ARTICLES PUBLISHED ONLINE: 6 APRIL 2014 | DOI: 10.1038/NMAT3941
Spatially resolving valley quantum interference of a donor in silicon J. Salfi1, J. A. Mol1, R. Rahman2, G. Klimeck2, M. Y. Simmons1, L. C. L. Hollenberg3 and S. Rogge1* Electron and nuclear spins of donor ensembles in isotopically pure silicon experience a vacuum-like environment, giving them extraordinary coherence. However, in contrast to a real vacuum, electrons in silicon occupy quantum superpositions of valleys in momentum space. Addressable single-qubit and two-qubit operations in silicon require that qubits are placed near interfaces, modifying the valley degrees of freedom associated with these quantum superpositions and strongly influencing qubit relaxation and exchange processes. Yet to date, spectroscopic measurements have only probed wavefunctions indirectly, preventing direct experimental access to valley population, donor position and environment. Here we directly probe the probability density of single quantum states of individual subsurface donors, in real space and reciprocal space, using scanning tunnelling spectroscopy. We directly observe quantum mechanical valley interference patterns associated with linear superpositions of valleys in the donor ground state. The valley population is found to be within 5% of a bulk donor when 2.85 ± 0.45 nm from the interface, indicating that valley-perturbation-induced enhancement of spin relaxation will be negligible for depths greater than 3 nm. The observed valley interference will render two-qubit exchange gates sensitive to atomic-scale variations in positions of subsurface donors. Moreover, these results will also be of interest for emerging schemes proposing to encode information directly in valley polarization. abrication of devices1,2 , spin readout1 and quantum control of spins3,4 in silicon has been accomplished at the singledonor level. However, addressable control and coupling within qubit arrays requires local gates and control interfaces, whose atomic-scale potentials strongly influence electronic valley degrees of freedom5–16 . Although these unconventional orbital degrees of freedom play no role in conventional silicon microelectronics, they invariably arise in quantized states in indirect gap materials, and are pervasive in quantum electronics. In silicon, valley physics determines qubit relaxation rates13,17,18 and are predicted to strongly influence two-qubit gates6,7 . Moreover, valley degrees of freedom in AlAs (ref. 19), silicon20 , diamond21 and graphene22 can play a role similar to spin in condensed matter systems. Encoding of information within the valley polarization has been proposed in AlAs (ref. 23), silicon24,25 and diamond21 , and within the polarization of chiral valley pseudospin in carbon-based nanostructures26–28 . Here, we report the measurement of individual states of subsurface donors using cryogenic scanning tunnelling spectroscopy (STS). Quantum mechanical valley interference patterns were observed in real space, associated with linear quantum superpositions of wavevectors in the six conduction band valleys of silicon. Enabled by the high-accuracy empirical determination of donor depth and electric field, we perform a parameterfree comparison with atomistic theory, establishing that the z-valley population is ∼38 ± 2% for a 2.85 ± 0.45 nm deep donor, perturbed by only ∼5% from the value of ∼33.3% for a donor in bulk silicon. Consequently, donors more than 3 nm deep should not experience a significant valley-repopulation-induced increase in spin-lattice relaxation. Moreover, the nearly bulk-like valley interference observed will render two-qubit exchange gates sensitive to atomic variations in donor position6,7 , even for subsurface donors. Measurements of valley quantum interference presented
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herein address spatial aspects of the established theory of shallow impurities in silicon7,10,29,30 that are essential for engineering spin qubit arrays leveraging silicon’s exceptional coherence31,32 . We employed unconventional STS-based spatially resolved single-electron transport to measure, in real space, valley interference in single quantum states of subsurface arsenic donors. Ultra-high vacuum annealing, with parameters targeted to deplete the upper ∼10 nm of donors33 , was performed on a highly arsenic-doped silicon wafer (see Methods). Spatially well-isolated residual donors (density ∼1011 cm−2 ) were found in the depletion region by subsurface imaging34 at U = −1.25 V. The electronic isolation of donor-bound states within the depletion region was quantified, as illustrated in Fig. 1a, by single-electron transport spectroscopy employing the scanning tunnelling microscope (STM) tip at 4.2 K. The measured differential conductance dI /dU above donor 1 (Fig. 1b, blue squares) exhibited single-electron tunnelling peaks, from the substrate’s impurity band35 , to the subsurface donor brought into the bias window by tip-induced band bending, to the tip. Fitting of the first two peaks to single-electron transport theory36,37 (Fig. 1b, red line) established a coupling h(Γin + Γout ) kB T ∼ 350 µeV for donor 1, similar to states in single-atom transistors11 . Here, h is Planck’s constant, kB is Boltzmann’s constant, T is the temperature, and Γin and Γout are tunnel-in and tunnel-out rates, respectively. The lowest energy peak (U ≈ −0.8 V) was found to be bound to donor 1 in the spatially resolved dI /dU (Fig. 1c). Higher energy peaks (U < −1.05 V) in Fig. 1b,c are two-electron states (Supplementary Information). Spatially resolved measurements of donor-bound states were carried out using an unconventional scheme. Constant-current imaging of single dopants (see refs 34,38) was avoided because it necessarily measures multiple states. On the other hand, measurement of dI /dU versus U on a two-dimensional spatial grid
1 Centre
for Quantum Computation and Communication Technology, School of Physics, The University of New South Wales, Sydney, New South Wales 2052, Australia, 2 Purdue University, West Lafayette, Indiana 47906, USA, 3 Centre for Quantum Computation and Communication Technology, School of Physics, University of Melbourne, Parkville, Victoria 3010, Australia. *e-mail: [email protected] NATURE MATERIALS | VOL 13 | JUNE 2014 | www.nature.com/naturematerials
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Figure 1 | Spatial measurement of single quantum states of a subsurface donor. a, Illustration of single-electron tunnelling from the conduction impurity band (CIB; ref. 35), to the subsurface donor in the depletion region with electron probability density |9(r)|2 , to the tip. b, Top: Measured dI/dU (blue squares) for a tip position above the subsurface donor, and least-squares fit (red line). Bottom: Schematic band diagram, whereby the sample bias U < 0 brings the donor-bound states into resonance with the Fermi energy in the CIB, which is spatially separated from the donor by a depletion region. c, Measured differential conductance dI/dU versus x and sample bias U, along a line passing over a single donor. Scale (from blue to red): 0–0.25 nA V−1 . The tip location for measurement in b is indicated by the vertical dashed line (green square). d, Constant current (I0 = 150 pA) topography z(x, y) recorded during the first line scan at U = −1.45 V, a bias where there is minimal evidence of a buried donor. Scale (from blue to white): 0–100 pm. e, Spatially resolved single-electron tunnelling current I(r) ∝ |9(r)|2 through the single donor-bound state, for the same area as d, but with U = −0.85 V and the feedback loop off. The signal vanishes away from the donor, as expected for a true single-state measurement. White dashed line (red diamond) denotes where data in c was obtained. Scale (from blue to white): 0–0.35 nA.
would be impractically time consuming for the spatial frequency range |kx |, |ky | ≥ 8π/a0 necessary to avoid frequency aliasing, and the desired resolution ∼2π/50a0 to resolve band structure components. We therefore employed an unconventional two-pass scan to spatially measure individual donor-bound states. In the first pass of each line, the surface topography z(x, y) was measured at U = −1.45 V. During the second pass of each line, a current I (r) proportional to the probability density |9(r)|2 for a single bound state was measured, at a bias (U2 = −0.85 V for donor 1) ensuring one state in the bias window for single-electron transport. The feedback loop was turned off for the second pass, and the tip position set to r = (x, y, z(x, y) − δz); δz = 0.25 nm closer to the sample relative to the first line (topography). The value of δz was chosen such that I (U2 ) ∼ exp(κz) ∼ Γout , or equivalently, Γout Γin . In this regime, I (r) ∝ |9(r)|2 , where r is the central coordinate of the tip apex orbital39 . For donor 1, the measured z(x, y) and I (r) ∝ |9(r)|2 are shown in Fig. 1d,e, respectively. In Fig. 1e, |9(r)|2 vanishes away from the donor, as expected for a measurement of a single bound quantum state. To investigate the valley interference of donor 1, we evaluated numerically the two-dimensional Fourier representation of the measured I (r) ∝ |9(r)|2 , shown in Fig. 2a. The vertices of the outer dashed box are reciprocal lattice vectors G2 = 2π/a0 (p, q) with p = ±1, q = ±1 and a0 = 0.543 nm. Figure 2a contains ellipsoid-shaped structures, highlighted in green, at positions k = kµ xˆ and k = kµ yˆ (where kµ = 0.85(2π/a0 )) of silicon’s conduction band minima, indicated by white crosses. We term the structure outlined in blue within k . 0.5(2π/a0 ) as the probability envelope, because it contains the lowest spatial frequencies of the probability density. Dashed outlined features centred at 2π/a0 (±1/2, ∓ 1/2) and 2π/a0 (±1/2, ±1/2) are created by the 2 × 1 reconstruction, and are related to the probability envelope and ellipsoids, respectively, as evidenced by their 2π/a0 (∓1/2, ±1/2) displacement (white arrows, Fig. 2a) relative to them. A depth z0 = (5 ± 1)a0 (20 ± 4 lattice planes) for donor 1 was empirically obtained by fitting the spatially 606
resolved spectral shift of the conduction band edge to a dielectric screened Coulomb potential37,40,41 (Supplementary Information). Shown for comparison in Fig. 2b is the Fourier transform of the vacuum tail of the ground state probability density |91 (r)|2 obtained by sp3 d 5 s∗ tight-binding (Supplementary Information). The predicted ellipsoids and probability envelope features are in very good agreement with measurements in Fig. 2a. As follows from the discussion below, the ellipsoids and probability envelope features are produced by quantum interference processes associated with quantum superpositions of wavevectors in six different valleys in the donor state. Such a R P ground state can be written as7 9i (r) = µ αµi dk03 Fµ (k0 )φkµ +k0 (r), where αµi (µ = 1...6) are the valley quantum numbers, and as illustrated in Fig. 2c, Fµ (k0 ) is the the reciprocal space distribution of the wavefunction over the Bloch functions φkµ +k0 (r) about the minima kµ . The full Fourier representation of the state is obtained by substitutingPthePrepresentation for the Bloch R functions, such that7 9i (r) = G µ dk0 3 9i (k) exp(ik · r), where 9i (k) = αµi Ak−G,G Fµ (k0 ), k = G + kµ + k0 , and G = [p, q, r] = [0, 0, 0], [1, 1, 1], [2, 0, 0], [2, 2, 0], [3, 1, 1], ... (in units of 2π/a0 ) and their equivalents are the reciprocal lattice vectors with nonzero Bloch amplitudes Ak−G,G in silicon42 . As 9(kµ ) ∝ αµ Fµ (0), the ellipsoid features in |9(k)| at k = G2 ± xk ˆ µ and k = G2 ± yˆ kµ (green ellipsoids) are present only i i when α±x 6 = 0 and α±y 6 = 0, respectively, whereas the projected i ellipsoid at G2 (yellow circle) is present only when α±z 6 = 0. For a donor in bulk silicon, the ground state is a spin degenerate valley singlet with α 1 = 6−1/2 [1, 1, 1, 1, 1, 1]. Hence, the calculated ground state Fourier amplitudes |91 (k)| presented in Fig. 2d for a depth 6.25a0 are in good agreement with the expected reciprocal space representation for a bulk-like donor with valley configuration similar to α 1 . In bulk silicon the singlet, whose ionization energy is 52 meV for an arsenic donor, lies 22 meV below a spindegenerate excited valley triplet with α 2 = 2−1/2 [1, −1, 0, 0, 0, 0], α 3 = 2−1/2 [0, 0, 1, −1, 0, 0], and α 4 = 2−1/2 [0, 0, 0, 0, 1, −1]. The NATURE MATERIALS | VOL 13 | JUNE 2014 | www.nature.com/naturematerials
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NATURE MATERIALS DOI: 10.1038/NMAT3941 a
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Figure 2 | Valley interference of a single quantum state in reciprocal space. a, Fourier amplitudes of measured I(r) ∝ |9(r)|2 for donor-bound electrons in Fig. 1. The corners of the outer dashed square are reciprocal lattice vectors 2π/a0 (p, q) with p = ±1 and q = ±1. White crosses denote kµ = 0.85(2π/a0 )(±1, 0) and kµ = 0.85(2π/a0 )(0, ±1). Ellipsoid structures are found within the green boundaries, the probability envelope is found within the blue boundaries, and 2 × 1 reconstruction-induced features are found within the grey dashed boundaries. Scale [0,0.5] (normalized). b, Calculated Fourier amplitude of the single-electron (D0 ) ground state probability density |9(r)|2 for a z0 = 6.25a0 ≈ 3.4 nm deep donor, with a 2 × 1 surface reconstruction. Scale [0,1] (normalized). c, Distribution of wavevectors about valley minima in the plane of the surface (green ellipsoids) and perpendicular to it (yellow ellipsoids), and magnification of a single valley distribution Fx (k0 ). d, Overlay of the reciprocal space model for the D0 ground state wavefunction on the atomistic calculation of the |9(k)| using the tight-binding method. Labelled wavevectors have units 2π/a0 . e, Overlay of the reciprocal space model for the D0 ground state probability density on the atomistic calculation using the tight-binding method. For d and e, a 1 × 1 reconstruction was assumed for simplicity.
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constructive valley interference of the singlet at the arsenic site (the central cell) is responsible for the energy difference29,30 . Two quantum interference processes are of particular interest in the Fourier representation of Fig. 2a and b, which maps components exp(−ik0 · r) in 9i∗ (r) and exp(ik00 · r) in 9i (r) to exp(i(k00 − k0 ) · r) in |9i (r)|2 = 9i∗ (r)9i (r). First, ellipsoids at k ≈ ±xk ˆ µ (k ≈ ±ˆy kµ ) in |9i (r)|2 , overlaying atomistic calculations in Fig. 2e, arise from cross-product terms of z valleys and x (y) valleys. Consequently, the ellipsoids in the Fourier decomposition of |9(r)|2 are direct evidence for the presence of all three valleys in the measured (Fig. 2a) and atomistically calculated (Fig. 2b) subsurface donor ground state. From the same mapping, it follows that x, y and z valleys contribute to the peak at k = 0, whereas only x and y valleys contribute to the side peak at k = 0.15(2π/a0 )(±1, ±1). The relative heights of the central and side peaks in the measured and calculated probability envelopes were compared to characterize the deviation of donor 1 from the α 1 (bulk) valley configuration. The electric field in experiments, estimated in the Supplementary Information to be Ez = −0.3 ± 1.9 MV m−1 using tip-heightdependent dI /dU measurements, is low enough to be safely neglected in calculations of donor-bound states43 . Calculations for decreasing depths 10.25a0 to 5.25a0 , presented in Fig. 3, demonstrate an increasing contribution of the central peak, signalling an increase in the z-valley population relative to the x and y valleys. This arises from the competition between the interface, which introduces a valley-orbit potential8,12,14 and depopulates the x and y valleys with lighter masses parallel to the interface, and the donor ion’s valley-orbit potential, which equalizes all six valleys. The side-peak amplitude saturates at 7.25a0 , indicating that the valleys approach the bulk configuration. Shown in Fig. 3 (green diamonds), the measured donor’s profile agrees well with calculations for 6.25a0 at zero field, in excellent agreement with the empirically determined depth z0 = (5 ± 1)a0 of donor 1. We estimate that the z-valley configuration of the measured donor differs by only ∼5% from a donor in bulk silicon (33.3%), because states calculated for 7.25a0 and 6.25a0 depths have z-valley populations of 36% and 40%, respectively (Supplementary Information). The wider central peak in measurements could be the result of enhanced lateral localization due to image charges associated with dielectric mismatch for the 3 nm deep donor, not taken into account in the theory. Overall,
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Figure 3 | Valley population. Comparison of measured (green diamonds) and theoretical (solid lines) reciprocal space profiles of |91 (r)|2 for the donor ground state, along the 110 direction in reciprocal space. The height of the central peak (k = 0) relative to the side peaks (k = 0.15(2π/a0 )(±1, ±1)) increases with decreasing depth from 10.25a0 (black line), 7.25a0 (green line), 6.25a0 (red line), to 5.25a0 (blue line), indicating repopulation from the x and y valleys into z valleys with decreasing depth. The calculated first excited state, shown for 5.25a0 (blue dashed line), populates only the z valley, and hence has no side lobes.
the agreement of the reciprocal space profile with the theoretical description from the calculations is remarkable. Note that the lowpass response of the STM tip orbital is not expected for such low spatial frequencies k . 0.21(2π/a0 ) (λ & 2.6 nm). The real-space representation of the ellipsoids, shown in Fig. 4a and obtained by digital Fourier filtering, resembles a low-frequency s-like envelope modulated by x- and y-directed standing wave patterns with wavelength λµ ≈ 2π/kµ = 0.65 nm. As described above, the ellipsoids in |91 (r)|2 arise from cross terms between the x (y) valleys, which oscillate at k = kµ xˆ (k = kµ yˆ ), and the z valley. Consequently, their real-space representation can be easily shown (Supplementary Information) to haveR the form Fz (r)(Fx (r) cos(kµ x) + Fy (r) cos(kµ y)), where Fi (r) = d3 k0 Fi (k0 ) exp(ik0 · r) is the donor ground state envelope
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NATURE MATERIALS DOI: 10.1038/NMAT3941
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Figure 4 | Valley interference of a single quantum state in real space. a, Measured real-space representation of the relative probability (normalized to the maximum) of four ellipsoid features at k ≈ kµ for donor 1 in Fig. 2, obtained by a frequency selective inverse Fourier transform of Fig. 2a. Arrow denotes the [110] dimer direction. Red cross denotes the ion position determined from the real-space ellipsoid pattern, as discussed in the Supplementary Information. b, Predicted real-space representation of four ellipsoid features based on atomistic calculations. The best match was obtained for a z0 = 6.25a0 ≈ 3.4 nm deep donor. c, Measured real-space representation of the probability envelope. d, Predicted real-space representation of a type-A probability envelope. e, Same as a, but for the measured donor 2 with a type-B probability envelope. f, Predicted real-space representation of four ellipsoid features based on atomistic calculations. The best match was obtained for a z0 = 6.75a0 ≈ 3.7 nm deep donor. g, Same as c, but for a type-B probability envelope. h, Predicted real-space representation of the probability envelope based on an atomistic calculation for f. All insets: Corresponding region in the first Brillouin zone enclosed within dashed boundary.
function. The maximum of the real-space oscillation pattern, labelled with a red cross in Fig. 4, identifies the position of the ion. Notably, the appearance of elongation along [100] and [010] directions (Fig. 4a) is also predicted by atomistic theory (Fig. 4b). The oscillations observed in Fig. 4a,b are associated with superpositions of wavevectors, not unlike Friedel oscillations of metallic surface states observed by STM at impurities and step edges44,45 . The present case differs because the superpositions occupy six valleys along three orthogonal [001] directions, rather than a two-dimensional Fermi surface. Like the ellipsoids, the probability envelope feature introduced in the discussion of Fig. 2 arises from valley interference. Therefore, it is not surprising that the real-space representation of the probability envelope in Fig. 4c has an unusual shape, characterized by a node along x = −y. This node, predicted by calculations (Fig. 4d), contrasts the simple envelope of oscillations in Fig. 4a,b. Notably, a survey of fifty subsurface donors with indistinguishable spectral signatures revealed 24 ‘type-A’ envelopes typified by donor 1 in Fig. 4c,d, and 26 ‘type-B’ envelopes typified by donor 2 in Fig. 4g,h. Whereas the ellipsoid interference pattern of the type-B donor (Fig. 4e) resembles that of the type-A donor (Fig. 4a), the probability envelope of the type-B donor (Fig. 4g) has a protrusion along x = −y. The depth of donor 2 was not measured owing to a technical problem. The type-A and type-B probability envelopes are at first peculiar. An examination of tight-binding calculations of |9(r)|2 for donors occupying successively deeper planes revealed a sequence A, A, B, B, A, A, B, B for donor depths 5.00a0 , 5.25a0 , 5.50a0 , 5.75a0 , 6.00a0 , 6.25a0 , 6.50a0 and 6.75a0 . Both the calculated ellipsoids 608
(Fig. 4f) and probability envelope (Fig. 4h) of the 6.75a0 type-B donor are in excellent agreement with measurements in Fig. 4e and Fig. 4g, respectively. Moreover, the calculated probability envelope for a fixed donor depth was found to depend sensitively on the lattice plane below the surface where it was evaluated. The two different probability envelopes are therefore a manifestation of rapid spatial variation of |9(r)|2 along z with lattice and valley spatial frequencies (and their harmonics), and the relative sensitivity of the tip to the topmost atomic planes. Nevertheless, the spatial average of type-A and type-B probability envelopes is s-like, as expected from the effective mass approach29 . We found an uncertainty of approximately a0 in the depth determination, considerably larger than the uncertainty < 0.25a0 required to assign a donor to a crystal lattice plane. Nevertheless, the empirically observed 48:52 distribution ratio is consistent with theory, assuming (as expected) a random distribution of donor depths. Any interface-induced mixing with excited valley-orbit states enhances the spin-lattice relaxation rate17,18 . The nearly bulk-like valley interference pattern observed for ∼3 nm deep donors is a direct indication of small mixing with valley-orbit excited states. We therefore expect that arsenic donors whose depth exceeds 3 nm (approximately one effective Bohr radius) would have spin-lattice relaxation rates T1−1 only slightly larger than bulk donors. For magnetic fields of practical interest1 , T1 of subsurface donors will still greatly exceed the spin coherence time of electrons in isotopepurified silicon31 . However, proximity to non-ideal interfaces could introduce other relaxation or decoherence processes. Moreover, the observed interference in Fig. 4 is precisely the phenomenon predicted to render two-qubit gates sensitive to atomic-scale NATURE MATERIALS | VOL 13 | JUNE 2014 | www.nature.com/naturematerials
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NATURE MATERIALS DOI: 10.1038/NMAT3941 variations in donor position6,7 . This can be thought of as a direct consequence of interfering terms in the exchange interaction R J (R) = 9 ∗ (r1 − R)9 ∗ (r2 )Vee (|r1 − r2 |)9(r1 )9(r2 − R) between donors separated by a displacement R, each having latticeincommensurate (λ = 2π/kµ = 0.65 nm) spatial oscillations. The spatially resolved single-electron transport we have demonstrated provides a novel level of access to valley physics. We have observed electronic valley quantum interference for a single donor atom whose depth and electric field have been independently empirically determined; capabilities not available in single-electron transport spectroscopy of donors in nanoscale transistors2,11,15 . We have identified that a donor 2.85 ± 0.45 nm from an interface and in a low field Ez = 0.3 ± 1.9 MVm−1 has an electronic valley population deviating by only ∼5% from a donor in bulk silicon. This level of understanding is essential for the engineering of addressable single-qubit and two-qubit operations in silicon donor devices6,7,9,10,46 to exploit silicon’s remarkable coherence31,32 .
Methods Samples where single-electron tunnelling through donor-bound electronic states was observed were prepared by flash annealing a commercial n-type (arsenic-doped) silicon wafer with resistivity 0.004–0.001 -cm to a temperature ∼1,050 ◦ C for 10 s, a total of three times. After the final flash anneal, the temperature was rapidly quenched to 800 ◦ C, followed by slow (1 ◦ C s−1 ) cooling to 340 ◦ C, producing a 2 × 1 surface reconstruction. Hydrogen passivation was carried out by dosing with atomic hydrogen at a temperature of 340 ◦ C and pressure of 5 × 10−7 mbar, for twelve minutes. This flash anneal procedure is known from secondary ion mass spectroscopy to deplete the upper ∼10 nm of the wafer of arsenic dopants33 , shallow enough to maintain sufficient coupling to the conduction impurity band to obtain a measurable single-electron tunnelling current through donor-bound states. Donors found by subsurface dopant imaging, as described in the main text, were residual impurities statistically distributed throughout the depletion region33,38 . No implantation was performed. No donors were found for samples flashed three times at 1,200 ◦ C, which are expected to have too deep (∼100 nm) a surface depletion and too few residual arsenic dopants33 . Measurements were performed using an Omicron low-temperature scanning tunnelling microscope (LT-STM) operating in ultrahigh vacuum at a temperature of 4.2 K. Current I was measured as a function of the sample voltage U using ultralow noise electronics, and dI /dU was obtained by numerical differentiation. Spatially resolved measurements of donors were obtained on frames 20 nm × 20 nm in size, containing a single subsurface dopant, with a spatial resolution of 0.02–0.04 nm. Fine calibration of reciprocal lattice vector positions was carried out by Fourier transforming topographies (Supplementary Information) acquired simultaneously with quantum states, using the multi-line scan technique.
Received 20 November 2013; accepted 7 March 2014; published online 6 April 2014
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ARTICLES 12. Saraiva, A., Calderón, M., Hu, X., Das Sarma, S. & Koiller, B. Physical mechanisms of interface-mediated intervally coupling in Si. Phys. Rev. B 80, 081305 (2009). 13. Morton, J. J. L., McCamey, D. R., Eriksson, M. A. & Lyon, S. A. Embracing the quantum limit in silicon computing. Nature 479, 345–353 (2011). 14. Yang, C. H. et al. Spin-valley lifetimes in a silicon quantum dot with tunable valley splitting. Nature Commun. 4, 2069 (2013). 15. Roche, B. et al. Detection of a large valley-orbit splitting in silicon with two-donor spectroscopy. Phys. Rev. Lett. 108, 206812 (2012). 16. Zwanenburg, F. A. et al. Silicon quantum electronics. Rev. Mod. Phys. 85, 961–1019 (2013). 17. Roth, L. g factor and donor spin-lattice relaxation for electrons in germanium and silicon. Phys. Rev. 118, 1534–1540 (1960). 18. Hasegawa, H. Spin-lattice relaxation of shallow donor states in Ge and Si through a direct phonon process. Phys. Rev. 118, 1523–1534 (1960). 19. Gunawan, O. et al. Valley susceptibility of an interacting two-dimensional electron system. Phys. Rev. Lett. 97, 186404 (2006). 20. Takashina, K. et al. Impact of valley polarization on the resistivity in two dimensions. Phys. Rev. Lett. 106, 196403 (2011). 21. Isberg, J. et al. Generation, transport and detection of valley-polarized electrons in diamond. Nature Mater. 12, 760–764 (2013). 22. Young, A. F. et al. Spin and valley quantum Hall ferromagnetism in graphene. Nature Phys. 8, 550–556 (2012). 23. Gunawan, O., Habib, B., De Poortere, E. & Shayegan, M. Quantized conductance in an AlAs two-dimensional electron system quantum point contact. Phys. Rev. B 74, 155436 (2006). 24. Soykal, Ö., Ruskov, R. & Tahan, C. Sound-based analogue of cavity quantum electrodynamics in silicon. Phys. Rev. Lett. 107, 235502 (2011). 25. Culcer, D., Saraiva, A., Koiller, B., Hu, X. & Das Sarma, S. Valley-based noise-resistant quantum computation using Si quantum dots. Phys. Rev. Lett. 108, 126804 (2012). 26. Rycerz, A., Tworzydło, J. & Beenakker, C. W. J. Valley filter and valley valve in graphene. Nature Phys. 3, 172–175 (2007). 27. Tombros, N. et al. Quantized conductance of a suspended graphene nanoconstriction. Nature Phys. 7, 697–700 (2011). 28. Pei, F., Laird, E. A., Steele, G. A. & Kouwenhoven, L. P. Valley-spin blockade and spin resonance in carbon nanotubes. Nature Nanotech. 7, 630–634 (2012). 29. Kohn, W. & Luttinger, J. Theory of donor states in silicon. Phys. Rev. 98, 915–922 (1955). 30. Pantelides, S. & Sah, C. Theory of localized states in semiconductors. I. New results using an old method. Phys. Rev. B 10, 621–637 (1974). 31. Tyryshkin, A. M. et al. Electron spin coherence exceeding seconds in high-purity silicon. Nature Mater. 11, 143–147 (2011). 32. Steger, M. et al. Quantum information storage for over 180 s using donor spins in a 28 Si ‘semiconductor vacuum’. Science 336, 1280–1283 (2012). 33. Pitters, J. L., Piva, P. G. & Wolkow, R. A. Dopant depletion in the near surface region of thermally prepared silicon (100) in UHV. J. Vacuum Sci. Technol. B 30, 021806 (2012). 34. Koenraad, P. M. & Flatté, M. E. Single dopants in semiconductors. Nature Materials 10, 91–100 (2011). 35. Van Mieghem, P. Theory of band tails in heavily doped semiconductors. Rev. Mod. Phys. 64, 755–793 (1992). 36. Foxman, E. et al. Effects of quantum levels on transport through a Coulomb island. Phys. Rev. B 47, 10020–10023 (1993). 37. Mol, J. A., Salfi, J., Miwa, J. A., Simmons, M. Y. & Rogge, S. Interplay between quantum confinement and dielectric mismatch for ultrashallow dopants. Phys. Rev. B 87, 245417 (2013). 38. Sinthiptharakoon, K. et al. Investigating individual arsenic dopant atoms in silicon using low-temperature scanning tunnelling microscopy. J. Phys.: Condens. Matter 26, 012001 (2014). 39. Chen, C. J. Tunneling matrix elements in three-dimensional space: the derivative rule and the sum rule. Phys. Rev. B 42, 8841–8857 (1990). 40. Teichmann, K. et al. Controlled charge switching on a single donor with a scanning tunneling microscope. Phys. Rev. Lett. 101, 076103 (2008). 41. Lee, D.-H. & Gupta, J. A. Tunable field control over the binding energy of single dopants by a charged vacancy in GaAs. Science 330, 1807–1810 (2010). 42. Cohen, M. & Bergstresser, T. Band structures and pseudopotential form factors for fourteen semiconductors of the diamond and zinc-blende structures. Phys. Rev. 141, 789–796 (1966). 43. Rahman, R. et al. Engineered valley-orbit splittings in quantum-confined nanostructures in silicon. Phys. Rev. B 83, 195323 (2011). 44. Crommie, M. F., Lutz, C. P. & Eigler, D. M. Imaging standing waves in a 2-dimensional electron-gas. Nature 363, 524–527 (1993). 45. Sprunger, P. T., Petersen, L., Plummer, E., Leasgsgaard, E. & Besenbacher, F. Giant Friedel oscillations on the beryllium(0001) surface. Science 275, 1764–1767 (1997).
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ARTICLES 46. Rahman, R. et al. Gate-induced g -factor control and dimensional transition for donors in multivalley semiconductors. Phys. Rev. B 80, 155301 (2009).
Acknowledgements The authors would like to thank J. Verduijn for helpful discussions. This work is supported by the European Commission Future and Emerging Technologies Proactive Project MULTI (317707) and the ARC Centre of Excellence for Quantum Computation and Communication Technology (CE110001027), and in part by the US Army Research Office (W911NF-08-1-0527). This work is part of the research program of the Foundation for Fundamental Research on Matter (FOM), which is part of the Netherlands Organization for Scientific Research (NWO). S.R. acknowledges a Future Fellowship (FT100100589). M.Y.S. acknowledges a Laureate Fellowship. The use of nanoHUB.org computational resources operated by the Network for Computational Nanotechnology funded by the US National Science Foundation under grant EEC-0228390 is gratefully acknowledged.
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Author contributions J.S., J.A.M. and S.R. designed and conducted the experiments. R.R. performed the multi-million atom calculations. J.S., L.C.L.H. and S.R. made the key contributions to the Fourier analysis. All the authors contributed to analysis and writing the paper.
Additional information Supplementary information is available in the online version of the paper. Reprints and permissions information is available online at www.nature.com/reprints. Correspondence and requests for materials should be addressed to S.R.
Competing financial interests The authors declare no competing financial interests.
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Spatially resolving valley quantum interference of a donor in silicon J. Salfi1, J. A. Mol1, R. Rahman2, G. Klimeck2, M. Y. Simmons1, L. C. L. Hollenberg3 and S. Rogge1* Electron and nuclear spins of donor ensembles in isotopically pure silicon experience a vacuum-like environment, giving them extraordinary coherence. However, in contrast to a real vacuum, electrons in silicon occupy quantum superpositions of valleys in momentum space. Addressable single-qubit and two-qubit operations in silicon require that qubits are placed near interfaces, modifying the valley degrees of freedom associated with these quantum superpositions and strongly influencing qubit relaxation and exchange processes. Yet to date, spectroscopic measurements have only probed wavefunctions indirectly, preventing direct experimental access to valley population, donor position and environment. Here we directly probe the probability density of single quantum states of individual subsurface donors, in real space and reciprocal space, using scanning tunnelling spectroscopy. We directly observe quantum mechanical valley interference patterns associated with linear superpositions of valleys in the donor ground state. The valley population is found to be within 5% of a bulk donor when 2.85 ± 0.45 nm from the interface, indicating that valley-perturbation-induced enhancement of spin relaxation will be negligible for depths greater than 3 nm. The observed valley interference will render two-qubit exchange gates sensitive to atomic-scale variations in positions of subsurface donors. Moreover, these results will also be of interest for emerging schemes proposing to encode information directly in valley polarization. abrication of devices1,2 , spin readout1 and quantum control of spins3,4 in silicon has been accomplished at the singledonor level. However, addressable control and coupling within qubit arrays requires local gates and control interfaces, whose atomic-scale potentials strongly influence electronic valley degrees of freedom5–16 . Although these unconventional orbital degrees of freedom play no role in conventional silicon microelectronics, they invariably arise in quantized states in indirect gap materials, and are pervasive in quantum electronics. In silicon, valley physics determines qubit relaxation rates13,17,18 and are predicted to strongly influence two-qubit gates6,7 . Moreover, valley degrees of freedom in AlAs (ref. 19), silicon20 , diamond21 and graphene22 can play a role similar to spin in condensed matter systems. Encoding of information within the valley polarization has been proposed in AlAs (ref. 23), silicon24,25 and diamond21 , and within the polarization of chiral valley pseudospin in carbon-based nanostructures26–28 . Here, we report the measurement of individual states of subsurface donors using cryogenic scanning tunnelling spectroscopy (STS). Quantum mechanical valley interference patterns were observed in real space, associated with linear quantum superpositions of wavevectors in the six conduction band valleys of silicon. Enabled by the high-accuracy empirical determination of donor depth and electric field, we perform a parameterfree comparison with atomistic theory, establishing that the z-valley population is ∼38 ± 2% for a 2.85 ± 0.45 nm deep donor, perturbed by only ∼5% from the value of ∼33.3% for a donor in bulk silicon. Consequently, donors more than 3 nm deep should not experience a significant valley-repopulation-induced increase in spin-lattice relaxation. Moreover, the nearly bulk-like valley interference observed will render two-qubit exchange gates sensitive to atomic variations in donor position6,7 , even for subsurface donors. Measurements of valley quantum interference presented
F
herein address spatial aspects of the established theory of shallow impurities in silicon7,10,29,30 that are essential for engineering spin qubit arrays leveraging silicon’s exceptional coherence31,32 . We employed unconventional STS-based spatially resolved single-electron transport to measure, in real space, valley interference in single quantum states of subsurface arsenic donors. Ultra-high vacuum annealing, with parameters targeted to deplete the upper ∼10 nm of donors33 , was performed on a highly arsenic-doped silicon wafer (see Methods). Spatially well-isolated residual donors (density ∼1011 cm−2 ) were found in the depletion region by subsurface imaging34 at U = −1.25 V. The electronic isolation of donor-bound states within the depletion region was quantified, as illustrated in Fig. 1a, by single-electron transport spectroscopy employing the scanning tunnelling microscope (STM) tip at 4.2 K. The measured differential conductance dI /dU above donor 1 (Fig. 1b, blue squares) exhibited single-electron tunnelling peaks, from the substrate’s impurity band35 , to the subsurface donor brought into the bias window by tip-induced band bending, to the tip. Fitting of the first two peaks to single-electron transport theory36,37 (Fig. 1b, red line) established a coupling h(Γin + Γout ) kB T ∼ 350 µeV for donor 1, similar to states in single-atom transistors11 . Here, h is Planck’s constant, kB is Boltzmann’s constant, T is the temperature, and Γin and Γout are tunnel-in and tunnel-out rates, respectively. The lowest energy peak (U ≈ −0.8 V) was found to be bound to donor 1 in the spatially resolved dI /dU (Fig. 1c). Higher energy peaks (U < −1.05 V) in Fig. 1b,c are two-electron states (Supplementary Information). Spatially resolved measurements of donor-bound states were carried out using an unconventional scheme. Constant-current imaging of single dopants (see refs 34,38) was avoided because it necessarily measures multiple states. On the other hand, measurement of dI /dU versus U on a two-dimensional spatial grid
1 Centre
for Quantum Computation and Communication Technology, School of Physics, The University of New South Wales, Sydney, New South Wales 2052, Australia, 2 Purdue University, West Lafayette, Indiana 47906, USA, 3 Centre for Quantum Computation and Communication Technology, School of Physics, University of Melbourne, Parkville, Victoria 3010, Australia. *e-mail: [email protected] NATURE MATERIALS | VOL 13 | JUNE 2014 | www.nature.com/naturematerials
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Figure 1 | Spatial measurement of single quantum states of a subsurface donor. a, Illustration of single-electron tunnelling from the conduction impurity band (CIB; ref. 35), to the subsurface donor in the depletion region with electron probability density |9(r)|2 , to the tip. b, Top: Measured dI/dU (blue squares) for a tip position above the subsurface donor, and least-squares fit (red line). Bottom: Schematic band diagram, whereby the sample bias U < 0 brings the donor-bound states into resonance with the Fermi energy in the CIB, which is spatially separated from the donor by a depletion region. c, Measured differential conductance dI/dU versus x and sample bias U, along a line passing over a single donor. Scale (from blue to red): 0–0.25 nA V−1 . The tip location for measurement in b is indicated by the vertical dashed line (green square). d, Constant current (I0 = 150 pA) topography z(x, y) recorded during the first line scan at U = −1.45 V, a bias where there is minimal evidence of a buried donor. Scale (from blue to white): 0–100 pm. e, Spatially resolved single-electron tunnelling current I(r) ∝ |9(r)|2 through the single donor-bound state, for the same area as d, but with U = −0.85 V and the feedback loop off. The signal vanishes away from the donor, as expected for a true single-state measurement. White dashed line (red diamond) denotes where data in c was obtained. Scale (from blue to white): 0–0.35 nA.
would be impractically time consuming for the spatial frequency range |kx |, |ky | ≥ 8π/a0 necessary to avoid frequency aliasing, and the desired resolution ∼2π/50a0 to resolve band structure components. We therefore employed an unconventional two-pass scan to spatially measure individual donor-bound states. In the first pass of each line, the surface topography z(x, y) was measured at U = −1.45 V. During the second pass of each line, a current I (r) proportional to the probability density |9(r)|2 for a single bound state was measured, at a bias (U2 = −0.85 V for donor 1) ensuring one state in the bias window for single-electron transport. The feedback loop was turned off for the second pass, and the tip position set to r = (x, y, z(x, y) − δz); δz = 0.25 nm closer to the sample relative to the first line (topography). The value of δz was chosen such that I (U2 ) ∼ exp(κz) ∼ Γout , or equivalently, Γout Γin . In this regime, I (r) ∝ |9(r)|2 , where r is the central coordinate of the tip apex orbital39 . For donor 1, the measured z(x, y) and I (r) ∝ |9(r)|2 are shown in Fig. 1d,e, respectively. In Fig. 1e, |9(r)|2 vanishes away from the donor, as expected for a measurement of a single bound quantum state. To investigate the valley interference of donor 1, we evaluated numerically the two-dimensional Fourier representation of the measured I (r) ∝ |9(r)|2 , shown in Fig. 2a. The vertices of the outer dashed box are reciprocal lattice vectors G2 = 2π/a0 (p, q) with p = ±1, q = ±1 and a0 = 0.543 nm. Figure 2a contains ellipsoid-shaped structures, highlighted in green, at positions k = kµ xˆ and k = kµ yˆ (where kµ = 0.85(2π/a0 )) of silicon’s conduction band minima, indicated by white crosses. We term the structure outlined in blue within k . 0.5(2π/a0 ) as the probability envelope, because it contains the lowest spatial frequencies of the probability density. Dashed outlined features centred at 2π/a0 (±1/2, ∓ 1/2) and 2π/a0 (±1/2, ±1/2) are created by the 2 × 1 reconstruction, and are related to the probability envelope and ellipsoids, respectively, as evidenced by their 2π/a0 (∓1/2, ±1/2) displacement (white arrows, Fig. 2a) relative to them. A depth z0 = (5 ± 1)a0 (20 ± 4 lattice planes) for donor 1 was empirically obtained by fitting the spatially 606
resolved spectral shift of the conduction band edge to a dielectric screened Coulomb potential37,40,41 (Supplementary Information). Shown for comparison in Fig. 2b is the Fourier transform of the vacuum tail of the ground state probability density |91 (r)|2 obtained by sp3 d 5 s∗ tight-binding (Supplementary Information). The predicted ellipsoids and probability envelope features are in very good agreement with measurements in Fig. 2a. As follows from the discussion below, the ellipsoids and probability envelope features are produced by quantum interference processes associated with quantum superpositions of wavevectors in six different valleys in the donor state. Such a R P ground state can be written as7 9i (r) = µ αµi dk03 Fµ (k0 )φkµ +k0 (r), where αµi (µ = 1...6) are the valley quantum numbers, and as illustrated in Fig. 2c, Fµ (k0 ) is the the reciprocal space distribution of the wavefunction over the Bloch functions φkµ +k0 (r) about the minima kµ . The full Fourier representation of the state is obtained by substitutingPthePrepresentation for the Bloch R functions, such that7 9i (r) = G µ dk0 3 9i (k) exp(ik · r), where 9i (k) = αµi Ak−G,G Fµ (k0 ), k = G + kµ + k0 , and G = [p, q, r] = [0, 0, 0], [1, 1, 1], [2, 0, 0], [2, 2, 0], [3, 1, 1], ... (in units of 2π/a0 ) and their equivalents are the reciprocal lattice vectors with nonzero Bloch amplitudes Ak−G,G in silicon42 . As 9(kµ ) ∝ αµ Fµ (0), the ellipsoid features in |9(k)| at k = G2 ± xk ˆ µ and k = G2 ± yˆ kµ (green ellipsoids) are present only i i when α±x 6 = 0 and α±y 6 = 0, respectively, whereas the projected i ellipsoid at G2 (yellow circle) is present only when α±z 6 = 0. For a donor in bulk silicon, the ground state is a spin degenerate valley singlet with α 1 = 6−1/2 [1, 1, 1, 1, 1, 1]. Hence, the calculated ground state Fourier amplitudes |91 (k)| presented in Fig. 2d for a depth 6.25a0 are in good agreement with the expected reciprocal space representation for a bulk-like donor with valley configuration similar to α 1 . In bulk silicon the singlet, whose ionization energy is 52 meV for an arsenic donor, lies 22 meV below a spindegenerate excited valley triplet with α 2 = 2−1/2 [1, −1, 0, 0, 0, 0], α 3 = 2−1/2 [0, 0, 1, −1, 0, 0], and α 4 = 2−1/2 [0, 0, 0, 0, 1, −1]. The NATURE MATERIALS | VOL 13 | JUNE 2014 | www.nature.com/naturematerials
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Figure 2 | Valley interference of a single quantum state in reciprocal space. a, Fourier amplitudes of measured I(r) ∝ |9(r)|2 for donor-bound electrons in Fig. 1. The corners of the outer dashed square are reciprocal lattice vectors 2π/a0 (p, q) with p = ±1 and q = ±1. White crosses denote kµ = 0.85(2π/a0 )(±1, 0) and kµ = 0.85(2π/a0 )(0, ±1). Ellipsoid structures are found within the green boundaries, the probability envelope is found within the blue boundaries, and 2 × 1 reconstruction-induced features are found within the grey dashed boundaries. Scale [0,0.5] (normalized). b, Calculated Fourier amplitude of the single-electron (D0 ) ground state probability density |9(r)|2 for a z0 = 6.25a0 ≈ 3.4 nm deep donor, with a 2 × 1 surface reconstruction. Scale [0,1] (normalized). c, Distribution of wavevectors about valley minima in the plane of the surface (green ellipsoids) and perpendicular to it (yellow ellipsoids), and magnification of a single valley distribution Fx (k0 ). d, Overlay of the reciprocal space model for the D0 ground state wavefunction on the atomistic calculation of the |9(k)| using the tight-binding method. Labelled wavevectors have units 2π/a0 . e, Overlay of the reciprocal space model for the D0 ground state probability density on the atomistic calculation using the tight-binding method. For d and e, a 1 × 1 reconstruction was assumed for simplicity.
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constructive valley interference of the singlet at the arsenic site (the central cell) is responsible for the energy difference29,30 . Two quantum interference processes are of particular interest in the Fourier representation of Fig. 2a and b, which maps components exp(−ik0 · r) in 9i∗ (r) and exp(ik00 · r) in 9i (r) to exp(i(k00 − k0 ) · r) in |9i (r)|2 = 9i∗ (r)9i (r). First, ellipsoids at k ≈ ±xk ˆ µ (k ≈ ±ˆy kµ ) in |9i (r)|2 , overlaying atomistic calculations in Fig. 2e, arise from cross-product terms of z valleys and x (y) valleys. Consequently, the ellipsoids in the Fourier decomposition of |9(r)|2 are direct evidence for the presence of all three valleys in the measured (Fig. 2a) and atomistically calculated (Fig. 2b) subsurface donor ground state. From the same mapping, it follows that x, y and z valleys contribute to the peak at k = 0, whereas only x and y valleys contribute to the side peak at k = 0.15(2π/a0 )(±1, ±1). The relative heights of the central and side peaks in the measured and calculated probability envelopes were compared to characterize the deviation of donor 1 from the α 1 (bulk) valley configuration. The electric field in experiments, estimated in the Supplementary Information to be Ez = −0.3 ± 1.9 MV m−1 using tip-heightdependent dI /dU measurements, is low enough to be safely neglected in calculations of donor-bound states43 . Calculations for decreasing depths 10.25a0 to 5.25a0 , presented in Fig. 3, demonstrate an increasing contribution of the central peak, signalling an increase in the z-valley population relative to the x and y valleys. This arises from the competition between the interface, which introduces a valley-orbit potential8,12,14 and depopulates the x and y valleys with lighter masses parallel to the interface, and the donor ion’s valley-orbit potential, which equalizes all six valleys. The side-peak amplitude saturates at 7.25a0 , indicating that the valleys approach the bulk configuration. Shown in Fig. 3 (green diamonds), the measured donor’s profile agrees well with calculations for 6.25a0 at zero field, in excellent agreement with the empirically determined depth z0 = (5 ± 1)a0 of donor 1. We estimate that the z-valley configuration of the measured donor differs by only ∼5% from a donor in bulk silicon (33.3%), because states calculated for 7.25a0 and 6.25a0 depths have z-valley populations of 36% and 40%, respectively (Supplementary Information). The wider central peak in measurements could be the result of enhanced lateral localization due to image charges associated with dielectric mismatch for the 3 nm deep donor, not taken into account in the theory. Overall,
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Figure 3 | Valley population. Comparison of measured (green diamonds) and theoretical (solid lines) reciprocal space profiles of |91 (r)|2 for the donor ground state, along the 110 direction in reciprocal space. The height of the central peak (k = 0) relative to the side peaks (k = 0.15(2π/a0 )(±1, ±1)) increases with decreasing depth from 10.25a0 (black line), 7.25a0 (green line), 6.25a0 (red line), to 5.25a0 (blue line), indicating repopulation from the x and y valleys into z valleys with decreasing depth. The calculated first excited state, shown for 5.25a0 (blue dashed line), populates only the z valley, and hence has no side lobes.
the agreement of the reciprocal space profile with the theoretical description from the calculations is remarkable. Note that the lowpass response of the STM tip orbital is not expected for such low spatial frequencies k . 0.21(2π/a0 ) (λ & 2.6 nm). The real-space representation of the ellipsoids, shown in Fig. 4a and obtained by digital Fourier filtering, resembles a low-frequency s-like envelope modulated by x- and y-directed standing wave patterns with wavelength λµ ≈ 2π/kµ = 0.65 nm. As described above, the ellipsoids in |91 (r)|2 arise from cross terms between the x (y) valleys, which oscillate at k = kµ xˆ (k = kµ yˆ ), and the z valley. Consequently, their real-space representation can be easily shown (Supplementary Information) to haveR the form Fz (r)(Fx (r) cos(kµ x) + Fy (r) cos(kµ y)), where Fi (r) = d3 k0 Fi (k0 ) exp(ik0 · r) is the donor ground state envelope
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Figure 4 | Valley interference of a single quantum state in real space. a, Measured real-space representation of the relative probability (normalized to the maximum) of four ellipsoid features at k ≈ kµ for donor 1 in Fig. 2, obtained by a frequency selective inverse Fourier transform of Fig. 2a. Arrow denotes the [110] dimer direction. Red cross denotes the ion position determined from the real-space ellipsoid pattern, as discussed in the Supplementary Information. b, Predicted real-space representation of four ellipsoid features based on atomistic calculations. The best match was obtained for a z0 = 6.25a0 ≈ 3.4 nm deep donor. c, Measured real-space representation of the probability envelope. d, Predicted real-space representation of a type-A probability envelope. e, Same as a, but for the measured donor 2 with a type-B probability envelope. f, Predicted real-space representation of four ellipsoid features based on atomistic calculations. The best match was obtained for a z0 = 6.75a0 ≈ 3.7 nm deep donor. g, Same as c, but for a type-B probability envelope. h, Predicted real-space representation of the probability envelope based on an atomistic calculation for f. All insets: Corresponding region in the first Brillouin zone enclosed within dashed boundary.
function. The maximum of the real-space oscillation pattern, labelled with a red cross in Fig. 4, identifies the position of the ion. Notably, the appearance of elongation along [100] and [010] directions (Fig. 4a) is also predicted by atomistic theory (Fig. 4b). The oscillations observed in Fig. 4a,b are associated with superpositions of wavevectors, not unlike Friedel oscillations of metallic surface states observed by STM at impurities and step edges44,45 . The present case differs because the superpositions occupy six valleys along three orthogonal [001] directions, rather than a two-dimensional Fermi surface. Like the ellipsoids, the probability envelope feature introduced in the discussion of Fig. 2 arises from valley interference. Therefore, it is not surprising that the real-space representation of the probability envelope in Fig. 4c has an unusual shape, characterized by a node along x = −y. This node, predicted by calculations (Fig. 4d), contrasts the simple envelope of oscillations in Fig. 4a,b. Notably, a survey of fifty subsurface donors with indistinguishable spectral signatures revealed 24 ‘type-A’ envelopes typified by donor 1 in Fig. 4c,d, and 26 ‘type-B’ envelopes typified by donor 2 in Fig. 4g,h. Whereas the ellipsoid interference pattern of the type-B donor (Fig. 4e) resembles that of the type-A donor (Fig. 4a), the probability envelope of the type-B donor (Fig. 4g) has a protrusion along x = −y. The depth of donor 2 was not measured owing to a technical problem. The type-A and type-B probability envelopes are at first peculiar. An examination of tight-binding calculations of |9(r)|2 for donors occupying successively deeper planes revealed a sequence A, A, B, B, A, A, B, B for donor depths 5.00a0 , 5.25a0 , 5.50a0 , 5.75a0 , 6.00a0 , 6.25a0 , 6.50a0 and 6.75a0 . Both the calculated ellipsoids 608
(Fig. 4f) and probability envelope (Fig. 4h) of the 6.75a0 type-B donor are in excellent agreement with measurements in Fig. 4e and Fig. 4g, respectively. Moreover, the calculated probability envelope for a fixed donor depth was found to depend sensitively on the lattice plane below the surface where it was evaluated. The two different probability envelopes are therefore a manifestation of rapid spatial variation of |9(r)|2 along z with lattice and valley spatial frequencies (and their harmonics), and the relative sensitivity of the tip to the topmost atomic planes. Nevertheless, the spatial average of type-A and type-B probability envelopes is s-like, as expected from the effective mass approach29 . We found an uncertainty of approximately a0 in the depth determination, considerably larger than the uncertainty < 0.25a0 required to assign a donor to a crystal lattice plane. Nevertheless, the empirically observed 48:52 distribution ratio is consistent with theory, assuming (as expected) a random distribution of donor depths. Any interface-induced mixing with excited valley-orbit states enhances the spin-lattice relaxation rate17,18 . The nearly bulk-like valley interference pattern observed for ∼3 nm deep donors is a direct indication of small mixing with valley-orbit excited states. We therefore expect that arsenic donors whose depth exceeds 3 nm (approximately one effective Bohr radius) would have spin-lattice relaxation rates T1−1 only slightly larger than bulk donors. For magnetic fields of practical interest1 , T1 of subsurface donors will still greatly exceed the spin coherence time of electrons in isotopepurified silicon31 . However, proximity to non-ideal interfaces could introduce other relaxation or decoherence processes. Moreover, the observed interference in Fig. 4 is precisely the phenomenon predicted to render two-qubit gates sensitive to atomic-scale NATURE MATERIALS | VOL 13 | JUNE 2014 | www.nature.com/naturematerials
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NATURE MATERIALS DOI: 10.1038/NMAT3941 variations in donor position6,7 . This can be thought of as a direct consequence of interfering terms in the exchange interaction R J (R) = 9 ∗ (r1 − R)9 ∗ (r2 )Vee (|r1 − r2 |)9(r1 )9(r2 − R) between donors separated by a displacement R, each having latticeincommensurate (λ = 2π/kµ = 0.65 nm) spatial oscillations. The spatially resolved single-electron transport we have demonstrated provides a novel level of access to valley physics. We have observed electronic valley quantum interference for a single donor atom whose depth and electric field have been independently empirically determined; capabilities not available in single-electron transport spectroscopy of donors in nanoscale transistors2,11,15 . We have identified that a donor 2.85 ± 0.45 nm from an interface and in a low field Ez = 0.3 ± 1.9 MVm−1 has an electronic valley population deviating by only ∼5% from a donor in bulk silicon. This level of understanding is essential for the engineering of addressable single-qubit and two-qubit operations in silicon donor devices6,7,9,10,46 to exploit silicon’s remarkable coherence31,32 .
Methods Samples where single-electron tunnelling through donor-bound electronic states was observed were prepared by flash annealing a commercial n-type (arsenic-doped) silicon wafer with resistivity 0.004–0.001 -cm to a temperature ∼1,050 ◦ C for 10 s, a total of three times. After the final flash anneal, the temperature was rapidly quenched to 800 ◦ C, followed by slow (1 ◦ C s−1 ) cooling to 340 ◦ C, producing a 2 × 1 surface reconstruction. Hydrogen passivation was carried out by dosing with atomic hydrogen at a temperature of 340 ◦ C and pressure of 5 × 10−7 mbar, for twelve minutes. This flash anneal procedure is known from secondary ion mass spectroscopy to deplete the upper ∼10 nm of the wafer of arsenic dopants33 , shallow enough to maintain sufficient coupling to the conduction impurity band to obtain a measurable single-electron tunnelling current through donor-bound states. Donors found by subsurface dopant imaging, as described in the main text, were residual impurities statistically distributed throughout the depletion region33,38 . No implantation was performed. No donors were found for samples flashed three times at 1,200 ◦ C, which are expected to have too deep (∼100 nm) a surface depletion and too few residual arsenic dopants33 . Measurements were performed using an Omicron low-temperature scanning tunnelling microscope (LT-STM) operating in ultrahigh vacuum at a temperature of 4.2 K. Current I was measured as a function of the sample voltage U using ultralow noise electronics, and dI /dU was obtained by numerical differentiation. Spatially resolved measurements of donors were obtained on frames 20 nm × 20 nm in size, containing a single subsurface dopant, with a spatial resolution of 0.02–0.04 nm. Fine calibration of reciprocal lattice vector positions was carried out by Fourier transforming topographies (Supplementary Information) acquired simultaneously with quantum states, using the multi-line scan technique.
Received 20 November 2013; accepted 7 March 2014; published online 6 April 2014
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Acknowledgements The authors would like to thank J. Verduijn for helpful discussions. This work is supported by the European Commission Future and Emerging Technologies Proactive Project MULTI (317707) and the ARC Centre of Excellence for Quantum Computation and Communication Technology (CE110001027), and in part by the US Army Research Office (W911NF-08-1-0527). This work is part of the research program of the Foundation for Fundamental Research on Matter (FOM), which is part of the Netherlands Organization for Scientific Research (NWO). S.R. acknowledges a Future Fellowship (FT100100589). M.Y.S. acknowledges a Laureate Fellowship. The use of nanoHUB.org computational resources operated by the Network for Computational Nanotechnology funded by the US National Science Foundation under grant EEC-0228390 is gratefully acknowledged.
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Author contributions J.S., J.A.M. and S.R. designed and conducted the experiments. R.R. performed the multi-million atom calculations. J.S., L.C.L.H. and S.R. made the key contributions to the Fourier analysis. All the authors contributed to analysis and writing the paper.
Additional information Supplementary information is available in the online version of the paper. Reprints and permissions information is available online at www.nature.com/reprints. Correspondence and requests for materials should be addressed to S.R.
Competing financial interests The authors declare no competing financial interests.
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