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Accurate band structures from DFT and simple phonon-limited mobility calculations

Troels Markussen

QuantumHagen 2-7-2014

Part I III-V-MOS project Case study: InAs

» Simple model for conduction band» Effects of confinement

Part 2 Phonon-limited mobility calculations from combined Molecular Dynamics and

Green’s function transport

Outline

QuantumWise Slide 1

III-V-MOS project

The III-V-MOS Project is a European Collaborative Project ” Objective: Enabling fast and effective design of new transistors for high

performance electronics and with greatly reduced power consumption.”

QuantumWise Slide 2

III-V-MOS project



First-principles (DFT) Tight-binding, effective mass TCAD models

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Increasing accuracy and computational complexity

Increasing model completeness

III-V-MOS project



First-principles (DFT) Tight-binding, effective mass TCAD models

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Increasing accuracy and computational complexity

Increasing model completeness

Material parameters (InAs, InGaAs):Band gaps, effective mass,Strain dependenceConfinements effects

Advanced device simulations

DFT is strictly not a theory of quasi-particles, e.g. band structures The usual approximations (LDA, GGA) typically under estimate band

gaps

The infamous band gap problem

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Meta-GGA improves the band gap significantly [1] Meta-GGA: 𝐸𝑥𝑐 𝑛, 𝛻𝑛, 𝜏 Exchange potential:

c-parameter:

Meta-GGA

QuantumWise Slide 6

[1]: Tran & Blaha Phys. Rev. Lett. 102, 226401 (2009)

Meta-GGA improves the band gap significantly [1] Meta-GGA: 𝐸𝑥𝑐 𝑛, 𝛻𝑛, 𝜏 Exchange potential:

c-parameter:

ATK: 1. Self-consistent calculation of c. Only for bulk calculation without vacuum

regions.2. User-specified c (constant). Band gap increases with increasing c.

Meta-GGA

QuantumWise Slide 7

[1]: Tran & Blaha Phys. Rev. Lett. 102, 226401 (2009)

Meta-GGA improves the agreement significantly Self-consistent c-parameter.

Meta-GGA results are much better!

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InAs, bulk band structure (Meta-GGA, with spin-orbit coupling)

InAs

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InAs

InAs, bulk band structure (Meta-GGA, with spin-orbit coupling)

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InAs conduction band effective mass: m = 0.023 me (exp. 0.024 me)

Parabolic/effective mass model:

Non-parabolic model:

Bulk InAs conduction band


m* = 0.023 me

a = 3.13 eV-1

Question:

Can we describe the conduction band structure of an InAs slab based only on the bulk band structure parameters m, a ?


W

InAs slab

Effective mass is increased compared to bulk InAs Agreement with non-parabolic model


Parabolic/effective mass model Dispersion

Effective mass


Effective mass model vs. Non-parabolic model

Parabolic/effective mass model Dispersion

Effective mass

Non-parabolic model Dispersion

Effective mass


Effective mass model vs. Non-parabolic model

InAs slab: Effective mass vs. thickness


InAs slab: Band gap vs. thickness


InAs nanowire


Good agreement with non-parabolic model

III-V-MOS: Device calculations…work in progress

DFT device calculations, Meta-GGA. ~2300 atoms

We are calculating IV-curves (Vsd and Vsg) Band profiles Comparison with OMEN


20 nmElectrode extension

10 nmChannel

20 nmElectrode extension

Gate electrodes

Part II: Phonon limited mobility calculations from combined molecular dynamics and Green’s functions


Electron-phonon coupling

NEGF; SCBA

Problems: » Numerically very demanding» Needs approximations for the self-energies for large systems due to memory issues


Electron-phonon coupling

NEGF; SCBA

Problems: » Numerically very demanding» Needs approximations for the self-energies for large systems due to memory issues

This work: Lowest order expansion (LOE) of SCBA (available in ATK-2014) Molecular dynamics combined with (elastic) Landauer transmission


MD+Landauer: Workflow in VNL


MD-Landauer: Method


Transmissions (elastic) for MD calculations at 300 K

MD-Landauer: Method


Transmissions for different MD temperatures Increasing temperature increased scattering

Increasing temperature

Resistance:

Increases linearly with length of MD region

MD-Landauer: Method


Resistivity

Resistance:

Increases linearly with length of MD region

Conductivity:

Electron density

Mobility

MD-Landauer: Method


Resistivity

Density of states (separate calculation)

Results

CNT (7,0) mobility vs. temperature Fitted formula to Boltzmann Eq. results [1]:

𝜇0 = 𝜇1300𝐾

𝑇

𝑑

1 𝑛𝑚

2.26, 𝜇1 = 12,000

𝑐𝑚2

𝑉𝑠


[1]: PRL 94, 086802 (2005)

Lowest order expansion (LOE) of SCBA

Current formula [1] » All matrices evaluated at the Fermi energy» Non-interacting GFs


[1]: Paulsson, Frederiksen, Brandbyge, Phys. Rev. B 72, 201101(R) (2005)


Current formula [1] » All matrices evaluated at the Fermi energy» Non-interacting GFs


Electron-phonon coupling matrix for phonon mode l

Sum over phonon modes



Current formula [1]» All matrices evaluated at the Fermi energy» Non-interacting GFs

Effective, temperature dependent transmission function:


Electron-phonon coupling matrix for phonon mode l

Sum over phonon modes


Results from LOE

Close agreement between LOE and “Landauer+MD” and Boltzmann[1]


[1]: PRL 94, 086802 (2005)

Results

Calculated vs. experimental[1] mobilities


[1]: http://www.ioffe.rssi.ru/SVA/NSM/Semicond/index.html

Results

Calculated vs. experimental[1] mobilities


Low-doped silicon

[1]: http://www.ioffe.rssi.ru/SVA/NSM/Semicond/index.html

Discussion of MD+GF method

Not very rigorous Cannot account for finite biases Cannot describe heating effects

Very simple to implement Calculations can be run on a PC Conceptually simple, intuitive Includes anharmonic effects for the

vibrations Seems to capture some correct physics

and gives promising results. In agreement with LOE


Summary

Part I Accurate band structures can be obtained with DFT DFT results for InAs slabs and nanowire can be

accurately reproduced by the non-parabolic model. This enables an easy description of confined InAs

systems for device modelling.Part II: Conceptually very simple method for calculating

low-field phonon limited mobilities Good agreement with experimental data, Boltzmann

eq. data, and LOE results.


Acknowledgement

The III-V-MOS Project is a European Collaborative Project (2013-2016) funded by the European Commission under the 7th Framework Program

Collaborators in III-V-MOS project


Acknowledgement

The III-V-MOS Project is a European Collaborative Project (2013-2016) funded by the European Commission under the 7th Framework Program

Collaborators in III-V-MOS project

Thank you for your attention!


Accurate band structures from DFT and simple phonon ...€¦ · Accurate band structures from DFT...

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