. In addition, recent experimental observations suggest that surface spin clusters and/or correlated fluctuations of effective magnetic moments contribute significantly to flux noise [13,14]. Several plausible mechanisms for coupling of localized spin states to SQs consistent with experimentally observed noise spectral density have been proposed [15–18]. Koch et al. proposed that flux noise is generated by electrons with fixed moments hopping between localized traps [15]. De Sousa attributed the noise to spin flips of paramagnetic dangling bonds interacting with tunneling two level systems [16]. Faoro and Ioffe have shown that spin diffusion facilitated by the RKKY interactions at the superconductor-insulator interface could account for the observed noise [17], while Yu et al. have shown that the presence of a spin glass [18] or electron spin exchange via the hyperfine interaction [19] could be responsible. 0031-9007=14=112(1)=017001(5)
While fluctuations of the surface spins or their interactions give a successful description of 1=f flux noise, the microscopic origin of the electron spins remains unclear. Choi et al. proposed that metal induced gap states (MIGS) at the interface between the superconducting lead and the dielectric substrate could be a source of magnetic spin fluctuation [20]. While MIGS represent a plausible mechanism contributing to flux noise and several experimental efforts to modify the metal-substrate interface have shown improved noise performance [21–23], there are remaining limits to the observed coherence time that indicate the existence of additional significant contributors to flux noise [23,24]. In this work, we focus on building a comprehensive picture of possible microscopic magnetic noise sources on the substrate surface, independent of a specific mechanism for spin flips, in order to identify fabrication and design procedures with the potential to further reduce flux noise in these systems. Sapphire (-Al2 O3 ) is commonly used as a substrate for fabrication of SQs since it provides several advantages, namely, (i) high thermal conductivity, (ii) high dielectric constant, (iii) chemical inertness, and (iv) a coherent interface with Al, which has a workable superconducting transition temperature (Tc ¼ 1:2 K) [25]. In addition, aluminum oxide naturally grows at the surface of the Al superconductor during the fabrication of SQs. We have therefore carried out a systematic theoretical study of possible microscopic origins of surface spin states associated with the surface of -Al2 O3 and investigate atomic configurations and chemical effects that lead to nearly degenerate spin states on the surface. Both intrinsic and extrinsic defects are considered to identify sources of noise that either limit or cause temporal degradation of SQ coherence. Bulk -Al2 O3 has the R3c crystal structure including 30 atoms in a nonprimitive hexagonal unit cell. Six oxygen
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octahedral cages are stacked along the longitudinal direction and the octahedra are filled in the sequence of Al–Al– vacancy, as shown in Fig. 1. The -Al2 O3 (001) surface is modeled using density functional theory (DFT) with a slab consisting of six layers of the oxygen plane [26]. The plane between two Al layers, identified as the lowest energy cleavage plane [27], is chosen for the surface termination. ˚ of A 4 4 supercell was constructed with at least 12 A vacuum space to prevent interactions between periodic images. Further details of our simulation methods are provided in the Supplemental Material [28]. Figure 1 shows the relaxed atomic structure of the -Al2 O3 (001) surface. Surface reconstruction drives the less coordinated surface Al ions toward the oxygen plane, yielding a nearly flat surface with the Al ions sitting in the center of oxygen triangles. To evaluate the magnetic state of a given surface (i.e., whether it introduces a paramagnetic state that can contribute to flux noise), we calculate the energy difference between the two lowest magnetic states using spinconstrained DFT. For any given system, either a spin degenerate or spin nondegenerate state may be the ground state. A large difference in energy between magnetized and unmagnetized electronic states indicates that the state is magnetically stable. These states can only contribute significantly to 1=f flux noise in the presence of an additional spin-spin correlation such as RKKY or hyperfine interaction [17–19]. In contrast, a small energy difference indicates that relatively small changes in the microscopic environment could lead to near degeneracies between nearby magnetic states leading to 1=f noise from direct tunneling processes. This is in contrast to direct thermal excitation processes, which decay monotonically toward the low device operating temperatures. In our analysis, we have assumed that states with energy differences below 0:05 eV can be taken to be within the accuracy of the methods and the material model we have employed, and
FIG. 1 (color online). Schematic view of the -Al2 O3 surface. (a) Side view showing stacking of O octahedra with Al-Alvacant occupation along each column. (b) Top view showing both Al and O atoms on the surface layer. (c) Schematic view of structural location of each vacancy site.
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thus can contribute to tunneling between nearly degenerate magnetic states. The DFT calculations predict that the nondefective -Al2 O3 (001) surface is magnetically stable; the energy difference between the lowest two spin states is 2.21 eV, which is very high compared to the operating temperature of the SQ. Thus, the bare surface does not contribute flux noise to the SQ, and it is reasonable to seek the physical origin of magnetic instabilities from defect states near the surface. As a binary system, -Al2 O3 may exhibit two types of vacancies or interstitials as intrinsic defects. Prior work [29] has reported that defect formation energies (DFEs) of intrinsic defects are quite high in bulk -Al2 O3 , and thus these defects are not present in significant concentrations. However, the DFEs may vary considerably at the surface compared to the bulk. Both experiment and calculation suggest that the Schottky defect pair is more stable than either Frenkel pair [30]; thus, here we focus on investigating the effect of surface vacancies on the magnetic state of the -Al2 O3 surface. The two simple surface vacancies VAl and VO are investigated, which in their fully charged states are 000 and doubly positively charged triply negatively charged VAl VO , respectively. We also consider three defect complexes: 000 þ V , the ‘‘reconfigured Al vacancy’’ the ‘‘divacancy’’ VAl O ::: 000 000 þ 2VAl þ Ali , and the ‘‘reconfigured divacancy’’ 2VAl ::: Ali þ VO . [See Fig. 1(c) and Fig. S2 in [28]]. The thermodynamic stability of each surface defect is evaluated by computing the DFEs at the surface, using the Zhang-Northrup [31,28] formalism. Since the PBE functional significantly underestimates the band gap of -Al2 O3 (6.18 eV compared to the experimental value of 8.7 eV [32]), we employed the PBE0 hybrid functional, which mixes in 25% Hartree-Fock exchange, to calculate the DFEs without the band gap error. (The PBE0 band gap is 8.82 eV; see Fig. S3 [28] for a comparison of PBE and PBE0. Also note that the surface band gap is 1:8 eV smaller than the bulk band gap.) Figure 2(a) shows the DFEs of the different surface vacancies as a function of Fermi energy for the O-rich condition. Although lower than in the bulk [29], moderately high DFEs are observed for the 000 , various surface vacancies. Even the reconfigured VAl 000 whose DFE is 2.13 eV lower than that of VAl , still shows more than 1 eVof formation energy in the middle of the gap. Since vacancies are sufficiently stabilized at the surface to exist in appreciable concentrations there, the effect of these intrinsic defects on the magnetic state of -Al2 O3 surface warrants further investigation. Figure 2(b) shows the energy difference between the two lowest magnetic states of the -Al2 O3 surface with each intrinsic defect present, both in their fully charged states as well as several 000 and additional charge states. The results show that both VAl VO in the fully charged states preserve large energy differ000 ences between the two magnetic states: 0.62 eV for VAl and 1.14 eV for VO ; much too large for thermal fluctuations to
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FIG. 2 (color online). (a) Defect formation energy (DFE) and (b) energy difference between two lowest magnetic states for various intrinsic defects on the -Al2 O3 (001) surface. The DFEs are calculated at oxygen rich condition and T ¼ 0 K using PBE0. Charge states are indicated in parentheses. For defect complexes, the total charge is indicated.
cause flux noise in a SQ. Variation in the energy difference is evident for the different charge states, and reversal of the magnetic nature of the ground state is seen for odd numbers of electrons, but in all cases the energy difference is still large enough (> 0:26 eV) to prevent thermal excitation. 000 þ VO Similarly, the surface with a divacancy defect VAl also preserves the stability of the magnetic state of the pristine -Al2 O3 regardless of its charge state. However, the surface with the reconfigured Al vacancy (2VAl þ Ali ) behaves differently. The nonmagnetic ground state of the reconfigured Al vacancy in the fully charged state (3) shifts to a paramagnetic ground state in the less charged states regardless of the number of electrons. As a result, the significant energy difference between the two magnetic states for the fully charged case (1.04 eV) is greatly reduced in the other charge states (only 0.02– 0.07 eV), which is within the expected accuracy of the DFT calculations. Thus, the reconfigured VAl can provide low-energy states to which the qubit couples on the -Al2 O3 surface at cryogenic temperatures if there is variation in its charge state. Similar behavior is observed for the reconfigured divacancy (2VAl þ Ali þ VO ): the substantial energy difference of 1.48 eV for the fully charged 1 state is lowered to 0.08 eV for the neutral charge state. Electronic density of states (DOS) are analyzed to gain insight into the paramagnetic nature of the
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reconfigured VAl . We find that the reconfigured VAl introduces a nearly continuous partially occupied DOS near the valence band edge (or near the Fermi level at low temperatures) and thus results in magnetic noise by allowing very low energy spin flips. (See Fig. S4 in [28] for more details). However, as shown in Fig. 2(a), the energetically stable defects throughout the entire Fermi energy range are either 000 the fully charged reconfigured VAl , the fully charged VO , or x the neutral VAl , all of which are magnetically stable. Although the reconfigured Al vacancy can introduce low-temperature magnetic noise into the -Al2 O3 surface when not fully charged, the thermodynamically stable fully charged state ½2VAl þ Ali 3 does not. The other thermox dynamically stable defects (VO , VAl ) are also magnetically stable. Therefore, intrinsic defects cannot account for appreciable magnetic flux noise in SQs without an additional spin-spin coupling mechanism. Moreover, the intrinsic vacancy cannot explain the observed time-dependent fluctuation of decoherence [21], as fluctuations in the concentration of surface vacancies could only decrease over time if equilibrium is approached slowly with thermal cycling. Extrinsic defects such as OH [33,34], in contrast, have been postulated as a noise source and could account for these fluctuations. To begin our study of extrinsic defects, we first consider the adsorption of H2 O molecules on the -Al2 O3 surface, since ambient exposure is known to promote termination of the surface with hydroxyls [26]. The H2 O molecule on -Al2 O3 easily dissociates into H on the O site and OH on the Al site [35], in the geometry shown in Fig. 3(a). Our calculations show that the dissociative chemisorbed state is 0.43 eV more favorable than the molecular physisorbed state. For chemisorbed H2 O, the energy difference between the magnetic and nonmagnetic state is 1.9 eV and is, therefore, magnetically stable. However, we find that H can desorb from the surface [36], so the case when only OH is adsorbed must be considered. In fact, we find that when a single H is removed, the surface no longer preserves the stability of magnetic state with only 0.04 eV energy difference between the two magnetic states. To understand the change in the magnetic stability of the hydroxylated surface, the DOS [shown on Fig. 3(a)] is analyzed for the two surfaces. The dissociative chemisorption of H2 O introduces a completely filled gap state near the valence band edge. With the removal of H, this gap state loses its spin degeneracy and only an unoccupied down spin state is observed near the VBM. Since the empty gap state is very close to the VBM, an electron can easily excite into the spin-unpaired state even at low temperatures and induce magnetic instability—ultimately resulting in SQ flux noise. To understand the physical origin of the gap state, Bader’s atoms-in-molecules charge analysis [37] is performed to decompose the electron density and analyze approximate partial charges of the atoms. We find that the OH adsorption on -Al2 O3 (without neighboring H)
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FIG. 3 (color online). (a) Electronic density of states of -Al2 O3 (001) surface during H2 O adsorption process: perfect surface (top), surface with H2 O dissociatively chemisorbed (middle), and surface with only OH attached (bottom). (b) Difference of valence electron charge density for OH attached case with respect to the isolated noninteracting systems; the isosurfaces are plotted at 0:05e (blue) and þ0:3e (orange) (c) Spatial distribution of gap state created by OH adsorption; the isosurface of the gap state is shown at the level of 0:1e. The spatial location of electron deficiency on oxygen ions [blue isosurface in (b)] is identical to that of the gap state [cyan isosurface in (c)].
reduces the electron population of the first nearest neighbor O atoms by 0:1e and of the second nearest neighbor O atoms by 0:04e. Moreover, the spatial distribution of the electron deficiency is identical to that corresponding to the gap state [see Figs. 3(b) and 3(c)], indicating a correlation between electron deficiency on O atoms and surface magnetism. Expanding on this observation, we investigate the effects of 10 additional adsorbates capable of introducing different amounts of electron deficiency on the surface O atoms. We find that different terminating functional groups on the -Al2 O3 surface show noticeable differences in both stability of magnetic state and the electron deficiency on O ions. Figure 4 shows a general inverse correlation between the energy difference of the two lowest magnetic states and the average induced electron deficiency on the nearest O atoms upon adsorption of the different species. (Note that in each case, there is no terminating H on the adjacent O atom, for comparison to the OH case discussed above.) Weak electrophilic groups, such as SiH3 , CH3 and NH2 , withdraw less electron density from the -Al2 O3 surface and thus lead to lower levels of electron deficiency on the O atoms (0:02–0:04e) resulting in an enhanced stability of magnetic state of 0.2 eV or higher. In contrast, adsorbates showing strong electrophilic functionality lead to small energy differences between the two magnetic states (< 0:05 eV). Thus, strong correlation between electron deficiency on O neighbors and the stability of the magnetic state is observed. Our results indicate that magnetic instability arises from a gap state created by electron deficiency on O 2p orbitals,
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FIG. 4 (color online). Stability of magnetic state of adsorbate functional groups on -Al2 O3 (001) surface show an inverse correlation with the induced electron deficiency on neighboring O atoms.
whose depth relative to the VBM depends on the electrophilicity of the adsorbed functional group. Hence, weakly electrophilic functional groups can alleviate magnetic instability even when the adjacent H is missing and thus might serve as a surface passivation strategy to mitigate flux noise. Furthermore, since the presence and density of extrinsic defects (adsorbates) on the surface depends on external environmental conditions and can fluctuate with thermal cycles, which is not the case for intrinsic defects, the properties of surface adsorbates and their coverage can also explain the experimentally observed time dependent fluctuation of coherence time with operation of SQ devices [38]. These observations suggest that several methods may be possible to mitigate magnetic flux noise of SQs on an -Al2 O3 substrate. First, one can attempt to fully passivate the surface with weak electrophilic molecules. If chosen so that the binding energy of the passivant is high compared to temperatures exposed during thermal cycling and higher than potentially competing strong electrophilic molecules in the environment, such as H2 O, then the surface can further be stabilized against degradation over the operating lifetime. Our DFT study predicts that NH2 may be a good candidate for a substitutional passivant of environmental OH. (See Fig. 4 and Fig. S7 in [28].) Second, we note that an external electrical bias applied normal to the surface would reduce the electron deficiency at the surface through surface band bending. The field can accumulate electrons at the surface and thus alleviate the effect of electrophilic molecules. Finally, chemical means to inactivate the surface for dissociative adsorption of external molecules may be considered. In particular, previous work has reported that water can etch away surface Al ions and create a completely H/OH terminated surface [39]. Our calculations predict that the fully H/OH terminated -Al2 O3 (001) surface is both thermodynamically stable (with respect to desorption) and magnetically stable with an energy difference of 2.14 eV between the two lowest
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magnetic states. Therefore, magnetic flux noise in SQs may be mitigated by using a deionized water etch to prepare a fully H/OH terminated surface, although longterm stability of the surface passivation may still remain an issue with thermal cycling in this case. In summary, our study sheds light on the physical origin of a likely magnetic noise source on the -Al2 O3 (001) surface used for fabrication of SQs. Intrinsic vacancies are found not to create low-energy magnetic states that can couple to applied magnetic flux. Various surface adsorbates, if strongly electrophilic, do introduce low-energy magnetic states to the surface by introducing a shallow gap state associated with electron deficiency on neighboring O 2p orbitals. In addition, the origin of flux noise decoherence from these adsorbates can explain experimental observations of both disparities of coherence times and degradation or fluctuations of coherence time over the life of a SQ device. These results suggest practical chemical means to mitigate the magnetic noise and resolve present limitations of coherence time in SQs by controlling the surface chemistry and termination during SQ fabrication and operation. This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract No. DE-AC52-07NA27344, funded by the Laboratory Directed Research and Development Program at LLNL under project tracking code 12ERD-020. The LLNL Computing Grand Challenge Program is acknowledged for computational support.
*Corresponding author. [email protected] † Corresponding author. [email protected] [1] J. Q. You and F. Nori, Phys. Today 58, No. 11, 42 (2005). [2] J. Clarke and F. K. Wilhelm, Nature (London) 453, 1031 (2008). [3] J. M. Martinis, Quantum Inf. Process. 8, 81 (2009). [4] J. Zmuidzinas, Annu. Rev. Condens. Matter Phys. 3, 169 (2012). [5] B. A. Mazin, P. K. Day, J. Zmuidzinas, and H. G. Leduc, AIP Conf. Proc. 605, 309 (2002). [6] P. K. Day, H. G. LeDuc, B. A. Mazin, A. Vayonakis, and J. Zmuidzinas, Nature (London) 425, 817 (2003). [7] D. R. Schmidt, C. S. Yung, and A. N. Cleland, Appl. Phys. Lett. 83, 1002 (2003). [8] M. Hatridge, R. Vijay, D. H. Slichter, J. Clarke, and I. Siddiqi, Phys. Rev. B 83, 134501 (2011). [9] F. C. Wellstood, C. Urbina, and J. Clarke, Appl. Phys. Lett. 50, 772 (1987). [10] R. C. Bialczak et al., Phys. Rev. Lett. 99, 187006 (2007). [11] J. Bylander, S. Gustavsson, F. Yan, F. Yoshihara, K. Harrabi, G. Fitch, D. G. Cory, Y. Nakamura, J.-S. Tsai, and W. D. Oliver, Nat. Phys. 7, 565 (2011).
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[12] S. Sendelbach, D. Hover, A. Kittel, M. Mu¨ck, J. Martinis, and R. McDermott, Phys. Rev. Lett. 100, 227006 (2008). [13] S. Sendelbach, D. Hover, M. Muck, and R. McDermott, Phys. Rev. Lett. 103, 117001 (2009). [14] S. M. Anton et al., Phys. Rev. Lett. 110, 147002 (2013). [15] R. H. Koch, D. P. DiVincenzo, and J. Clarke, Phys. Rev. Lett. 98, 267003 (2007). [16] R. de Sousa, Phys. Rev. B 76, 245306 (2007). [17] L. Faoro and L. B. Ioffe, Phys. Rev. Lett. 100, 227005 (2008). [18] Z. Chen and C. C. Yu, Phys. Rev. Lett. 104, 247204 (2010). [19] J. S. Wu and C. C. Yu, Phys. Rev. Lett. 108, 247001 (2012). [20] S. Choi, D. H. Lee, S. G. Louie, and J. Clarke, Phys. Rev. Lett. 103, 197001 (2009). [21] A. Megrant et al., Appl. Phys. Lett. 100, 113510 (2012). [22] M. P. Weides, J. S. Kline, M. R. Vissers, M. O. Sandberg, D. S. Wisbey, B. R. Johnson, T. A. Ohki, and D. P. Pappas, Appl. Phys. Lett. 99, 262502 (2011). [23] J. B. Chang et al., Appl. Phys. Lett. 103, 012602 (2013). [24] H. Bluhm, J. A. Bert, N. C. Koshnick, M. E. Huber, and K. A. Moler, Phys. Rev. Lett. 103, 026805 (2009). [25] J. F. Cochran and D. E. Mapother, Phys. Rev. 111, 132 (1958). [26] X. G. Wang, A. Chaka, and M. Scheffler, Phys. Rev. Lett. 84, 3650 (2000). [27] J. Guo, D. E. Ellis, and D. J. Lam, Phys. Rev. B 45, 13 647 (1992). [28] See Supplemental Material at http://link.aps.org/ supplemental/10.1103/PhysRevLett.112.017001 for computational details and additional data. [29] N. D. M. Hine, K. Frensch, W. M. C. Foulkes, and M. W. Finnis, Phys. Rev. B 79, 024112 (2009). [30] S. K. Mohapatra and F. A. Kroger, J. Am. Ceram. Soc. 61, 106 (1978). [31] S. B. Zhang and J. E. Northrup, Phys. Rev. Lett. 67, 2339 (1991). [32] R. DeSalvo, A. A. Said, D. J. Hagan, E. W. Van Stryland, and M. Sheik-Bahae, IEEE J. Quantum Electron. 32, 1324 (1996). [33] J. M. Schneider, A. Anders, B. Hjorvarsson, I. Petrov, K. Macak, U. Helmersson, and J.-E. Sundgren, Appl. Phys. Lett. 74, 200 (1999). [34] J. M. Martinis et al., Phys. Rev. Lett. 95, 210503 (2005). [35] K. C. Hass, W. F. Schneider, A. Curioni, W. Andreoni, Science 282, 265 (1998). [36] Our DFT results show that the formation of H2 molecule from two adsorbed H atoms is 0.06 eV exothermic with 0.67 eV of energy barrier on -Al2 O3 surface. See Fig. S6 for details. [37] R. F. W. Bader and T. T. Nguyendang, Adv. Quantum Chem. 14, 63 (1981). [38] Our case study on completely OH passivated -Al2 O3 surface shows that multiple OH adsorbates create nearly degenerate gap states near the VBM and lead to lowenergy fluctuation of spin state over the range of coverages. [39] K. C. Hass, W. F. Schneider, A. Curioni, and W. Andreoni, J. Phys. Chem. B 104, 5527 (2000).
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Identification of the Local Sources of Paramagnetic Noise in Superconducting Qubit Devices Fabricated on -Al2 O3 Substrates Using Density-Functional Calculations Donghwa Lee,* Jonathan L. DuBois, and Vincenzo Lordi† Condensed Matter and Materials Division, Lawrence Livermore National Laboratory, Livermore, California 94550, USA (Received 20 May 2013; published 7 January 2014) Effective methods for decoupling superconducting qubits (SQs) from parasitic environmental noise sources are critical for increasing their lifetime and phase fidelity. While considerable progress has been made in this area, the microscopic origin of noise remains largely unknown. In this work, first principles density functional theory calculations are employed to identify the microscopic origins of magnetic noise sources in SQs on an -Al2 O3 substrate. The results indicate that it is unlikely that the existence of intrinsic point defects and defect complexes in the substrate are responsible for low frequency noise in these systems. Rather, a comprehensive analysis of extrinsic defects shows that surface aluminum ions interacting with ambient molecules will form a bath of magnetic moments that can couple to the SQ paramagnetically. The microscopic origin of this magnetic noise source is discussed and strategies for ameliorating the effects of these magnetic defects are proposed. DOI: 10.1103/PhysRevLett.112.017001
PACS numbers: 74.78.Na, 03.65.Yz, 71.15.Mb, 74.50.+r
Superconducting qubits (SQs) represent a promising route to achieving a scalable quantum computer [1,2]. However, the coupling between SQs and (as yet largely unidentified) parasitic noise sources has so far limited the functionality of current SQs by reducing the coherence time of the quantum states below a practical threshold for measurement and manipulation [3]. These noise sources also more generally limit the performance of a variety of devices based on superconducting circuits, including high energy-resolution cryogenic radiation detectors, sensitive magnetic detectors (superconducting quantum interference devices, or SQUIDs), nanomechanical motion sensors, and quantum-limited parametric amplifiers [4–8]. Magnetic flux noise with an approximate 1=f spectral density is known to be one of the dominant contributors to loss of fidelity in both SQUIDs [9] and SQs [10,11]. A number of experimental studies [10,12] clearly implicate surface spins as a significant contributing factor to this noise. In addition, recent experimental observations suggest that surface spin clusters and/or correlated fluctuations of effective magnetic moments contribute significantly to flux noise [13,14]. Several plausible mechanisms for coupling of localized spin states to SQs consistent with experimentally observed noise spectral density have been proposed [15–18]. Koch et al. proposed that flux noise is generated by electrons with fixed moments hopping between localized traps [15]. De Sousa attributed the noise to spin flips of paramagnetic dangling bonds interacting with tunneling two level systems [16]. Faoro and Ioffe have shown that spin diffusion facilitated by the RKKY interactions at the superconductor-insulator interface could account for the observed noise [17], while Yu et al. have shown that the presence of a spin glass [18] or electron spin exchange via the hyperfine interaction [19] could be responsible. 0031-9007=14=112(1)=017001(5)
While fluctuations of the surface spins or their interactions give a successful description of 1=f flux noise, the microscopic origin of the electron spins remains unclear. Choi et al. proposed that metal induced gap states (MIGS) at the interface between the superconducting lead and the dielectric substrate could be a source of magnetic spin fluctuation [20]. While MIGS represent a plausible mechanism contributing to flux noise and several experimental efforts to modify the metal-substrate interface have shown improved noise performance [21–23], there are remaining limits to the observed coherence time that indicate the existence of additional significant contributors to flux noise [23,24]. In this work, we focus on building a comprehensive picture of possible microscopic magnetic noise sources on the substrate surface, independent of a specific mechanism for spin flips, in order to identify fabrication and design procedures with the potential to further reduce flux noise in these systems. Sapphire (-Al2 O3 ) is commonly used as a substrate for fabrication of SQs since it provides several advantages, namely, (i) high thermal conductivity, (ii) high dielectric constant, (iii) chemical inertness, and (iv) a coherent interface with Al, which has a workable superconducting transition temperature (Tc ¼ 1:2 K) [25]. In addition, aluminum oxide naturally grows at the surface of the Al superconductor during the fabrication of SQs. We have therefore carried out a systematic theoretical study of possible microscopic origins of surface spin states associated with the surface of -Al2 O3 and investigate atomic configurations and chemical effects that lead to nearly degenerate spin states on the surface. Both intrinsic and extrinsic defects are considered to identify sources of noise that either limit or cause temporal degradation of SQ coherence. Bulk -Al2 O3 has the R3c crystal structure including 30 atoms in a nonprimitive hexagonal unit cell. Six oxygen
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octahedral cages are stacked along the longitudinal direction and the octahedra are filled in the sequence of Al–Al– vacancy, as shown in Fig. 1. The -Al2 O3 (001) surface is modeled using density functional theory (DFT) with a slab consisting of six layers of the oxygen plane [26]. The plane between two Al layers, identified as the lowest energy cleavage plane [27], is chosen for the surface termination. ˚ of A 4 4 supercell was constructed with at least 12 A vacuum space to prevent interactions between periodic images. Further details of our simulation methods are provided in the Supplemental Material [28]. Figure 1 shows the relaxed atomic structure of the -Al2 O3 (001) surface. Surface reconstruction drives the less coordinated surface Al ions toward the oxygen plane, yielding a nearly flat surface with the Al ions sitting in the center of oxygen triangles. To evaluate the magnetic state of a given surface (i.e., whether it introduces a paramagnetic state that can contribute to flux noise), we calculate the energy difference between the two lowest magnetic states using spinconstrained DFT. For any given system, either a spin degenerate or spin nondegenerate state may be the ground state. A large difference in energy between magnetized and unmagnetized electronic states indicates that the state is magnetically stable. These states can only contribute significantly to 1=f flux noise in the presence of an additional spin-spin correlation such as RKKY or hyperfine interaction [17–19]. In contrast, a small energy difference indicates that relatively small changes in the microscopic environment could lead to near degeneracies between nearby magnetic states leading to 1=f noise from direct tunneling processes. This is in contrast to direct thermal excitation processes, which decay monotonically toward the low device operating temperatures. In our analysis, we have assumed that states with energy differences below 0:05 eV can be taken to be within the accuracy of the methods and the material model we have employed, and
FIG. 1 (color online). Schematic view of the -Al2 O3 surface. (a) Side view showing stacking of O octahedra with Al-Alvacant occupation along each column. (b) Top view showing both Al and O atoms on the surface layer. (c) Schematic view of structural location of each vacancy site.
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thus can contribute to tunneling between nearly degenerate magnetic states. The DFT calculations predict that the nondefective -Al2 O3 (001) surface is magnetically stable; the energy difference between the lowest two spin states is 2.21 eV, which is very high compared to the operating temperature of the SQ. Thus, the bare surface does not contribute flux noise to the SQ, and it is reasonable to seek the physical origin of magnetic instabilities from defect states near the surface. As a binary system, -Al2 O3 may exhibit two types of vacancies or interstitials as intrinsic defects. Prior work [29] has reported that defect formation energies (DFEs) of intrinsic defects are quite high in bulk -Al2 O3 , and thus these defects are not present in significant concentrations. However, the DFEs may vary considerably at the surface compared to the bulk. Both experiment and calculation suggest that the Schottky defect pair is more stable than either Frenkel pair [30]; thus, here we focus on investigating the effect of surface vacancies on the magnetic state of the -Al2 O3 surface. The two simple surface vacancies VAl and VO are investigated, which in their fully charged states are 000 and doubly positively charged triply negatively charged VAl VO , respectively. We also consider three defect complexes: 000 þ V , the ‘‘reconfigured Al vacancy’’ the ‘‘divacancy’’ VAl O ::: 000 000 þ 2VAl þ Ali , and the ‘‘reconfigured divacancy’’ 2VAl ::: Ali þ VO . [See Fig. 1(c) and Fig. S2 in [28]]. The thermodynamic stability of each surface defect is evaluated by computing the DFEs at the surface, using the Zhang-Northrup [31,28] formalism. Since the PBE functional significantly underestimates the band gap of -Al2 O3 (6.18 eV compared to the experimental value of 8.7 eV [32]), we employed the PBE0 hybrid functional, which mixes in 25% Hartree-Fock exchange, to calculate the DFEs without the band gap error. (The PBE0 band gap is 8.82 eV; see Fig. S3 [28] for a comparison of PBE and PBE0. Also note that the surface band gap is 1:8 eV smaller than the bulk band gap.) Figure 2(a) shows the DFEs of the different surface vacancies as a function of Fermi energy for the O-rich condition. Although lower than in the bulk [29], moderately high DFEs are observed for the 000 , various surface vacancies. Even the reconfigured VAl 000 whose DFE is 2.13 eV lower than that of VAl , still shows more than 1 eVof formation energy in the middle of the gap. Since vacancies are sufficiently stabilized at the surface to exist in appreciable concentrations there, the effect of these intrinsic defects on the magnetic state of -Al2 O3 surface warrants further investigation. Figure 2(b) shows the energy difference between the two lowest magnetic states of the -Al2 O3 surface with each intrinsic defect present, both in their fully charged states as well as several 000 and additional charge states. The results show that both VAl VO in the fully charged states preserve large energy differ000 ences between the two magnetic states: 0.62 eV for VAl and 1.14 eV for VO ; much too large for thermal fluctuations to
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FIG. 2 (color online). (a) Defect formation energy (DFE) and (b) energy difference between two lowest magnetic states for various intrinsic defects on the -Al2 O3 (001) surface. The DFEs are calculated at oxygen rich condition and T ¼ 0 K using PBE0. Charge states are indicated in parentheses. For defect complexes, the total charge is indicated.
cause flux noise in a SQ. Variation in the energy difference is evident for the different charge states, and reversal of the magnetic nature of the ground state is seen for odd numbers of electrons, but in all cases the energy difference is still large enough (> 0:26 eV) to prevent thermal excitation. 000 þ VO Similarly, the surface with a divacancy defect VAl also preserves the stability of the magnetic state of the pristine -Al2 O3 regardless of its charge state. However, the surface with the reconfigured Al vacancy (2VAl þ Ali ) behaves differently. The nonmagnetic ground state of the reconfigured Al vacancy in the fully charged state (3) shifts to a paramagnetic ground state in the less charged states regardless of the number of electrons. As a result, the significant energy difference between the two magnetic states for the fully charged case (1.04 eV) is greatly reduced in the other charge states (only 0.02– 0.07 eV), which is within the expected accuracy of the DFT calculations. Thus, the reconfigured VAl can provide low-energy states to which the qubit couples on the -Al2 O3 surface at cryogenic temperatures if there is variation in its charge state. Similar behavior is observed for the reconfigured divacancy (2VAl þ Ali þ VO ): the substantial energy difference of 1.48 eV for the fully charged 1 state is lowered to 0.08 eV for the neutral charge state. Electronic density of states (DOS) are analyzed to gain insight into the paramagnetic nature of the
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reconfigured VAl . We find that the reconfigured VAl introduces a nearly continuous partially occupied DOS near the valence band edge (or near the Fermi level at low temperatures) and thus results in magnetic noise by allowing very low energy spin flips. (See Fig. S4 in [28] for more details). However, as shown in Fig. 2(a), the energetically stable defects throughout the entire Fermi energy range are either 000 the fully charged reconfigured VAl , the fully charged VO , or x the neutral VAl , all of which are magnetically stable. Although the reconfigured Al vacancy can introduce low-temperature magnetic noise into the -Al2 O3 surface when not fully charged, the thermodynamically stable fully charged state ½2VAl þ Ali 3 does not. The other thermox dynamically stable defects (VO , VAl ) are also magnetically stable. Therefore, intrinsic defects cannot account for appreciable magnetic flux noise in SQs without an additional spin-spin coupling mechanism. Moreover, the intrinsic vacancy cannot explain the observed time-dependent fluctuation of decoherence [21], as fluctuations in the concentration of surface vacancies could only decrease over time if equilibrium is approached slowly with thermal cycling. Extrinsic defects such as OH [33,34], in contrast, have been postulated as a noise source and could account for these fluctuations. To begin our study of extrinsic defects, we first consider the adsorption of H2 O molecules on the -Al2 O3 surface, since ambient exposure is known to promote termination of the surface with hydroxyls [26]. The H2 O molecule on -Al2 O3 easily dissociates into H on the O site and OH on the Al site [35], in the geometry shown in Fig. 3(a). Our calculations show that the dissociative chemisorbed state is 0.43 eV more favorable than the molecular physisorbed state. For chemisorbed H2 O, the energy difference between the magnetic and nonmagnetic state is 1.9 eV and is, therefore, magnetically stable. However, we find that H can desorb from the surface [36], so the case when only OH is adsorbed must be considered. In fact, we find that when a single H is removed, the surface no longer preserves the stability of magnetic state with only 0.04 eV energy difference between the two magnetic states. To understand the change in the magnetic stability of the hydroxylated surface, the DOS [shown on Fig. 3(a)] is analyzed for the two surfaces. The dissociative chemisorption of H2 O introduces a completely filled gap state near the valence band edge. With the removal of H, this gap state loses its spin degeneracy and only an unoccupied down spin state is observed near the VBM. Since the empty gap state is very close to the VBM, an electron can easily excite into the spin-unpaired state even at low temperatures and induce magnetic instability—ultimately resulting in SQ flux noise. To understand the physical origin of the gap state, Bader’s atoms-in-molecules charge analysis [37] is performed to decompose the electron density and analyze approximate partial charges of the atoms. We find that the OH adsorption on -Al2 O3 (without neighboring H)
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FIG. 3 (color online). (a) Electronic density of states of -Al2 O3 (001) surface during H2 O adsorption process: perfect surface (top), surface with H2 O dissociatively chemisorbed (middle), and surface with only OH attached (bottom). (b) Difference of valence electron charge density for OH attached case with respect to the isolated noninteracting systems; the isosurfaces are plotted at 0:05e (blue) and þ0:3e (orange) (c) Spatial distribution of gap state created by OH adsorption; the isosurface of the gap state is shown at the level of 0:1e. The spatial location of electron deficiency on oxygen ions [blue isosurface in (b)] is identical to that of the gap state [cyan isosurface in (c)].
reduces the electron population of the first nearest neighbor O atoms by 0:1e and of the second nearest neighbor O atoms by 0:04e. Moreover, the spatial distribution of the electron deficiency is identical to that corresponding to the gap state [see Figs. 3(b) and 3(c)], indicating a correlation between electron deficiency on O atoms and surface magnetism. Expanding on this observation, we investigate the effects of 10 additional adsorbates capable of introducing different amounts of electron deficiency on the surface O atoms. We find that different terminating functional groups on the -Al2 O3 surface show noticeable differences in both stability of magnetic state and the electron deficiency on O ions. Figure 4 shows a general inverse correlation between the energy difference of the two lowest magnetic states and the average induced electron deficiency on the nearest O atoms upon adsorption of the different species. (Note that in each case, there is no terminating H on the adjacent O atom, for comparison to the OH case discussed above.) Weak electrophilic groups, such as SiH3 , CH3 and NH2 , withdraw less electron density from the -Al2 O3 surface and thus lead to lower levels of electron deficiency on the O atoms (0:02–0:04e) resulting in an enhanced stability of magnetic state of 0.2 eV or higher. In contrast, adsorbates showing strong electrophilic functionality lead to small energy differences between the two magnetic states (< 0:05 eV). Thus, strong correlation between electron deficiency on O neighbors and the stability of the magnetic state is observed. Our results indicate that magnetic instability arises from a gap state created by electron deficiency on O 2p orbitals,
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FIG. 4 (color online). Stability of magnetic state of adsorbate functional groups on -Al2 O3 (001) surface show an inverse correlation with the induced electron deficiency on neighboring O atoms.
whose depth relative to the VBM depends on the electrophilicity of the adsorbed functional group. Hence, weakly electrophilic functional groups can alleviate magnetic instability even when the adjacent H is missing and thus might serve as a surface passivation strategy to mitigate flux noise. Furthermore, since the presence and density of extrinsic defects (adsorbates) on the surface depends on external environmental conditions and can fluctuate with thermal cycles, which is not the case for intrinsic defects, the properties of surface adsorbates and their coverage can also explain the experimentally observed time dependent fluctuation of coherence time with operation of SQ devices [38]. These observations suggest that several methods may be possible to mitigate magnetic flux noise of SQs on an -Al2 O3 substrate. First, one can attempt to fully passivate the surface with weak electrophilic molecules. If chosen so that the binding energy of the passivant is high compared to temperatures exposed during thermal cycling and higher than potentially competing strong electrophilic molecules in the environment, such as H2 O, then the surface can further be stabilized against degradation over the operating lifetime. Our DFT study predicts that NH2 may be a good candidate for a substitutional passivant of environmental OH. (See Fig. 4 and Fig. S7 in [28].) Second, we note that an external electrical bias applied normal to the surface would reduce the electron deficiency at the surface through surface band bending. The field can accumulate electrons at the surface and thus alleviate the effect of electrophilic molecules. Finally, chemical means to inactivate the surface for dissociative adsorption of external molecules may be considered. In particular, previous work has reported that water can etch away surface Al ions and create a completely H/OH terminated surface [39]. Our calculations predict that the fully H/OH terminated -Al2 O3 (001) surface is both thermodynamically stable (with respect to desorption) and magnetically stable with an energy difference of 2.14 eV between the two lowest
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magnetic states. Therefore, magnetic flux noise in SQs may be mitigated by using a deionized water etch to prepare a fully H/OH terminated surface, although longterm stability of the surface passivation may still remain an issue with thermal cycling in this case. In summary, our study sheds light on the physical origin of a likely magnetic noise source on the -Al2 O3 (001) surface used for fabrication of SQs. Intrinsic vacancies are found not to create low-energy magnetic states that can couple to applied magnetic flux. Various surface adsorbates, if strongly electrophilic, do introduce low-energy magnetic states to the surface by introducing a shallow gap state associated with electron deficiency on neighboring O 2p orbitals. In addition, the origin of flux noise decoherence from these adsorbates can explain experimental observations of both disparities of coherence times and degradation or fluctuations of coherence time over the life of a SQ device. These results suggest practical chemical means to mitigate the magnetic noise and resolve present limitations of coherence time in SQs by controlling the surface chemistry and termination during SQ fabrication and operation. This work was performed under the auspices of the U.S. Department of Energy by Lawrence Livermore National Laboratory under Contract No. DE-AC52-07NA27344, funded by the Laboratory Directed Research and Development Program at LLNL under project tracking code 12ERD-020. The LLNL Computing Grand Challenge Program is acknowledged for computational support.
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