physics-based device simulation - tu dresden · 2015-12-16 · pros & cons 4 i realistic device...

Physics-based Device Simulation

C. Jungemann

Institut für Theoretische ElektrotechnikRWTH Aachen University

52056 Aachen

2OutlineI MotivationI Pros & ConsI Semiclassical transportI ResultsI Conclusions

3MotivationSimulation hierarchy

I Quantum transportI Semiclassical transportI TCADI Compact models

plexity

Why semiclassical transport simulation (Boltzmann equation)?

I Complex band structuresI Detailed scattering modelsI Only material-dependent parametersI DC, AC and noise analysisI No adjustable parameters in the RF noise model

plexity

4Pros & ConsI Realistic device structureI Microscopic, accurate physics (Boltzmann equation)I Detailed physicsI Consistent modeling of transport and noiseI Numerically challengingI Very slow (a CPU day per bias point for a 2D device)I Limited device detail (2D, only few parasitics)I Only as good as the models

Semiclassical transport

6Semiclassical transportPath of an electron in a semiconductor

0 100 200 300 400

X [nm]

Newton’s equations of motion

= ~F ,d~rdt

Two first order ODEsTwo initial conditions (position, momentum)

0 100 200 300 400

X [nm]

= ~F ,d~rdt

0 100 200 300 400

X [nm]

= ~F ,d~rdt

0 100 200 300 400

X [nm]

= ~F ,d~rdt

0 100 200 300 400

X [nm]

= ~F ,d~rdt

7Semiclassical TransportParticle ensemble described by a distribution function (~p = ~~k )

f (~r , ~k , t)

Number of particles in a volume of phase space

(2π)3 f (~r , ~k , t)d3rd3k

Equilibrium (Boltzmann distribution)

f0 = c exp

(−ε(~r , ~k)

f (~r , ~k , t)

(2π)3 f (~r , ~k , t)d3rd3k

f0 = c exp

(−ε(~r , ~k)

f (~r , ~k , t)

(2π)3 f (~r , ~k , t)d3rd3k

f0 = c exp

(−ε(~r , ~k)

8Semiclassical TransportBoltzmann Transport Equation

∂f(~r , ~k , t

− q~~E(~r) · ∇~k f

(~r , ~k , t

(~k)· ∇~r f

(~r , ~k , t

(2π)3

∫W(~r , ~k |~k ′

)f(~r , ~k ′, t

(~r , ~k ′|~k

)f(~r , ~k , t

)d3k ′

Integro-differential equation

Direct discretization of the 6D phase space is not possible

Spherical Harmonics Expansion solver (SHE) for the BTE(Bologna group, Maryland group et al.)

∂f(~r , ~k , t

− q~~E(~r) · ∇~k f

(~r , ~k , t

(~k)· ∇~r f

(~r , ~k , t

(2π)3

∫W(~r , ~k |~k ′

)f(~r , ~k ′, t

(~r , ~k ′|~k

)f(~r , ~k , t

)d3k ′

∂f(~r , ~k , t

− q~~E(~r) · ∇~k f

(~r , ~k , t

(~k)· ∇~r f

(~r , ~k , t

(2π)3

∫W(~r , ~k |~k ′

)f(~r , ~k ′, t

(~r , ~k ′|~k

)f(~r , ~k , t

)d3k ′

∂f(~r , ~k , t

− q~~E(~r) · ∇~k f

(~r , ~k , t

(~k)· ∇~r f

(~r , ~k , t

(2π)3

∫W(~r , ~k |~k ′

)f(~r , ~k ′, t

(~r , ~k ′|~k

)f(~r , ~k , t

)d3k ′

9Spherical Harmonics ExpansionSpherical coordinates for k -space

(kx , ky , kz) → (k , ϑ, ϕ) → (ε, ϑ, ϕ)

Dependence on angles is expanded with spherical harmonics Yl,m(ϑ, ϕ)

Y0,0 =1√4π

Y1,−1 =

4πsinϑ sinϕ

Y1,0 =

4πcosϑ

Y1,1 =

4πsinϑ cosϕ

∮Yl,m(ϑ, ϕ)Yl ′,m′(ϑ, ϕ)dΩ = δl,l ′δm,m′ , dΩ = sinϑdϑdϕ

(kx , ky , kz) → (k , ϑ, ϕ) → (ε, ϑ, ϕ)

Y0,0 =1√4π

Y1,−1 =

4πsinϑ sinϕ

Y1,0 =

4πcosϑ

Y1,1 =

4πsinϑ cosϕ

(kx , ky , kz) → (k , ϑ, ϕ) → (ε, ϑ, ϕ)

Y0,0 =1√4π

Y1,−1 =

4πsinϑ sinϕ

Y1,0 =

4πcosϑ

Y1,1 =

4πsinϑ cosϕ

(kx , ky , kz) → (k , ϑ, ϕ) → (ε, ϑ, ϕ)

Y0,0 =1√4π

Y1,−1 =

4πsinϑ sinϕ

Y1,0 =

4πcosϑ

Y1,1 =

4πsinϑ cosϕ

(kx , ky , kz) → (k , ϑ, ϕ) → (ε, ϑ, ϕ)

Y0,0 =1√4π

Y1,−1 =

4πsinϑ sinϕ

Y1,0 =

4πcosϑ

Y1,1 =

4πsinϑ cosϕ

10Semiclassical transportSpherical harmonics expansion of the distribution function

gl,m(~r , ε, t) =1

(2π)3

∫δ(ε− ε(~k)

)Yl,m(ϑ, ϕ)f (~r , ~k , t)d3k

g(~r , ~k , t) = g(~r , ε, ϑ, ϕ, t) ≈lmax∑l=0

m=l∑m=−l

gl,m(~r , ε, t)Yl,m(ϑ, ϕ)

Electron density

n(~r , t) =2

(2π)3

∫f (~r , ~k , t)d3k = 2

√4π∫ ∞

0g0,0(~r , ε, t)dε

Electron current density (spherical bands)

~j(~r , t) = 2

√4π3

∫ ∞0

g1,1(~r , ε, t)g1,−1(~r , ε, t)g1,0(~r , ε, t)

gl,m(~r , ε, t) =1

(2π)3

∫δ(ε− ε(~k)

)Yl,m(ϑ, ϕ)f (~r , ~k , t)d3k

m=l∑m=−l

Electron density

n(~r , t) =2

(2π)3

∫f (~r , ~k , t)d3k = 2

√4π∫ ∞

0g0,0(~r , ε, t)dε

~j(~r , t) = 2

√4π3

∫ ∞0

g1,1(~r , ε, t)g1,−1(~r , ε, t)g1,0(~r , ε, t)

gl,m(~r , ε, t) =1

(2π)3

∫δ(ε− ε(~k)

)Yl,m(ϑ, ϕ)f (~r , ~k , t)d3k

m=l∑m=−l

Electron density

n(~r , t) =2

(2π)3

∫f (~r , ~k , t)d3k = 2

√4π∫ ∞

0g0,0(~r , ε, t)dε

~j(~r , t) = 2

√4π3

∫ ∞0

g1,1(~r , ε, t)g1,−1(~r , ε, t)g1,0(~r , ε, t)

gl,m(~r , ε, t) =1

(2π)3

∫δ(ε− ε(~k)

)Yl,m(ϑ, ϕ)f (~r , ~k , t)d3k

m=l∑m=−l

Electron density

n(~r , t) =2

(2π)3

∫f (~r , ~k , t)d3k = 2

√4π∫ ∞

0g0,0(~r , ε, t)dε

~j(~r , t) = 2

√4π3

∫ ∞0

g1,1(~r , ε, t)g1,−1(~r , ε, t)g1,0(~r , ε, t)

11Semiclassical transportSpherical harmonics expansion of the Boltzmann equation

1(2π)3

∫δ(ε− ε(~k)

)Yl,m(ϑ, ϕ) BEd3k

yields balance equations over energy/real space

∂gl,m

∂t− q~E ·

∂~jl,m∂ε

+∇~r ·~jl,m − Γl,m = Wl,mg

~jl,m(~r , ε, t) = v(ε)∑l ′,m′

~al,m,l ′,m′gl ′.m′(~r , ε, t)

The balance equations are coupled

1(2π)3

∫δ(ε− ε(~k)

)Yl,m(ϑ, ϕ) BEd3k

∂gl,m

∂t− q~E ·

∂~jl,m∂ε

~jl,m(~r , ε, t) = v(ε)∑l ′,m′

~al,m,l ′,m′gl ′.m′(~r , ε, t)

1(2π)3

∫δ(ε− ε(~k)

)Yl,m(ϑ, ϕ) BEd3k

∂gl,m

∂t− q~E ·

∂~jl,m∂ε

~jl,m(~r , ε, t) = v(ε)∑l ′,m′

~al,m,l ′,m′gl ′.m′(~r , ε, t)

1(2π)3

∫δ(ε− ε(~k)

)Yl,m(ϑ, ϕ) BEd3k

∂gl,m

∂t− q~E ·

∂~jl,m∂ε

~jl,m(~r , ε, t) = v(ε)∑l ′,m′

~al,m,l ′,m′gl ′.m′(~r , ε, t)

12Semiclassical transportH-transform

H(~r , ~k

(~k)− qΨ

(~r)⇒ g′(~r ,H, t) = g(~r , ε, t)

Cancels the energy derivative for the stationary state

HHHHHH

−q~E ·∂~j ′l,m∂ε

+∇~r · ~j ′l,m − Γ′l,m = Wl,mg′

Captures ballistic transport

H(~r , ~k

(~k)− qΨ

(~r)⇒ g′(~r ,H, t) = g(~r , ε, t)

HHHHHH

−q~E ·∂~j ′l,m∂ε

+∇~r · ~j ′l,m − Γ′l,m = Wl,mg′

H(~r , ~k

(~k)− qΨ

(~r)⇒ g′(~r ,H, t) = g(~r , ε, t)

HHHHHH

−q~E ·∂~j ′l,m∂ε

+∇~r · ~j ′l,m − Γ′l,m = Wl,mg′

H(~r , ~k

(~k)− qΨ

(~r)⇒ g′(~r ,H, t) = g(~r , ε, t)

HHHHHH

−q~E ·∂~j ′l,m∂ε

+∇~r · ~j ′l,m − Γ′l,m = Wl,mg′

13Semiclassical transportNumerics

I H-transformI Maximum Entropy Dissipation SchemeI Dimensional splittingI Box Integration (exact particle number conservation)I First order expansion has property MI Newton-Raphson method to solve BE and PE self-consistently

14Semiclassical transport

Complex band structure effects

First conduction band of Si Density of states of Si

0 1 2 3 4 5 6

3·1028

Energy [eV]

Calculated by the nonlocal empirical pseudopotential method

Only DOS and v2 are required for SHE

Complex band structure effects

First conduction band of Si Density of states of Si

0 1 2 3 4 5 6

3·1028

Energy [eV]

Calculated by the nonlocal empirical pseudopotential method

Only DOS and v2 are required for SHE

15Semiclassical transportElectron-phonon scattering rate of silicon at 300K

0 1 2 3 4 5 6

3·1014

Energy [eV]

tterin

te[1/s

All parameters determined by theory or basic experiments

16Semiclassical transportElectron mobility

0 100 200 300 400 500

Temperature [K]

BECanali

Electron velocity at 300Kin < 111 > direction

10−1 100 101 102105

Electric field [kV/cm]D

BECanali

Good agreement between simulation and experiment for silicon

Impact ionization

Ionization coefficient

2 4 6 8 10100

Inverse electric field [cm/kV]

BEExp.

Quantum yield

1 1.5 2 2.5 3

10−8

10−6

10−4

10−2

Energy [eV]

DiMaria

Good agreement between simulation and experiment for silicon

Results

Bipolar transistor

20Bipolar transistor

Doping profile Collector current

SHE can handle high doping concentrations without problems

SHE can handle small currents without problems

Doping profile Collector current

SHE can handle high doping concentrations without problems

SHE can handle small currents without problems

VBE = 0.55V , VCE = 0.5VEnergy distribution function

VCE = 0.5VElectron density

SHE can handle huge variations in the density without problems

Dependence on the maximum order of SHEVBE = 0.85V , VCE = 0.5V

Error in current Electron velocity

Transport in nanometric devices requires at least 5th order SHE

N+NN+ structure

24N+NN+ structure

25N+NN+ structure

26N+NN+ structure

27N+NN+ structure

28N+NN+ structure

-0.30 -0.20 -0.10 0.000 0.10 0.20 0.30

Silicon

Bottom oxide

Top oxide

Partially depleted SOI NMOSFET

30PDSOI NMOSFET

VSource

VDrain

Impactionization

n−MOSFET

L Γ X

[111] [100]

Every impact ionization event generates a secondary electron/hole pair

31PDSOI NMOSFET

0 0.2 0.4 0.6 0.8 10

Drain voltage [V]

with IIw/o II

Kink effect due to impact ionization (II) (Vgate = 1.0V )

Exact stationary solutions

31PDSOI NMOSFET

0 0.2 0.4 0.6 0.8 10

Drain voltage [V]

with IIw/o II

Kink effect due to impact ionization (II) (Vgate = 1.0V )

Exact stationary solutions

32PDSOI NMOSFET

0 0.2 0.4 0.6 0.8 10

Drain voltage [V]

]ID with IIID w/o II

10−16

10−15

10−14

10−13

10−12

10−11

10−10

II-curr.

About 17 orders of magnitude difference in currents at kink

pn Junction

34pn Junction

Simulation of avalanche breakdown

35 36 37 38 39

10−17

10−14

10−11

10−8

10−5

Reverse bias [V]

35 36 37 38 39

·10−5

Reverse bias [V]

Extremely steep increase of current with voltage (20µV/dec)

34pn Junction

Simulation of avalanche breakdown

35 36 37 38 39

10−17

10−14

10−11

10−8

10−5

Reverse bias [V]

35 36 37 38 39

·10−5

Reverse bias [V]

Extremely steep increase of current with voltage (20µV/dec)

35pn JunctionTransient DD simulation of switching a silicon pn-diode from 0 to -39V

100 101 102 10310−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

Time [ps]

Avalanche breakdown is slow!

35pn JunctionTransient DD simulation of switching a silicon pn-diode from 0 to -39V

100 101 102 10310−10

10−9

10−8

10−7

10−6

10−5

10−4

10−3

Time [ps]

Avalanche breakdown is slow!

36pn Junction

Simulation for 37.172V and 10pA/µm2

]0 1 2 3 4 510−14

10−5

Position [µm]

elec.holesPot.

0 1 2 3 4 5101

Position [µm]

[cm−

NANDelec.hole

II generation increases number of particles in the space charge region

37pn Junction

Increase of current by a factor of 1000 (dashed lines)

0 1 2 3 4 5101

Position [µm]

[cm−

0 1 2 3 4 510−16

10−7

Position [µm]

[µm−

1 C−

1 ]Particle density in the space charge region increases 1000 times

Relevant II generation rate scales with current

Hot electron/hole effects scale with current

37pn Junction

0 1 2 3 4 5101

Position [µm]

[cm−

0 1 2 3 4 510−16

10−7

Position [µm]

[µm−

1 C−

37pn Junction

0 1 2 3 4 5101

Position [µm]

[cm−

0 1 2 3 4 510−16

10−7

Position [µm]

[µm−

1 C−

37pn Junction

0 1 2 3 4 5101

Position [µm]

[cm−

0 1 2 3 4 510−16

10−7

Position [µm]

[µm−

1 C−

38pn Junction

Scaled electron/hole distribution function in the middle of the pn junction

0 1 2 3 4 5 6 710−4

10−1

Energy [eV]

1 eV−

1 A−

38pn Junction

Scaled electron/hole distribution function in the middle of the pn junction

0 1 2 3 4 5 6 710−4

10−1

Energy [eV]

1 eV−

1 A−

Vertical power transistor

40Power MOSFETAvalanche breakdown of a power transistor

BE simulation for 35.83V and 10pA/µm2

Electrostatic Potential [V]

-0.6 6.8 14 22 36

e-Imp-Ioniz [cm^-3*s^-1]

2.6e+12 2.6e+16 2.6e+22

h-Imp-Ioniz [cm^-3*s^-1]

5.2e+13 5.2e+17 5.2e+23

45 · 106 variables, 2 CPU hours

41Power MOSFET

BE simulation for 35.83V and 10pA/µm2

Electrostatic Potential [V]

-0.6 6.8 14 22 36

0.3 0.4 0.5 0.6 0.7 0.8 0.9 110−2

Normalized path-length

3 ]1eV2eV3eV

Injection of electrons and holes into the oxide can be calculated

Structure, DC and AC behaviorof the npn THz SiGe HBT

43Comparison with experiments

Comparison for an ST-Microelectronics SiGe HBTat 10GHz and VCE = 1.2V

Cutoff frequency

10−1 100 101 102

Collector current [mA/µm2]

BESchröter

Minimum noise figure

100 1010

BESchröter

Good agreement between simulation and measurement

44Device structure

100.00.0 10.0 20.0 30.0 40.0 50.0

OxideEmitter

Collector

Emitter-window width: 2 × 25nm

I 149 nodes in x-directionI 20 nodes in y -directionI 5th order SHE expansion (21

spherical harmonics)I Up to 1.42 · 108 unknown

variables (spacing in energyis 5.17meV and number ofenergy grid points dependson bias)

I Over 200 GB of memoryrequired (for AC)

I Up to 105 CPU-seconds for aDC point

44Device structure

100.00.0 10.0 20.0 30.0 40.0 50.0

OxideEmitter

Collector

44Device structure

100.00.0 10.0 20.0 30.0 40.0 50.0

OxideEmitter

Collector

44Device structure

100.00.0 10.0 20.0 30.0 40.0 50.0

OxideEmitter

Collector

44Device structure

100.00.0 10.0 20.0 30.0 40.0 50.0

OxideEmitter

Collector

44Device structure

100.00.0 10.0 20.0 30.0 40.0 50.0

OxideEmitter

Collector

44Device structure

100.00.0 10.0 20.0 30.0 40.0 50.0

OxideEmitter

Collector

45Doping and germanium profile

0.01 0.02 0.03 0.04·10−2

x [µm]

[cm−

NANDGe

Base thickness: 7nm

46Input characteristic

0.4 0.5 0.6 0.7 0.8 0.9 1

10−8

10−6

10−4

10−2

Base/emitter bias [V]

[mA/µ

ICIBIC/IB

VCE = 1.0VNo SRH recombination

47Output characteristic

10−3

10−2

10−1

0 0.5 1 1.5 2 2.5 30

Collector/emitter bias [V ]

IC|IB|

VBE = 0.7V

BVCE0 = 1.4V

48Gains

108 109 1010 1011 1012 1013

Frequency [Hz]

|HCB|2UMSG/MAG

108 109 1010 1011 1012 1013

Frequency [Hz]

|HCB|2UMSG/MAG

VBE = 0.88V, VCE = 1.0V, IC = 18.2mA/µm2

49Cutoff and max. oscil. frequencies

10−1 100 101 102

fT QSfT [email protected] [email protected]

VCE = 1.0V , fmax by extrapolation of Upeak fT = 1.2THz, peak fmax = 1.45THz

49Cutoff and max. oscil. frequencies

10−1 100 101 102

fT QSfT [email protected] [email protected]

VCE = 1.0V , fmax by extrapolation of Upeak fT = 1.2THz, peak fmax = 1.45THz

50Local contribution to the transit timefT =

12πτ

, τ =

∫dτdx

5 10 15 20 25 30 35

20Base

x [nm]

dτ/dx

IC = 54.6mA/µm2

IC = 19.6mA/µm2

VCE = 1.0V

51Minimum noise figure

10−1 100 101 102

] NFmin @ 1GHzNFmin @ 100GHzNFmin @ 300GHz

VCE = 1.0V

52Electrothermal Transport

High power densities lead to self-heating

Heating is non-local!

High power densities lead to self-heating

Heating is non-local!

Dirichlet boundary conditions

Scattering between optical and acoustic phonons goes both ways

Junction temperature determines the current

Self-heating can lead to thermal runaway

2D electrothermal simulation with SHE

Exact thermal simulation requires 3D simulation including metalization

2D electrothermal simulation with SHE

Exact thermal simulation requires 3D simulation including metalization

Conclusions

57ConclusionsI Semiclassical simulations give detailed physics at the price of high

CPU times and a 2D real spaceI Various effects can be included at a microscopic levelI DC, AC and noise analysisI Device degradation, self-heating, mechanical strain, ...I Transport at the nanoscale is very complex

physics-based device simulation - tu dresden · 2015-12-16 · pros & cons 4 i realistic device...

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