Silicon quantum electronics

Floris A. Zwanenburg, Andrew S. Dzurak, Andrea Morello, Michelle Y. Simmons, Lloyd C. L. Hollenberg, Gerhard Klimeck, Sven Rogge, Susan N. Coppersmith, and Mark A. Eriksson

Rev. Mod. Phys. 85, 961 – Published 10 July 2013

Abstract

This review describes recent groundbreaking results in Si, $Si / SiGe$ , and dopant-based quantum dots, and it highlights the remarkable advances in Si-based quantum physics that have occurred in the past few years. This progress has been possible thanks to materials development of Si quantum devices, and the physical understanding of quantum effects in silicon. Recent critical steps include the isolation of single electrons, the observation of spin blockade, and single-shot readout of individual electron spins in both dopants and gated quantum dots in Si. Each of these results has come with physics that was not anticipated from previous work in other material systems. These advances underline the significant progress toward the realization of spin quantum bits in a material with a long spin coherence time, crucial for quantum computation and spintronics.

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Received 20 June 2012

DOI:https://doi.org/10.1103/RevModPhys.85.961

Published by the American Physical Society

Authors & Affiliations

Floris A. Zwanenburg^*

NanoElectronics Group, MESA+ Institute for Nanotechnology, University of Twente, PO Box 217, 7500 AE Enschede, The Netherlands and Centre of Excellence for Quantum Computation and Communication Technology, The University of New South Wales, NSW 2052 Sydney, Australia

Andrew S. Dzurak, Andrea Morello, and Michelle Y. Simmons

Centre of Excellence for Quantum Computation and Communication Technology, The University of New South Wales, NSW 2052 Sydney, Australia

Lloyd C. L. Hollenberg

Centre of Excellence for Quantum Computation and Communication Technology, University of Melbourne, VIC 3010 Melbourne, Australia

Gerhard Klimeck

School of Electrical and Computer Engineering, Birck Nanotechnology Center, Network for Computational Nanotechnology, Purdue University, West Lafayette, Indiana 47907, USA

Sven Rogge

Centre of Excellence for Quantum Computation and Communication Technology, The University of New South Wales, NSW 2052 Sydney, Australia and Kavli Institute of Nanoscience, Delft University of Technology, 2628 CJ Delft, The Netherlands

Susan N. Coppersmith and Mark A. Eriksson

University of Wisconsin–Madison, Madison, Wisconsin 53706, USA

^*f.a.zwanenburg@utwente.nl

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Vol. 85, Iss. 3 — July - September 2013

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Figure 1
Combining material and electrostatic confinement to create single-electron transistors. First column: Schematic of dopants, 0D, 1D, and 2D structures. Second column: In the corresponding confinement potentials in $x$ , $y$ , and $z$ directions electron states are occupied up to the Fermi energy $E_{F}$ (dashed gray lines). Occupied and unoccupied electron states are indicated as straight and dashed lines, respectively. Third column: Schematic of the silicon nanostructure integrated into a transport device with source, drain, and gate electrodes. Fourth column: The potential landscape of the single-electron transistor is made up of a potential well which is tunnel coupled to source and drain reservoir and electrostatically coupled to gates which can move the ladder of electrochemical potentials, as described in Sec. 2b.Reuse & Permissions
Figure 2
Schematic diagrams of the electrochemical potential of a single-electron transistor. (a) There is no available level in the bias window between $μ_{S}$ and $μ_{D}$ , the electrochemical potentials of the source and the drain, so the electron number is fixed at $N$ due to Coulomb blockade. (b) The $μ_{N}$ level aligns with source and drain electrochemical potentials, and the number of electrons alternates between $N$ and $N - 1$ , resulting in a single-electron tunneling current.Reuse & Permissions
Figure 3
Zero-bias and finite-bias spectroscopy. (a) Zero-bias conductance $G$ of transport vs gate voltage $V_{G}$ at both $T ≫ T_{K}$ (solid line) and $T ≪ T_{K}$ (dashed line). In the first regime, the full width at half maximum (FWHM) of the Coulomb peaks corresponds to the level broadening $h Γ$ . In the Kondo regime ( $T ≪ T_{K}$ ), Coulomb blockade is overcome by coherent second-order tunneling processes (see text). (b) Stability diagram showing Coulomb diamonds in differential conductance $d I / d V_{SD}$ vs $e V_{SD}$ and $e V_{G}$ at $T = 0 K$ . The edges of the diamond-shaped regions correspond to the onset of the current. Diagonal lines of increased conductance emanating from the diamonds indicate transport through excited states. The indicated internal energy scales $E_{C}$ , $Δ E$ , $h Γ$ , and $T_{K}$ define the boundaries between different transport regimes. Cotunneling lines can appear when the applied bias exceeds $Δ E$ (see text). Adapted from 222.Reuse & Permissions
Figure 4
The five separate transport regimes in a three-terminal quantum device. (a) Schematic depiction of the regimes in which transport through a localized state takes place as a function of the external energy scales $k_{B} T$ and $V_{SD}$ . The transitions between regimes take place on the order of the internal energy scales $E_{C}$ , $Δ E$ , $h Γ$ , and $T_{K}$ . (b) Potential landscape of the three-terminal geometry, where the quantum states and the electrochemical potential of the leads are shown together with $k_{B} T$ , $V_{SD}$ and $E_{C}$ , $Δ E$ .Reuse & Permissions
Figure 5
Silicon crystal in real and reciprocal space. (a) 3D plot of the unit cell of the bulk silicon crystal in real space, showing the diamond or face-centered-cubic lattice, which has cubic symmetry. (b) Silicon crystal in reciprocal space. Brillouin zone of the silicon crystal lattice. It is the Wigner-Seitz cell of the body-centered-cubic lattice. $Γ$ is the center of the polyhedron. From 74.Reuse & Permissions
Figure 6
Band structure of bulk silicon. (a) The conduction band has six degenerate minima or valleys at $0.85 k_{0}$ . Results supplied by G. P. Srivastava, University of Exeter. From 74. (b) Zoom-in on the bottom of the conduction band and the top of the valence band (schematic, not exact). The band gap in bulk Si is 1.12 eV at room temperature, increasing to 1.17 eV at 4 K (135). The heavy and light hole bands are degenerate for $k = 0$ . The split-off band is separated from the other subbands by the spin-orbit splitting $Δ_{SO}$ of 44 meV.Reuse & Permissions
Figure 7
Valley splitting of dopants and of quantum dots in silicon quantum wells. (a) For a quantum well, in which a thin silicon layer is sandwiched between two layers of ${Si}_{x} {Ge}_{1 - x}$ , with $x$ typically $\sim 0.25 - 0.3$ , the sixfold valley degeneracy of bulk silicon is broken by the large in-plane tensile strain in the quantum well so that two $Γ$ levels are about 200 meV below the four $Δ$ levels (355). The remaining twofold degeneracy is broken by the confinement in the quantum well and by electric fields, with the resulting valley splitting typically $\sim 0.1 - 1 meV$ . (b) For phosphorus dopants, strong central-cell corrections near the dopant break the sixfold valley degeneracy of bulk silicon so that the lowest-energy valley state is nondegenerate (except for spin degeneracy), lowered by an energy 11.7 meV. The degeneracies of higher-energy levels are broken by lattice strain and by electric fields.Reuse & Permissions
Figure 8
Sketch of the two lowest-energy eigenstates in an infinite square well of the two-band model presented in the Supplemental Material 474. The envelopes of the two eigenfunctions are very similar to each other and to the sine behavior obtained in the absence of valley degeneracy; the effects of the valley degeneracy give rise to fast oscillations within this envelope. For a square well, one eigenfunction is symmetric and the other is antisymmetric; the symmetries are different because the fast oscillations have different phases as measured from the quantum well boundaries. This sensitive dependence of valley splitting on the atomic-scale physics near the well boundary is the source of the sensitive dependence of the valley splitting on disorder at the quantum well interfaces.Reuse & Permissions
Figure 9
Valley-orbit mixing. (a), (b) If the valley splitting $E_{V}$ and orbital level spacing $Δ E$ have very different values, the orbital and valley quantum numbers are well defined and there will be no mixing of orbital and valleylike behavior. (c) When $E_{V} \approx Δ E$ the valleys and orbits can hybridize in single-particle levels separated by the valley-orbit splitting $E_{VO}$ .Reuse & Permissions
Figure 10
Valley-orbit coupling from interface steps. Top: Gray-scale visualization of wave function oscillations in the presence of a perfectly smooth interface, oriented perpendicular to $\hat{z}$ . Middle: The relationship between the phase of the wave function oscillations and the interface is different on the two sides of an interface step. When the steps are close together, the phase does not adjust to the individual steps, and the valley splitting is suppressed. Bottom: When steps are far enough apart, the oscillations line up with the interface location on both sides of the steps, which causes the phase of the oscillations to depend on the transverse coordinate. This coupling between the behavior of the wave function in the $z$ direction and in the $x − y$ plane, which arises even when the well is atomically thin, is known as valley-orbit coupling.Reuse & Permissions
Figure 11
A silicon-based nuclear-spin quantum computer. (a) Schematic of Kane’s proposal for a scalable quantum computer in silicon using a linear array of ${}^{31}P$ donors in a silicon host. $J$ gates and $A$ gates control, respectively, the exchange interaction $J$ and the wave function, as shown in (b). From 180.Reuse & Permissions
Figure 12
Relative Stark shift of the contact hyperfine interaction for different donor depths ( $z$ ) calculated for a uniform field in the $z$ direction. (a) Using the tight-binding approach (258); (b) direct diagonalization in momentum space (442). Agreement in overall trends is reasonable, and for the $z = 10.86 nm$ case both methods predict ionization at $\sim 6 MV / m$ .Reuse & Permissions
Figure 13
Low-field Stark shift of the hyperfine interaction for momentum space diagonalization (BMB) and tight-binding (TB) methods. (a) Electric field response of hyperfine coupling at various donor depths (BMB and TB). (b) Quadratic (left-hand axis) and linear (right-hand axis) Stark coefficients as a function of donor depth (TB). (c) Shift of the ground state electron distribution (dipole moment) as a function of the electric field (TB). (d) The electric field gradient of the dipole moments as a function of donor depth (TB). From 328.Reuse & Permissions
Figure 14
$J$ oscillations in the exchange coupling. Calculated exchange coupling between two phosphorus donors in Si (solid lines) and Ge (dashed lines) along high-symmetry directions for the diamond structure. Values appropriate for impurities at substitutional sites are given by the circles (Si) and diamonds (Ge). Off-lattice displacements by 10% of the nearest-neighbor distance lead to the perturbed values indicated by the squares (Si) and crosses (Ge). From 208.Reuse & Permissions
Figure 15
Smoothing out the exchange oscillations—the exchange coupling $J$ as a function of donor separation along [110]. Top curve: Calculation using the effective-mass wave function. Middle curve: Calculation of $J$ based on wave functions obtained using direct momentum diagonalization over a large basis of Bloch states (BMB) with no core correction of the impurity potential ( $η = 0$ ). Bottom curve: BMB calculation of $J$ with a core correction ( $η = 5.8$ ) that reproduces the donor ground state and valley splitting. Note that the points refer to substitutional sites in the silicon matrix. Although the donor separations are relatively small in this case, the spatial variation of the exchange interaction appears to be significantly damped compared to the effective-mass treatment. All $J$ values are calculated in the Heitler-London approximation. From 442.Reuse & Permissions
Figure 16
Gate control of the two-donor system. Averaged charge distribution along the interdonor axis for various strengths of the $J$ -gate potential ( $μ$ ) for the (a) singlet and (b) triplet states (fixed donor separation at $10 a_{B}$ ). From 97.Reuse & Permissions
Figure 17
Band structure of the $1 / 4$ monolayer phosphorus $δ$ -doped layer. (a) The calculation by 321: the solid lines show the band structure without exchange correlation and short-range effects, while the dotted lines show the band structure obtained in the full model. (b) The DFT calculation in a supercell with 200 cladding layers by 56. The plane projected bulk band structure of Si is represented by the gray continuum. The Fermi level is indicated by a horizontal dashed line. From 321, and 56.Reuse & Permissions
Figure 18
Calculated electronic spectrum of a single-atom transistor. Top left: Calculated energies of the $D^{0}$ and $D^{-}$ ground states (GS) as a function of the applied gate voltage $V_{G}$ . The difference in the energy of these two ground states gives a charging energy of $E_{C} \approx 46.5 meV$ , which is in excellent agreement with the measurement in this device. Potential profiles between source and drain electrodes calculated for $V_{G} = 0.45 V$ (top middle) and 0.72 V (bottom left). The calculated orbital probability density of the ground state for the $D^{0}$ potential (top right) is more localized around the donor than for the $D^{-}$ potential (bottom right), which is screened by the bound electron. From 116.Reuse & Permissions
Figure 19
Self-assembled nanocrystals. (a) STM image of a $Ge / Si (001)$ cluster with a height of 2.8 nm. Scan area is $40 \times 40 nm$ . From 272. (b) Band diagram for a $Si / Ge / Si$ heterostructure, showing the accumulation of holes owing to the valence band offset between Ge and Si. (c) Schematic of a quantum-dot device obtained by contacting a single SiGe nanocrystal to aluminum source or drain electrodes. The heavily doped substrate is used as a backgate for the measurements in (d) where $I_{SD}$ is plotted as a function of $V_{G}$ and $V_{SD}$ . (c), (d) From 183.Reuse & Permissions
Figure 20
Bottom-up grown nanowires. (a) TEM image of a Si nanowire; crystalline material (the Si core) appears darker than amorphous material ( ${SiO}_{x}$ sheath) in this imaging mode. Scale bar, 10 nm. (b) High-resolution TEM image of the crystalline Si core and amorphous ${SiO}_{x}$ sheath. The (111) planes (black arrows) are oriented perpendicular to the growth direction (white arrow). (a), (b) Adapted from 275. (c) Stability diagram of a $p − Si$ nanowire quantum dot. From 469. (d) SEM image of a nanowire quantum dot with NiSi Schottky contacts. From 472.Reuse & Permissions
Figure 21
Layer design and corresponding band diagram of a $Si / SiGe$ modulation-doped heterostructure used to form top-gated quantum dots. From 26.Reuse & Permissions
Figure 22
(a) Scanning electron micrograph of the Schottky gates used to form a gated quantum dot in $Si / SiGe$ . (b) Coulomb diamonds: Conductance of the dot as a function of the voltage $V_{G}$ applied to gates $G 1$ and $G 2$ and of the drain-source voltage $V_{DS}$ . From 25.Reuse & Permissions
Figure 23
Si-MOS quantum dot with large-area top gate. (a) Cross-sectional schematic, showing two oxide and two gate layers, formed on a silicon substrate. The lower ${SiO}_{2}$ layer is thermally grown, while the upper oxide layer is formed using plasma deposition. The large-area upper gate induces a 2DEG at the $Si / {SiO}_{2}$ interface, while the lower gates locally deplete the 2DEG to form a quantum dot. (b) Top-view schematic, showing lower depletion gates (black) and induced electron layer (gray). (c) Normalized spacings $δ$ between Coulomb peaks in dot conductance as a function of upper gate voltage. Inset: Raw Coulomb oscillations in dot conductance as a function of upper gate voltage. From 373.Reuse & Permissions
Figure 24
Si-MOS quantum dot with compact multilayer gate stack. (a) Scanning electron microscope image of device. (b) Cross-sectional schematic, showing three oxide layers and three Al gate layers, formed on a silicon substrate. The ${SiO}_{2}$ layer is thermally grown in a high-temperature process, while the thin ${Al}_{2} O_{3}$ layers between the gates are formed by low-temperature oxidation of the aluminum. (c) Stability map obtained by plotting differential conductance through the device as a function of source-drain bias $V_{SD}$ and plunger ( $P$ ) gate voltage $V_{P}$ . The first diamond opens up completely, indicating that the dot has been fully depleted of electrons. (d) Coulomb oscillations as a function of plunger gate voltage $V_{P}$ for the first 23 electrons in the dot. From 239.Reuse & Permissions
Figure 25
Gated quantum dot formed from a $Si / SiGe$ heterostructure with a global accumulation gate. (a) Cross-sectional view of the heterostructure and the two layers of gates. (b) Top-view SEM image of the gates with a numerical simulation of the electron density superimposed. From 261.Reuse & Permissions
Figure 26
Multigated quantum dot in etched silicon nanowire. (a) Schematic top view and cross-sectional view of the device. Three lower “wrap-around” gates (LGS, LGC, LGD) are used to form tunnel barriers in an etched silicon nanowire. (b) Top-view scanning electron microscope image of the device before the upper gate is deposited. (c) Equivalent circuit of the device. (d) Coulomb blockade oscillations in device conductance as a function of central gate voltage $V_{LGC}$ when the two outer gates (LGS, LGD) are biased to set each tunnel barrier to $G = 1 μ S$ . Inset: Coulomb oscillations for a range of values of barrier conductance from 20 nS to $8 μ S$ . From 119.Reuse & Permissions
Figure 27
Single-gated quantum dot in etched silicon nanowire. (a) SEM images and (b) cross-sectional schematics taken perpendicular to the nanowire (upper) and along the nanowire (lower). (c) Stability map (Coulomb diamonds) obtained by plotting differential current through the device as a function of source-drain bias $V_{d}$ and wrap-gate voltage $V_{g}$ . From 362, and 152.Reuse & Permissions
Figure 28
Noninvasive charge sensing of a $Si / SiGe$ quantum dot using a quantum point contact (QPC) sensor. (a) SEM device image. (b) (Top) Derivative of the QPC current $d I_{QPC} / d V_{G}$ as a function of gate voltage $V_{G}$ . The peaks correspond to changes in the number of electrons in the dot. (Bottom) Current $I_{dot}$ through the quantum dot as a function of $V_{G}$ . (c) QPC sensor output in the few-electron limit. No further transitions occur for $V_{G} < 1.68 V$ , indicating an empty quantum dot. From 374.Reuse & Permissions
Figure 29
Noninvasive charge sensing of a Si-MOS quantum dot using a single-electron transistor (SET) sensor. (a) SEM device image, showing a Si-MOS SET sensor (upper device) that is capacitively coupled to a Si-MOS quantum dot (lower device). (b) Transport current $I_{D}$ through the quantum dot shows Coulomb peaks as a function of dot plunger gate voltage $V_{PD}$ . The changing potential on the dot is detected by monitoring the uncompensated current $I_{S}$ through the SET sensor, which shows charge transfer events superimposed on a rising background, due to the coupling of the SET to $V_{PD}$ . This background can be largely removed by adding a linear correction (fixed compensation) to the SET gate voltage $V_{PS}$ , and then further enhanced by plotting the derivative $d I_{S} / d V_{PD}$ . From 462.Reuse & Permissions
Figure 30
Gate design enabling few-electron occupation. The gate design in (a) is a natural way to form a quantum dot tunnel coupled to two reservoirs, as shown by the arrows. As the dot becomes smaller, however, it is very difficult to maintain a high tunnel rate to both reservoirs. The gate design in (b), based on Fig. 1 of 58, enables a small dot to be coupled to both reservoirs.Reuse & Permissions
Figure 31
Schematic diagram of a few-electron quantum dot formed from a $Si / SiGe$ heterostructure with a double quantum well and an accumulation gate contacted by an air bridge. Inset: SEM micrograph of the gate region of a corresponding device. From 34.Reuse & Permissions
Figure 32
Transport data showing the last hole in a Si nanowire-based quantum dot. Inset: SEM image of the device showing the NiSi contacts and a $Cr / Au$ side gate device. From 473.Reuse & Permissions
Figure 33
Excited state magnetospectroscopy in Si quantum dots. (a) Zeeman splitting at the 0-1 and 1-2 transition in a few-hole Si nanowire quantum dot; (b) the corresponding magnetic field dependence of the Zeeman energy. From 473. (c) Anisotropic $g$ factors in SiGe nanocrystals; (d) the corresponding excited-state magnetospectroscopy. From 183.Reuse & Permissions
Figure 34
Ground state magnetospectroscopy. Three examples of even-odd hole spin filling. (a) A many-hole Si nanowire quantum dot (469); (b) a few-hole Si nanowire quantum dot (473); (c) a many-hole $Ge / Si$ nanowire quantum dot (336).Reuse & Permissions
Figure 35
Spin filling in valleys in a planar MOS Si quantum dot. (a) Magnetospectroscopy of the first two electrons entering the quantum dot. The circle $2 a$ marks a kink in the second Coulomb peak at $\sim 0.86 T$ . The arrows in the boxes ( $V O_{1}$ for valley orbit 1 and $V O_{2}$ for valley orbit 2) represent the spin filling of electrons in the quantum dot. (b) For $B < 0.86 T$ , the first two electrons fill with opposite spins in the same valley-orbit level (top panel). The Zeeman energy at the kink is equal to the valley-orbit splitting (0.10 meV). From 239.Reuse & Permissions
Figure 36
Schematic stability diagrams for a double dot system. Maps are shown for (a) small, (b) intermediate, and (c) large interdot couplings. The equilibrium charge on each dot in each domain is denoted by ( $N 1$ , $N 2$ ). (e) Region within the dotted square of (b), corresponding to the unit cell of the double dot stability diagram at finite-bias voltage. The solid lines separate the charge domains. Classically, the regions of the stability diagram where current flows are given by the gray triangles. From 420.Reuse & Permissions
Figure 37
Evolution from a single dot to a double quantum dot in a gated silicon nanowire device. (a) Equivalent circuit. (b)–(e) Contour plots of the drain current as a function of the outer barrier gate voltages $V_{LGS}$ and $V_{LGD}$ . The central barrier gate voltages used were (b) $V_{LGC} = - 0.75$ , (c) $- 1.13$ , (d) $- 1.18$ , and (e) $- 1.284 V$ . From 119.Reuse & Permissions
Figure 38
Gate tunable double quantum dots. (a) SEM image of a $Ge / Si$ nanowire-based hole quantum dot. The $Ge / Si$ nanowire at top (white in image) is gated by metal gates to form a double dot. (b), (c) Charge stability maps of the conductance as a function of plunger gate voltages. (d) SEM image of an electron quantum dot defined by electrostatic top gates in a $Si / SiGe$ heterostructure. (e) Charge-sensing measurement showing the difference in the charge detection signal from the dot farthest from the QPC (4 small steps in $I_{QPC}$ ) and the dot closest to the QPC (single large step) as a function of gate voltage. (e) Two-dimensional plot of the charge-sensing current showing the sequential addition of electrons to the left and right dots. (a)–(c) From 160. (d)–(f) From 375.Reuse & Permissions
Figure 39
Bias spectroscopy of silicon double quantum dots. (a) Stability map with a source-drain bias $V_{SD} = 1 mV$ for a silicon nanowire double dot, depicted in Fig. 26, obtained by plotting source-drain current $I$ as a function of two barrier gate voltages. The triple points have clearly evolved into bias triangles. (b) Bias triangles for two triple points at $V_{SD} = 1 mV$ , obtained in a Si-MOS double dot. (c) Line trace of $I_{SD}$ , taken along arrow in (b), showing resonances corresponding to excited states in the double dot. From 241, and 238.Reuse & Permissions
Figure 40
Single-electron occupancy in a $Si / SiGe$ double quantum dot. (a) SEM of the device. (b) Charge stability map of the double dot, obtained by plotting the QPC charge sensor output as a function of the control gate voltages $V_{L}$ and $V_{R}$ . The charge configurations $(n, m)$ are marked, showing depletion to the (0, 0) state. From 409.Reuse & Permissions
Figure 41
Pauli spin blockade in a silicon MOS double quantum dot. (a) SEM image; (b) cross-sectional schematic of the Si-MOS device. Gates $L 1$ and $L 2$ induce electron reservoirs at the $Si / {SiO}_{2}$ interface, while barrier gates $B 1 - B 3$ define the double dot potential. Plunger gates $P 1$ and $P 2$ control the occupancy of each dot. (c), (d) Current $I_{SD}$ as a function of $V_{P 1}$ and $V_{P 2}$ for $B = 0 T$ . (c) For $V_{SD} = + 2.5 mV$ , the ground state and excited states of a full bias triangle are observed. The current flows freely at the $S (0, 2) − S (1, 1)$ transition, as illustrated in the box marked by the dot. (b) The same configuration at $V_{SD} = - 2.5 mV$ . Here the current between the singlet and triplet states is fully suppressed by spin blockade (box marked by star). (e) The measured singlet-triplet splitting $Δ_{S T}$ , plotted as a function of magnetic field $B$ . From 221.Reuse & Permissions
Figure 42
Spin blockade and lifetime-enhanced transport in a $Si / SiGe$ double quantum dot. (a) Measured and (b) schematic charge stability map of current $I$ through the double dot with a source-drain bias of $V_{SD} = + 0.2 mV$ . The dotted trapezoids in (a) and (b) mark the zero current regions due to spin blockade, as depicted in the schematics in (c). (d) Measured and (e) schematic charge stability map of current $I$ with a source-drain bias of $V_{SD} = - 0.3 mV$ . In this bias direction there is no blockade and current flows throughout the entire bias triangle; however, additional tails are observed due to lifetime-enhanced transport, as depicted schematically in (f) and described in the text. From 363.Reuse & Permissions
Figure 43
Conductance in micron-scale silicon MOSFETs. (a) Typical low-temperature conductance pattern of a 1980s generation MOSFET around the threshold regime. The strongly oscillating but chaotic pattern that appears at low temperature is associated with localized states in the channel region. (b) Schematic representation of the three major conduction mechanisms through the channel. From 105.Reuse & Permissions
Figure 44
The importance of discrete dopants in nanoscale MOSFETs. (a) The transition from continuously ionized dopant charge and smooth boundaries and interfaces to (b) a 4-nm MOSFET where there are less than 10 Si atoms along the channel. From 13.Reuse & Permissions
Figure 45
Transport through dopants ion implanted in a nano-FET. (a) Schematic of a nano-FET where roughly three donors have been implanted into the $50 \times 30 nm$ active area of the device. (b) The stability diagram showing the differential conductance as a function of the barrier gate and dc source-drain bias, highlighting the resonant tunneling peaks $a_{1}$ , $b_{1}$ , and $c_{1}$ of the three donors. From 403.Reuse & Permissions
Figure 46
Three examples of device layouts that illustrate different transport regimes for the detection of a single dopant. (a) Capacitive coupling to the channel which leads to a modification of the channel current due to the charge state of a dopant. (b) Tunneling through a dopant in the access region in series with transport through the channel. (c) Direct tunneling through a dopant in the channel in the subthreshold regime. From (a) 297, (b) 151, and (c) 362.Reuse & Permissions
Figure 47
Three-dopant transport regimes in a transistor geometry. (a) An example of the dopant detection regime based on the capacitive coupling of the channel for an undoped (left) and doped (right) double-gate sample. The signature of a single acceptor charging event is evident in the doped sample. From 297. (b) An example of the second regime where the dopant is in the barrier of the access region in series with a quantum dot. The top line represents the room temperature FET characteristics and the line below the low-temperature Coulomb peaks From 151. (c) The third regime with direct transport through a dopant in the subthreshold limit. From 362.Reuse & Permissions
Figure 48
Direct tunneling through a dopant in a short-channel FET. (a) Illustration of a Monte Carlo simulation of the doping profile in a 20 nm channel where some dopants diffused into the channel region from the source and drain. (b) The dashed curve shows the current averaged over many devices where the black line indicates the threshold. Two devices show a drastically lower threshold linked to resonant transport at low temperature as indicated in (c) for the device with the lowest $V_{th}$ . These data show the clear connection between the low threshold of these devices at room temperature and the resonant transport at low temperature, both mediated by a single dopant. From 309.Reuse & Permissions
Figure 49
Few-electron quantum dot. (a) An STM image of the central device region of a few-electron single-crystal quantum dot acquired during hydrogen lithography, showing a four terminal device with source ( $S$ ), drain ( $D$ ), and two in-plane gates ( $G 1$ , $G 2$ ). The bright regions correspond to areas where phosphorus donors will be incorporated. (b) A close-up showing the central quantum dot containing $6 \pm 3$ donors. (c) Stability diagram showing the conductance $d I / d V_{SD}$ through the dot as a function of gate voltage $V_{G}$ and bias voltage $V_{SD}$ . (d) A close-up of the transition [white square in (c)] reveals a high density of conduction resonances with an average energy spacing of $\approx 100 μ eV$ . (e) The sixfold degeneracy of the conduction band minima of bulk silicon is lifted by confining the electrons vertically to two dimensions and is then split again by abrupt, lateral confinement. From 115.Reuse & Permissions
Figure 50
A single-atom transistor. (a) 3D perspective STM image of a hydrogenated silicon surface. Phosphorus will incorporate in the bright shaded regions selectively desorbed with an STM tip to form electrical leads to a single phosphorus atom patterned precisely in the center. (b) The source ( $S$ ), drain ( $D$ ), and two gate leads ( $G 1$ , $G 2$ ) to the central donor, which is incorporated into the dotted square region. (c) The electronic spectrum of the single-atom transistor, showing the drain current $I_{SD}$ as a function of source-drain bias $V_{SD}$ and gate voltage $V_{G}$ applied to both gates. (d) The differential conductance $d I_{SD} / d V_{SD}$ as a function of $V_{SD}$ and $V_{G}$ in the region of the $D^{0}$ diamond shown in (c). (e) A comparison of the potential profile between the source and drain electrodes in this device (straight line) to an isolated bulk phosphorus donor (dashed line), where the $D^{0}$ state resides 45.6 meV below $E_{cb}$ . In contrast, the $D^{0}$ state in the single-atom transistor resides closer to the top of the potential barrier. From 116.Reuse & Permissions
Figure 51
Excited state spectroscopy of single-gated donors. (a) Differential conductance of a dopant in a FET. Excited states are indicated by the dots and arrows. Inset in (a) shows current $I_{SD}$ as a function of gate voltage at $V_{b} = 40 mV$ where each plateau indicates the addition of a quantum channel due to an orbital. (b) Simulations of the gated donors eigenstates: wave function density of the $D^{0}$ -ground state ( $| Ψ_{GS} |^{2}$ ) located 4.3 nm below the interface in three different electric field regimes: Coulomb confinement regime $0 MV m^{- 1}$ (left), hybridized regime $20 MV m^{- 1}$ (middle), and interfacial confinement regime $40 MV m^{- 1}$ (right). The gray plane indicates the $Si / {SiO}_{2}$ interface. From 225.Reuse & Permissions
Figure 52
Sequential transport through a double donor device with independent gate control. The left panel shows the two opposing gates similar to a conventional FinFET geometry but with a split gate. The channel received a background doping of $10^{18} P / {cm}^{3}$ and this device demonstrates independent gate control of two dopants. The right panel shows a finite-bias stability diagram revealing bulk-like excited states of the dopant. From 335.Reuse & Permissions
Figure 53
A donor-based double quantum dot in silicon. (a) An overview STM image of the device showing the two quantum dots, tunnel coupled to the source and drain ( $S / D$ ) leads and capacitively coupled to the gates $G_{1 (2)}$ . (b) Close-up of the two quantum dots $\sim 4 nm$ in diameter. The double quantum dot angle $α = 60 ° \pm 3 °$ has been optimized for maximum electrostatic control while suppressing parallel leakage through the dots. (c) Modeled and (d) measured charge stability diagrams show excellent agreement, demonstrating independent electrostatic control of the individual dots. From 433.Reuse & Permissions
Figure 54
Charge sensing using a donor-based single-electron transistor coupled to a small donor dot. (a) Filled-state STM image of the overall device pattern, showing (in lighter contrast) the regions where the hydrogen resist monolayer has been desorbed to create the source ( $S$ ) and drain ( $D$ ) contacts of the single-electron transistor, and the two gates ( $G_{1}$ , $G_{2}$ ). (b) High-resolution image of the device pattern within the white box in (a), showing the SET island ( $D_{1}$ ) and the quantum dot ( $D_{2}$ ) (c) Charge stability plot showing the dependence of $I_{SD}$ on the gate voltages ( $V_{G 1}$ , $V_{G 2}$ ), for a constant $V_{SD} = - 50 μ V$ . The high current lines correspond to the Coulomb peaks of the SET. Inset: High-resolution map of a small section of (c) showing discontinuity of a current line, due to a particular charge transition of $D 2$ . The triangles in the main map indicate a total of seven such transitions of $D 2$ . From 254.Reuse & Permissions
Figure 55
(a) Spin-lattice relaxation rate $T_{1}^{- 1}$ of P donors in bulk Si, at $B \approx 0.3 T$ and $T = 1.2 K$ , as a function of the field orientation. The angular dependence allows the separation of “valley repopulation” and “single valley” contributions. From 448. (b) $T_{1}^{- 1} (B)$ for single P donors in two different devices. Both show a $T_{1}^{- 1} \propto B^{5}$ contribution, but device A also exhibits a $B$ -independent plateau, attributed to dipolar flip-flops with nearby donors. Also shown is $T_{1}^{- 1} (3.3 T)$ in bulk Si:P. From 277. (c) $T_{1}^{- 1} (B)$ in a gate-defined $Si / SiGe$ dot (●), compared to data for a InGaAs dot (▪,⧫). From 144. (d) $T_{1} (B)$ in a gate-defined Si-MOS dot, for the one-electron (▪) and two-electron (○) states. From 456.Reuse & Permissions
Figure 56
(a) Experimental echo decay and cluster-expansion theory for ${}^{nat}{Si} ∶ P$ at different angles of the magnetic field with respect to the crystallographic [001] axis. Notice the echo envelope modulation arising from anisotropic hyperfine coupling between donor electron and ${}^{29}{Si}$ nuclei. From 453. (b) Decoherence time $T_{2}$ for Si:P as a function of ${}^{29}{Si}$ concentration $C_{N}$ for different dopant concentrations $C_{E}$ . Symbols are experimental data points. From 450.Reuse & Permissions
Figure 57
(a) Sketch of the spin-dependent transition between a donor-bound electron and an interface trap, following the creation of free carriers through illumination. (b) Schematics of an EDMR device. P donors close to charge traps at the $Si / {SiO}_{2}$ interface contribute a spin-dependent scattering mechanism for the electrons traveling between the Au contacts. A resonant microwave excitation alters the polarization of the donor-bound electrons, causing a measurable change of the overall device resistance (c) Electrically detected Rabi oscillations of P-donor electrons at different values of the driving power. From 388.Reuse & Permissions
Figure 58
(a) Schematics of a single-charge trap coupled to the channel of a Si transistor. (b) Single-electron spin resonance measurement, obtained by monitoring the average current through the transistor as a function of magnetic field, while applying a microwave excitation at 45 GHz. The excess current at the resonance frequency arises from the change in charge occupancy of the trap, made possible by the driven flipping of its electron spin. From 458.Reuse & Permissions
Figure 59
(a) Spin-to-charge conversion scheme for a single donor tunnel coupled to the island of an SET. The presence of quantized states inside the SET island can be ignored if the single-particle energy level spacing is smaller than the thermal broadening. From 276. (b) Single-shot readout of a donor electron spin. The individual traces show the evolution of the readout signal as a function of the donor electrochemical potential with respect to the Fermi level. From 277.Reuse & Permissions
Figure 60
Single-shot readout of singlet-triplet states in a $Si / SiGe$ double quantum dot. (a), (b) QPC current traces $I_{QPC}$ while pulsing the detuning with a square wave. Singlet states are identified when $I_{QPC}$ returns to a high value as in (b). (c) Charge stability diagram and pulsing levels. (d)–(f) Time traces of $I_{QPC}$ at different magnetic fields as indicated. Increasing $B$ extends the lifetime of the $T_{11}$ (constant current) state. (g) Control sequence, pulsing outside the spin-blockade region. From 314.Reuse & Permissions
Figure 61
Coherent manipulation of singlet-triplet states in a $Si / SiGe$ double quantum dot. (a) Charge stability diagram of the double dot system. Arrows describe the trajectory in gate space during the pulsing sequence shown in (b). The (0,2) singlet state is prepared at point $F$ . Adiabatically moving to point $S$ , where the exchange coupling is very weak, brings the system to the (1,1) singlet. Pulsing to point $E$ turns on the exchange and causes the two-spin state to oscillate between the (1,1) singlet and triplet. $M$ is the measurement point where the electrons recombine in the (0,2) state if in a singlet state. (c) Rabi oscillations of the singlet probability as a function of the exchange pulse duration (time spent at point $E$ ) and (0,2)-(1,1) detuning $ϵ$ . (d) Bloch sphere representation of the trajectories of the two-spin states for different initial values of the hyperfine fields. From 261.Reuse & Permissions
Figure 62
Single-atom electron spin qubit based on an implanted ${}^{31}P$ donor. (a) Optimized design of an on-chip planar transmission line capable of delivering coherent microwave pulses at frequencies up to 50 GHz. From 76 (b) Scanning electron micrograph of the spin qubit device. (c) Rabi oscillations of the electron spin state with 10 dBm driving power at 30 GHz. (d) Measurement of spin coherence with Hahn echo and $X Y X Y$ dynamical decoupling. (b)–(d) From 310.Reuse & Permissions

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