Letter pubs.acs.org/NanoLett
Valley Splitting in a Silicon Quantum Device Platform Jill A. Miwa,† Oliver Warschkow,‡ Damien J. Carter,§ Nigel A. Marks,§ Federico Mazzola,∥ Michelle Y. Simmons,⊥ and Justin W. Wells*,∥ †
Department of Physics and Astronomy, Interdisciplinary Nanoscience Center (iNANO), University of Aarhus, 8000 Aarhus C, Denmark ‡ Centre for Quantum Computation and Communication Technology, School of Physics, The University of Sydney, Sydney, NSW 2006, Australia § Nanochemistry Research Institute, Curtin University, P.O. Box U1987, Perth WA 6845 ∥ Department of Physics, Norwegian University of Science and Technology (NTNU), N-7491 Trondheim, Norway ⊥ Centre of Excellence for Quantum Computation and Communication Technology, School of Physics, University of New South Wales, Sydney, NSW 2052, Australia S Supporting Information *
ABSTRACT: By suppressing an undesirable surface Umklapp process, it is possible to resolve the two most occupied states (1Γ and 2Γ) in a buried two-dimensional electron gas (2DEG) in silicon. The 2DEG exists because of an atomically sharp profile of phosphorus dopants which have been formed beneath the Si(001) surface (a δ-layer). The energy separation, or valley splitting, of the two most occupied bands has critical implications for the properties of δ-layer derived devices, yet until now, has not been directly measurable. Density functional theory (DFT) allows the 2DEG band structure to be calculated, but without experimental verification the size of the valley splitting has been unclear. Using a combination of direct spectroscopic measurements and DFT we show that the measured band structure is in good qualitative agreement with calculations and reveal a valley splitting of 132 ± 5 meV. We also report the effective mass and occupation of the 2DEG states and compare the dispersions and Fermi surface with DFT. KEYWORDS: δ-layer, quantum computer, valley-splitting, ARPES, Umklapp, silicon
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integer multiples of the surface reciprocal lattice momentum to be exchanged between a surface and a photoemitted electron, thus, regardless whether an electron originates from a bulk, surface or δ-layer state, periodicities corresponding to the surface reciprocal lattice are not only to be expected but have already been observed.21 Here, we present measurements on two samples, identical except for their surface reconstructions, thus allowing the surface momentum exchange to be disentangled, and the valley splitting and band dispersions to be accessible for the first time. We have recently demonstrated that despite the assumed short mean free path of low kinetic energy photoelectrons, it is nonetheless possible to probe the bandstructure of buried twodimensional layers using angle-resolved photoemission spectroscopy (ARPES).21 The photoemission intensity from the buried states is resonantly enhanced such that the small fraction of its wave function, which exists near the surface, is strong enough to be directly measured.22 This enhancement occurs at photon energies corresponding to electron emission at bulk
uantum devices, such as gated quantum dots, have been fabricated successfully by numerous means, and have advanced to the ultimate limit where individual atoms dictate device behavior.1−3 A method that has shown steady progress in recent years utilizes phosphorus δ-doped Si(001) as the foundation from which devices are derived.3−6 A common prediction for such δ-doped layers is the formation of confined states derived from the bulk conduction band minima.7 The lowest lying states, called 1Γ and 2Γ are nondegenerate and their separation energy, that is, valley splitting, facilitates an alternative approach to performing operations in quantum electronic devices8−10 and hence attracts much interest.4,11,12 The magnitude of the valley splitting has implications for critical device properties, such as decoherence and transport. Because a direct measure has hitherto not been possible, a reliance on various calculations, offering values from 6 to 270 meV,13−19 has been necessary. Here we demonstrate an experimental approach involving removal of a surface Umklapp processes that otherwise obfuscate these states, such that a valley splitting of 132 ± 5 meV is directly measured. When an electron is photoemitted from a solid, the surface Umklapp process is one of the few mechanisms that allows the exchange of parallel momentum, and hence parallel momentum is otherwise conserved.20 The surface Umklapp process allows © 2014 American Chemical Society
Received: December 20, 2013 Revised: February 21, 2014 Published: February 26, 2014 1515
dx.doi.org/10.1021/nl404738j | Nano Lett. 2014, 14, 1515−1519
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ance of the bulk and δ-layer bandstructures are essentially unchanged, except for the manifestations of the surface Umklapp process.24 The electronic states at the Fermi level (EF), due to the buried δ-layer, are seen to be affected by the surface Umklapp process; for the (1 × 1) surface, the constant energy map collected at the EF shows a feature around kx = ky = 0 only (Figure 1b), whereas the (2 × 1) surface (Figure 1a), shows additional repetitions of the state at kx = ± 0.8 Å−1 and ky = ± 0.8 Å−1, corresponding to the 2 × surface reciprocal lattice vector. Thus there is no evidence of long-range ordering in the δ-layer itself, because the periodic features are associated with the surface Umklapp process and not the δ-layer. Once the additional Umklapp features have been removed, the electronic structure of the δ-layer state can be seen more clearly. ARPES measurements acquired in the bulk ⟨100⟩ direction with a photon energy of hν = 113 eV are shown in Figure 1c and provide an overview of the position of the δ-layer state, which appears as a small bright feature near the Fermi level, with respect to the Si bulk bands. A corresponding schematic (based on density functional theory (DFT) calculations) is given in Figure 1d with both the δ-layer states, (i.e., 1Γ, 2Γ, and 1Δ) and Si bulk bands marked. Detailed ARPES measurements in the same direction, but performed at a temperature of 90 K and photon energy hν = 36 eV, reveal that the δ-layer spectral feature has the appearance of a filled parabola (Figure 2a). In fact, this feature can be fully described by the presence of two parabolic states, separated in binding energy. This is exemplified in Figure 2b,c for a vertical (i.e.,
high symmetry points.20,23 However, since the surface is known to be (2 × 1) reconstructed with an equivalent (1 × 2) rotational domain giving the appearance of (2 × 2), it was previously not possible to conclude whether the observed 2 × periodicity (see Figure 1a) originates in the buried layer, or is
Figure 1. Direct comparison of the δ-doped Si(001)-(2 × 1) and δdoped Si(001)-(1 × 1) samples. Overview ARPES measurements, taken at hν = 113 eV, showing the constant energy surface at the Fermi level for an extended region of k||, and corresponding LEED patterns collected at Ek = 250 eV (insets) for exemplary (a) δ-doped Si(001)-(2 × 1) and (b) δ-doped Si(001)-(1 × 1) samples. The yellow squares overlaid indicate the surface (1 × 1) unit cell, and serve as a reference. (c) ARPES measurement made in the bulk ⟨100⟩ direction with a photon energy of hν = 113 eV is shown for the (1 × 1) case. The δ-layer states appear as a small bright feature located near the Fermi level. (d) Schematic illustration indicating the position of the δlayer states with respect to the bulk Si bands based on DFT (combining details of Supporting Information Figures S1b and S2b24). Figure 2. Detail of the δ-layer state of the δ-doped Si(001)-(1 × 1) sample. (a) Low-temperature (90 K) false-color ARPES measurements, taken at hν = 36 eV, of the δ-layer states with the fitted positions of the components overlaid (black × and blue +). Detail of the individual fitted points relative to the parabolic trend is shown as an inset. Note that the small modulations in the positions of fitted peak positions arise from intensity variations in the detector. (b) A constant momentum slice (orange), extracted at ky = 0 fitted with two Gaussian components modified by a Fermi function (black dashed line). (c) A constant energy slice (orange) extracted at the Fermi level together with three Gaussian components. Overall fits are represented by the black lines.
rather a result of the surface Umklapp process. Furthermore, step edges are inclined toward either the ⟨11̅0⟩ or ⟨110⟩ axes thus giving rise to the observed one-dimensional streaking that further obfuscates the measurement and frustrates the direct measurement of the valley splitting and band dispersions. δ-doped Si(001) samples are made in situ with a typical (2 × 1) reconstructed surface and the following measurements are treated to alter the surface termination22 to (1 × 1), as confirmed by the low energy electron diffraction (LEED) images in the insets of Figure 1a,b. Because the two alternate preparations change only the surface periodicity, the appear1516
dx.doi.org/10.1021/nl404738j | Nano Lett. 2014, 14, 1515−1519
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calculational approach used. The preparation recipe used here is known to reliably produce a dense (2.4 × 1014 cm−2) and narrow (
