Electron spin coherence of shallow donors in natural and isotopically enriched germanium
Germanium is a widely used material for electronic and optoelectronic devices and recently it has become an important material for spintronics and quantum computing applications. Donor spins in silicon have been shown to support very long coherence times () when the host material is isotopically enriched to remove any magnetic nuclei. Germanium also has non-magnetic isotopes so it is expected to support long s while offering some new properties. Compared to Si, Ge has a strong spin-orbit coupling, large electron wavefunction, high mobility, and highly anisotropic conduction band valleys which will all give rise to new physics. In this Letter, the first pulsed electron spin resonance (ESR) measurements of and the spin-lattice relaxation () times for As and P donors in natural and isotopically enriched germanium are presented. We compare samples with various levels of isotopic enrichment and find that spectral diffusion due to Ge nuclear spins limits the coherence in samples with significant amounts of Ge. For the most highly enriched samples, we find that limits to . We report an anisotropy in and the ensemble linewidths for magnetic fields oriented along different crystal axes but do not resolve any angular dependence to the spectral-diffusion-limited in samples with Ge.
Germanium was the original material for transistors, and is now being developed for the latest semiconductor electronics Lee et al. (2005). Recently, it has become a key material for spintronics Li et al. (2013); Dushenko et al. (2015); Shen et al. (2010) and quantum computingVrijen et al. (2000); Rahman et al. (2009); Witzel et al. (2012) devices. Compared to silicon, donor electrons in Ge have higher mobility ( times)Lee et al. (2005), larger wavefunctions ( nm compared to nm), Wilson (1964); Feher (1959), stronger spin-orbit couplingLiu (1962), and highly anisotropic conduction band valleys Rahman et al. (2009). Much of silicon’s success in the quantum computing community has hinged on the attainability of long coherence times () exceeding seconds when Si is isotopically enriched to have no magnetic nuclei Morton et al. (2011); Tyryshkin et al. (2012); Wolfowicz et al. (2013); Itoh and Watanabe (2014). Germanium also has non-magnetic isotopes so it has been expected to support long coherence times. In this Letter, we report the first electron spin coherence measurements for donor electron spins in Ge. We find that spectral diffusion due to Ge limits in natural Ge samples whereas the spin-lattice relaxation time, , limits in isotopically enriched Ge. The longest we measured is ms at 350 mK in a magnetic field () of 0.44 T. The low-temperature fits the temperature dependence theorized by Roth Roth (1960) and Hasegawa Hasegawa (1960) which also predicts . This suggests that considerably longer coherence times are possible at lower fields.
While for donors in Ge is shorter than the times demonstrated for Si, Ge-based qubits have some important advantages. For example the larger electron wavefunctions relax the lithographic requirements for exchange coupling two donors which is important for most donor-based quantum computing schemes Kane (1998). This is advantageous considering Ge is compatible with most of the same nanofabrication techniques as silicon and single-donor devices are achievable Scappucci et al. (2011). Another useful feature of Ge is the large spin-orbit coupling and shallow donor depth which leads to a very large spin-orbit Stark shift in Ge (nearly 5 orders of magnitude larger than in silicon) Rahman et al. (2009) meaning that Ge based qubits are extremely tuneable. This will be important for gated quantum devicesKane (1998).
Despite these features, the spin coherence of donor electrons in Ge has remained mostly unstudied. The first experiments were conducted over fifty years ago by Feher, Wilson, and Gere Feher et al. (1959); Wilson (1964), but their measurements were limited to continuous wave (CW) ESR spectroscopy. They estimated for As and P donors based on power saturation measurements, but experimental errors were large. These experiments are difficult because wavefunction overlap occurs for densities as low as donors/cm such that only lightly doped samples with correspondingly weak signals are useful for isolated donor experiments. Some limited experiments on Sb Pontinen and Sanders (1966); Hale et al. (1975) and P Morigaki and Mitsuma (1963) donors in highly strained Ge were also reported. More recently, pulsed nuclear magnetic resonance studies were conducted on Ge nuclear spins Verkhovskii et al. (1999, 2003); Panich et al. (2007) which found that the Ge nuclear spin coherence in germanium can be .
The samples discussed in this Letter include commercially available, natural Ge doped either As/cm or P/cm. Ge is the only naturally occurring isotope of Ge (7.75% abundance) with a nuclear spin and is thus expected to be a limiting factor in the donor spin coherence at low temperatures. Three isotopically enriched samples were prepared at Lawrence Berkeley National Laboratory. The first is a piece of neutron transmutation doped Ge described in Ref.Itoh et al. (1993, 1994). This sample is uniformly doped with As to a density of donors/cm and contains a residual 3.8% Ge. The other two samples are 96% Ge crystal (0.1% Ge) and a 99.99% Ge crystal (0.01% Ge). They have P concentrations of donors/cm and donors/cm, respectively and are described in Itoh et al. (1993); Asen-Palmer et al. (1997). The crystallographic orientation of the samples was determined using X-ray diffraction. The sample details are summarized in Table 1.
|Sample Name||Ge||Ge||Ge||Ge||Ge||Doping (cm|| Linewidth (mT)||(ns)|
Percent abundances for the natural germanium samples were taken from Ref. Berglund and Wieser (2011)
The experiments down to 1.65 K were performed in a pumped He cryostat (H.S. Martin), and lower temperature data were obtained in a He cryostat (Janis Research). All data were taken at X-band (9.65 GHz) in a Bruker dielectric resonator (MD5). The ESR spectra were measured via echo-detected field sweeps using a standard Hahn-echo pulse sequence (/2 - - - - echo). Typical spectra are shown in Fig. 1(a) for phosphorus donors in the 0.1% Ge:P sample and in Fig. 1(b) for arsenic donors in the 3.8% Ge:As sample. From these plots we extract a hyperfine coupling constant of 3.55 mT for As and 2.04 mT for P.
The ESR linewidth depends strongly on the sample orientation and the abundance of Ge present in the sample, as noted by Wilson Wilson (1964). With oriented along one of the directions, the linewidth is narrowest and is limited primarily by hyperfine interactions with Ge. At this orientation the line broadening from spin-orbit strain effects is suppressed by valley symmetry about the as explained in Refs.Wilson (1964); Roth (1960); Hasegawa (1960)). To give a sense of the strain-induced line broadening for away from 001 equivalent directions, Fig.1(c) shows the angular dependence of the linewidth for select samples rotated in the (10) plane relative to the axis. There is also an isotopic dependence of the linewidth away from the direction and we presume this is due to isotopic strain Stegner et al. (2010). The strong dependence of the linewidth on field orientation conveniently allows for accurate in situ orientation of the crystals. Unless otherwise noted, all data presented in this manuscript assumes is oriented along a axis.
One can predict the effect of Ge on the ESR linewidth through the hyperfine interaction using a second moment calculation Kittel and Abrahams (1953), which gives , where is the linewidth, and is the percent abundance of Ge. The measured ESR linewidths for samples of various isotopic enrichment with is shown in Fig. 1(d). The point at = 0.8% was taken from Wilson Wilson (1964). The solid curve in Fig. 1(d) gives the expected dependence for broadening of the line due to Ge hyperfine interactions for As. The solid curve fits the data well, implying that Ge is indeed the dominant mechanism for line broadening in this orientation. The linewidth can be interpreted as an ensemble dephasing time, , which is also shown in Table 1.
was measured using an inversion-recovery pulse sequence ( - t - /2 - - - - echo). The values of are plotted in Fig. 2 for P(a) and As(b) donors. The same two mechanisms limit for all of the samples. At higher temperatures, is limited by a highly temperature () dependent process. The theory of Roth and Hasegawa Roth (1960); Hasegawa (1960) predicted a Raman process to dominate at these temperatures but this dependence does not fit our data well. An Orbach process does fit the data as shown in Fig. 2. The Orbach process is of the form , where is a prefactor that can be calculated using Ref. Castner (1967), is the valley-orbit splitting, and is the Boltzmann constant. The valley-orbit splittings extracted from the fits in Fig. 2 agree well with the values measured by Ramdas ( meV for P and meV for As Ramdas and Rodriguez (1981)). Likewise, the values of extracted from our fits agree with the values calculated using Castner’s theory Castner (1967) to within a factor of 2.
At lower temperatures, a single-phonon process with a dependence appears to dominate. This relaxation process is a result of the multivalley structure of germanium. In the unperturbed ground state, there are four degenerate valleys located along the equivalent crystallographic axes. Each valley has an axially symmetric g-tensor, but the effective g-tensor, , is given as a weighted average over all four valleys. In the electron ground state, each valley has equal amplitude, and, by symmetry, is isotropic Roth (1960). When strain is applied, valley energies shift relative to each other, leading to valley repopulation and a change in . The strain from phonons near the Larmor frequency modulates , effectively mixing the spin up and down states. This gives a as calculated by Roth Roth (1960) and Hasegawa Hasegawa (1960) which agrees well with our experimental data. The calculated estimates for at 350 mK are within 10% for Ge:As and 30% for Ge:P. The theory predicts that due to this single-phonon process should scale with the square of the anisotropy. The valley anisotropy of Ge was measured to be 3 orders of magnitude larger than in Si Wilson (1964), implying that the single-phonon process should be 6 orders of magnitude stronger in germanium. This accounts for the short times observed for donors in germanium as compared with silicon.
An interesting property of the single-phonon spin-lattice relaxation mechanism is an anisotropy in predicted by the Roth-Hasegawa theoryRoth (1960); Hasegawa (1960). The 3.8% Ge:As crystal was rotated in the plane at 1.8 K, and the resulting is plotted in Fig.3. The theory predicts that, for rotation in this plane, the spin-lattice relaxation is given by:
where is a scaling factor which can be calculated following HasegawaHasegawa (1960), and is the field orientation relative to . Hasegawa calculated for arsenic in Ge, but a fit to the data reveals . We observe that for oriented along a axis, becomes 3 times longer than along .
We note that for donors in highly enriched samples is shorter than it is for donors in the natural material as seen in Fig. 2(b). This effect is still under investigation, but one possible mechanism is the presence of isotopic strain in the natural germaniumStegner et al. (2010). Wilson Wilson (1964) demonstrated the use of large strains to partially lift the valley degeneracy, thus disrupting the single-phonon relaxation mechanism. Modelling the effects of strain can be complex, as strain not only modulates , but can also modify the form of Eq.(1). Nevertheless, controlled strain may be beneficial for future quantum devices based on germanium.
We also measured the electron spin coherence time, , for each of the samples using the standard Hahn-echo pulse sequence. The decay curves at 1.8 K for are shown in Fig. 4(a) for Ge:P and in Fig. 4(b) for Ge:As. These decays are fit to an exponential decay of the form , where scales the amplitude, is the delay between the and pulses in the Hahn echo sequence, and is a fitting parameter that depends on the decoherence mechanism. The 0.1% Ge:P sample decays with =1 over the measured temperature range. For this sample it was found that (representing the absolute limit Schweiger and Jeschke (2001)) down to 350 mK temperatures, meaning that decoherence due to Ge is negligibly small with this level of isotopic enrichment at these temperatures. For samples with , we find that varies from 1 at high temperatures to 2.1 at low temperatures. This is a characteristic of Ge spectral diffusion limiting the coherence. At 1.8 K, the Ge:As, Ge:P, and 3.8% Ge:As samples decay with this form.
The temperature dependence of is also plotted in Fig. 2 and fit to , where is the (temperature independent) spectral-diffusion-limited coherence time. For the natural germanium samples, limits the coherence to 57 s whereas the 3.8% Ge:As sample is limited to 113 s. From similar work in silicon Abe et al. (2010); de Sousa and Das Sarma (2003), one might expect an orientation dependence to . We measured the orientation dependence of for the 3.8% Ge:As sample at 1.8 K and fit the decays with a curve of the form to separate the component from Mims et al. (1961). No angular dependence of could be resolved.
While coherence times of over one millisecond for isotopically enriched material open the possibility of using donor electrons in Ge for quantum computing devices, these coherence times are much shorter than those for donors in isotopically enriched silicon (seconds) Tyryshkin et al. (2012); Wolfowicz et al. (2013). To extend the Ge donor coherence, one must either overcome the limit or use nuclear spins which may support longer coherence times. There are several promising techniques to extend the limit. One approach is to take advantage of the anisotropy, which will allow for up to a factor of 3 increase in when devices are oriented with , but this enhancement comes at the expense of a shorter ensemble . A simple alternative is to operate devices at lower temperatures, since . Perhaps the most effective technique is to operate devices at lower frequencies since theory predicts . More complicated strategies are also available. In particular, one can apply a large strain, as demonstrated by Wilson Wilson (1964) which shifts the valley energy levels, thus suppressing valley repopulation and the associated relaxation mechanisms. Another recent proposal suggests patterning Ge in a periodic structure to open a phononic bandgap at the Larmor frequencySmelyanskiy et al. (2014). Such a structure would suppress the single phonon process.
In summary, we have measured the ESR linewidths, coherence times, and spin-lattice relaxation times for donors in natural and isotopically enriched germanium at X-band microwave frequencies. We find that the linewidths are primarily broadened by hyperfine interactions with Ge spins when is oriented along the 001 axis and by strain in other orientations. We find that donor electron spin coherence is limited by spectral diffusion due to hyperfine interactions with Ge nuclei for the Ge () and 3.8% Ge:As () samples, thus scales approximately as which is similar to siliconde Sousa and Das Sarma (2003). For the more highly enriched 0.1% Ge:P sample, was limited to down to 350 mK, the lowest temperature we have measured ( ms for ). We observe a large anisotropy in , which is explained by the theory of Roth and HasegawaRoth (1960); Hasegawa (1960), with the longest occuring for . It is predicted that at lower magnetic fields and thus should become substantially longer.
Acknowledgements.Work at Princeton was supported by the NSF through the Materials World Network and MRSEC Programs (Grant No. DMR-1107606 and DMR-01420541), and the ARO (Grant No. W911NF-13-1-0179). The work at Keio has been supported by the Core-to-Core Program by JSPS, and the Grants-in-Aid for Scientific Research, and Project for Developing Innovation Systems by MEXT.
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