Development of a Massively Parallel Nano-electronic Modeling Tool and its Application to Quantum Computing Devices Sunhee Lee Network for Computational Nanotechnology Electrical and Computer Engineering sunnyleekr@purdue.edu

Building block for quantum computing device Quantum dot (QD) Confinement (particle-in-a-box) s- p- d- like orbitals (“artificial atom”) Optical applications (LED/PD) Applications for quantum computers (QC) Carry electron/nucleus spin info. n=1 n=2 n=3 QDOT Lab @nanoHUB.org Light absorption 6~7 ionized P in Si M. Füchsle et.al., Nature Nanotechnology, 2010 Hanson and Awschalom, Nature 453, 2008

Phosphorus quantum dot in Si Ionized P impurity QD Phosphorus quantum dot in Si Promising candidate for QC device Long spin coherence times Naturally uniform Store electron/nucleus spin info. Fabrication challenges First single donor QD system !! STM+MBE technology 2D dopant patterning QD images adopted with permission from Simmons’ group 1 2 3 4 5 3

Single Donor Quantum Dot: Experiment Experimental Work (UNSW): a single donor QD ! Questions: is this real?? How can we explain the coupling of the channel donor to the Si:P leads ? Can we quantify the controllability of plane Si:P leads on the channel confinement ? Why are there the conductance streaks at the Coulomb diamond edges ? Prove it is real! (Purdue) 4

Modeling Si:P QD : Need for atomistic modeling 150 nm 16 nm 10 nm 15 nm Si SiGe Kharche et al. Appl. Phys. Lett. 90, 092109 (2007) Alloy disorder Rough steps Predicting valley splitting in Si (First excited state) – (GND state) Important measure in QC 10 ueV ~ 1 meV Random alloy disorder Sample variation (Error bars) ex) disorders in the 2D Si:P layer, published in PRB Individual dopant spectrum Single impurity QD in finFET Atomistic treatment with localized basis set sp3d5s* atomistic tight-binding Lansbergen et al. Nat. Phys. 4, 656 (2008)

Modeling Si:P QD : Experience Modeling Work (Purdue): Single Donor QD system Strength Single impurity physics (R. Rahman & S. Rogge) Realistic modeling of Si:P contacts Strong connections to experiment Single electron charging energy, transition points, gate controllability & Coulomb diamond 6

Modeling Si:P QD : NEMO3D-peta Localized orbital basis (sp3d5s*) Atomistic structure (~106 atoms) NEMO3D (physics) (Schrödinger solver) Solving an eigenvalue problem Atomistic grid 𝐻Ψ=𝐸Ψ dim 𝐻 = (106~107) (atoms) X (10~20) (basis/atom) = 107~108 !! NEMO3D-peta (2008~) Atomistic tight-binding, million atom simulation tool For QD-like simulations Inherits the physics aspect of NEMO3D Schrödinger-Poisson self-consistency module 3D spatial parallelization Useful in self-consistent simulations NEMO3D-peta (Schrödinger-Poisson solver) + NEMO3D

Parallelization engine in NEMO3D-peta Why do we need “better” parallel computing?  To reduce simulation time even more! NEMO3D: 1D slices NEMO3D-peta: 2D/3D slices 𝐻= 1 2 4 8 16 Bent’s work NEMO3D : single shot eigenvalue problem NEMO3D-peta: Self-consistent simulation !! (10~30 iterations) time # procs.

NEMO3D-peta Highlights (2008~present) 90,000+ lines of code (from scratch!!) 3.5+ years of development ~8 applications implemented Expandable and maintainable 15,000,000 compute hours awarded Capable of utilizing 32,000 processors Released to Intel (2010) Top of the Barrier / bandstructure app. 1 nanoHUB tool 1d-hetero 15 Publications in line 9 journal and conference papers (3 experimental) 2 journal publication accepted (1 B. Weber et al. Science) 4 journal publications ready for submission (1 M. Fuechsle et al.)

NEMO3D-peta for QD simulation NEMO3D-peta development Localized orbital basis (sp3d5s*) Atomistic structure (~106 atoms) NEMO3D (physics) (Schrödinger solver) 3D spatial parallelization QD Device modeling Potential-charge self-consistency

Single Donor Quantum Dot: Questions Experimental Work (UNSW): a single donor QD ! Questions: is this real?? How can we explain the coupling of the channel donor to the Si:P leads ? Can we quantify the controllability of plane Si:P leads on the channel confinement ? Why are there the conductance streaks at the Coulomb diamond edges ? Prove it is real !! 11

Modeling: Domain Domain 3D schematic Top view Doping plane 2D (n++ doped) 3D distribution of charge 3D schematic p-type substrate (1015cm-3) 56 nm 128 nm 360 nm δ-doping plane S D G1 G2 Top view [001] [1-10] [110]

Modeling: Background potential Semi-classical calculation Background potential WITHOUT impurity QD Leads (n++) doping region, ND=1021 (cm-3) Background doping (p-) NA=1015(cm-3) VSD = 0 VG1=VG2=VG Device geometry (top view) Semi-classical region [110] [1-10] SRC DRN G1 G2

Modeling: Impurity QD potential Empty QD (ionized donor QD) Binding energy data of P in Si (Rep. Prog. Phys., Vol. 44, 1981) Coulombic (1/r) + TB param. fitting (Work by R. Rahman @ Nat. Phys.) “D+” state QD changes shape with electron filling !! Single electron filled QD QD potential “screened”  Shallower potential Self-consistent calculation Next ground state “floats up” “D0” state Before applying gate bias

Modeling: Potential profile Superposition Background potential QD potential Equilibrium potential profile [110] (nm) 15

Modeling: Charge filling (Ack: H. Ryu) Device geometry (top view) Quantum region [110] [1-10] SRC DRN G1 G2 Quantum region Channel region 12x60x20 (nm3) Compute ground eigenstate at each Vg Determine charge filling Does Ground state hit EF(SRC)? D+

Modeling: Charge filling (Ack: H. Ryu) VDS = 0 V, sweep VG Plot Ground state eigenvalue (1s(A)) EF VG = 0.0 V Channel empty (D+) D+ -5 5 - -5 5 [110] (nm) [110] (nm) Ground eigenstate Acknowledgment: Dr. Hoon Ryu

Modeling: Charge filling (Ack: H. Ryu) VDS = 0 V, sweep VG Plot Ground state eigenvalue (1s(A)) EF VG = 0.2 V Channel empty (D+) D+ -5 5 - -5 5 [110] (nm) [110] (nm) Ground eigenstate

Modeling: Charge filling (Ack: H. Ryu) VDS = 0 V, sweep VG Plot Ground state eigenvalue (1s(A)) EF VG ≈ 0.45 V 1s(A) hits EF D+  D0 transition Screened QD ! (impose D0 potential) D+ -5 5 - -5 5 [110] (nm) [110] (nm) Ground eigenstate

Modeling: Charge filling (Ack: H. Ryu) VDS = 0 V, sweep VG Plot Ground state eigenvalue (1s(A)) EF VG ≈ 0.55 V Channel filled by one electron (D0) D0 -5 5 - -5 5 [110] (nm) [110] (nm) Ground eigenstate

Modeling: Charge filling (Ack: H. Ryu) VDS = 0 V, sweep VG Plot Ground state eigenvalue (1s(A)) EF VG ≈ 0.72 V 1s(A) hits EF D0  D- transition D0 -5 5 - -5 5 [110] (nm) [110] (nm) Ground eigenstate

Modeling: Charge filling (Ack: H. Ryu) Simulation vs. Experiment: How close are we ? 3. EC = 46.3 meV [110] (nm) 5 -5 - D+ 0e 1e Experiment Theory 1. 0e  1e Transition VG (V) 0.40 0.45 2. 1e  2e Transition VG (V) 0.80 0.72 3. Charging energy EC (meV) 47 ± 2 46.3 4. Gate Lever-arm 0.11 0.15

Modeling: Coulomb diamond (Ack: Y.H.M. Tan) Lead DOS profiles Methodology, S. Lee, PRB 2011 Si:P wire, H. Ryu, PhD dissertation, 2011 B. Weber, Science 2011 Extract results from NEMO3D-peta Channel states Lead DOS profiles Rate equation tool Transition points ( 0.42, 0.72V) Charging energy (Ec = 46.3 meV) Gate controllability (slope a = 0.15)  Lead DOS profiles (streaks) Channel states, EF

Single Donor Quantum Dot: Answers Experimental Work (UNSW): a single donor QD ! How can we explain the coupling of the channel donor to the Si:P leads ? Semi-classical treatment of gate biasing No stark effect (parallel shift of ground state) Can we quantify the controllability of plane Si:P leads on the channel confinement ? Transition points / Charging energy Why are there the conductance streaks at the Coulomb diamond edges ? Excited states + DOS of the leads 24

Quantitative match with experiment Conclusion Quantitative match with experiment Transition point / charging energy / in-plane gate modulation  A strong support for single impurity QD Methodology applicable for future Si:P QD devices Experiment Theory 1. 0e  1e Transition VG (V) 0.40 0.45 2. 1e  2e Transition VG (V) 0.80 0.72 3. Charging energy EC (meV) 47 ± 2 46.3 4. Gate Lever-arm 0.11 0.15

Focused on the electrostatic modeling of single donor QD Summary Focused on the electrostatic modeling of single donor QD Gate modulation and charge filling A quantitative match with the experimental results Methodology can be extended to future Si:P QD system Transition phase (Y.H.M Tan) Double Donor QD (D-168) Understanding the two-electron operations in multiple QD systems Find new methods to efficiently model QDs Double Quantum Dot

Acknowledgment Committee members Special thanks to … Thanks to … Prof. Gerhard Klimeck Prof. Mark Lundstrom, Prof. Leonid Rokhinson, Prof. Alejandro Strachan & Prof. Michelle Simmons Special thanks to … Dr. Hoon Ryu Matthias Tan, Zhengping Jiang & Junzhe Geng Dr. Abhijeet Paul Changwook Jeong, Seokmin Hong & Jayoung Park Thanks to … Dr. Mathieu Luisier, Dr. Honghyun Park, Dr. Jim Fonseca & Dr. Michael Povolotskyi Sunggeun Kim, Parijat Sengupta, Mehdi Salmani, Saumitra Mehrotra & Yahua Tan Quantum dot subgroup CQC2T Collaborators Dr. Lloyd Hollenberg Dr. Suddhasatta Mahapatra, Dr. Jill Miwa, Dr. Martin Fuechsle and Bent Weber Cheryl Haines & Vicki Johnson Funding agencies: NSF, ARO, MSD, SRC …

List of publications S. Lee, H. Ryu, Z. Jiang, and G. Klimeck, “Million atom electronic structure and device calculations on peta-scale computers,” in 13th International Workshop on Computational Electronics, 2009 (IWCE '09), May 2009 H. Ryu, S. Lee, and G. Klimeck, “A study of temperature-dependent properties of n-type delta-doped Si band-structures in equilibrium,” in 13th International Workshop on Computational Electronics, 2009 (IWCE '09), May 2009 S. Lee, H. Ryu, G. Klimeck, H. Campbell, S. Mahapatra, M. Y. Simmons, and L. C. L. Hollenberg, “Equilibrium bandstructure of a phosphorus delta-doped layer in silicon using a tight-binding approach,” IEEE Proceedings of NANO 2010, 2010 H. Ryu, S. Lee, B. Weber, S. Mahapatra, M. Simmons, L. Hollenberg, and G. Klimeck, “Quantum transport in ultra-scaled phosphorous-doped silicon nanowires,” in Silicon Nanoelectronics Workshop (SNW), Jun. 2010 B. Weber, S. Mahapatra, W. R. Clarke, R. H., L. S., G. Klimeck, L. C. L. Hollenberg, and M. Y. Simmons, “Quantum transport in atomic-scale silicon nanowires,” in Silicon Nanoelectronics Workshop (SNW), Jun. 2010 G. Tettamanzi, A. Paul, G. Lansbergen, J. Verduijn, S. Lee, N. Collaert, S. Biesemans, G. Klimeck, and S. Rogge, “Thermionic emission as a tool to study transport in undoped n-FinFETs,” IEEE Electron Device Letters, vol. 31, Feb. 2010 G. Tettamanzi, A. Paul, S. Lee, S. Mehrotra, N. Collaert, S. Biesemans, G. Klimeck, and S. Rogge, “Interface trap density metrology of state-of-the-art undoped Si n-FinFETs,” IEEE Electron Device Letters, vol. 32, Apr. 2011 A. Paul, G. C. Tettamanzi, S. Lee, S. Mehrotra, N. Colleart, S. Biesemans, S. Rogge, and G. Klimeck, “Interface trap density metrology from sub-threshold transport in highly scaled undoped Si n -FinFETs,” accepted for publication in Journal of Applied Physics 2011 A. G. Akkala, S. Steiger, J. M. D. Sellier, S. Lee, M. Povolotskyi, T. C. Kubis, H. Park, S. Agarwal, and G. Klimeck, “1d heterostructure tool,” https://nanohub.org/resources/5203, Sep. 2008 (Now replaced by NEMO 5) S. Lee, H. Ryu, H. Campbell, L. C. L. Hollenberg, M. Y. Simmons and G. Klimeck, “Electronic structure of realistically extended atomistically resolved disordered Si:P δ-doped layers,” Physical Review B, 84 205309, 2011 B. Weber, S. Mahapatra, H. Ryu, S. Lee, A. Fuhrer, T. C. G. Reusch, D. L. Thompson, W.C.T. Lee, G. Klimeck, L. C. L. Hollenberg, M.Y. Simmons, “Ohm’s law Survives to the Atomic Scale,”, accepted for publication in Science 2011 Three other publications ready for submission, one in preparation

Result 4 : Coulomb diamond Basic Features Ground states Ground + excited states Coupling DOS in leads

Electrostatically defined QD (UNSW) Si MOS QD Electrostatically defined QD (UNSW) MOS fabrication technology Dit = 5x1010 cm-2eV-1 (x 0.1~0.01) Nelectron = 0, 1, 2, … !! Lateral confinement Vertical confinement Electron charging [001] [110] [110]

Six-valley degeneracy Challenges Six-valley degeneracy Valley splitting (Δ) = First excited eigenstate – GND state In this QD : ~100 ueV Questions What are the possible factors that influence VS ? Does our results compare experimental results ? Typical quantum well case example

Self-consistent simulation Method [110] [001] [1-10] Simulation domain Size = 60x90x30 nm3, 8 million atoms Self-consistent simulation Input 1: Barrier height (VB1=VB2) Input 2: Plunger gate size (30xWc) Wc= 30,40,50 & 60 nm Input 3: Assume 1 electron filled Output 1: VP Output 2: VS

Smaller dot, Large lateral barrier  Stronger confinement Results Small lateral barrier height Large lateral barrier height Weak vertical confinement Strong vertical confinement Smaller dot, Large lateral barrier  Stronger confinement Eigenstates float up  Deeper vertical confinement required VS range : 100~500 ueV (100 ueV exp.) VS tunable but sensitive to QD geometry and lateral barrier height

Work is still in progress Conclusion VS in Si MOS QD 100~500 ueV (100 ueV exp.) VS can be tunable Controlling barrier height Adjusting QD size Sensitive to electrostatics Work is still in progress Excited state spectrum @ N electron regime Compare VS with SiGe-Si-SiGe QD