Simulation of Silicon Nanodevices at Cryogenic …...device as an input to a quantum-mechanical solver [11, 12], which in turn calculates precise donor energies and other parameters

Fahd A. Mohiyaddin1,2 · Franklin G. Curtis1,2 · M. Nance Ericson1,3 ·Travis S. Humble1,2

Simulation of Silicon Nanodevices at CryogenicTemperatures for Quantum Computing

Abstract Cryogenic nanoscale semiconductor devicesare crucial for a wide variety of applications. The ac-curate design of such devices involve solving their elec-trostatics from fundamental semiconductor equationsat low temperatures. We employ COMSOL to model aprototype cryogenic nanostructure, that can be used toreadout the spin of a single electron in silicon, for quan-tum computing applications. By achieving convergencedown to 15 Kelvin (K), we provide a guideline of tech-niques that aid to enhance convergence at cryogenictemperatures. We further compare the device electro-statics at different temperatures, which aids us to es-timate the accuracy with temperature.

1 Introduction

Electronic devices operating at cryogenic temperaturesare critical for a range of applications including spacesatellites, medicine, fundamental physics research and

This manuscript has been authored by UT-Battelle, LLC,under Contract No. DE-AC0500OR22725 with the U.S. De-partment of Energy. The United States Government re-tains and the publisher, by accepting the article for pub-lication, acknowledges that the United States Governmentretains a non-exclusive, paid-up, irrevocable, world-wide li-cense to publish or reproduce the published form of thismanuscript, or allow others to do so, for the United StatesGovernment purposes. The Department of Energy will pro-vide public access to these results of federally sponsoredresearch in accordance with the DOE Public Access Plan(http://energy.gov/downloads/doe-public-access-plan).

1Quantum Computing InstituteOak Ridge National LaboratoryOak Ridge, TN

2Computational Sciences and Engineering DivisionOak Ridge National LaboratoryOak Ridge, TN

3Electrical and Electronics Systems Research Divi-sionOak Ridge National LaboratoryOak Ridge, TN

E-mail: [email protected]

quantum computing. It is well known that cryogenicsemiconductor devices exhibit fast operation speeds,low power dissipation, small leakage currents, reducednoise and thermal degradation, when compared to theirroom temperature counterparts [1,2]. Advances in ma-terials, superconductivity, electronic integration, andcooling techniques have further reinforced the signifi-cance and applications of low temperature nanoscaledevices [3–5].

Optimal realization of these emerging cryogenic cir-cuits and systems requires modeling and simulationtools with suitable accuracy at cryogenic temperatures.Simulation of these devices requires numerically solv-ing the fundamental semiconductor equations [6] atlow temperature, and estimating vital electrostatic pa-rameters such as electric fields, currents, conductionband energies and carrier densities. However, the nu-merical modeling of semiconductor devices at cryo-genic temperatures poses significant convergence issuessince several parameters, such as carrier density, scaleexponentially at such temperatures. This leads to in-termediate solutions with sharp discontinuities in theelectric fields and carrier densities, thereby failing toconverge.

We report on techniques to ease convergence andextend modeling to the case of silicon based devicesat very low temperatures. We illustrate this with anexample of a 3-dimensional prototype device whichcan be used to readout the spin state of a phospho-rus electron in a silicon quantum computer. We specif-ically employ the Semiconductor Module of the COM-SOL MultiPhysics Software [7], since it offers the com-bined advantage of flexibility and capability to solve forseveral independent physical equations and parame-ters self-consistently. With proper choices of the finite-element solver, equations, initial values for the electrondensity, and an efficient mesh, we achieve convergencedown to 15 K for the nanostructure. Our results for thisspecific device also directly fit into a broader contextof developing a computational workflow to accuratelydesign silicon devices for quantum computing [8].

The remainder of the paper is organized as follows:in Section 2, we outline the details of the nanostructure

Excerpt from the Proceedings of the 2017 COMSOL Conference in Boston

500 nm

Donor

Fig. 1 COMSOL model of the MOS-based nanostructureemployed in this work. The device can be used to readoutthe spin state of an electron bound to a phosphorus donoratom, which is implanted into the silicon substrate. Alu-minum gate electrodes in the vicinity of the donor atomare highlighted in blue. The top, barrier and donor gateelectrodes are biased at 1 V, 0 V and 0 V respectively.

used in this work. We then summarize the semiconduc-tor equations solved by COMSOL, and highlight con-vergence issues at low temperature, in Section 3. Sec-tion 4 reviews the techniques that have aided in achiev-ing convergence down to 15 K for this nanostructure.Section 5 describes the electrostatics observed in thedevice at 15 K. Finally in Section 6, we compare theresults obtained at higher temperatures and quantifythe accuracy of the simulations with temperature.

2 Test Nanostructure Model

The COMSOL model of the device is shown in Fig-ure 1, and is comprised of a metal-oxide-semiconductor(MOS) silicon nanostructure with aluminum gate elec-trodes and SiO2 dielectric. The top and barrier gateelectrodes in Figure 1 are appropriately biased to cre-ate a single-electron-transistor (SET) at the Si-SiO2interface (shown later in Figure 4a) [9]. The source anddrain regions correspond to heavily doped n+ regionshaving a phosphorus (31P) concentration of 1020 cm−3.Ohmic contacts are assigned to these regions with typ-ical source-drain voltages (VDS) ∼ 10 µV. The siliconsubstrate is lightly n doped with a 31P concentrationof 1013 cm−3, and is grounded with an ohmic contactbeneath it.

A 31P donor atom is implanted next to the SETinto the silicon substrate, as illustrated in Figure 1.Tuning the energy levels of the electron bound to thedonor aids in reading out its spin state, and is elab-orated later in Section 5 [10]. In a previous work, weshowed that a semi-classical estimate of the conductionband energy is sufficient to estimate the donor electron

energy and simulate spin-readout in such devices, towithin a reasonable accuracy (∼ 1 meV). We can alsouse the electrostatic potential and electric fields in thedevice as an input to a quantum-mechanical solver [11,12], which in turn calculates precise donor energies andother parameters relevant to quantum computing [13].

To characterize the device electrostatics, includ-ing the electron density, conduction band energy andelectric fields, we invoke the COMSOL Semiconduc-tor module, which solves the semiconductor equationsoutlined in Section 3.

3 COMSOL Semiconductor Module at LowTemperature

Given an input device layout and gate voltages, theSemiconductor module solves the following Poisson andcurrent continuity equations in the semiconductor, toestimate the carrier densities, currents and electrostaticpotential [6,7].

∇ · (�∇V ) = −q (p− n+ND+ −NA−) , (1a)

∂n

∂t=

1

q(∇ · Jn)− Un, (1b)

∂p

∂t= −1

q(∇ · Jp)− Up, (1c)

where � is the permittivity, V is the potential, q isthe elementary charge, n and p are the electron andhole densities respectively, ND+ and NA− are the ion-ized donor and acceptor densities respectively, t is thetime, Jn and Jp are the electron and hole current den-sities respectively, and Un and Up are the net recom-bination rates for the electrons and holes respectively.

The carrier densities in Equation 1 are related totheir quasi-Fermi energy levels, and the valence andconduction band energies, using the two well knownequations below:

n = NC F1/2

(EFn − EckBT

), (2a)

p = NV F1/2

(Ev − EFpkBT

), (2b)

where NC and NV correspond to the density ofstates, F1/2(η) is the Fermi integral of order 1/2 andapproaches eη when η → −∞, Ec and Ev are the con-duction and valence band edges respectively, EFn andEFp are the electron and hole quasi- Fermi levels re-spectively, kB is Boltzmann constant, and T is the tem-perature. The band energies in Equations 2 are relatedto the electrostatic potential V , as follows:

Ec = −χ− qV, (3a)



Donor

50 nm 50 nm

(a) (b)

Fig. 2 (a) Mesh at the surface of the oxide, illustratinghigh mesh densities in the vicinity of gates. (b) Mesh atthe corner of the device, highlighting a swept mesh in theoxide and in the substrate near the Si-SiO2 interface.

Ev = −χ− Eg − qV, (3b)

where χ is the electron affinity of silicon, and Eg isthe band gap. Note that COMSOL considers the Fermipotential as the reference zero potential for V .

To calculate the charge densities arising from ion-ized donors, we employ the following incomplete ion-ization model:

ND+ =ND

1 + gD exp(EFn−EDkBT

) , (4a)NA− =

NA

1 + gA exp(EA−EFpkBT

) , (4b)where ED(A) and ND(A) are the impurity energy

levels and doping densities respectively, gD = 2 andgA = 4 are the degeneracy factors of the impuritiesconsidering spin. Equations 2, 3 and 4 are fed intoEquations 1a, 1b and 1c, which are self-consistentlysolved with appropriate boundary conditions over afinite mesh to estimate the n, p, V and their dependentvariables in the semiconductor.

Voltages applied to the source, drain and substrateare modeled with an Ohmic boundary condition. Thisenforces charge neutrality for the electron and holedensities neq and peq respectively at the boundary, i.e.

neq − peq +N−a −N+d = 0. (5)Substituting Equation 5 into Equations 2, 3 and 4,

the Ohmic boundary condition yields

neq =1

2

(N+d −N

−a

)+

1

2

√(N+d −N

−a

)2+ 4γnγpn2i ,

(6a)

peq = −1

2

(N+d −N

−a

)+

1

2

√(N+d −N

−a

)2+ 4γnγpn2i ,

(6b)

Veq = V0−χ−Eg2q

+kBT

q

(log

(neqγnni

)+

1

2log

(NvNc

)),

(6c)

where V0 is the voltage applied at the Ohmic con-tact, Veq is the electrostatic potential at the boundary,ni =

√NcNvexp (−Eg/2kBT ) is the intrinsic carrier

density, and γn and γp are defined by the following:

γn =F1/2

(EFn−EckBT

)exp

(EFn−EckBT

) , (7a)γp =

F1/2

(Ev−EFpkBT

)exp

(Ev−EFpkBT

) . (7b)All of the equations shown above are solved by

COMSOL in the silicon substrate. The Poisson equa-tion is solved to obtain electrostatics in the remainderof the device. Since the oxide and silicon substrate havedifferent sets of equations that need to be solved, weensure that the oxide and immediate substrate beneathit have the same lateral mesh. A tetrahedral mesh isthen used for the remaining substrate. Figure 2 illus-trates the mesh used for this nanostructure.

Note that the carrier densities n and p in Equation2 have an exponential dependence on temperature. As-suming Ec > EF , Figure 3a plots the spatial gradientof the logarithm of electron density n, as a functionof electric field and temperature. For sufficiently lowtemperatures and large electric fields, we note that theelectron densities vary sharply with position. For ex-ample, the electron density varies by over 2 orders ofmagnitude per nm, when the electric field approaches∼ 5 MV/m at 15 K. This hampers convergence as ex-tremely fine meshes are required to capture the steepexponential gradients of the carrier densities at suchtemperatures. We also highlight that extremely smallcarrier densities at low temperatures cannot be calcu-lated with the numerical precision available in COM-SOL. This will lead to divide-by-zero errors, such as inEquation 6c. Figure 3b shows the COMSOL conver-gence plots at different temperatures, when the carrierdensities are directly solved for, using the finite ele-ment method. The plot clearly indicates that obtainingconvergence at temperatures below 50 K is challenging,and requires additional constraints and methods.

4 Guidelines for low temperature convergence

The nanostructure employed in this work is purely n-doped, and we are primarily interested in the electrondensities, along with the potential. Hence, we first sim-plify the model and approximate the hole densities asp = n2i /n. This eases convergence by reducing thenumber of degrees of freedom (∼ 1.5 million) to besolved. This approximation has negligible effect on thepotential, electric fields and electron densities.

The source-drain voltage VDS is 10 µV, which im-plies that the Fermi level EF varies by 10 µeV in the



10 50 100 150 200 250 3000

2

4

6

8

10

0

1

2

3

4

5

Ele

ctr

ic F

ield

Ex

(MV

/m)

Temperature (K)

(nm-1)

𝒅

𝒅𝒙𝐥𝐨𝐠𝟏𝟎(𝒏)

10-5

10-4

10-3

10-2

10-1

100

101

102

103

Err

or

Iterations for Different Temperatures

15K 50K 100K 300K

Finite Element Linear Solver

(a) (b)

Iterations for Different Temperatures

10-4

10-3

10-2

10-1

100

101

102

10-5

Err

or

15K 20K 30K 50K 100K 200K 300K

10-5

10-3

10-1

101

103

1018 m-3 1019 m-3 1020m-3 1021m-3 1022m-3Temperature = 300 K

Iterations for Different Initial Electron Densities10-5

10-3

10-1

101

103

1011 m-3 1012 m-3 1013m-3 1014m-3 1015m-3

Temperature = 15 K

Err

or

Err

or

(c) (d)Finite Element Logarithmic Solver

Fig. 3 (a) Spatial gradient of the logarithm of the electron density n as a function of electric field and temperature. Atlow temperatures and large electric fields, sharp gradients in n hamper convergence. (b)-(d) COMSOL convergence plotsobtained for (b) different temperatures, when the finite-element method is used for solving the carrier density directly, (c)different initial starting values ninit of the electron density, at 300 K and 15 K, and (d) different temperatures by followingthe guidelines presented in Section 4. Each point in the convergence plots represents a single Newton iteration.

device. The variation of EF causes significant conver-gence issues at low temperatures. Since, we are inter-ested in the device potential, rather than currents, weapproximate VDS as 0 V. This condition limits the ac-curacy of the potential and conduction band energy(Ec) to within tens of µeV. We will see in Figure 4cthat Ec varies by several meV in such devices, and thatthe above approximation is reasonable.

While the sharp exponential variation in the elec-tron density hampers convergence, we can circumventthe issue by solving for the logarithm of the electrondensity n directly. This is implemented in the COM-SOL Semiconductor module and results in better con-vergence at low temperatures, as the spatial variationof log(n) is more gradual than that of n.

By default, COMSOL uses the intrinsic carrier den-sity (ni) as an initial starting value (ninit) for the elec-tron density while solving the semiconductor equa-tions. ninit sets the scaling factors in the Jacobian ma-trix. At ultra low temperatures, ni apporoaches zero,resulting in extreme values in the Jacobian, hamperingconvergence. Hence, it is crucial to choose an appropri-ate value of ninit at low temperatures. Figure 3c showsthe COMSOL convergence plots for different ninit, at

300 K and 15 K. While different choices of ninit havenegligible effect on convergence at 300 K, we highlightthat even an order of magnitude deviation in ninit re-sults in failed convergence at 15 K.

In Equation 6, Veq varies logarithmically with theintrinsic carrier density ni, which in turn has an expo-nential dependence on the temperature. The expres-sion for Veq can therefore be simplified to Equation 8 toyield the same physical equation, promoting smootherconvergence. We note that this modification is criticalwhen modeling the device below 25 K.

Veq = V0 − χ+kBT

q

(log

(neqγnNc

)). (8)

By following the above steps, we achieve conver-gence for temperatures down to 15 K, as illustrated inthe convergence plots in Figure 3d. We can potentiallyobtain convergence at lower temperatures with auxil-iary temperature sweeps, where the solution at eachtemperature step acts as an initial guess for the subse-quent step. However, since the electron density variessignificantly between subsequent steps, this techniquehas only aided us to further reduce the temperatureby ∼ 3 K.



Electron Density

(cm-3)

10-40

10-30

10-20

10-10

100

1010

1020

DonorSET Island

Donor Gate Top Gate

-60 -40 -20 0 20

-10

-20

-30

-40

-50

z (

nm

)

y (nm)

Electron Density

(cm-3)

1020

1010

100

10-10

10-20

SiO2

Donor Gate Top Gate

-60 -40 -20 0 20

-10

-20

-30

-40

-50

z (

nm

)

y (nm)

Conduction Band

Energy Ec (meV)

120

100

80

60

40

SiO2

20

0

Donor Gate Top Gate

-60 -40 -20 0 20

-10

-20

-30

-40

-50

z (

nm

)

y (nm)

Electric Field

|F| (MV/m)

7

6

5

4

3

2

1

SiO2

(a) (b)

(c) (d)

SET Island

SET Island SET Island

| ۧ↑

| ۧ↓

Ec – EF= 45.6meV

(EF)

100 nm

Fig. 4 (a) Electron density (n) in the device highlighting the single-electron-transistor (SET) island. (b) Electron densityalong an x-slice cut through the centre of the SET island. (c) Conduction band energy (Ec) variation illustrating thelocations (dashed line) where the donor electron has the appropriate energy for its spin to be read out. Inset : Spinreadout principle showing that the donor electron can preferentially tunnel to the SET island depending on its spin. (d)Magnitude of the electric field (|F|) along the x-slice. All the above electrostatic parameters have been estimated with theSemiconductor Module in COMSOL.

Simplifications of the Fermi integral in Equation 2,and truncating extremely low carrier densities to finitevalues, may also aid in convergence at lower tempera-tures. This has been illustrated in previous works [14]and with other software [15]. We are yet to quantify theaccuracy obtained with such techniques in COMSOL,and leave this for future work.

5 Device Electrostatics at 15K

We will now analyze the electrostatics in the deviceat 15 K. Figures 4a and 4b show the electron den-sity (n) from the top and along a slice of the de-vice respectively. The position of the single-electron-transistor (SET) island is evident in Figure 4a, wherethe barrier gates (biased at 0 V) completely diminishelectrons beneath them. We also note that n varies byover 60 orders of magnitude from the SET island to theregion under the donor and barrier gates. Such decaysin the electron densities can be expected from Equa-tion 2. However, at high electron density regions, weobserved that n was fairly constant at 1020 cm−3 with

temperature. This density is primarily determined bythe gate potential, band bending at the interface andscreening from existing electrons in these regions, andtemperature has a much smaller effect.

Recall that the nanostructure also includes phos-phorus donors implanted near the SET island. TheSET aids to readout the spin of the donor electronusing the following mechanism. The potential seen bythe donor electron, including the Coulomb potential ofits nucleus, is shown in the inset of Figure 4c. For spinreadout, the energy of the donor electron is alignedwith the Fermi Level of the SET island. In an exter-nally applied magnetic field B0, the Zeeman energysplitting between the spin states results in preferentialtunneling of the spin-up electron to the island, whichmodifies the current passing through the SET [10].

In a previous work for such devices, we computedthat the energy of the donor electron is 45.6 ± 1 meVbeneath the conduction band energy Ec [16]. We there-fore use Ec obtained from COMSOL to model thedonor electron energy to within an accuracy of ∼ 1meV. Figure 4c plots Ec with respect to the Fermilevel EF , along a slice of the device. The dashed line



(d)

(b)Donor Gate Top Gate

-60 -40 -20 0 20

-10

-20

-30

-40

-50

z (

nm

)

y (nm)

Ec(20K) -Ec(15K)

(meV)

0.4

0

-0.4

-0.8

-1.2

SiO2

SET Island

Donor Gate Top Gate

-60 -40 -20 0 20

-10

-20

-30

-40

-50

z (

nm

)

y (nm)

0.5

0.4

0.3

0.2

0.1

SiO2

SET Island

0

(a)Donor Gate Top Gate

-60 -40 -20 0 20

-10

-20

-30

-40

-50

z (

nm

)

y (nm)

Ec(300K) -Ec(15K)

(meV)

160

140

120

100

80

SiO2

SET Island

60

Donor Gate Top Gate

-60 -40 -20 0 20

-10

-20

-30

-40

-50

z (n

m)

y (nm)

|F(300K)| - |F(15K)|(MV/m)

10

8

6

4

2

SiO2SET Island

0

-2

(c)|F(20K)| - |F(15K)|

(MV/m)

Fig. 5 (a)-(b) Difference between the conduction band energy Ec obtained with COMSOL at (a) 300 K and 15 K, and(b) 20 K and 15 K. (c)-(d) Difference between the magnitude of electric field |F| estimated at (c) 300 K and 15 K, and(d) 20 K and 15 K. The dashed line corresponds to spin-readout locations extracted from Figure 4c.

in Figure 4c shows locations where Ec = 45.6 meV.For the gate voltages used in the model, this line cor-responds to positions where the donor electron has theappropriate energy (∼ EF ) for its spin to be read out.

Figure 4d shows the magnitude of the electric field|F| along a slice of the device. The electric fields ofseveral MV/m in Figure 4d are indeed large enough tosignificantly modify parameters for quantum comput-ing, such as hyperfine [17] and exchange couplings [18].Finally, we emphasize that the magnitude and varia-tion of the COMSOL electron densities, electric fieldsand conduction band energies are consistent with thosefrom Sentaurus TCAD [15] on similar devices [16].

6 Comparison with Higher Temperatures

Quantum computing experiments performed on suchdevices are at milli-Kelvin (mK) temperatures. Whilewe have only achieved convergence at temperatures farfrom the mK regime, comparison of results at highertemperatures can aid to quantify the accuracy of thesimulations with temperature.

In Figure 5a, we plot the difference between theconduction band energies Ec obtained at 300 K and

15 K, and note significant discrepancies of ∼ 100 meV.The magnitude of the electric fields |F| are also dif-ferent by several MV/m at the two temperatures, asindicated in Figure 5c. Hence, it is essential to simulatesuch devices at low enough temperatures.

Figures 5b and 5d plot the discrepancies of the con-duction band energy and electric field, between 20 Kand 15 K. The variation of both parameters with tem-perature are sensitive to the position. However, we em-phasize that Ec and |F| differ by at most 0.5 meV and0.1 MV/m respectively, at the spin readout locationsestimated previously from Figure 4c. With Ec being ∼45.6 meV and the donor experiencing electric fields ofseveral MV/m (Figure 4d), the spin readout locus willonly be slightly different at the two temperatures, bywell within a nm.

In Figure 6, we show the variation of Ec and |F|with temperature for different depths along the spinreadout locus. For a range of temperatures between 15K and 50 K, the plots indicate that the Ec and |F| willvary by less than 2 meV and 0.1 MV/m respectively,when the temperature is modified by 5 K. These wouldtranslate to inaccuracies of ∼ 1 nm in the spin readoutlocus, for temperature variations of 5 K.



15 20 25 30 35 40 45-0.02

0

0.02

0.04

0.06

0.08

0.10

Temperature T (K)

|F(T

+ 5

)|–

|F(T

)|(M

V/m

)

(b)15 20 25 30 35 40 45

-0.5

0

0.5

1.0

1.5

2.0

Temperature T (K)

Ec(T

+ 5

)–

Ec(T

)(m

eV

)

(a)

-15 -20-25 -30-35 -40

-45 -50

z(nm)

-15 -20-25 -30-35 -40

-45 -50

z(nm)

Fig. 6 Variation of (a) conduction band energy Ec, and (b)electric field |F| with temperature, obtained from COM-SOL simulations. We specifically consider positions alongthe spin readout locus obtained from Figure 4c.

7 Summary

Electronic devices operating at low temperatures arebecoming increasingly important in many modern ap-plications in science and computing. Modeling and sim-ulation of such devices at low temperatures is therebycritical for accurate device design and characteriza-tion. We have provided a guideline for simulating theelectrostatics at low temperature with the COMSOLSemiconductor module, using an example of a siliconquantum computing device. We have analyzed the de-vice electrostatics including the electron density, con-duction band energy, electric fields and donor loca-tions for spin readout at 15 K, and compared the re-sults with higher temperatures. Future work will con-centrate on obtaining convergence at lower tempera-tures (mK) using the additional techniques mentionedin Section 4, benchmarking COMSOL simulation re-sults with measurements obtained from experimentaldevices, and also extending modeling to estimate cur-rents and mobilities at low temperatures.

References

1. Gutierrez-D, E. A., Deen, J., and Claeys, C., Low tem-perature electronics: physics, devices, circuits, and ap-

plications, Academic Press, 2000.2. Balestra, F. and Ghibaudo, G., “Physics and perfor-

mance of nanoscale semiconductor devices at cryogenictemperatures,” Semiconductor Science and Technol-ogy , Vol. 32, No. 2, Jan 2017, pp. 023002.

3. Balestra, F. and Ghibaudo, G., Device and circuitcryogenic operation for low temperature electronics,Springer Science & Business Media, 2013.

4. Deaver, J., Deaver, B. S., and Ruvalds, J., Advances insuperconductivity , Springer Science & Business Media,2013.

5. Cabrera, B., Gutfreund, H., and Kresin, V. Z., FromHigh-Temperature Superconductivity to Microminia-ture Refrigeration, Springer Science & Business Media,2012.

6. Sze, S. M. and Ng, K. K., Physics of semiconductordevices, John wiley & sons, 2006.

7. “Semiconductor Module User’s Guide,” COMSOLMultiphysics® v. 5.2a, COMSOL AB, Stockholm,Sweden, 2016.

8. Humble, T. S., Ericson, M. N., Jakowski, J., Huang,J., Britton, C., Curtis, F. G., Dumitrescu, E. F., Mo-hiyaddin, F. A., and Sumpter, B. G., “A computationalworkflow for designing silicon donor qubits,” Nanotech-nology , Vol. 27, No. 42, Sep 2016, pp. 424002.

9. Angus, S., Ferguson, A., Dzurak, A., and Clark, R.,“Gate-Defined Quantum Dots in Intrinsic Silicon,”Nano Letters, Vol. 7, No. 7, Jun 2007, pp. 2051–2055.

10. Morello, A., Pla, J., Zwanenburg, F., Chan, K., Tan,K., Huebl, H., Möttönen, M., Nugroho, C., Yang, C.,van Donkelaar, J., et al., “Single-shot readout of anelectron spin in silicon,” Nature, Vol. 467, No. 7316,Oct 2010, pp. 687–691.

11. Klimeck, G., Ahmed, S. S., Bae, H., Kharche, N., Rah-man, R., Clark, S., Haley, B., Lee, S., Naumov, M.,Ryu, H., Saied, F., Prada, M., Korkusinski, M., andBoykin, T. B., “Atomistic simulation of realisticallysized nanodevices using NEMO 3-D - Part I: Mod-els and benchmarks,” IEEE Trans. Electron Devices,Vol. 54, No. 9, Sep 2007, pp. 2079–2089.

12. Klimeck, G., Ahmed, S. S., Kharche, N., Korkusinski,M., Usman, M., Prada, M., and Boykin, T. B., “Atom-istic simulation of realistically sized nanodevices usingNEMO 3-D - Part II: Applications,” IEEE Trans. Elec-tron Devices, Vol. 54, No. 9, Sep 2007, pp. 2090 – 2099.

13. Zwanenburg, F. A., Dzurak, A. S., Morello, A., Sim-mons, M. Y., Hollenberg, L. C. L., Klimeck, G., Rogge,S., Coppersmith, S. N., and Eriksson, M. A., “Sili-con quantum electronics,” Reviews of Modern Physics,Vol. 85, No. 3, Jul 2013, pp. 961–1019.

14. Nielsen, E., Gao, X., Kalashnikova, I., Muller, R. P.,Salinger, A. G., and Young, R. W., “QCAD simula-tion and optimization of semiconductor double quan-tum dots,” Tech. rep., Sandia National Laboratories,2013.

15. “Sentaurus Device User Guide,” Synopsys ® v. J-2014.09, Synopsys Inc., Mountain View, California,United States of America, 2014.

16. Mohiyaddin, F. A., Rahman, R., Kalra, R., Klimeck,G., Hollenberg, L. C. L., Pla, J. J., Dzurak, A. S., andMorello, A., “Noninvasive Spatial Metrology of Single-Atom Devices,” Nano Lett., Vol. 13, No. 5, May 2013,pp. 1903–1909.

17. Rahman, R., Wellard, C., Bradbury, F., Prada, M.,Cole, J., Klimeck, G., and Hollenberg, L., “High Pre-cision Quantum Control of Single Donor Spins in Sili-con,” Phys. Rev. Lett., Vol. 99, Jul 2007, pp. 036403.

18. Wellard, C. J., Hollenberg, L. C. L., Parisoli, F., Kettle,L. M., Goan, H.-S., McIntosh, J. A. L., and Jamieson,D. N., “Electron exchange coupling for single-donorsolid-state spin qubits,” Phys. Rev. B , Vol. 68, Nov2003, pp. 195209.



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Simulation of Silicon Nanodevices at Cryogenic …...device as an input to a quantum-mechanical solver [11, 12], which in turn calculates precise donor energies and other parameters