富岳成果創出加速プログラム

省エネルギー次世代半導体デバイス開発のための量子論マルチシミュレーション

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富岳物質材料系合同研究会 Dec 2021

Computics Approach in Science of Power Electronics

Why Power Electronics?
– Semiconductor Technology to Sustain Our Planet– Need Science based on Quantum Theory in Nanoworld
An example in science and technology of power electronics– Leakage current in GaN pn junction and dislocation-acceptor complex
Nakano et al., APL 117, 012105 (2020)

An example in computics:– Neural-network assisted order-N orbital-free DFT
Imoto, Imada, Oshiyama: Phys Rev Research 3, 033198 (2021)

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富岳物質材料系合同研究会 Dec 2021

Widegap Semiconductors such as GaN or SiC Save Energies

SiC Limit

3

GaN Limit
high robustness

Si Limitsave energy

Widegap
Semiconductors
SiSiCGaN

富岳物質材料系合同研究会 Dec 2021

Superior Physical Properties are not All: Prescription of Fabrication Required

SiC-device already in the market
in
SiC power module

Shinkansen

Tokyo Metro

Automobile

However, the device performance is much lower than expected
Fabrication Processes of the Devices should be Optimized

VerticalMOSFET

4

epitaxial films

device interfaces

SiO2

Nanowire
MOSFET

channelgate insulator

gate insulator

富岳物質材料系合同研究会 Dec 2021

Screw Dislocations Suspected for the Leakage in pn Junctions of GaN
Usami et al., APL 112, 182106 (2018); 114, 232105 (2019)

Dislocation thatcauses theleakage!!

Dislocation that
does notLeakage Positions byLight Emission
Microscope
under -550V

Forward and Reverse I-V Characteristics

ReverseForward5

Dislocation
positions:etch-pit image

LACBED determines the Burgers vector b: It is 𝒃𝒃 = 0001 i.e., Screw Dislocation

富岳物質材料系合同研究会 Dec 2021

Structural Identification of Screw Dislocation
Possible 16 types of dislocation cores(view from [0001])

Double(n|m) vs Single(n|m)

b=[0001]

GaN

dislocation line Position of the dislocation line?
Center of the hexagon (single)
orCenter of two hexagons (double)

GaN

 Empty or filled with Ga and/or N?
How many atoms(n|m): # of Ga and N atoms, respectively, removed from full-core structure6

富岳物質材料系合同研究会 Dec 2021

Energetics of the Dislocation Cores
Temperature T and Pressure P in experiment→ Chemical potential µ
Typical growth condition( µGa + µ N =ε GaN )Dislocation cores with more Ga are stable!
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Structural characteristics:Ga-Ga bond
富岳物質材料系合同研究会 Dec 2021

Pair of Screw Dislocation and Mg AcceptorMg@ Ga site aroundthe D(0|2) Screw Dislocation

Binding Energy (eV)Binding Energy of the dislocation and Mg

X: dislocation line: Mg

Screw Dislocation Attracts the Acceptor Mg!
8富岳物質材料系合同研究会 Dec 2021

Screw Dislocation that Converts p-type to n-type: Mg-Dislocation Complex

ε(+/0)

Cylindrical n region

𝐸𝐸𝑓𝑓

Upper shift
In Mg-dislc

Density of States
D(0|2)+Mg
9Hybridization between the Dislocation (Ga-Ga) State and Mg
State→ Acceptor Level disappears and Donor Level Emerges→ pn junction collapses → cylindrical n regions → Leakage
富岳物質材料系合同研究会 Dec 2021

大規模計算が欲しい！汎関数がわかればいい！
Density ρ (r) Functional Theory → Kohn-Sham Schemeρ (r) = ∑ ϕi (r )Hohenberg, Kohn: PR 136, B864 (1964)
Kohn, Sham: PR 140, A1133 (1965)

2i

Original Single Euler equation in DFT becomes N-Kohn-Sham equations
which require O(N3) computation cost.

Orbital-Free Density-Functional (OFDFT) Scheme=E[ ρ ]

∫vnucl(r ) ρ (r) dr + Ts [ ρ ] + ∫ ∫ρ (r ) ρ (r ′)r − r′drdr ′ + E XC [ ρ ]Euler equation

δ Ts [ ρ ]δ EH [ ρ ] δ EXC [ ρ ]
TF
T[ρ]
=
τ(r ) F [ ρ ] dr
µ with s+ vnucl (r ) ++=∫δρ (r )δρ (r )δρ (r )

We need to play with Single Euler equation which require

Enhancement factor

only O(N) computation cost!
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富岳物質材料系合同研究会 Dec 2021

Neural-Network Assisted Kinetic Energy Density Functional

TsNN [n] = ∫ τ TF (r ) F NeuralNetwork (r ; n) dr
何を学ばせるか
1. Ts[n]の値？

δ Ts [n]δ EH [n] δ EXC [n]+ vnucl (r ) ++=µδ n( r )δ n( r )δ n( r )2. その汎関数微分？
3. あるいは？

我々はEuler方程式を解かねばならない。そこに現れるのはエネルギーの汎関数微分…. そしてエネルギーの一次微分が構造的性質などの物理量を決めている→ Kinetic Energy Density Functional （KEDF）の一次汎関数微分(Kinetic Energy FunctionalDerivbative: KEFD)を学ばせよう。具体的には 実空間上の各点でのその値を学ばせ、そして、
Nt

1  δ TsNN [n]1
コスト関数を最小化=L∑ Nt m=1 2 11δn(rm ) −δ TsKS [n]δn(rm ) 2

富岳物質材料系合同研究会 Dec 2021

Ts [n] = ∫ τTF(r ) FNN(r ; n) dr
δ TsKS [n]11  δ TsNN [n]=−L(r)(r)
Neural-Network Training∑ mm 
Nt m=1 2  δ nδnNt2σ ( x)

Output from j-th neuron in l-th layer:xz (jl ) = σ (l ) (a (jl ) )l −1
Input a j is the output(l )a j = ∑ W jk(l ) zk(l −1)
from the previous layer:
k =0

D(l )

Descriptors

z1(0) = s = ∇n  2(3π 2 )1 3 n 4 3 Outputz2(0) = q = ∇ 2 n  4(3π 2 ) 2 3 n5 3 

DN −1
DN −2(N )NN 2(N )( N −1)( N − 2)=z1
F=
( s , q ) ∑ W1i σ  ∑ Wijσ  ∑ W jk σ ( ⋅⋅⋅)  
j 0 =(In the present computation, # hidden layers=3,=i 0=
k0



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 L12

富岳物質材料系合同研究会 Dec 2021

Transferrable Kinetic Energy Functional Trained only by some of
diamond data and Evidenced by 24 Distinct Systems
11 Solids: diamond, graphene, ds-Si, fcc-Si, β-tin Si, 3C-SiC, bcc-Li, fcc-Al, fcc-Cu, bcc-Na, NaCl
6 diatomic molecules: Li2, C2, Na2, Al2, SI2, Cl2
7 atoms: Li, C, Na, Al, Si, Cl, Cu
Fcc-CuElectronDensity
ds-Si

Overall,In unit ofbohr-3 x 10-213

富岳物質材料系合同研究会 Dec 2021

Structural Propertiesof 11 solids

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Order N - OFDFT Computations
Imoto, Imada, Oshiyama: Phys Rev Research 3, 033198 (2021)

Computational time measured for
SiC 576～4704-atom systems

In a single SCF iteration,
t1
subspace diagonalization+ conjugate-gradient minimization+ Gram-Schmidt orthonormalizationt2
density update and total energy
t3
mixing procedures
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富岳物質材料系合同研究会 Dec 2021