* Wu et al. [2022b] Yixuan Wu, Bo Zheng, Jintai Chen, Danny Z Chen, and Jian Wu.  Self-learning and One-Shot Learning Based Single-Slice Annotation for 3D Medical Image Segmentation.  In _Medical Image Computing and Computer Assisted Intervention–MICCAI 2022: 25th International Conference, Singapore, September 18–22, 2022, Proceedings, Part VIII_ , pages 244–254. Springer, 2022b.
  * Yu et al. [2021] Qinji Yu, Kang Dang, Nima Tajbakhsh, Demetri Terzopoulos, and Xiaowei Ding.  A location-sensitive local prototype network for few-shot medical image segmentation.  In _2021 IEEE 18th International Symposium on Biomedical Imaging (ISBI)_ , pages 262–266. IEEE, 2021.
  * Yuan et al. [2021] Zheng Yuan, Andre Esteva, and Ran Xu.  MetaHistoSeg: A Python Framework for Meta Learning in Histopathology Image Segmentation.  In _Deep Generative Models, and Data Augmentation, Labelling, and Imperfections_ , pages 268–275. Springer, 2021.
  * Zhao et al. [2019] Amy Zhao, Guha Balakrishnan, Fredo Durand, John V Guttag, and Adrian V Dalca.  Data augmentation using learned transformations for one-shot medical image segmentation.  In _Proceedings of the IEEE/CVF conference on computer vision and pattern recognition_ , pages 8543–8553, 2019.
  * Zhao et al. [2022] Ziyuan Zhao, Fangcheng Zhou, Zeng Zeng, Cuntai Guan, and S Kevin Zhou.  Meta-hallucinator: Towards Few-Shot Cross-Modality Cardiac Image Segmentation.  In _Medical Image Computing and Computer Assisted Intervention–MICCAI 2022: 25th International Conference, Singapore, September 18–22, 2022, Proceedings, Part V_ , pages 128–139. Springer, 2022.
  * Zhou et al. [2021] Hong-Yu Zhou, Hualuo Liu, Shilei Cao, Dong Wei, Chixiang Lu, Yizhou Yu, Kai Ma, and Yefeng Zheng.  Generalized Organ Segmentation by Imitating One-shot Reasoning using Anatomical Correlation.  In _International Conference on Information Processing in Medical Imaging_ , pages 452–464. Springer, 2021.
  * Ali et al. [2020] Sharib Ali, Binod Bhattarai, Tae-Kyun Kim, and Jens Rittscher.  Additive angular margin for few shot learning to classify clinical endoscopy images.  In _International Workshop on Machine Learning in Medical Imaging_ , pages 494–503. Springer, 2020.
  * Cai et al. [2020] Aihua Cai, Wenxin Hu, and Jun Zheng.  Few-shot learning for medical image classification.  In _International Conference on Artificial Neural Networks_ , pages 441–452. Springer, 2020.
  * Cai et al. [2022] Anping Cai, Leiting Chen, Yongqi Chen, Jiahao Fang, Mengqi Sun, and Zhou Chuan.  Pre-MocoDiagnosis: Few-Shot Ophthalmic Diseases Recognition using Contrastive Learning.  In _2022 IEEE International Conference on Bioinformatics and Biomedicine (BIBM)_ , pages 2059–2066. IEEE, 2022.
  * Cano and Cruz-Roa [2020] Fabian Cano and Angel Cruz-Roa.  An exploratory study of one-shot learning using Siamese convolutional neural network for histopathology image classification in breast cancer from few data examples.  In _15th International Symposium on Medical Information Processing and Analysis_ , volume 11330, pages 66–73. SPIE, 2020.
  * Chen et al. [2019b] Jiaojiao Chen, Jianbo Jiao, Shengfeng He, Guoqiang Han, and Jing Qin.  Few-shot breast cancer metastases classification via unsupervised cell ranking.  _IEEE/ACM Transactions on Computational Biology and Bioinformatics_ , 18(5):1914–1923, 2019b.  Publisher: IEEE.
  * Chou et al. [2022] Yiyu Chou, Samuel W Remedios, John A Butman, and Dzung L Pham.  Automatic classification of MRI contrasts using a deep Siamese network and one-shot learning.  In _Medical Imaging 2022: Image Processing_ , volume 12032, pages 110–114. SPIE, 2022.
  * Dai et al. [2023] Zhiyong Dai, Jianjun Yi, Lei Yan, Qingwen Xu, Liang Hu, Qi Zhang, Jiahui Li, and Guoqiang Wang.  PFEMed: Few-shot medical image classification using prior guided feature enhancement.  _Pattern Recognition_ , 134:109108, 2023.  Publisher: Elsevier.
  * Huang et al. [2022] Weikai Huang, Yijin Huang, and Xiaoying Tang.  AugPaste: One-Shot Anomaly Detection for Medical Images.  In _Ophthalmic Medical Image Analysis: 9th International Workshop, OMIA 2022, Held in Conjunction with MICCAI 2022, Singapore, Singapore, September 22, 2022, Proceedings_ , pages 1–11. Springer, 2022.
  * Jiang et al. [2022] Hongyang Jiang, Mengdi Gao, Heng Li, Richu Jin, Hanpei Miao, and Jiang Liu.  Multi-Learner Based Deep Meta-Learning for Few-Shot Medical Image Classification.  _IEEE Journal of Biomedical and Health Informatics_ , 27(1):17–28, 2022.  Publisher: IEEE.
  * Jin et al. [2021] Yufei Jin, Huijuan Lu, Wenjie Zhu, Ke Yan, Zhigang Gao, and Zhao Li.  CTFC: A Convolution and Visual Transformer Based Classifier for Few-Shot Chest X-ray Images.  In _2021 2nd International Conference on Artificial Intelligence and Computer Engineering (ICAICE)_ , pages 616–622. IEEE, 2021.
  * Mahapatra et al. [2022] Dwarikanath Mahapatra, Zongyuan Ge, and Mauricio Reyes.  Self-Supervised Generalized Zero Shot Learning For Medical Image Classification Using Novel Interpretable Saliency Maps.  _IEEE Transactions on Medical Imaging_ , 2022.  Publisher: IEEE.
  * Maicas et al. [2019] Gabriel Maicas, Andrew P Bradley, Jacinto C Nascimento, Ian Reid, and Gustavo Carneiro.  Pre and post-hoc diagnosis and interpretation of malignancy from breast DCE-MRI.  _Medical Image Analysis_ , 58:101562, 2019.  Publisher: Elsevier.
  * Mohan [2021] Vishnu Mohan.  Detection of COVID-19 from Chest X-ray Images: A Deep Learning Approach.  In _2021 Ethics and Explainability for Responsible Data Science (EE-RDS)_ , pages 1–7. IEEE, 2021.
  * Moukheiber et al. [2022] Dana Moukheiber, Saurabh Mahindre, Lama Moukheiber, Mira Moukheiber, Song Wang, Chunwei Ma, George Shih, Yifan Peng, and Mingchen Gao.  Few-Shot Learning Geometric Ensemble for Multi-label Classification of Chest X-Rays.  In _Data Augmentation, Labelling, and Imperfections: Second MICCAI Workshop, DALI 2022, Held in Conjunction with MICCAI 2022, Singapore, September 22, 2022, Proceedings_ , pages 112–122. Springer, 2022.
  * Naren et al. [2021] Tarun Naren, Yuanda Zhu, and May Dongmei Wang.  COVID-19 diagnosis using model agnostic meta-learning on limited chest X-ray images.  In _Proceedings of the 12th ACM Conference on Bioinformatics, Computational Biology, and Health Informatics_ , pages 1–9, 2021.
  * Ouahab et al. [2022] Achraf Ouahab, Olfa Ben-Ahmed, and Christine Fernandez-Maloigne.  A Self-attentive Meta-learning Approach for Image-Based Few-Shot Disease Detection.  In _Resource-Efficient Medical Image Analysis: First MICCAI Workshop, REMIA 2022, Singapore, September 22, 2022, Proceedings_ , pages 115–125. Springer, 2022.
  * Paul et al. [2020] Angshuman Paul, Yu-Xing Tang, and Ronald M Summers.  Fast few-shot transfer learning for disease identification from chest x-ray images using autoencoder ensemble.  In _Medical Imaging 2020: Computer-Aided Diagnosis_ , volume 11314, pages 33–38. SPIE, 2020.
  * Paul et al. [2021a] Angshuman Paul, Thomas C Shen, Yifan Peng, Zhiyong Lu, and Ronald M Summers.  Learning Few-Shot Chest X-Ray Diagnosis Using Images From The Published Scientific Literature.  In _2021 IEEE 18th International Symposium on Biomedical Imaging (ISBI)_ , pages 344–348. IEEE, 2021a.
  * Paul et al. [2021b] Angshuman Paul, Thomas C Shen, Sungwon Lee, Niranjan Balachandar, Yifan Peng, Zhiyong Lu, and Ronald M Summers.  Generalized zero-shot chest x-ray diagnosis through trait-guided multi-view semantic embedding with self-training.  _IEEE Transactions on Medical Imaging_ , 40(10):2642–2655, 2021b.  Publisher: IEEE.
  * Singh et al. [2021] Rishav Singh, Vandana Bharti, Vishal Purohit, Abhinav Kumar, Amit Kumar Singh, and Sanjay Kumar Singh.  MetaMed: Few-shot medical image classification using gradient-based meta-learning.  _Pattern Recognition_ , 120:108111, 2021.  Publisher: Elsevier.
  * Vétil et al. [2022] Rebeca Vétil, Clément Abi-Nader, Alexandre Bône, Marie-Pierre Vullierme, Marc-Michel Rohé, Pietro Gori, and Isabelle Bloch.  Learning shape distributions from large databases of healthy organs: applications to zero-shot and few-shot abnormal pancreas detection.  In _Medical Image Computing and Computer Assisted Intervention–MICCAI 2022: 25th International Conference, Singapore, September 18–22, 2022, Proceedings, Part II_ , pages 464–473. Springer, 2022.
  * Xiao et al. [2021] Junsheng Xiao, Huahu Xu, DiKai Fang, Chen Cheng, and HongHao Gao.  Boosting and rectifying few-shot learning prototype network for skin lesion classification based on the internet of medical things.  _Wireless Networks_ , pages 1–15, 2021.  Publisher: Springer.
  * Yan et al. [2022] Jin Yan, Kaiyuan Feng, Hongyu Zhao, and Kai Sheng.  Siamese-Prototypical Network with Data Augmentation Pre-training for Few-shot Medical Image Classification.  In _2022 2nd International Conference on Frontiers of Electronics, Information and Computation Technologies (ICFEICT)_ , pages 387–391. IEEE, 2022.
  * Yarlagadda et al. [2019] Dig Vijay Kumar Yarlagadda, Praveen Rao, Deepthi Rao, and Ossama Tawfik.  A system for one-shot learning of cervical cancer cell classification in histopathology images.  In _Medical Imaging 2019: Digital Pathology_ , volume 10956, pages 216–221. SPIE, 2019.
  * Zhang et al. [2022] Ce Zhang, Qingshan Cui, and Shaolong Ren.  Few-shot Medical Image Classification with MAML Based on Dice Loss.  In _2022 IEEE 2nd International Conference on Data Science and Computer Application (ICDSCA)_ , pages 348–351. IEEE, 2022.
  * Zhu et al. [2020] Wei Zhu, Haofu Liao, Wenbin Li, Weijian Li, and Jiebo Luo.  Alleviating the incompatibility between cross entropy loss and episode training for few-shot skin disease classification.  In _International Conference on Medical Image Computing and Computer-Assisted Intervention_ , pages 330–339. Springer, 2020.
  * Fechter and Baltas [2020] Tobias Fechter and Dimos Baltas.  One-shot learning for deformable medical image registration and periodic motion tracking.  _IEEE transactions on medical imaging_ , 39(7):2506–2517, 2020.  Publisher: IEEE.
  * Ferrante et al. [2018] Enzo Ferrante, Ozan Oktay, Ben Glocker, and Diego H Milone.  On the adaptability of unsupervised CNN-based deformable image registration to unseen image domains.  In _International Workshop on Machine Learning in Medical Imaging_ , pages 294–302. Springer, 2018.
  * He et al. [2022b] Yuting He, Tiantian Li, Rongjun Ge, Jian Yang, Youyong Kong, Jian Zhu, Huazhong Shu, Guanyu Yang, and Shuo Li.  Few-shot learning for deformable medical image registration with perception-correspondence decoupling and reverse teaching.  _IEEE Journal of Biomedical and Health Informatics_ , 26(3):1177–1187, 2022b.  Publisher: IEEE.
  * Zhang et al. [2021] Yunlu Zhang, Xue Wu, H Michael Gach, Harold Li, and Deshan Yang.  GroupRegNet: a groupwise one-shot deep learning-based 4D image registration method.  _Physics in Medicine & Biology_, 66(4):045030, 2021.  Publisher: IOP Publishing.
  * Kavur et al. [2021] A Emre Kavur, N Sinem Gezer, Mustafa Barış, Sinem Aslan, Pierre-Henri Conze, Vladimir Groza, Duc Duy Pham, Soumick Chatterjee, Philipp Ernst, Savaş Özkan, and others.  CHAOS challenge-combined (CT-MR) healthy abdominal organ segmentation.  _Medical Image Analysis_ , 69:101950, 2021.  Publisher: Elsevier.
  * Zhuang [2018] Xiahai Zhuang.  Multivariate mixture model for myocardial segmentation combining multi-source images.  _IEEE transactions on pattern analysis and machine intelligence_ , 41(12):2933–2946, 2018.  Publisher: IEEE.
  * Wang et al. [2017] Xiaosong Wang, Yifan Peng, Le Lu, Zhiyong Lu, Mohammadhadi Bagheri, and Ronald M Summers.  Chestx-ray8: Hospital-scale chest x-ray database and benchmarks on weakly-supervised classification and localization of common thorax diseases.  In _Proceedings of the IEEE conference on computer vision and pattern recognition_ , pages 2097–2106, 2017.


# Electrically driven electron spin resonance mediated by spin-valley-orbit
coupling in a silicon quantum dot111Published on npj Quantum Information 4,6
(2018), doi:10.1038/s41534-018-0059-1. License CC BY 4.0

Andrea Corna  CEA, INAC-PHELIQS, 17 Rue des Martyrs, F-38000 Grenoble, France
University Grenoble Alpes, Grenoble, France  Léo Bourdet  CEA, INAC-MEM, 17
Rue des Martyrs, F-38000 Grenoble, France  University Grenoble Alpes,
Grenoble, France  Romain Maurand  CEA, INAC-PHELIQS, 17 Rue des Martyrs,
F-38000 Grenoble, France  University Grenoble Alpes, Grenoble, France
Alessandro Crippa  CEA, INAC-PHELIQS, 17 Rue des Martyrs, F-38000 Grenoble,
France  University Grenoble Alpes, Grenoble, France  Dharmraj Kotekar-Patil
CEA, INAC-PHELIQS, 17 Rue des Martyrs, F-38000 Grenoble, France  University
Grenoble Alpes, Grenoble, France  Heorhii Bohuslavskyi  CEA, INAC-PHELIQS, 17
Rue des Martyrs, F-38000 Grenoble, France  CEA, LETI-MINATEC, 17 Rue des
Martyrs, F-38000 Grenoble, France  University Grenoble Alpes, Grenoble, France
Romain Laviéville  CEA, LETI-MINATEC, 17 Rue des Martyrs, F-38000 Grenoble,
France  University Grenoble Alpes, Grenoble, France  Louis Hutin  CEA, LETI-
MINATEC, 17 Rue des Martyrs, F-38000 Grenoble, France  University Grenoble
Alpes, Grenoble, France  Sylvain Barraud  CEA, LETI-MINATEC, 17 Rue des
Martyrs, F-38000 Grenoble, France  University Grenoble Alpes, Grenoble, France
Xavier Jehl  CEA, INAC-PHELIQS, 17 Rue des Martyrs, F-38000 Grenoble, France
University Grenoble Alpes, Grenoble, France  Maud Vinet  CEA, LETI-MINATEC, 17
Rue des Martyrs, F-38000 Grenoble, France  University Grenoble Alpes,
Grenoble, France  Silvano De Franceschi  CEA, INAC-PHELIQS, 17 Rue des
Martyrs, F-38000 Grenoble, France  University Grenoble Alpes, Grenoble, France
Yann-Michel Niquet  CEA, INAC-MEM, 17 Rue des Martyrs, F-38000 Grenoble,
France  University Grenoble Alpes, Grenoble, France  Marc Sanquer  CEA, INAC-
PHELIQS, 17 Rue des Martyrs, F-38000 Grenoble, France  University Grenoble
Alpes, Grenoble, France

The ability to manipulate electron spins with voltage-dependent electric
fields is key to the operation of quantum spintronics devices such as spin-
based semiconductor qubits. A natural approach to electrical spin control
exploits the spin-orbit coupling (SOC) inherently present in all materials. So
far, this approach could not be applied to electrons in silicon, due to their
extremely weak SOC. Here we report an experimental realization of electrically
driven electron-spin resonance in a silicon-on-insulator (SOI) nanowire
quantum dot device. The underlying driving mechanism results from an interplay
between SOC and the multi-valley structure of the silicon conduction band,
which is enhanced in the investigated nanowire geometry. We present a simple
model capturing the essential physics and use tight-binding simulations for a
more quantitative analysis. We discuss the relevance of our findings to the
development of compact and scalable electron-spin qubits in silicon.

##  I Introduction

Silicon is a strategic semiconductor for quantum spintronics, combining long
spin coherence and mature technology Zwanenburg13 . Research on silicon-based
spin qubits has seen a tremendous progress over the past five years. In
particular, very long coherence times have been achieved with the introduction
of devices based on the nuclear-spin-free 28Si isotope, enabling the
suppression of hyperfine coupling, the main source of spin decoherence
Tyryshkin12 . Single qubits with fidelities exceeding 99% as well as a first
demonstration of a two-qubit gate have been reported Veldhorst14 ; Laucht15 ;
Veldhorst15b .

finding a viable pathway towards large-scale integration is the next step. To
this aim, access to electric-field-mediated spin control would facilitate
device scalability, circumventing the need for more demanding control schemes
based on magnetic-field-driven spin resonance. Electric-field control requires
a mechanism coupling spin and motional degrees of freedom. This so-called
spin-orbit coupling (SOC) is generally present in atoms and solids – due to a
relativistic effect, electrons moving in an electric-field gradient experience
in their reference frame an effective magnetic field. In the case of electrons
in silicon, however, SOC is intrinsically very weak.

Possible approaches to circumvent this limitation have so far relied either on
the introduction of micromagnets, generating local magnetic field gradients
and hence an artificial SOC Pioro-Ladriere08 ; Kawakami14 ; Rancic16 ;
Takeda16 , or on the use of hole spins Maurand16 , for which SOC is strong. In
both cases, relatively fast coherent spin rotations could be achieved through
resonant radio-frequency (RF) modulation of a control gate voltage. While the
actual scalability of these two solutions remains to be investigated, other
valuable opportunities may emerge from the rich physics of electrons in
silicon nanostructures Nestoklon06 ; Zwanenburg13 ; Hao14 ; Schoenfield17 ;
Jock17 ; Veldhorst15 ; Ruskov2017 .

Silicon is indeed an indirect band-gap semiconductor with six degenerate
conduction-band valleys. This degeneracy is lifted in quantum dots (QD) where
quantum confinement leaves only two low-lying valleys that can be coupled by
potential steps at Si/SiO2 interfaces. The resulting valley eigenstates, which
we label $v_{1}$ and $v_{2}$, are separated by an energy splitting $\Delta$
ranging from a few tens of $\mu$eV to a few meV Sham79 ; Saraiva09 ; Friesen10
; Culcer10 . $\Delta$ depends on the confinement potential, and can hence be
tuned by externally applied electric fields Goswami07 ; Yang13 ; Laucht15 .
Even if weak Rashba08 , SOC can couple valley and spin degrees of freedom
when, following the application of a magnetic field, $E_{Z}\sim\Delta$, where
$E_{Z}$ is the Zeeman energy splitting. It has been shown that this operating
regime can result in enhanced spin relaxation Yang13 ; Veldhorst14 ; Huang2014
.

Here we demonstrate that it can be exploited to perform electric-dipole spin
resonance (EDSR) Golovach06 ; Nowack07 ; Nadj-Perge10 ; Pribiag13 . This
functionality is enabled by the use of a quantum dot with a low-symmetry
confinement potential. We discuss the implications of these results for the
development of silicon spin qubits.

##  II Results

Figure 1: Device schematics. a, Sketch of the sample and measurement setup.
The silicon nanowire with the source/drain extensions is colored in yellow,
the two top gates in green, and the side gate in violet. The gate oxides are
colored in orange and the buried oxide (BOX) of the SOI in blue. b, Colorized
device top view obtained by scanning electron microscopy before the deposition
of the spacers of a device similar to the one used in the experiment. False-
colors as in a. c, Sketch of the transverse cross-section of the device along
the dashed line in b, with the spacer in gray. The cross-section of the
silicon nanowire (yellow background) shows a color map of the square of the
tight-binding wave function of the lowest conduction band state.

The experiment is carried out on a silicon nanowire device fabricated on a 300
mm diameter silicon-on-insulator wafer using an industrial-scale fabrication
line Maurand16 . The device, shown in the schematic of fig. 1a and in the
scanning electron micrograph of fig. 1b, consists of an undoped, $30$ nm-wide
and $12$ nm-thick silicon channel oriented along $[110]$, with $n$-doped
contacts. Two $35$ nm-wide top-gates (gate 1 and gate 2), spaced by $30$ nm,
partially cover the channel. An additional gate (gate 3) is located on the
opposite side at a distance of $50$ nm from the nanowire. Electron transport
measurements were performed in a dilution refrigerator with a base temperature
$T=15$ mK. At this temperature, two QDs in series, labeled as QD1 and QD2, can
be defined by the accumulation voltages $V_{g1}$ and $V_{g2}$ applied to gate
1 and gate 2, respectively. The two QDs are confined against the nanowire edge
covered by the gates, forming so-called “corner” dots Voisin14 ; Gonzalez15 ,
as confirmed by tight-binding simulations of the lowest energy states, whose
wave-functions are shown in fig. 1c. We tune the electron filling of QD1 and
QD2 down to relatively small occupation numbers $n_{1}$ and $n_{2}$,
respectively ($n_{1},n_{2}<10$, as inferred from the threshold voltage at room
temperature and the charging energy Maurand16 ). The side gate is set to a
negative $V_{g3}=-0.28$ V in order to further push the QD wave-functions
against the opposite nanowire edges.

In the limit of vanishing inter-dot coupling and odd occupation numbers, both
QD1 and QD2 have a spin-1/2 ground state. At finite magnetic field,
$\mathbf{B}$, the respective spin degeneracies are lifted by the Zeeman energy
$E_{Z}=g\mu_{\rm B}B$, where $\mu_{\rm B}$ is the Bohr magneton and $g$ is the
Landé $g$-factor, which is close to the bare electron value ($g\simeq 2$) for
electrons in silicon Feher59b . In essence, our experiment consists in
measuring electron transport through the double dot while driving EDSR in QD2.
The polarized spin in QD1 acts as an effective “spin filter” regulating the
current flow as a function of the spin admixture induced by EDSR in QD2. This
Pauli blockade regime can be achieved only when the double dot is biased in a
charge/spin configuration where inter-dot tunneling is forbidden by spin
conservation Hanson07 . The simplest case involves the inter-dot charge
transition $(n_{1}=1,n_{2}=1)\to(n_{1}=0,n_{2}=2)$, where one electron tunnels
from QD1 into QD2. The two electrons may indeed form singlet (S) or triplet
(T) states. While the singlet $S(1,1)$ and triplet $T(1,1)$ states are only
weakly split by exchange interations and magnetic field and may both be
loaded, the triplet $T(0,2)$ states remain typically out of reach because they
must involve some orbital excitation of QD2. The system may hence be trapped
for long times in the $T(1,1)$ states since tunneling from $T(1,1)$ to the
$S(0,2)$ ground-state is forbidden by Pauli exclusion principle Hanson07 .
This scenario can be generalized to the $(n_{1},n_{2})\to(n_{1}-1,n_{2}+1)$
transitions where $n_{1}$ and $n_{2}$ are odd integers. The current is
strongly suppressed unless EDSR mixes $T(1,1)$ and $S(1,1)$ by rotating the
spin in QD2.

Because the opposite $(0,2)\to(1,1)$ transition (or, more generically,
$(n_{1}-1,n_{2}+1)\to(n_{1},n_{2})$) is never blocked (there is always a (1,1)
spin singlet to tunnel to), the Pauli blockade regime can be revealed by
source-drain current rectification Ono02 . fig. 2 presents measurements of the
source-drain current, $I_{ds}$, as a function of $(V_{g1},V_{g2})$ in a charge
configuration exhibiting Pauli rectification. fig. 2a corresponds to a source-
drain bias voltage $V_{ds}=-2.5$ mV and a magnetic field $B=0.7$ T. Current
flows within characteristic triangular regions Hanson07  where the
electrochemical potential of dot 1, $\mu_{1}(n_{1},n_{2})$, is lower than the
electrochemical potential of dot 2, $\mu_{2}(n_{1}-1,n_{2}+1)$. The energy
detuning $\varepsilon$ between the two electrochemical potentials increases
when moving along the red arrow. Current contains contributions from both
elastic (i.e. resonant) and inelastic inter-dot tunneling. fig. 2b shows that
reversing the bias voltage (i.e. $V_{ds}=2.5$ mV) yields the desired Pauli
rectification characterized by truncated current triangles (In Supplementary
Note 1 we discuss the presence of a concomitant valley-blockade effect similar
to the one shown by Hao et al. Hao14 ).

The extent of the spin-blockade region measured along the detuning axis
corresponds to the energy splitting, $\Delta_{\rm ST}$, between singlet and
triplet states in the $(n_{1}-1,n_{2}+1)$ charge configuration (which is
equivalent to $(0,2)$), basically the singlet-triplet splitting in QD2 222in
the case of non-degenerate triplet states at finite $B$, $\Delta_{\rm ST}$ is
the splitting with respect to the triplet state, $T_{0}$, with zero spin
projection along $\mathbf{B}$. We find $\Delta_{\rm ST}=1.9$ meV. figures 2-c)
and 2-d) show $I_{ds}$ as a function of $B$ and $\varepsilon$ for negative and
positive $V_{ds}$, respectively. As expected Ono02 , in the non-spin blocked
polarity (fig. 2c) $I_{ds}$ shows essentially no dependence on $B$. In the
opposite polarity, spin blockade is lifted at low field ($B\lesssim 0.1$ T),
due to spin-flip cotunneling Lai11 ; Yamahata12 , as well as at $B=0.31$ T.
This unexpected feature will be discussed later.

Figure 2: Pauli Spin Blockade. a, Map of the current $I_{ds}$ as a function of
$V_{g1}$ and $V_{g2}$ at finite $V_{ds}=-2.5$ mV around a pair of triangles
showing current rectification. The magnetic field $B=0.7$ T is parallel to
$\mathbf{x}-\mathbf{y}$ (see axes on fig. 1) The number of electrons in each
dot is given between parentheses, with $n$, $m$ integers. b, Same as a at
opposite $V_{ds}=+2.5$ mV. c, Current as a function of detuning and magnetic
field. The detuning axis is highlighted by a red arrow in a. d, Same as c at
opposite $V_{ds}=+2.5$ mV with detuning axis defined in b.

We now focus on the spin resonance experiment. To manipulate the spin
electrically, we set $V_{g1}$ and $V_{g2}$ in the spin blocked region and
apply a microwave excitation of frequency $\nu$ on gate 2. fig. 3a displays
$I_{ds}$ as a function of $B$ and $\nu$ at constant power at the microwave
source (The power at the sample depends on $\nu$ and is estimated to be
$-53\,\text{dBm}\simeq 0.7\,\text{mV}$ peak at $\nu\approx$ 9.6 GHz). Several
lines of increased current are visible in this plot, highlighting resonances
along which Pauli spin blockade is lifted. They are labeled A, B, C and V. In
the simplest case, spin resonance occurs when the microwave photon energy
matches the Zeeman splitting between the two spin states of a doublet, i.e.
when $h\nu=E_{Z}=g\mu_{\rm B}B$. We assign such a resonance to line A, because
line A extrapolates to the origin ($B=0$, $\nu=0$). Its slope gives $g_{\rm
A}=1.980\pm 0.005$, which is compatible with the $g$-factor expected for
electrons in silicon Feher59b . Also line C extrapolates to the origin but
with approximately half the slope (i.e. $g_{\rm C}=0.96\pm 0.01$). We
attribute this line to a second-harmonic driving process Scarlino15 .

We now focus on resonances B and V. The slope of line B is also compatible
with the electron $g$-factor ($g_{\rm B}=2.00\pm 0.01$). However, line B
crosses zero-frequency at $B_{\rm V}=0.314\pm 0.001$ T, corresponding exactly
to the magnetic field at which the non-dispersive resonance V appears.
Consequently, line B can be assigned to transitions between spin states
associated with two distinct orbitals. When these spin states cross at $B_{\rm
V}$, Pauli spin blockade is lifted independently of the microwave excitation
leading to the non-dispersive resonance V (see Supplementary Note 1 for
details on the lifting of spin blockade at $B_{\rm V}$).

Figure 3: Spin Resonance. a, Color plot of the measured drain current $I_{ds}$
as function of the magnetic field $B$ and microwave frequency $\nu$. The gates
are biased in the spin-blockade regime. Three different measurements are
gathered for clarity. EDSR transitions are revealed by oblique straight lines,
labeled with letters A, B, and C. A vertical line (V) is also present. Line C
is the weak feature that starts around $\approx 0.63$ T close to line B in the
smallest inset. The faint vertical line at $\approx 0.61$T is an artifact. b,
Diagram of the energy levels of a system with spin $S=1/2$ and two valleys
($v_{1}$, $v_{2}$) as function of magnetic field. At low magnetic field, the
two lowest (highest) levels belongs to valley $v_{1}$ ($v_{2}$). Near the
anti-crossing field $B_{\rm V}$, the states $\lvert v_{1},\uparrow\rangle$ and
$\lvert v_{2},\downarrow\rangle$ hybridize due to SOC. The spin and valley
composition of the hybridized states is quantified by the color scale on the
right. The possible EDSR transitions are marked with green arrows (within
$v_{1}$ or $v_{2}$) and orange/violet arrows (between $v_{1}$ and $v_{2}$). c,
Sketch of the possible EDSR transitions in panel b as function of magnetic
field and microwave frequency.

In order to understand the experimental EDSR spectrum of fig. 3a, we neglect
in a first approximation the hybridization between the two QDs and consider
only QD2 filled with one electron. We have developed a model that accounts for
the mixing between spin and valley states due to SOC. In our silicon nanowire
geometry, the confinement is strongest along the $z$ direction (normal to the
SOI substrate), so that the low-energy levels belong to the $\Delta_{\pm z}$
valleys. Valley coupling at the Si/SiO2 interface lifts the twofold valley
degeneracy Sham79 ; Saraiva09 ; Friesen10 ; Culcer10 ; Zwanenburg13 ,
resulting in two spin-degenerate valley eigenstates $v_{1}$ and $v_{2}$ with
energies $E_{1}$ and $E_{2}$, respectively, and a valley splitting
$\Delta=E_{2}-E_{1}$. The expected energy diagram of the one-electron spin-
valley states is plotted as a function of $B$ in fig. 3b (see Supplementary
Note 2). The lowest spin-valley states can be identified as $\lvert
v_{1},\downarrow\rangle$, $\lvert v_{1},\uparrow\rangle/\lvert
v_{2},\downarrow\rangle$ anti-crossing at $B=B_{\rm V}$, and $\lvert
v_{2},\uparrow\rangle$ (the spin being quantized along $\mathbf{B}$). From
this energy diagram we assign the resonant transitions observed in fig. 3a as
follows: line A corresponds to EDSR between states $\lvert
v_{1},\downarrow\rangle$ and $\lvert v_{1},\uparrow\rangle$ (and between
states $\lvert v_{2},\downarrow\rangle$ and $\lvert v_{2},\uparrow\rangle$);
line B arises from EDSR between states $\lvert v_{1},\uparrow\rangle$ and
$\lvert v_{2},\downarrow\rangle$ Scarlino17 ; line V is associated with the
anti-crossing between states $\lvert v_{1},\uparrow\rangle$ and $\lvert
v_{2},\downarrow\rangle$ when $E_{Z}=\Delta$. We can thus measure
$\Delta=g\mu_{B}B_{V}=36\,\mu$eV. Note that the experimental EDSR spectrum
does not capture all possible transitions since some of them fall out of the
scanned $(B,\nu)$ range. fig. 3c shows the expected EDSR spectrum starting
from $\nu=0$ and $B=0$. The measured region is indicated in light blue (lower
values of $\nu$ and $B$ could not be explored due to the onset of photon-
assisted charge pumping and to the lifting of spin blockade, respectively).

The RF magnetic field associated with the microwave excitation on gate 2 is
too weak to drive conventional ESR Golovach06 . Since a pure electric field
cannot couple opposite spin states, SOC must be involved in the observed EDSR.
The atomistic spin-orbit Hamiltonian primarily couples the different $p$
orbitals of silicon Chadi77 ; the $\Delta_{\pm z}$ states are, however, linear
combinations of $s$ and $p_{z}$ orbitals with little admixture of $p_{x}$ and
$p_{y}$, which explains why the SOC matrix elements are weak in the conduction
band of silicon. Yet the mixing between $\lvert v_{1},\uparrow\rangle$ and
$\lvert v_{2},\downarrow\rangle$ by “inter-valley” SOC can be strongly
enhanced when the splitting between these two states is small enough. We can
capture the main physics and identify the relevant parameters using the
simplest perturbation theory in the limit $B\ll B_{\rm V}$. The states
$\lvert\Downarrow\rangle\equiv\lvert v_{1},\downarrow\rangle$ and
$\lvert\Uparrow\rangle\equiv\lvert v_{1},\uparrow\rangle$ indeed read to first
order in the spin-orbit Hamiltonian $H_{\rm SOC}$:

$\displaystyle\lvert\Downarrow\rangle$ $\displaystyle=\lvert
v_{1},\downarrow\rangle-\frac{C_{v_{1}v_{2}}}{\Delta+g\mu_{B}B}\lvert
v_{2},\uparrow\rangle+...$ (1a) $\displaystyle\lvert\Uparrow\rangle$
$\displaystyle=\lvert
v_{1},\uparrow\rangle+\frac{C_{v_{1}v_{2}}^{*}}{\Delta-g\mu_{B}B}\lvert
v_{2},\downarrow\rangle+...\,,$ (1b)

where:

$C_{v_{1}v_{2}}=\langle v_{2},\uparrow\rvert H_{\rm SOC}\lvert
v_{1},\downarrow\rangle=-\langle v_{1},\uparrow\rvert H_{\rm SOC}\lvert
v_{2},\downarrow\rangle\,.$ (2)

Therefore, $\lvert v_{1},\uparrow\rangle$ admixes a significant fraction of
$\lvert v_{2},\downarrow\rangle$ when the splitting $\Delta-g\mu_{B}B$ between
these two states decreases. As $\lvert v_{2},\downarrow\rangle$ can be coupled
to $\lvert v_{1},\downarrow\rangle$ by the RF electric field, this allows for
Rabi oscillations between $\lvert\Uparrow\rangle$ and
$\lvert\Downarrow\rangle$. Along line A, the Rabi frequency at resonance
($h\nu=g_{A}\mu_{\rm B}B$) reads:

$hf=e\delta V_{g2}\left|\langle\Uparrow\rvert
D\lvert\Downarrow\rangle\right|\simeq 2eg\mu_{B}B\delta
V_{g2}\frac{|D_{v_{1}v_{2}}||C_{v_{1}v_{2}}|}{\Delta^{2}}\,,$ (3)

where $\delta V_{g2}$ is the amplitude of the microwave modulation on gate 2,
$D(\textbf{r})=\partial{V_{t}(\textbf{r})}/\partial{V_{g2}}$ is the derivative
of the total potential $V_{t}(\mathbf{r})$ in the device with respect to the
gate potential $V_{g2}$, and:

$D_{v_{1}v_{2}}=\langle v_{1},\uparrow\rvert D\lvert
v_{2},\uparrow\rangle=\langle v_{1},\downarrow\rvert D\lvert
v_{2},\downarrow\rangle$ (4)

is the matrix element of $D(\textbf{r})$ between valleys $v_{1}$ and $v_{2}$.
The gate-induced electric field essentially drives motion in the $(yz)$ plane.
$D_{v_{1}v_{2}}$ is small yet non negligible in SOI nanowire devices because
the $v_{1}$ and $v_{2}$ wavefunctions show out of phase oscillations along
$z$, and can hence be coupled by the vertical electric field. The field along
$y$ does not result in a sizable $D_{v_{1}v_{2}}$ unless surface roughness
disorder couples the motions along $z$ and in the $(xy)$ plane Gamble13 ;
Boross16 . Although $C_{v_{1}v_{2}}$ is weak in silicon, SOC opens a path for
an electrically driven spin resonance
$\lvert\Downarrow\rangle\to\lvert\Uparrow\rangle$ through a virtual transition
from $\lvert v_{1},\downarrow\rangle$ to $\lvert v_{2},\downarrow\rangle$,
mediated by the microwave field, and then from $\lvert
v_{2},\downarrow\rangle$ to $\lvert v_{1},\uparrow\rangle$, mediated by SOC.
Note, however, that the above equations are only valid at small magnetic
fields where perturbation theory can be applied. A non-perturbative model
valid at all fields is introduced in the Supplementary Note 2. It explicitly
accounts for the anti-crossing (and strong hybridization by SOC) of states
$\lvert v_{1},\uparrow\rangle$ and $\lvert v_{2},\downarrow\rangle$ near
$B=B_{\rm V}$ Yang13 ; Hao14 ; Huang2014 , but features the same matrix
elements as above. This model shows that the Rabi frequency is maximal near
$B=B_{\rm V}$, and confirms that there is a concurrent spin resonance $\lvert
v_{1},\uparrow\rangle\leftrightarrow\lvert v_{2},\downarrow\rangle$, shifted
by the valley splitting $\Delta=36\,\mu$eV (lines B/B’ on fig. 3c), as well
as, in principle, a possible resonance $\lvert
v_{1},\downarrow\rangle\leftrightarrow\lvert v_{2},\uparrow\rangle$ (line D).

We have validated the above interpretation against $sp^{3}d^{5}s^{*}$ tight-
binding (TB) calculations. TB is well suited to that purpose as it accounts
for valley and spin-orbit coupling at the atomistic level. We consider a
simplified single-gate device model capturing the essential geometry (fig.
4a). A realistic surface roughness disorder with rms amplitude $\Delta_{\rm
SR}=0.4$ nm Bourdet16  is included in order to reduce the valley splitting
down to the experimental value Culcer10 . A detailed description of the TB
calculations is given in the Supplementary Note 3. The top panel of fig. 4b
shows the $B$ dependence of the energy of the first four TB states $\lvert
1\rangle...\lvert 4\rangle$. An anti-crossing is visible between states
$\lvert 2\rangle$ and $\lvert 3\rangle$ at $B_{\rm V}=0.3$ T. We calculate
$|C_{v_{1}v_{2}}|=1.8\,\mu$eV and $|D_{v_{1}v_{2}}|=70\,\mu$V/V. The $B$
dependence of the TB Rabi frequency $f$ on line A is shown in the bottom panel
of fig. 4b. There is a prominent peak near $B=B_{\rm V}$ where $\lvert
2\rangle$ and $\lvert 3\rangle$ have a mixed $\lvert
v_{1},\uparrow\rangle$/$\lvert v_{2},\downarrow\rangle$ character. The maximum
Rabi frequency $f_{\rm max}\simeq e\delta V_{g2}|D_{v_{1}v_{2}}|/(\sqrt{2}h)$
is limited by $D_{v_{1}v_{2}}$ while the full width at half maximum of the
peak, $\Delta B_{\rm FWHM}\simeq 12|C_{v_{1}v_{2}}|/(\sqrt{7}g\mu_{B})=0.07$ T
is controlled by the SOC matrix element $C_{v_{1}v_{2}}$ (see Supplementary
Note 3). The Rabi frequency remains however sizable over a few $\Delta B_{\rm
FWHM}$. We point out that $C_{v_{1}v_{2}}$ and $D_{v_{1}v_{2}}$ may depend on
the actual roughness at the Si/SiO2 interface.

The calculated Rabi frequencies compare well against those reported for
alternative silicon based systems. For example, the expected Rabi frequency is
around 4.2 MHz for $B=0.35$ T, close to anticrossing field $B_{\rm V}$, and
for a microwave excitation amplitude $\delta V_{g2}=0.7$ mV, close to the
experimental value (see fig. 4b). This frequency is comparable with those
achieved with coplanar antennas Pla2012 ; Veldhorst14  and in some experiments
with micromagnets Kawakami14 ; Takeda16 .

One of the most salient fingerprint of the above EDSR mechanism is the
dependence of $f$ on the magnetic field orientation. Indeed, it must be
realized that $C_{v_{1}v_{2}}$ may vary with the orientation of the magnetic
field (as the spin is quantized along $\mathbf{B}$ in Eq. (2)). Actually,
symmetry considerations supported by TB calculations show that
$C_{v_{1}v_{2}}$ and hence $f$ are almost zero when $\mathbf{B}$ is aligned
with the nanowire axis, due to the existence of a $(yz)$ mirror plane
perpendicular to that axis (see Supplementary Note 4). As a simple hint of
this result, we may consider a generic Rashba SOC Hamiltonian of the form
$H_{\rm SOC}\propto(\mathbf{E}\times\mathbf{p})\cdot\bm{\sigma}$, where
$\mathbf{E}$ is the electric field, $\mathbf{p}$ the momentum, and
$\bm{\sigma}$ the Pauli matrices. Symmetric atoms on each side of the $(yz)$
plane contribute to $C_{v_{1}v_{2}}$ with opposite $E_{x}$ and $p_{x}$
components. Therefore, only the $\propto(E_{y}p_{z}-E_{z}p_{y})\sigma_{x}$
component of $H_{\rm SOC}$ makes a non-zero contribution to $C_{v_{1}v_{2}}$,
but does not couple opposite spins when $\mathbf{B}\parallel\mathbf{x}$. The
current on line A is, to a first approximation, proportional to $f^{2}$
Koppens07 ; Schroer11 . The TB $f^{2}$ is plotted in fig. 4c as a function of
the angle $\theta$ between an in-plane magnetic field
$\mathbf{B}\perp\mathbf{z}$ and the nanowire axis $\mathbf{x}$. It shows the
$\propto\sin^{2}\theta$ dependence expected from the above considerations. The
experimental $I_{ds}$, also plotted in fig. 4c, shows the same behavior,
supporting our interpretation. The fact that $I_{ds}$ remains finite for
$\mathbf{B}\parallel\mathbf{x}$ may be explained by the fact that the $(yz)$
symmetry plane is mildly broken by disorder and voltage biasing.

In a recent work, Huang et al. Huang17  proposed a mechanism for EDSR based on
electrically induced oscillations of an electron across an atomic step at a
Si/SiO2 or a Si/SiGe hetero-interface. The step enhances the SOC between the
ground and the excited state of the same valley. The Rabi frequency is,
however, limited by the maximal height of the step that the electron can
overcome (typically 1 nm). The EDSR reported here has a different origin. It
results from the finite SOC and dipole matrix elements between the ground-
states of valleys $v_{1}$ and $v_{2}$. These couplings are sizable only in a
low-symmetry dot structure such as a corner QD Voisin14 , where there is only
one mirror plane (see Supplementary Note 4). Indeed, we have verified that
more symmetric device structures with at least two symmetry planes show a
dramatic suppression of the SOC matrix element (since, as hinted above, each
mirror rules out two out of three components in $H_{\rm SOC}$, leaving no
possible coupling). This is the case of a typical nanowire field-effect
transistor with the gate covering three sides of the nanowire channel, for
which TB calculations give no EDSR since $C_{v_{1}v_{2}}=0$ whatever the spin
or magnetic field orientation. Conversely, the design of QDs without any
symmetry left should maximize the opportunities for EDSR.

Figure 4: TB calculations results. a, Electrostatic model of the device. The
silicon wire appears in yellow, SiO2 in blue, HfO2 in orange and the gate in
green. We did not include the lateral gate in the simulations. b, TB energy
levels as a function of the magnetic field $\mathbf{B}||\mathbf{y}$, and TB
Rabi frequency along line A (identified by the green arrows) as a function of
the magnetic field for a reference microwave amplitude $\delta V_{g2}=1$ mV.
The points are the TB data and the solid lines are the analytical model
developed in Supplementary Note 3. The TB valley splitting energy $\Delta\sim
36\,\mu$eV is in good agreement with the experimental value. The Rabi
frequency shows a prominent peak near the anti-crossing field $B_{\rm V}$. c,
Experimental measurement of EDSR current as a function of the angle $\theta$
between an in-plane magnetic field $\mathbf{B}\perp\mathbf{z}$ and the
nanowire axis $\mathbf{x}$ ($\theta=0^{\circ}$), and TB Rabi frequency $f^{2}$
as a function of $\theta$.

##  III Discussion

In conclusion, we have reported an experimental demonstration of electric-
dipole, spin-valley resonance mediated by intrinsic SOC in a silicon electron
double QD. Although SOC is weak in silicon, its effect can be enhanced in the
corner QDs of an etched SOI device, owing to their reduced symmetry. SOC
enables EDSR on the spin-split doublet of the first, lowest energy valley by
mixing the up-spin state of that valley with the down-spin state of the second
valley. The EDSR Rabi frequency is strongly enhanced near the corresponding
anti-crossing, namely when the valley and Zeeman splittings are close enough.
This enhancement comes with a price though, since we expect the spin
relaxation time $T_{1}$ (and presumably also the spin coherence time $T_{2}$)
to be simultaneously reduced Yang13 ; Huang2014 . Therefore, we anticipate
that the efficiency of the reported EDSR mechanism for spin qubit manipulation
will be conditioned by the possibility to tune the valley splitting $\Delta$,
in order to bring the qubit near the anti-crossing point for manipulation,
then away from the anti-crossing point to mitigate decoherence. Given the
strong dependence of the valley splitting on gate voltages in silicon-based
devices Goswami07 ; Takashina2006 , this possibility appears within reach and
will be addressed in future experiments.

## Methods

The silicon nanowire transistors are manufactured on a 300 mm Silicon-On-
Insulator (SOI) processing line Maurand16 . first, silicon nanowires are
etched from a SOI wafer with a 12 nm thick undoped silicon layer and a 145 nm
thick buried oxide. The nanowire channels are oriented along the $[110]$
direction. The width $W$ of the nanowires, initially defined by deep ultra-
violet (DUV) lithography, is trimmed down to about 30 nm by a well-controlled
etching process. Two parallel top-gates, $\simeq 35$ nm wide and spaced by
$\simeq 30$ nm are patterned with $e$-beam lithography in order to control the
double quantum dot. An additional side gate is also placed parallel to the
nanowire at a distance of 50 nm in order to strengthen confinement in the
corner dots of the Si nanowire. The gate stack consists in a 2.5 nm thick
layer of SiO2, a 1.9 nm thick layer of HfO2, a thin ($\simeq 5$ nm) layer of
TiN metal and a much thicker ($\simeq 50$ nm) layer of polysilicon. Then,
insulating SiN spacers are deposited all around the gates and are etched.
Their width is deliberately large ($\simeq 25$ nm) in order to cover
completely the nanowire channel between the two gates and protect it from
subsequent ion implantation. Arsenic and phosphorous are indeed implanted in
order to achieve low resistance source/drain contacts. The wide spacers also
limit dopant diffusion from the heavily implanted contact regions into the
channel. The dopants are activated by spike annealing followed by
silicidation. The devices are finalized with a standard microelectronics back-
end of line process.

The devices are first screened at room temperature. Those showing the best
performances (symmetrical characteristics for both top gates with no gate
leakage current, low subthreshold swing) are cleaved from the original 300-mm
wafer in order to be mounted on a printed-circuit-board chip carrier with
high-frequency lines. The sample is measured in a wet dilution fridge with a
base temperature $T=15$ mK. The magnetic field is applied by means of a 2D
superconducting vector magnet in the $(xy)$ plane parallel to the SOI wafer.

All terminals are connected with bonding wires to DC lines; gate 2 is also
connected with a bias-tee to a microwave line. The DC block is a low rise-time
Tektronik PSPL5501A, while the RF filter is made with a 10 k$\Omega$ SMD
resistance mounted on the chip carrier plus a wire acting as inductor. The DC
voltages are generated at room temperature by custom battery-powered opto-
isolated voltage sources. The microwave signal is generated by a commercial
analog microwave generator (Anritsu MG3693C). The RF line is equipped with a
series of attenuators at room temperature and in the cryostat for signal
thermalization ($1$ K), with a total attenuation of $\simeq 38$ dB at 10 GHz.
The current in the nanowire is measured by a custom transimpedance amplifier
with a gain of $10^{9}$ V/A and then digitized by a commercial multimeter
(Agilent 34410A).

The tight-binding calculations are performed with the $sp^{3}d^{5}s^{*}$ model
of Ref. Niquet09, . The potential in the device is calculated with a finite-
volume Poisson solver, then the eigenstates of the dot are computed with an
iterative Jacobi-Davidson solver. The Rabi frequencies are obtained from Eq.
(3).

## Data availability

The data that support the findings of this study are available from the
corresponding authors upon reasonable request.

###### Acknowledgements.

This work was supported by the European Union’s Horizon 2020 research and
innovation program under grant agreement No 688539 MOSQUITO. Part of the
calculations were run on the TGCC/Curie and CINECA/Marconi machines using
allocations from GENCI and PRACE. We thank Cosimo Orban for the 3D rendering
of the sample.

## Competing interests

The authors declare no competing financial interests.

## Author contributions

H.B., R.L., L.H., S.B., and M.V. led device fabrication. A. Corna performed
the experiment with help from R.M., A. Crippa and D.K.-P. under the
supervision of X.J., S.D.F. and M.S. L. B. and Y.-M.N. did the modeling and
simulations. M.V., X.J., S.D.F. and M.S led the all project. All authors co-
wrote the manuscript.

## Corresponding authors

Correspondence and requests of material should be addressed to Marc Sanquer
(marc.sanquer@cea.fr), Silvano De Franceschi<EMAIL_ADDRESS>or
Yann-Michel Niquet (yann-michel.niquet@cea.fr).

## References

  * (1) Zwanenburg, F. A. _et al._ Silicon quantum electronics.  _Review of Modern Physics_ 85, 961 (2013).  URL https://dx.doi.org/10.1103/RevModPhys.85.961.
  * (2) Tyryshkin, A. M. _et al._ Electron spin coherence exceeding seconds in high-purity silicon.  _Nature Materials_ 11, 143 (2012).  URL https://dx.doi.org/10.1038/nmat3182.
  * (3) Veldhorst, M. _et al._ An addressable quantum dot qubit with fault-tolerant control-fidelity.  _Nature Nanotechnology_ 9, 981 (2014).  URL https://dx.doi.org/10.1038/nnano.2014.216.
  * (4) Laucht, A. _et al._ Electrically controlling single-spin qubits in a continuous microwave field.  _Science Advances_ 1, e1500022 (2015).  URL https://dx.doi.org/10.1126/sciadv.1500022.
  * (5) Veldhorst, M. _et al._ A two-qubit logic gate in silicon.  _Nature_ 526, 410 (2015).  URL https://dx.doi.org/10.1038/nature15263.
  * (6) Pioro-Ladrière, M. _et al._ Electrically driven single-electron spin resonance in a slanting zeeman field.  _Nature Physics_ 4, 776 (2008).  URL https://dx.doi.org/10.1038/nphys1053.
  * (7) Kawakami, E. _et al._ Electrical control of a long-lived spin qubit in a Si/SiGe quantum dot.  _Nature Nanotechnology_ 9, 666 (2014).  URL https://dx.doi.org/10.1038/nnano.2014.153.
  * (8) Rančić, M. J. & Burkard, G.  Electric dipole spin resonance in systems with a valley-dependent $g$ factor.  _Physical Review B_ 93, 205433 (2016).  URL https://dx.doi.org/10.1103/PhysRevB.93.205433.
  * (9) Takeda, K. _et al._ A fault-tolerant addressable spin qubit in a natural silicon quantum dot.  _Science Advances_ 2, e1600694 (2016).  URL https://dx.doi.org/10.1126/sciadv.1600694.
  * (10) Maurand, R. _et al._ A cmos silicon spin qubit.  _Nature Communications_ 7, 13575 (2016).  URL https://dx.doi.org/10.1038/ncomms13575.
  * (11) Nestoklon, M. O., Golub, L. E. & Ivchenko, E. L.  Spin and valley-orbit splittings in $\mathrm{Si}\mathrm{Ge}∕\mathrm{Si}$ heterostructures.  _Physical Review B_ 73, 235334 (2006).  URL https://dx.doi.org/10.1103/PhysRevB.73.235334.
  * (12) Hao, X., Ruskov, R., Xiao, M., Tahan, C. & Jiang, H.  Electron spin resonance and spin-valley physics in a silicon double quantum dot.  _Nature Communications_ 5, 3860 (2014).  URL https://dx.doi.org/10.1038/ncomms4860.
  * (13) Schoenfield, J. S., Freeman, B. M. & Jiang, H.  Coherent manipulation of valley states at multiple charge configurations of a silicon quantum dot device.  _Nature Communications_ 8, 64 (2017).  URL https://dx.doi.org/10.1038/s41467-017-00073-x.
  * (14) Jock, R. M. _et al._ Probing low noise at the MOS interface with a spin-orbit qubit.  _arXiv_ 1707.04357 (2017).  URL https://arxiv.org/abs/1707.04357.
  * (15) Veldhorst, M. _et al._ Spin-orbit coupling and operation of multivalley spin qubits.  _Physical Review B_ 92, 201401(R) (2015).  URL https://dx.doi.org/10.1103/PhysRevB.92.201401.
  * (16) Ruskov, R., Veldhorst, M., Dzurak, A. S. & Tahan, C.  Electron g-factor of valley states in realistic silicon quantum dots.  _arXiv_ 1708.04555 (2017).  URL https://arxiv.org/abs/1708.04555.
  * (17) Sham, L. J. & Nakayama, M.  Effective-mass approximation in the presence of an interface.  _Physical Review B_ 20, 734 (1979).  URL https://dx.doi.org/10.1103/PhysRevB.20.734.
  * (18) Saraiva, A. L., Calderon, M. J., Hu, X., Das Sarma, S. & Koiller, B.  Physical mechanisms of interface-mediated intervalley coupling in si.  _Physical Review B_ 80, 081305(R) (2009).  URL https://dx.doi.org/10.1103/PhysRevB.80.081305.
  * (19) Friesen, M. & Coppersmith, S. N.  Theory of valley-orbit coupling in a Si/SiGe quantum dot.  _Physical Review B_ 81, 115324 (2010).  URL https://dx.doi.org/10.1103/PhysRevB.81.115324.
  * (20) Culcer, D., Hu, X. & Das Sarma, S.  Interface roughness, valley-orbit coupling, and valley manipulation in quantum dots.  _Physical Review B_ 82, 205315 (2010).  URL https://dx.doi.org/10.1103/PhysRevB.82.205315.
  * (21) Goswami, S. _et al._ Controllable valley splitting in silicon quantum devices.  _Nature Physics_ 3, 41 (2007).  URL https://dx.doi.org/10.1038/nphys475.
  * (22) Yang, C. H. _et al._ Spin-valley lifetimes in a silicon quantum dot with tunable valley splitting.  _Nature Communications_ 4, 2069 (2013).  URL https://dx.doi.org/10.1038/ncomms3069.
  * (23) Rashba, E. I.  Theory of electric dipole spin resonance in quantum dots: Mean field theory with gaussian fluctuations and beyond.  _Physical Review B_ 78, 195302 (2008).  URL https://dx.doi.org/10.1103/PhysRevB.78.195302.
  * (24) Huang, P. & Hu, X.  Spin relaxation in a si quantum dot due to spin-valley mixing.  _Physical Review B_ 90, 235315 (2014).  URL https://dx.doi.org/10.1103/PhysRevB.90.235315.
  * (25) Golovach, V. N., Borhani, M. & Loss, D.  Electric-dipole-induced spin resonance in quantum dots.  _Physical Review B_ 74, 165319 (2006).  URL https://dx.doi.org/10.1103/PhysRevB.74.165319.
  * (26) Nowack, K. C., Koppens, F. H. L., Nazarov, Y. V. & Vandersypen, L. M. K.  Coherent control of a single electron spin with electric fields.  _Science_ 318, 1430 (2007).  URL https://dx.doi.org/10.1126/science.1148092.
  * (27) Nadj-Perge, S., Frolov, S. M., Bakkers, E. P. A. M. & Kouwenhoven, L. P.  Spin-orbit qubit in a semiconductor nanowire.  _Nature_ 468, 1084 (2010).  URL https://dx.doi.org/10.1038/nature09682.
  * (28) Pribiag, V. S. _et al._ Electrical control of single hole spins in nanowire quantum dots.  _Nature Nanotechnology_ 8, 170 (2013).  URL https://dx.doi.org/10.1038/nnano.2013.5.
  * (29) Voisin, B. _et al._ Few-electron edge-state quantum dots in a silicon nanowire field- effect transistor.  _Nano Letters_ 14, 2094 (2014).  URL https://dx.doi.org/10.1021/nl500299h.
  * (30) Gonzalez-Zalba, M. F., Barraud, S., Ferguson, A. J. & Betz, A. C.  Probing the limits of gate-based charge sensing.  _Nature Communications_ 6, 6084 (2015).  URL https://dx.doi.org/10.1038/ncomms7084.
  * (31) Feher, G.  Electron spin resonance experiments on donors in silicon. i. electronic structure of donors by the electron nuclear double resonance technique.  _Physical Review_ 114, 1219 (1959).  URL https://dx.doi.org/10.1103/PhysRev.114.1219.
  * (32) Hanson, R., Petta, J. R., Tarucha, S. & Vandersypen, L. M. K.  Spins in few-electron quantum dots.  _Reviews of Modern Physics_ 79, 1217 (2007).  URL https://dx.doi.org/10.1103/RevModPhys.79.1217.
  * (33) Ono, K., Austing, D. G., Tokura, Y. & Tarucha, S.  Current rectification by pauli exclusion in a weakly coupled double quantum dot system.  _Science_ 297, 1313 (2002).  URL https://dx.doi.org/10.1126/science.1070958.
  * (34) Lai, N. S. _et al._ Pauli spin blockade in a highly tunable silicon double quantum dot.  _Scientific Reports_ 1, 110 (2011).  URL https://dx.doi.org/10.1038/srep00110.
  * (35) Yamahata, G. _et al._ Magnetic field dependence of pauli spin blockade: A window into the sources of spin relaxation in silicon quantum dots.  _Physical Review B_ 86, 115322 (2012).  URL https://dx.doi.org/10.1103/PhysRevB.86.115322.
  * (36) Scarlino, P. _et al._ Second-harmonic coherent driving of a spin qubit in a Si / SiGe quantum dot.  _Physical Review Letters_ 115, 106802 (2015).  URL https://dx.doi.org/10.1103/PhysRevLett.115.106802.
  * (37) Scarlino, P. _et al._ Dressed photon-orbital states in a quantum dot: Intervalley spin resonance.  _Physical Review B_ 95, 165429 (2017).  URL https://dx.doi.org/10.1103/PhysRevB.95.165429.
  * (38) Chadi, D. J.  Spin-orbit splitting in crystalline and compositionally disordered semiconductors.  _Physical Review B_ 16, 790 (1977).  URL https://dx.doi.org/10.1103/PhysRevB.16.790.
  * (39) Gamble, J. K., Eriksson, M. A., Coppersmith, S. N. & Friesen, M.  Disorder-induced valley-orbit hybrid states in Si quantum dots.  _Physical Review B_ 88, 035310 (2013).  URL https://dx.doi.org/10.1103/PhysRevB.88.035310.
  * (40) Boross, P., Széchenyi, G., Culcer, D. & Pályi, A.  Control of valley dynamics in silicon quantum dots in the presence of an interface step.  _Physical Review B_ 94, 035438 (2016).  URL https://dx.doi.org/10.1103/PhysRevB.94.035438.
  * (41) Bourdet, L. _et al._ Contact resistances in trigate and finfet devices in a non-equilibrium green’s functions approach.  _Journal of Applied Physics_ 119, 084503 (2016).  URL https://dx.doi.org/10.1063/1.4942217.
  * (42) Pla, J. J. _et al._ A single-atom electron spin qubit in silicon.  _Nature_ 489, 541 (2012).  URL https://dx.doi.org/10.1038/nature11449.
  * (43) Koppens, F. H. L. _et al._ Detection of single electron spin resonance in a double quantum dot.  _Journal of Applied Physics_ 101, 081706 (2007).  URL https://dx.doi.org/10.1063/1.2722734.
  * (44) Schroer, M. D., Petersson, K. D., Jung, M. & Petta, J. R.  Field tuning the g factor in InAs nanowire double quantum dots.  _Physical Review Letters_ 107, 176811 (2011).  URL https://dx.doi.org/10.1103/PhysRevLett.107.176811.
  * (45) Huang, W., Veldhorst, M., Zimmerman, N. M., Dzurak, A. S. & Culcer, D.  Electrically driven spin qubit based on valley mixing.  _Physical Review B_ 95, 075403 (2017).  URL https://dx.doi.org/10.1103/PhysRevB.95.075403.
  * (46) Takashina, K., Ono, Y., Fujiwara, A., Takahashi, Y. & Hirayama, Y.  Valley polarization in Si(100) at zero magnetic field.  _Physical Review Letters_ 96, 236801 (2006).  URL https://dx.doi.org/10.1103/PhysRevLett.96.236801.
  * (47) Niquet, Y. M., Rideau, D., Tavernier, C., Jaouen, H. & Blase, X.  Onsite matrix elements of the tight-binding hamiltonian of a strained crystal: Application to silicon, germanium, and their alloys.  _Physical Review B_ 79, 245201 (2009).  URL https://dx.doi.org/10.1103/PhysRevB.79.245201.

Supplementary notes for “Electrically driven electron spin resonance mediated
by spin-valley-orbit coupling in a silicon quantum dot”

## Supplementary note 1: Valleys and spin-valley blockade

In the main text, we have discussed the nature of the A, B, C and V line in a
one-particle picture. In this supplementary note, we introduce a two-particle
picture for the blockade, which accounts for the valley degree of freedom and
gives a better description of the V line.

Spin blockade can arise when the current flows through the sequence of charge
configurations
$(n_{1},n_{2})\equiv(2n+1,2m+1)\to(2n,2m+2)\to(2n,2m+1)\to(2n+1,2m+1)...$,
where $n_{1}$ and $n_{2}$ are the number of electrons in dots 1 and 2.Ono  _et
al._ (2002); Hanson  _et al._ (2007) Indeed, the $(2n+1,2m+1)$ states can be
mapped onto singlet $S(1,1)$ and triplet $T(1,1)$ states, while the
$(2n,2m+2)$ states can be mapped onto singlet $S(0,2)$ and triplet $T(0,2)$
states. While the $S(1,1)$ and $T(1,1)$ states are almost degenerate, the
$S(0,2)$ and $T(0,2)$ states can be significantly split because the $T(0,2)$
state must involve some orbital excitation. The $(2n+1)^{\rm th}$ electron may
enter in dot 1 through any of the $(1,1)$ configurations at high enough
source-drain bias. Once in a $T(1,1)$ state, the system may, however, get
trapped for a long time if the $T(0,2)$ states are still out of the bias
window, because tunneling from $T(1,1)$ to $S(0,2)$ requires a spin flip. The
current is hence suppressed. At reverse source-drain bias, the current flows
through the sequence of charge configurations
$(2n+1,2m+1)\to(2n,2m+1)\to(2n,2m+2)\to(2n+1,2m+1)...$, which can not be spin-
blocked, giving rise to current rectification (see Fig. 2 of main text).

The observation of inter-valleys resonances suggests that $m$ is even
(otherwise only transitions between $v_{2}$ states would be observed in dot
2). We assume from now on that $n$ is also even. As a matter of fact, the
absence of visible bias triangles for lower gate voltages suggests that
$m=n=0$, though we can not exclude the existence of extra triangles with
currents below the detection limit.

Figure S1: Scheme of the energy levels for a system with two dots, filled with
two electrons in the $(1,1)$ and $(0,2)$ charge configurations, as a function
of magnetic field. We assume different valley splittings in the two dots
(respectively $\Delta_{1}>\Delta_{2}$). Triplets in the $(0,2)$ configurations
with two electrons in the same valley are not represented, since they are
energetically far away. Spin-valley blocked states are marked with a red
cross.

We discuss below the role of valley blockade in the present experiments. We
assume the valley splitting is much larger in dot 1 ($\Delta_{1}$) than in dot
2 ($\Delta_{2}=36\,\mu$eV) due to disorder and bias conditions. The valley
splitting in dot 1 is actually beyond the bandwidth of the EDSR setup. This
reflects the stochastic variations from one dot to an other, as confirmed by
tight-binding simulations.

In the $(1,1)$ charge configuration, the low-energy states can be
characterized by their spin component [singlet (S) or triplet (T0, T-, T+)]
and by the valley occupied in each dot ($v_{1}$ or $v_{2}$). Sixteen states
can be constructed in this way (see Fig. S1). We neglect in a first
approximation the small exchange splitting between singlet and triplet states
with same valley indices. The magnetic field $\mathbf{B}=B\mathbf{b}$ splits
the T- (total spin $\langle S_{\mathbf{b}}\rangle=-1$) and T+ states ($\langle
S_{\mathbf{b}}\rangle=+1$) from the S and $T_{0}$ states ($\langle
S_{\mathbf{b}}\rangle=0$). The splitting between T+ and T- is
$E_{z}=2g\mu_{B}B$.

Similar states can be constructed in the $(0,2)$ charge configuration. The
$S^{v_{1}v_{1}}(0,2)$ and $S^{v_{1}v_{1}}(1,1)$ are detuned by the bias on
gates 1 and 2. We focus on detunings smaller than the orbital singlet-triplet
splitting $\Delta_{\rm ST}=1.9$ meV, so that neither the $v_{1}v_{1}$ nor the
$v_{2}v_{2}$ triplets can be reached from the $(1,1)$ states.

Given the small $\Delta_{2}=36\,\mu$eV extracted from spin resonance, we need,
however, to reconsider the mechanisms for current rectification. Indeed, the
system must be spin-and-valley blockedHao  _et al._ (2014) since the detuning
is typically much larger than $\Delta_{2}$ so that $T^{v_{1}v_{2}}(0,2)$
states are accessible in the bias window. Assuming that both spin and valley
are conserved during tunneling, the spin and valley blocked $(1,1)$ states are
actually $T_{-}^{v_{1}v_{1}}$, $T_{+}^{v_{1}v_{1}}$, $T_{-}^{v_{2}v_{2}}$ and
$T_{+}^{v_{2}v_{2}}$. Although $T_{0}^{v_{1}v_{1}}$ and $T_{0}^{v_{2}v_{2}}$
are, in principle, also spin and valley blocked, they may be mixed with the
nearly degenerate $S^{v_{1}v_{1}}$ and $S^{v_{2}v_{2}}$ states by, e.g., spin-
orbit coupling (SOC) and nuclear spin disorder,Koppens  _et al._ (2005); Nadj-
Perge  _et al._ (2010) and be therefore practically unblocked.

We can now refine the interpretation of the different lines. Each one
corresponds to a different set of transitions between blocked and unblocked
$(1,1)$ states. Line A corresponds to transitions between
$T_{\pm}^{v_{1}v_{1}}$ and $T_{0}^{v_{1}v_{1}}$/$S^{v_{1}v_{1}}$ states, and
to transitions between $T_{\pm}^{v_{2}v_{2}}$ and
$T_{0}^{v_{2}v_{2}}$/$S^{v_{2}v_{2}}$ states. Line B corresponds to
transitions between $T_{+}^{v_{1}v_{1}}$ and
$T_{0}^{v_{1}v_{2}}$/$S^{v_{1}v_{2}}$ states, and between $T_{-}^{v_{2}v_{2}}$
and $T_{0}^{v_{2}v_{1}}$/$S^{v_{2}v_{1}}$ states. The line D on Fig. 3 of the
main text would correspond to transitions between $T_{-}^{v_{1}v_{1}}$ and
$T_{0}^{v_{1}v_{2}}$/$S^{v_{1}v_{2}}$ states, and between $T_{+}^{v_{2}v_{2}}$
and $T_{0}^{v_{2}v_{1}}$/$S^{v_{2}v_{1}}$ states. These transitions give rise
to the same spectrum as in the one-particle picture.

Line V is independent on the microwave frequency and also appears when no
microwaves are applied. At the magnetic field $B_{\rm
V}\simeq\Delta_{2}/(g\mu_{B})$, the states $T_{+}^{v_{1}v_{1}}$ and
$T_{0}^{v_{1}v_{2}}/S^{v_{1}v_{2}}$, as well as the states
$T_{-}^{v_{2}v_{2}}$ and $T_{0}^{v_{2}v_{1}}/S^{v_{2}v_{1}}$ are almost
degenerate. The mixing of these near degenerate blocked and unblocked states
by SOC lifts spin and valley blockade of the $T_{+}^{v_{1}v_{1}}$ and
$T_{-}^{v_{2}v_{2}}$ states, giving rise to an excess of current at $B\simeq
B_{\rm V}$ independent on the microwave excitation.

Note that we also expect a horizontal line H (independent on the magnetic
field) corresponding to transitions between $T_{\pm}^{v_{1}v_{1}}$ and
$T_{\pm}^{v_{1}v_{2}}$, and between $T_{\pm}^{v_{2}v_{1}}$ and
$T_{\pm}^{v_{2}v_{2}}$, at frequency $\nu=\Delta/h=8.7$ GHz. Horizonal lines
are indeed visible in the expected frequency range on Fig. 3a (main text), but
they can not be unequivocally assigned due to the onset of photon-assisted
charge pumping resulting in parasitic features.

## Supplementary note 2: Theory

In this supplementary note, we propose a model for spin-orbit driven EDSR in
silicon quantum dots accounting for the strong hybridization of the spin and
valley states near the anti-crossing field $B=B_{\rm V}$.

We consider a silicon QD with strongest confinement along the $z$ direction so
that the low-lying conduction band levels belong to the $\Delta_{\pm z}$
valleys. This dot is controlled by a gate with potential $V_{g}$. In the
absence of valley and spin-orbit coupling, the ground-state is fourfold
degenerate (twice for spins and twice for valleys). Valley couplingSham and
Nakayama (1979); Saraiva  _et al._ (2009); Friesen and Coppersmith (2010);
Culcer  _et al._ (2010); Zwanenburg  _et al._ (2013) splits this fourfold
degenerate level into two spin-degenerate states $\lvert v_{1},\sigma\rangle$
and $\lvert v_{2},\sigma\rangle$ with energies $E_{1}$ and $E_{2}$, separated
by the valley splitting energy $\Delta=E_{2}-E_{1}$
($\sigma=\,\uparrow,\downarrow$ is the spin index). In the simplest
approximation, $\lvert v_{1},\sigma\rangle$ and $\lvert v_{2},\sigma\rangle$
are bonding and anti-bonding combinations of the $\Delta_{\pm z}$ states.

The remaining spin degeneracy can be lifted by a static magnetic field
$\mathbf{B}$. The energy of state $\lvert v_{n},\sigma\rangle$ is then
$E_{n,\sigma}=E_{n}\pm\frac{1}{2}g\mu_{B}B$ ($+$ for up states, $-$ for down
states, the spin being quantized along $\mathbf{B}$). Here $g$ is the gyro-
magnetic factor of the electrons, which is expected to be close to
$g_{0}=2.0023$ in silicon.Feher (1959) We may neglect the effects of the
magnetic field on the orbital motion of the electrons in a first
approximation. The wave functions
$\varphi_{n,\sigma}(\mathbf{r})=\langle\mathbf{r}|v_{n},\sigma\rangle$ can
then be chosen real.

The gate potential is modulated by a RF signal with frequency $\nu$ and
amplitude $\delta V_{g}$ in order to drive EDSR between states $\lvert
v_{1},\downarrow\rangle$ and $\lvert v_{1},\uparrow\rangle$. At resonance
$h\nu=g\mu_{B}B$, the Rabi frequency reads:

$hf=e\delta V_{g}\left|\langle v_{1},\uparrow\rvert D\lvert
v_{1},\downarrow\rangle\right|\,,$ (S1)

where $D(\mathbf{r})=\partial V_{t}(\mathbf{r})/\partial V_{g}$ is the
derivative of the total potential $V_{t}(\mathbf{r})$ in the device with
respect to $V_{g}$.333If the electrostatics of the device is linear,
$D(\mathbf{r})=V_{t}^{0}(\mathbf{r})/V_{g}$, where $V_{t}^{0}(\mathbf{r})$ is
the total potential in the empty device with the gate at potential $V_{g}$,
and all other terminals grounded. We discard the effects of the displacement
currents (concomitant ESR), which are negligible.Golovach  _et al._ (2006) In
the absence of SOC, $hf$ is zero as an electric field can not couple opposite
spins.

As discussed in the main text, spin-orbit couples the orbital and spin motions
of the electron.Nestoklon  _et al._ (2006); Yang  _et al._ (2013); Huang and
Hu (2014); Hao  _et al._ (2014); Veldhorst  _et al._ (2015); Huang  _et al._
(2017) “Intra-valley” SOC mixes spins within the $\Delta_{+z}$ or the
$\Delta_{-z}$ valley, while “inter-valley” SOC mixes spins between the
$\Delta_{+z}$ and $\Delta_{-z}$ valleys. Both intra-valley and inter-valley
SOC may couple $\lvert v_{1},\sigma\rangle$ with the excited states of valley
1. Inter-valley SOC may also couple $\lvert v_{1},\sigma\rangle$ with all
$v_{2}$ states. Its effects can be strongly enhanced if the valley splitting
$\Delta$ is small enough. Indeed, in the simplest non-degenerate perturbation
theory, the states $\lvert\Downarrow\rangle\equiv\lvert
v_{1},\downarrow\rangle$ and $\lvert\Uparrow\rangle\equiv\lvert
v_{1},\uparrow\rangle$ read to first order in the spin-orbit Hamiltonian
$H_{\rm SOC}$:

$\displaystyle\lvert\Downarrow\rangle$ $\displaystyle=\lvert
v_{1},\downarrow\rangle-\frac{C_{v_{1}v_{2}}}{\Delta+g\mu_{B}B}\lvert
v_{2},\uparrow\rangle-\frac{iR_{v_{1}v_{2}}}{\Delta}\lvert
v_{2},\downarrow\rangle$ (S2a) $\displaystyle\lvert\Uparrow\rangle$
$\displaystyle=\lvert
v_{1},\uparrow\rangle+\frac{C_{v_{1}v_{2}}^{*}}{\Delta-g\mu_{B}B}\lvert
v_{2},\downarrow\rangle+\frac{iR_{v_{1}v_{2}}}{\Delta}\lvert
v_{2},\uparrow\rangle\,,$ (S2b)

with

$\displaystyle C_{v_{1}v_{2}}$ $\displaystyle=\langle v_{2},\uparrow\rvert
H_{\rm SOC}\lvert v_{1},\downarrow\rangle=-\langle v_{1},\uparrow\rvert H_{\rm
SOC}\lvert v_{2},\downarrow\rangle$ (S3a) $\displaystyle R_{v_{1}v_{2}}$
$\displaystyle=-i\langle v_{2},\downarrow\rvert H_{\rm SOC}\lvert
v_{1},\downarrow\rangle=-i\langle v_{1},\uparrow\rvert H_{\rm SOC}\lvert
v_{2},\uparrow\rangle\,.$ (S3b)

The above equalities follow from time-reversal symmetry considerations for
real wave functions. $C_{v_{1}v_{2}}$ is complex and $R_{v_{1}v_{2}}$ is real.
We have neglected all mixing beyond the four $\lvert v_{n},\sigma\rangle$,
because the higher-lying excited states usually lie $\gtrsim 1$ meV above
$E_{1}$ and $E_{2}$ (as inferred from $\Delta_{\rm ST}$ on Fig. 2, main text).

Inserting Eqs. (S2) into Eq. (S1), then expanding in powers of $B$ yields to
first order in $B$ and $H_{\rm SOC}$:

$hf=e\delta V_{g}\left|\langle\Uparrow\rvert
D\lvert\Downarrow\rangle\right|=2eg\mu_{B}B\delta
V_{g}\frac{|C_{v_{1}v_{2}}||D_{v_{1}v_{2}}|}{\Delta^{2}}\,.$ (S4)

where

$D_{v_{1}v_{2}}=\langle v_{1},\sigma\rvert D\lvert v_{2},\sigma\rangle=\langle
v_{2},\sigma\rvert D\lvert v_{1},\sigma\rangle$ (S5)

is the electric dipole matrix element between valleys $v_{1}$ and $v_{2}$. As
expected, the Rabi frequency is proportional to $\delta V_{g}D_{v_{1}v_{2}}$,
$C_{v_{1}v_{2}}$, and to $B$ (as the contributions from the $\propto
C_{v_{1}v_{2}}$ terms in Eqs. (S2) cancel out if time-reversal symmetry is not
broken by the $\propto g\mu_{B}B$ terms of the denominators). It is also
inversely proportional to $\Delta^{2}$ ; namely the smaller the valley
splitting, the faster the rotation of the spin. $R_{v_{1}v_{2}}$ does not
contribute to lowest order because it couples states with the same spin.

$C_{v_{1}v_{2}}$ and $D_{v_{1}v_{2}}$ are known to be small in the conduction
band of silicon.Nestoklon  _et al._ (2006); Yang  _et al._ (2013); Hao  _et
al._ (2014); Veldhorst  _et al._ (2015); Huang  _et al._ (2017) Actually,
$D_{v_{1}v_{2}}$ is zero in any approximation that completely decouples the
$\Delta_{\pm z}$ valleys (such as the simplest effective mass approximation).
It is, however, finite in tight-bindingDi Carlo (2003); Delerue and Lannoo
(2004) or advanced $\mathbf{k}\cdot\mathbf{p}$ modelsSaraiva  _et al._ (2009);
Culcer  _et al._ (2010) for the conduction band of silicon. According to Eq.
(S4), the Rabi frequency can be significant if $E_{1,\uparrow}$ is close
enough to $E_{2,\downarrow}$ to enhance spin-valley mixing by $H_{\rm SOC}$.
This happens when $\Delta$ is small and/or when $g\mu_{B}B\approx\Delta$ (see
later discussion). The main path for EDSR is then the virtual transition from
$\lvert v_{1},\downarrow\rangle$ to $\lvert v_{2},\uparrow\rangle$ (mediated
by $H_{\rm SOC}$), then from $\lvert v_{2},\uparrow\rangle$ to $\lvert
v_{1},\uparrow\rangle$ (mediated by the RF field).

The above equations are valid only for very small magnetic fields $B$, as non-
degenerate perturbation theory breaks down near the anti-crossing between
$E_{1,\uparrow}$ and $E_{2,\downarrow}$ when $g\mu_{B}B\approx\Delta$ (see
Fig. 3b of the main text). We may deal with this anti-crossing using
degenerate perturbation theory in the $\\{\lvert v_{1},\uparrow\rangle,\lvert
v_{2},\downarrow\rangle\\}$ subspace, while still using Eq. (S2a) for state
$\lvert\Downarrow\rangle$. However, such a strategy would spoil the
cancellations between $\lvert\Downarrow\rangle$ and $\lvert\Uparrow\rangle$
needed to achieve the proper behavior $hf\to 0$ when $B\to 0$. We must,
therefore, treat SOC in the full $\\{\lvert v_{1},\downarrow\rangle,\lvert
v_{1},\uparrow\rangle,\lvert v_{2},\downarrow\rangle,\lvert
v_{2},\uparrow\rangle\\}$ subspace.Yang  _et al._ (2013); Huang and Hu (2014)

The total Hamiltonian then reads:

$H=\mathop{{}\begin{pmatrix}E_{1}-\frac{1}{2}g\mu_{B}B&0&-iR_{v_{1}v_{2}}&C_{v_{1}v_{2}}^{*}\\\
0&E_{1}+\frac{1}{2}g\mu_{B}B&-C_{v_{1}v_{2}}&iR_{v_{1}v_{2}}\\\
iR_{v_{1}v_{2}}&-C_{v_{1}v_{2}}^{*}&E_{2}-\frac{1}{2}g\mu_{B}B&0\\\
C_{v_{1}v_{2}}&-iR_{v_{1}v_{2}}&0&E_{2}+\frac{1}{2}g\mu_{B}B\\\
\end{pmatrix}}_{\textstyle\mathstrut}\,.$ (S6)

As discussed before, $R_{v_{1}v_{2}}$ is not expected to make significant
contributions to the EDSR as it mixes states with the same spin. We may
therefore set $R_{v_{1}v_{2}}=0$ for practical purposes; $H$ then splits into
two $2\times 2$ blocks in the $\\{\lvert v_{1},\uparrow\rangle,\lvert
v_{2},\downarrow\rangle\\}$ and $\\{\lvert v_{1},\downarrow\rangle,\lvert
v_{2},\uparrow\rangle\\}$ subspaces. The diagonalization of the $\\{\lvert
v_{1},\downarrow\rangle,\lvert v_{2},\uparrow\rangle\\}$ block yields
energies:

$E_{\pm}=\frac{1}{2}(E_{1}+E_{2})\pm\frac{1}{2}\sqrt{(\Delta+g\mu_{B}B)^{2}+4|C_{v_{1}v_{2}}|^{2}}\,$
(S7)

and eigenstates:

$\displaystyle\lvert\psi_{+}\rangle$ $\displaystyle=\alpha\lvert
v_{1},\downarrow\rangle+\beta\lvert v_{2},\uparrow\rangle$ (S8a)
$\displaystyle\lvert\psi_{-}\rangle$ $\displaystyle=\beta\lvert
v_{1},\downarrow\rangle-\alpha^{*}\lvert v_{2},\uparrow\rangle$ (S8b)

with:

$\displaystyle\alpha$
$\displaystyle=\frac{-2C_{v_{1}v_{2}}}{\left(4|C_{v_{1}v_{2}}|^{2}+W^{2}\right)^{1/2}}$
(S9a) $\displaystyle\beta$
$\displaystyle=\frac{W}{\left(4|C_{v_{1}v_{2}}|^{2}+W^{2}\right)^{1/2}}$ (S9b)

and:

$W=\Delta+g\mu_{B}B+\sqrt{(\Delta+g\mu_{B}B)^{2}+4|C_{v_{1}v_{2}}|^{2}}\,.$
(S10)

Likewise, the diagonalization of the $\\{\lvert v_{1},\uparrow\rangle,\lvert
v_{2},\downarrow\rangle\\}$ block yields energies:

$E_{\pm}^{\prime}=\frac{1}{2}(E_{1}+E_{2})\pm\frac{1}{2}\sqrt{(\Delta-g\mu_{B}B)^{2}+4|C_{v_{1}v_{2}}|^{2}}\,$
(S11)

and eigenstates:

$\displaystyle\lvert\psi_{+}^{\prime}\rangle$
$\displaystyle=\alpha^{\prime}\lvert
v_{1},\uparrow\rangle+\beta^{\prime}\lvert v_{2},\downarrow\rangle$ (S12a)
$\displaystyle\lvert\psi_{-}^{\prime}\rangle$
$\displaystyle=\beta^{\prime}\lvert
v_{1},\uparrow\rangle-\alpha^{\prime*}\lvert v_{2},\downarrow\rangle$ (S12b)

with:

$\displaystyle\alpha^{\prime}$
$\displaystyle=\frac{2C_{v_{1}v_{2}}^{*}}{\left(4|C_{v_{1}v_{2}}|^{2}+W^{\prime
2}\right)^{1/2}}$ (S13a) $\displaystyle\beta^{\prime}$
$\displaystyle=\frac{W^{\prime}}{\left(4|C_{v_{1}v_{2}}|^{2}+W^{\prime
2}\right)^{1/2}}$ (S13b)

and:

$W^{\prime}=\Delta-g\mu_{B}B+\sqrt{(\Delta-g\mu_{B}B)^{2}+4|C_{v_{1}v_{2}}|^{2}}\,.$
(S14)

We can finally compute the Rabi frequencies for the resonant transitions
between the ground-state $\lvert\psi_{-}\rangle$ and the mixed spin and valley
states $\lvert\psi_{\pm}^{\prime}\rangle$:

$\displaystyle hf_{-}$ $\displaystyle=e\delta
V_{g}\left|\langle\psi_{-}^{\prime}\rvert
D\lvert\psi_{-}\rangle\right|=e\delta
V_{g}|\alpha^{\prime}\beta+\alpha^{*}\beta^{\prime}||D_{v_{1}v_{2}}|$ (S15a)
$\displaystyle hf_{+}$ $\displaystyle=e\delta
V_{g}\left|\langle\psi_{+}^{\prime}\rvert
D\lvert\psi_{-}\rangle\right|=e\delta
V_{g}|\alpha\alpha^{\prime}-\beta\beta^{\prime}||D_{v_{1}v_{2}}|\,.$ (S15b)

The Rabi frequencies for the resonant transitions between
$\lvert\psi_{+}\rangle$ and $\lvert\psi_{\pm}^{\prime}\rangle$ are equivalent.
We can also compute the Rabi frequency between the states
$\lvert\psi^{\prime}_{\pm}\rangle$:

$hf_{<}=e\delta V_{g}\left|\langle\psi_{+}^{\prime}\rvert
D\lvert\psi_{-}^{\prime}\rangle\right|=e\delta
V_{g}|\alpha^{\prime}\beta^{\prime}||D_{v_{1}v_{1}}-D_{v_{2}v_{2}}|\,,$ (S16)

as well as the Rabi frequency between the states $\lvert\psi_{\pm}\rangle$:

$hf_{>}=e\delta V_{g}\left|\langle\psi_{+}\rvert
D\lvert\psi_{-}\rangle\right|=e\delta
V_{g}|\alpha\beta||D_{v_{1}v_{1}}-D_{v_{2}v_{2}}|\,,$ (S17)

where $D_{v_{1}v_{1}}=\langle v_{1},\sigma\rvert D\lvert v_{1},\sigma\rangle$
and $D_{v_{2}v_{2}}=\langle v_{2},\sigma\rvert D\lvert v_{2},\sigma\rangle$.
The expansion of Eq. (S15a) in powers of $B$ and $C_{v_{1}v_{2}}$ yields back
Eq. (S4) at low magnetic fields. Yet Eqs. (S15) are valid up to much larger
magnetic field (typically $g\mu_{B}B\lesssim E_{3}-E_{1}$, where $E_{3}$ is
the energy of the next-lying state).

The typical energy diagram is plotted as a function of $B$ in Fig. 3b of the
main text. From an experimental point of view, line A of Fig. 3b corresponds
to $f_{-}$ ($B<B_{\rm V}$) or $f_{+}$ ($B>B_{\rm V}$), while line B
corresponds to $f_{<}$. The line D on Fig. 3c would correspond to $f_{>}$.

## Supplementary note 3: Tight-binding modeling

###  III.1 Methodology and devices

We have validated the model of supplementary note 2 against tight-binding (TB)
calculations.Di Carlo (2003); Delerue and Lannoo (2004) TB is well suited to
the description of such devices as it accounts for valley and spin-orbit
coupling at the atomistic level (without the need for, e.g., extrinsic Rashba
or Dresselhaus-like terms in the Hamiltonian).

We consider the prototypical device of Fig. S2a. The $[110]$ silicon nanowire
(dielectric constant $\varepsilon_{\rm Si}=11.7$) is $W=30$ nm wide and $H=10$
nm thick. It is etched in a $(001)$ SOI film on top a 25 nm thick buried oxide
(BOX).444The BOX used in the simulations is thinner than in the experiments
(145 nm) for computational convenience. We have checked that this does not
have any sizable influence on the results. The QD is defined by the central,
30 nm long gate. This gate covers only part of the nanowire in order to
confine a well-defined “corner state”. The QD is surrounded by two lateral
gates that control the barrier height between the dots (periodic boundary
conditions being applied along the wire). The front gate stack is made of a
layer of SiO2 ($\varepsilon_{\rm SiO_{2}}=3.9$) and a layer of HfO2
($\varepsilon_{\rm HfO_{2}}=20$). The device is embedded in Si3N4
($\varepsilon_{\rm Si_{3}N_{4}}=7.5$). We did not include the lateral gate in
the simulations. All terminals are grounded except the central gate.

We compute the first four eigenstates $\lvert 1\rangle...\lvert 4\rangle$ of
this device using a $sp^{3}d^{5}s^{*}$ TB model.Niquet  _et al._ (2009) The
dangling bonds at the surface of silicon are saturated with pseudo-hydrogen
atoms. We include the effects of SOC and magnetic field. The SOC Hamiltonian
is written as a sum of intra-atomic termsChadi (1977)

$H_{\rm SOC}^{\rm TB}=2\lambda\sum_{i}\mathbf{L}_{i}\cdot\mathbf{S}\,,$ (S18)

where $\mathbf{L}_{i}$ is the angular momentum on atom $i$, $\mathbf{S}$ is
the spin and $\lambda$ is the SOC constant of silicon. The action of the
magnetic field on the spin is described by the bare Zeeman Hamiltonian $H_{\rm
z}=g_{0}\mu_{B}\mathbf{B}\cdot\mathbf{S}$, and the action of the magnetic
field on the orbital motion of the electrons is accounted for by Peierl’s
substitution.Graf and Vogl (1995) We can then monitor the different Rabi
frequencies

$hf_{ij}=e\delta V_{g}\left|\langle i\rvert D\lvert j\rangle\right|\,.$ (S19)

The wave function of the ground-state $\lvert 1\rangle$ is plotted in Fig. S2b
($V_{g}=0.1$ V). It is, as expected for etched SOI structures, confined in a
“corner dot” below the gate. Note that this TB description goes beyond the
model of supplementary note 2 in including the action of the magnetic field on
the orbital motion, and in dealing with all effects non-perturbatively.

Figure S2:  (left) Electrostatic model of the device. The silicon wire appears
in red, SiO2 in green, HfO2 in blue and the gate in gray. (right) Envelope of
the squared TB wave function of the lowest-lying eigenstate ($V_{g}=0.1$ V).

In the following, we define $x=[110]$ (nanowire axis), $y=[\bar{1}10]$
(perpendicular to the nanowire) and $z=[001]$ (perpendicular to the
substrate).

###  III.2 Dependence on the magnetic field amplitude

Figure S3:  (a) Energy levels as a function of the magnetic field
$\mathbf{B}\parallel\mathbf{y}$. (b) Reduced Rabi frequency $F=f/\delta V_{g}$
as a function of the magnetic field. The TB data are compared to the model of
supplementary note 2. The vertical dash-dotted line is the anti-crossing field
$B_{\rm V}=\Delta/(g\mu_{B})$. The horizontal dash-dotted line is the maximal
Rabi frequency $hF_{\rm max}=e|D_{v_{1}v_{2}}|$.

The energy of the first four eigenstates is plotted as a function of
$\mathbf{B}\parallel\mathbf{y}$ in Fig. S3a. States $\lvert 1\rangle$…$\lvert
4\rangle$ all belong to the $\Delta_{z}$ valleys, as confinement remains
stronger along $z$ than along $y$ in a very wide range of gate voltages. The
lowest eigenstate $\lvert 1\rangle$ can be identified with
$\lvert\psi_{-}\rangle\simeq\lvert v_{1},\downarrow\rangle$, the second one
$\lvert 2\rangle$ and the third one $\lvert 3\rangle$ with
$\lvert\psi^{\prime}_{\pm}\rangle$ ($\lvert v_{1},\uparrow\rangle$/$\lvert
v_{2},\downarrow\rangle$ anti-crossing at $B=B_{\rm V}=1.17$ T), and the
fourth one $\lvert 4\rangle$ with $\lvert\psi_{+}\rangle\simeq\lvert
v_{2},\uparrow\rangle$.

The reduced TB Rabi frequency $F=f/\delta V_{g}$ between $\lvert
v_{1},\downarrow\rangle$ and $\lvert v_{1},\uparrow\rangle$ is plotted as a
function of $\mathbf{B}\parallel\mathbf{y}$ in Fig. S3. Hence $F\equiv F_{-}$
before the anti-crossing between $\lvert v_{1},\uparrow\rangle$ and $\lvert
v_{2},\downarrow\rangle$ at $B=B_{\rm V}$, and $F\equiv F_{+}$ after that
anti-crossing. This transition corresponds to line A of the main text. The
dependence of $F$ on $B$ is strongly non-linear, with a prominent peak near
the anti-crossing.

We can then switch off SOC, recompute the TB eigenstates at
$\mathbf{B}=\mathbf{0}$, extract

$\displaystyle\Delta$ $\displaystyle=E_{2}-E_{1}=135.86\ \mu{\rm eV}$ (S20a)
$\displaystyle\left|C_{v_{1}v_{2}}\right|$ $\displaystyle=\left|\langle
v_{2},\uparrow\rvert H_{\rm SOC}\lvert v_{1},\downarrow\rangle\right|=3.25\
\mu{\rm eV}$ (S20b) $\displaystyle\left|D_{v_{1}v_{2}}\right|$
$\displaystyle=\left|\langle v_{1},\sigma\rvert D\lvert
v_{2},\sigma\rangle\right|=179.26\ \mu{\rm V/V}$ (S20c)
$\displaystyle\left|D_{v_{1}v_{1}}-D_{v_{2}v_{2}}\right|$
$\displaystyle=\left|\langle v_{1},\sigma\rvert D\lvert
v_{1},\sigma\rangle-\langle v_{2},\sigma\rvert D\lvert
v_{2},\sigma\rangle\right|=813.26\ \mu{\rm V/V}$ (S20d)

and input these into Eqs. (S15). As shown in Fig. S3, Eqs. (S15) perfectly
reproduce the TB data. There are, in particular, no virtual transitions
outside the lowest four $\lvert v_{n},\sigma\rangle$ that contribute
significantly to the Rabi frequency.

Figure S4:  Reduced Rabi frequency $F_{<}=f_{<}/\delta V_{g}$ as a function of
the magnetic field. The TB data (symbols) are compared to the model of
supplementary note 2 (line). The vertical dash-dotted line is the anti-
crossing field $B_{\rm V}=\Delta/(g\mu_{B})$.

The reduced TB Rabi frequency $F_{<}=f_{<}/\delta V_{g}$ is also plotted on
Fig. S4. This transition corresponds to line B of the main text. It also shows
a peak near the $B=B_{\rm V}$, and is pretty strong.

In the present device, the TB valley splitting is much larger than the
experiment ($\Delta\simeq 36\,\mu$eV). However, $\Delta$ decreases once
surface roughness is introducedCulcer  _et al._ (2010) and can range from a
few tens to a few hundreds of $\mu$eV depending on the bias conditions. The TB
data reported in Fig. 4 of the main text have been computed for a particular
realization of surface roughness disorder that reproduces the experimental
$\Delta\simeq 36\,\mu$eV. The surface roughness profiles used in these
simulations have been generated from a Gaussian auto-correlation function with
rms $\Delta_{\rm SR}=0.4$ nm and correlation length $\Lambda_{\rm SR}=1.5$
nm.Goodnick  _et al._ (1985); Bourdet  _et al._ (2016) Note that disorder also
reduces $|C_{v_{1}v_{2}}|$, $|D_{v_{1}v_{2}}|$, and
$|D_{v_{1}v_{1}}-D_{v_{2}v_{2}}|$. A strategy for the control of the valley
splitting in etched SOI devices will be reported elsewhere.

It is clear from Eqs. (S15) that $F_{\pm}<F_{\rm max}$, where $hF_{\rm
max}=e|D_{v_{1}v_{2}}|$ is limited only by the dipole matrix element
$D_{v_{1}v_{2}}$. In particular, near the anti-crossing field $B=B_{\rm V}$,

$hF_{+}\simeq hF_{-}=F_{\rm ac}=\frac{e|D_{v_{1}v_{2}}|}{\sqrt{2}}$ (S21)

only depends on $D_{v_{1}v_{2}}$. The SOC matrix element $C_{v_{1}v_{2}}$
actually controls the width of the peak around $B=B_{\rm V}$. The full width
at half-maximum $\Delta B_{\rm FWHM}$ of this peak ($F=F_{\rm ac}/2$) indeed
reads:

$g\mu_{B}\Delta B_{\rm FWHM}\simeq\frac{12|C_{v_{1}v_{2}}|}{\sqrt{7}}\,.$
(S22)

## Supplementary note 4: Role of symmetries

To highlight the role of symmetries, we plot the reduced TB Rabi frequency $F$
(corresponding to line A of the main text) and SOC matrix element
$|C_{v_{1}v_{2}}|$ as a function of the orientation of the magnetic field in
Fig. S5 (same bias conditions as in Fig. S3, $|\mathbf{B}|=1.1$ T). As
expected, $F$ shows the same trends as $|C_{v_{1}v_{2}}|$ – although there are
small discrepancies. These discrepancies result from the effects of the
magnetic field on the orbital motion of the electron, dismissed in the
supplementary note 2.

Figure S5: (a) Reduced Rabi frequency $F=f/\delta V_{g}$ and (b) SOC matrix
element $|C_{v_{1}v_{2}}|$ as a function of the magnetic field orientation
($|\mathbf{B}|=1.1$ T).

Strikingly, the SOC matrix element and Rabi frequency are almost zero when the
magnetic field is aligned with the nanowire axis ($x$). The dependence of
$C_{v_{1}v_{2}}$ on the magnetic field (or spin) orientation suggests that
only the $\propto\sigma_{x}$ term of the SOC Hamiltonian is relevant for this
matrix element. This can be supported by a symmetry analysis. Now quantifying
the spin along $z$, the TB SOC Hamiltonian [Eq. (S18)] reads:

$H_{\rm SOC}^{\rm TB}=\lambda\left({\cal L}_{x}\sigma_{x}+{\cal
L}_{y}\sigma_{y}+{\cal L}_{z}\sigma_{z}\right)\,,$ (S23)

where $\sigma_{\alpha}$ are the Pauli matrices, ${\cal
L}_{\alpha}=\sum_{i}L_{i,\alpha}$ and $L_{i,\alpha}$ is the component
$\alpha=x,y,z$ of the angular momentum on atom $i$. The device of Fig. S6 is
only invariant by reflection through the $(yz)$ plane (space group $C_{s}$).
The table of characters of this space group is reproduced in Table 1. Let us
remind that for any observable $O$, unitary transform (symmetry operation)
$R$, and wave functions $\lvert\psi_{1}\rangle$ and $\lvert\psi_{2}\rangle$,

$O_{12}=\langle\psi_{1}\rvert O\lvert\psi_{2}\rangle=\langle R\psi_{1}\rvert
ROR^{\dagger}\lvert R\psi_{2}\rangle$ (S24)

$C_{s}$ | $E$ | $\sigma(yz)$
---|---|---
$A_{1}$ | $1$ | $+1$
$A_{2}$ | $1$ | $-1$

$C_{2v}$ | $E$ | $\sigma(yz)$ | $\sigma(xz)$ | $C_{2}(z)$
---|---|---|---|---
$A_{1}$ | $1$ | $+1$ | $+1$ | $+1$
$A_{2}$ | $1$ | $-1$ | $-1$ | $+1$
$B_{1}$ | $1$ | $+1$ | $-1$ | $-1$
$B_{2}$ | $1$ | $-1$ | $+1$ | $-1$

Table 1: Table of characters of the $C_{s}$ and $C_{2v}$ groups. $L_{x}=\mathop{{}\begin{pmatrix}0&0&0\\\ 0&0&-i\\\ 0&i&0\end{pmatrix}}_{\textstyle\mathstrut}$ | $L_{y}=\mathop{{}\begin{pmatrix}0&0&i\\\ 0&0&0\\\ -i&0&0\end{pmatrix}}_{\textstyle\mathstrut}$ | $L_{z}=\mathop{{}\begin{pmatrix}0&-i&0\\\ i&0&0\\\ 0&0&0\end{pmatrix}}_{\textstyle\mathstrut}$
---|---|---
Table 2: Matrices of $L_{x}$, $L_{y}$ and $L_{z}$ in the atomic
$\\{p_{x},p_{y},p_{z}\\}$ basis set.

Both $\lvert v_{1}\rangle$ and $\lvert v_{2}\rangle$ states belong to the
irreducible representation $A_{1}$ of $C_{s}$. Therefore, $\sigma(yz)\lvert
v_{1}\rangle=\lvert v_{1}\rangle$ and $\sigma(yz)\lvert v_{2}\rangle=\lvert
v_{2}\rangle$. Also, $D(\mathbf{r})=\partial V_{t}(\mathbf{r})/\partial V_{g}$
is compatible with the $C_{s}$ symmetry, so that
$\sigma(yz)D\sigma(yz)^{\dagger}=D$. Therefore, Eq. (S24) does not set any
condition on $D_{v_{1}v_{2}}$. However, $\sigma(yz)$ transforms a $p_{x}$
orbital into a $-p_{x}$ orbital, hence transforms ${\cal L}_{y}$ into $-{\cal
L}_{y}$, ${\cal L}_{z}$ into $-{\cal L}_{z}$, but leaves ${\cal L}_{x}$
invariant (see Table 2). Eq. (S24) then imposes that ${\cal L}_{y}$ and ${\cal
L}_{z}$ can not couple the $\lvert v_{1}\rangle$ and $\lvert v_{2}\rangle$
states. This is why only the $\propto{\cal L}_{x}\sigma_{x}$ term of the
atomistic SOC Hamiltonian makes a non-zero contribution. As a result,
$|C_{v_{1}v_{2}}|$ and the Rabi frequency are proportional to $|\sin\theta|$,
where $\theta$ is the angle between $\mathbf{x}$ and an in-plane magnetic
field. The effective SOC Hamiltonian between $\lvert v_{1},\downarrow\rangle$
and $\lvert v_{2},\uparrow\rangle$ is likely dominated by Rashba-type
contributions of the form $H_{\rm SOC}^{\rm eff}\propto p_{y,z}\sigma_{x}$
(where $p_{y}$ and $p_{z}$ are the momenta along $y$ and $z$). These Rashba-
type contributions may arise from the electric field of the gate along $y$ and
$z$, and/or from the top and lateral Si/SiO2 interfaces bounding the corner
dot.

How far is the EDSR connected with the formation of corner dots in etched SOI
structures ? To answer this question, we have computed the Rabi frequency in a
standard “Trigate” device where the gate surrounds the channel on three sides
(see Fig. S6a). The dimensions are the same as in Fig. S2, but the central
gate now covers the totality of the nanowire. As shown in Fig. S6b, the low-
lying conduction band states are still confined at the top interface, but the
wave function is symmetric with respect to the $(xz)$ plane (no “corner”
effect at zero back gate voltage). It turns out that there is no EDSR whatever
the orientation of the magnetic field. Actually, the SOC matrix element
$C_{v_{1}v_{2}}$ is zero for all spin orientations.

Figure S6:  (left) Electrostatic model of the trigate device. At variance with
Fig. S2, the central gate covers all three sides of the nanowire. (right)
Envelope of the squared TB wave function of the lowest-lying eigenstate
($V_{g}=0.1$ V).

Therefore the confinement in a corner dot seems to be a pre-requisite for a
sizable EDSR. As a matter of fact, the devices of Figs. S6 and S2 have
different symmetries. Indeed, the device of Fig. S6 is invariant by
reflections through the $(yz)$ and $(xz)$ planes, and by the twofold rotation
around the $z$ axis (space group $C_{2v}$). The states $\lvert v_{1}\rangle$
and $\lvert v_{2}\rangle$ of this device again belong to the irreducible
representation $A_{1}$ of $C_{2v}$ (Table 1). For the same reasons as before,
Eq. (S24) does not set any condition on $D_{v_{1}v_{2}}$. The existence of a
$\sigma(yz)$ mirror still imposes that only ${\cal L}_{x}$ can couple $\lvert
v_{1}\rangle$ and $\lvert v_{2}\rangle$ states. Yet the introduction of a
$\sigma(xz)$ mirror, which leaves only ${\cal L}_{y}$ invariant, prevents such
a coupling. Therefore, none of the operators ${\cal L}_{x}$, ${\cal L}_{y}$
and ${\cal L}_{z}$ can couple the $\lvert v_{1}\rangle$ and $\lvert
v_{2}\rangle$ states. $C_{v_{1}v_{2}}$ is thus zero, and SOC-mediated EDSR is
not possible.

To conclude, the symmetry must be sufficiently low in order to achieve EDSR in
the conduction band of silicon. This condition is realized in corner dots
where there is only one $\sigma(yz)$ mirror left (yet preventing EDSR for a
magnetic field along the wire axis). The design of geometries without any
symmetry left would in principle maximize the opportunities for EDSR.

## References

  * Ono  _et al._ (2002) K. Ono, D. G. Austing, Y. Tokura, and S. Tarucha, Science 297, 1313 (2002).
  * Hanson  _et al._ (2007) R. Hanson, J. R. Petta, S. Tarucha, and L. M. K. Vandersypen, Reviews of Modern Physics 79, 1217 (2007).
  * Hao  _et al._ (2014) X. Hao, R. Ruskov, M. Xiao, C. Tahan, and H. Jiang, Nature Communications 5, 3860 (2014).
  * Koppens  _et al._ (2005) F. H. L. Koppens, J. A. Folk, J. M. Elzerman, R. Hanson, L. H. W. van Beveren, I. T. Vink, H. P. Tranitz, W. Wegscheider, L. P. Kouwenhoven, and L. M. K. Vandersypen, Science 309, 1346 (2005).
  * Nadj-Perge  _et al._ (2010) S. Nadj-Perge, S. M. Frolov, J. W. W. van Tilburg, J. Danon, Y. V. Nazarov, R. Algra, E. P. A. M. Bakkers, and L. P. Kouwenhoven, Physical Review B 81, 201305(R) (2010).
  * Sham and Nakayama (1979) L. J. Sham and M. Nakayama, Physical Review B 20, 734 (1979).
  * Saraiva  _et al._ (2009) A. L. Saraiva, M. J. Calderon, X. Hu, S. Das Sarma, and B. Koiller, Physical Review B 80, 081305(R) (2009).
  * Friesen and Coppersmith (2010) M. Friesen and S. N. Coppersmith, Physical Review B 81, 115324 (2010).
  * Culcer  _et al._ (2010) D. Culcer, X. Hu, and S. Das Sarma, Physical Review B 82, 205315 (2010).
  * Zwanenburg  _et al._ (2013) F. A. Zwanenburg, A. S. Dzurak, A. Morello, M. Y. Simmons, L. C. L. Hollenberg, G. Klimeck, S. Rogge, S. N. Coppersmith, and M. A. Eriksson, Review of Modern Physics 85, 961 (2013).
  * Feher (1959) G. Feher, Physical Review 114, 1219 (1959).
  * Golovach  _et al._ (2006) V. N. Golovach, M. Borhani, and D. Loss, Physical Review B 74, 165319 (2006).
  * Nestoklon  _et al._ (2006) M. O. Nestoklon, L. E. Golub, and E. L. Ivchenko, Physical Review B 73, 235334 (2006).
  * Yang  _et al._ (2013) C. H. Yang, A. Rossi, R. Ruskov, N. S. Lai, F. A. Mohiyaddin, S. Lee, C. Tahan, G. Klimeck, A. Morello, and A. S. Dzurak, Nature Communications 4, 2069 (2013).
  * Huang and Hu (2014) P. Huang and X. Hu, Physical Review B 90, 235315 (2014).
  * Veldhorst  _et al._ (2015) M. Veldhorst, R. Ruskov, C. H. Yang, J. C. C. Hwang, F. E. Hudson, M. E. Flatté, C. Tahan, K. M. Itoh, A. Morello, and A. S. Dzurak, Physical Review B 92, 201401(R) (2015).
  * Huang  _et al._ (2017) W. Huang, M. Veldhorst, N. M. Zimmerman, A. S. Dzurak, and D. Culcer, Physical Review B 95, 075403 (2017).
  * Di Carlo (2003) A. Di Carlo, Semiconductor Science and Technology 18, R1 (2003).
  * Delerue and Lannoo (2004) C. Delerue and M. Lannoo,  _Nanostructures: Theory and Modelling_ (Springer, New-York, 2004).
  * Niquet  _et al._ (2009) Y. M. Niquet, D. Rideau, C. Tavernier, H. Jaouen, and X. Blase, Physical Review B 79, 245201 (2009).
  * Chadi (1977) D. J. Chadi, Physical Review B 16, 790 (1977).
  * Graf and Vogl (1995) M. Graf and P. Vogl, Physical Review B 51, 4940 (1995).
  * Goodnick  _et al._ (1985) S. M. Goodnick, D. K. Ferry, C. W. Wilmsen, Z. Liliental, D. Fathy, and O. L. Krivanek, Physical Review B 32, 8171 (1985).
  * Bourdet  _et al._ (2016) L. Bourdet, J. Li, J. Pelloux-Prayer, F. Triozon, M. Cassé, S. Barraud, S. Martinie, D. Rideau, and Y.-M. Niquet, Journal of Applied Physics 119, 084503 (2016).