Computics Approach to Development of the Next-Generation
Semiconductor ScienceImportance of energy-saving semiconductor technology in sustainablesociety– Power electronics, Non-volatile memories– Necessity of quantum-theory based calculations

Computics: present capability of calculations based on density-functional
theory (DFT)– On current massively parallel computer architecture not with accelerators yet

Mechanism of the flash memory: Nitrogen vacancy (VN) in Silicon Nitride– Floating state distributed in internal space in the host sparse material– Multiple charge states which store multi-electrons around VN[F. Nanataki, K. Shiraishi & A. Oshiyama: Phys. Rev. B (2022)]
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30th Anniversary Symposium of CCS

Life and Electron Devices
PCs

Networking

CloudVR Gaming
https://weekly.ascii.jp/elem/000/002/629/2629133/
Industrial IoT

Automotive

MobileElectron Device
Assembly ofCPU, Memory,
NAND Flash ….
The material is Si

2However, for energy savingsociety, HEAT is the problem

30th Anniversary Symposium of CCS

Wide Bandgap Semiconductors such as GaN or SiC Save Energies
GaN
Tokyo MetroSiSiC

Power Device Performance

Automobile

However, the device performance is much
lower than expected
GaN Limit

Fabrication Processes should be Optimized
high robustness

Si Limit

SiC Limit
3Shinkansen

save energy

gate insulator
VerticalMOSFET

epitaxial films

device interfaces
30th Anniversary Symposium of CCS

Next-Generation Electron Devices by Replacing Si with Emerging Materials

Non-Volatile Flash Memory
Memory Cell= Charge Storing SpaceSi Floating GateSiN• Requirement of Integration• Multi-Level Cell

What is the memory cell in SiN?
What is the localized electron state there?
Physics is unknown.
NEED: Accurate Large-Scale Calculations Based on Quantum Theory for
10,000 – 100,000-Atom Systems
430th Anniversary Symposium of CCS

Equations in Density-Functional Theory (DFT)
Hohenberg, Kohn: PR 136, B864 (1964)

1𝑛𝑛(𝑟𝑟)𝑛𝑛(𝑟𝑟 ′ )𝐸𝐸[𝑛𝑛] = �𝑣𝑣nucl (𝑟𝑟)𝑛𝑛(𝑟𝑟)𝑑𝑑𝑑𝑑 + 𝑇𝑇[𝑛𝑛] + �
𝑑𝑑𝑑𝑑𝑑𝑑𝑟𝑟 ′ + 𝐸𝐸𝑋𝑋𝑋𝑋 [𝑛𝑛]′𝑟𝑟 − 𝑟𝑟
2- Total Energy of a material:

is a functional of the electron density,n(r=) N ∫ | Ψ (r , r2 , r3 , r4 , ⋅⋅⋅rN ) |2 dr2 dr3 ⋅⋅⋅ drN- Euler equation:

δ E[n]=µδnKohn, Sham: PR 140, A1133 (1965)

Kohn-Shamequation

 1 2

−∇+
rrvn(;())
eff
 2
 ϕi (r ) =ε i ϕi (r )
δ EXC [n]n(r′)′veff (r ) =vnucl (r ) + ∫dr +| r − r′ |
δ n(r )

universal functionallarge-scale
to nano and micro
world

accurate enoughbetter EXC

long-time simulation

Car, Parrinello: PRL 55, 2472 (1985)

Car-Parrinello
Molecular
Dynamics
equation
5µϕi =−δ E / δϕi∗ + ∑ Λ ijϕ jj

 = −∂E / ∂RMIRIDFTI

30th Anniversary Symposium of CCS

Need Cross-Disciplinary Approach (Computics)
between Physics/Chemistry and Computer Science & Technology
Theoretical Performance ofTOP1 Supercomputer

http://computics-material.jp/index.html

Ｏur real-space scheme: RSDFT code

https://github.com/j-iwata/RSDFTAll the quantities such as n(r), v(r) etc computed at eachreal-space mesh points, leading to that Fourier Transform isessentially unnecessary.

Currently,

 100,000-atom DFT calculations for Si-Nanowire FET (2011
Gordon-Bell Prize)
massively parallel architecture 10,000-atom DFT calculations with semilocal approximations (LDA/GGA) to EXC feasible 1,000-5,000-atom calculations with hybrid approximation
Computics:
also feasible
Collaboration between Physics/Chemistry & CPMD simulations for 1000 – 3000 atom systems up toComputer Sciencenano seconds feasible
methodology and algorithms suitableto current architecture

630th Anniversary Symposium of CCS

A Recent Example: Epitaxial Growth of Power Semiconductor GaN
Bui, Boero, Shiraishi, Oshiyama: J. Phys. Chem C (2017); J. Cryst Growth (2019); JJAP (2020); Appl Surf Sci. (2021, 2022)

DFT calculations has revealed:
Gas Source NH3 decomposed on the growth surface
Resulting NH diffusing to the surface step-edges

Car-Parrinello Molecular Dynamics
CPMD @ 1300 KGa adatom
GaNHs at the edge incorporated to the GaN film

Subsurface
NHowever, real growth surface seems to be aliquid
at growth temperature 1300 K.
7Ga adatom diffuses andforms a dimer
30th Anniversary Symposium of CCS

2-Dimensional Liquid Phase of Ga Adatom at Growth Temperature
Diffusion Coefficient D of Ga adatom
on GaN(0001)
1 t →∞ 1 Nv I (t ) v I (0) dt
D= ∫∑03
N I =18 Arrhenius behavior with 0.28 eV barrier
at low temperature Sudden increase in D at 1300 K by an order

Pair correlation function of Ga adatomon GaN(0001)

 At low temperature, peak at 6 Å, characteristic
to the adsorption structure At 1300K, peek at 2.6 Å, characteristic
to Ga-Ga bond30th Anniversary Symposium of CCS

Further Efforts to Larger-Scale Calculations: Order N Scheme Current scheme of DFT (Kohn-Sham Scheme) scales as cube of the target size N, O(N3) But solving Euler equation in DFT costs O(N) in itself, if we get rid of the orbitals {ϕi (r )}
 1 2

(;())
−∇+rrvneff
 2
 ϕi (r ) =ε i ϕi (r )i=1～O(N)δ EXC [n]n(r′)
veff (r ) =vnucl (r ) + ∫dr ′ +| r − r′ |
δ n(r )

δ Ts [n]δ EH [n] δ EXC [n]µ+ vnucl (r ) ++=δ n( r )δ n(r)
δ n( r ) {ϕi (r )} is unnecessary if we know the functional form of the kinetic energy

Ts [n] = ∫ τ TF (r ) F [n] drEnhancement factor

Orbital-Free DFT (OFDFT)

 We rely on the Neural Network to develop Ts [n]Imoto, Imada, Oshiyama:
Phys Rev Research 3, 033198 (2021)
930th Anniversary Symposium of CCS

Ts [n] = ∫ τ

TF(r ) FNN(r ; n) drNt


δ TsKS [n]11  δ TsNN [n]=−()()
Lrr
Neural-Network Training∑ mm 
Nt m=1 2  δ nδnσ ( x)

Output from j-th neuron in l-th layer:z (jl ) = σ (l ) (a (jl ) )2xD(l )

l −1
Input a j is the output(l )a j = ∑ W jk(l ) zk(l −1)
from the previous layer:
k =0

Descriptorsz1(0) = s = ∇n  2(3π 2 )1 3 n 4 3 Output

z2(0) = q = ∇ 2 n  4(3π 2 ) 2 3 n5 3 

DN −1
DN −2

NN 2(N )(N )( N −1)( N − 2)z1
F=
( s , q ) ∑ W1i σ  ∑ Wijσ  ∑ W jk σ ( ⋅⋅⋅)  =j 0 =(In the present computation, # hidden layers=3,k0=i 0=DN# neurons per layer =3)

Prepare n( r ) , s ( r ) and q (r ) from the training data (in carbon diamond).
(l )Construct F as functionals of n, q and s with the NN parameters Wij .
Train the NN (optimize NN parameters Wij(l ) ) so as to minimize the cost function L
1030th Anniversary Symposium of CCS

Structural Propertiesof 11 solids

1130th Anniversary Symposium of CCS

Neural-Network-Assisted Order N - OFDFT Scheme
Imoto, Imada, Oshiyama: Phys Rev Research 3, 033198 (2021)

Computational time measured for
SiC 576～4704-atom systems

DFT / CPMD calculations formillions of atoms are feasible
with moderate size ofcomputational resources

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30th Anniversary Symposium of CCS

Computics Approach to Development of the Next-Generation
Semiconductor ScienceImportance of energy-saving semiconductor technology in sustainablesociety– Power electronics, Non-volatile memories– Necessity of quantum-theory based calculations

Computics: present capability of calculations based on density-functional
theory (DFT)– On current massively parallel computer architecture not with accelerators yet

Mechanism of the flash memory: Nitrogen vacancy (VN) in Silicon Nitride– Floating state distributed in internal space in the host sparse material– Multiple charge states which store multi-electrons around VN[F. Nanataki, K. Shiraishi & A. Oshiyama: unpublished (2022)]
1330th Anniversary Symposium of CCSElectron States in Condensed Matters
Extended Statese.g. Bloch states in CrystalvsLocalized Statese.g. States induced by imperfection
dangling bond state

Trapping carriers inthe localized region
Vacancy V in xtal

Deterioration ofelectron devices

Grapheneplane

However, controlled electron trapping isstoring the information in the localized region:i.e. the memory
V in SiN

Interlayer stateorN

Floating state in sparse materiall

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30th Anniversary Symposium of CCS

Floating State and Band-gap Variation in SiC Polytypes
 Polytypes: Sequence of bilayers (A, B & C) have a variety as 2H, 3C, 4H, 6H, 8H, 10H, 15R,,,
3C structure from[101]structurefrom [2201][periodicity6Hn andsymmetry
Hexagonal/Cubic/Rombohedral]
4H structure
3C structure

Length = ∞2H structure

Length = 7 a0 / 2 2

 Band Gaps: 2.2 eV (3C), 2.86 eV (6H), 3.26 eV (4H),,, 40% difference!
Bonds and Bands: the bonding sp3 = the valence band, the anti-bonding sp3 = the conduction band Conduction Band State is FLOATING!

15Amplitude of the orbital

GGA Band Gap (eV)

Matsushita, Oshiyama: Phys. Rev. Lett (2012); ibid., (2014); Nano Lett (2017); JPhysSocJpn (2017)

Quantum Confinementin the channel

30th Anniversary Symposium of CCS

Floating State in Silicon Nitride
Consider β-Si3N4: packing efficiency =0.28

memorycell

Gateoxide

SiN

SiN
oxide
Sisubstrate

Conduction-band Stateis FLOATING

Then, is distributed in the internal space

Highest amplitude
is at back of Si,not along bonds

View from [0001] (c axis)
16

Side view perp to [0001]
30th Anniversary Symposium of CCS

Nitrogen Vacancy VN in Silicon Nitride
Chemically, VN induces 3 Si dangling bonds.
They are converted to a singlet a1
and a doublet e under C3v symmetry.

Level Structure near the gap for neutral VN
 e1 and e2: different
nodal planes
 Symmetry-lowering
Jahn-Teller splitting⇦ Covalencye2 Exchange spin splitting⇦ Ionicity, depending on its charge state
17e1JTupspin

EXby DFT:
HybridApprox

downspin
30th Anniversary Symposium of CCS

Jahn-Teller and Exchange Splitting ,,,,and Localized Floating Character(Kohn-Sham) Electronic Levels: from VN+1 to VN-5

e1 and e2

JTBut That’s NOT All
 Floating states hidden in the conduction
bands emerge in the gap (red bars),
being localization around VN

EXVN0

from 2 directions

Covalency
vsIonicity
18

vacant site

VN-5

30th Anniversary Symposium of CCS

Thermodynamic Level of VN in SiN

Thermodynamic level ε (q/q’): Fermi-energy position in the gap at which
the different charge states q and q’ has the same total energy

 This is the level experimentallydetected.
 Control of the Fermi-level position,
presumably by device design⇒ VN : multi-level memory cell ?

ε (0/-)
Further EI/E

ε (+/0)electronInjection/emission
Valence band
maximum19

ε (-/-3)Further EI/E

Covalency (Jahn-Teller)
Ionicity (Spin splitting) andSparsity (Floating)make it!

Conduction band
minimum
30th Anniversary Symposium of CCS

In Summary,Importance of Computics Approach based on Quantum Theory for
Accurate-Enough Large-Scale Long-Time First-Principles Calculations /Simulations, Contributing to the Development of Energy-Saving ElectronDevices
Ultimate Memory Cells Endorsed by Newly Discovered Floating States?
Lattice Vacancy in Silicon Nitride

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30th Anniversary Symposium of CCS