Accurate band structures from DFT and simple phonon ...€¦ · Accurate band structures from DFT...
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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
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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.”
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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.”
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
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.”
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
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[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
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[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 conduction band effective mass: m = 0.023 me (exp. 0.024 me)
Parabolic/effective mass model:
Non-parabolic model:
Bulk InAs conduction band
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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 ?
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W
InAs slab
Effective mass is increased compared to bulk InAs Agreement with non-parabolic model
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Parabolic/effective mass model Dispersion
Effective mass
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Effective mass model vs. Non-parabolic model
Parabolic/effective mass model Dispersion
Effective mass
Non-parabolic model Dispersion
Effective mass
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Effective mass model vs. 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
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20 nmElectrode extension
10 nmChannel
20 nmElectrode extension
Gate electrodes
Part II: Phonon limited mobility calculations from combined molecular dynamics and Green’s functions
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Electron-phonon coupling
NEGF; SCBA
Problems: » Numerically very demanding» Needs approximations for the self-energies for large systems due to memory issues
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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
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MD-Landauer: Method
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Transmissions for different MD temperatures Increasing temperature increased scattering
Increasing temperature
Resistance:
Increases linearly with length of MD region
MD-Landauer: Method
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Resistivity
Resistance:
Increases linearly with length of MD region
Conductivity:
Electron density
Mobility
MD-Landauer: Method
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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
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[1]: PRL 94, 086802 (2005)
Lowest order expansion (LOE) of SCBA
Current formula [1] » All matrices evaluated at the Fermi energy» Non-interacting GFs
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[1]: Paulsson, Frederiksen, Brandbyge, Phys. Rev. B 72, 201101(R) (2005)
Lowest order expansion (LOE) of SCBA
Current formula [1] » All matrices evaluated at the Fermi energy» Non-interacting GFs
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Electron-phonon coupling matrix for phonon mode l
Sum over phonon modes
[1]: Paulsson, Frederiksen, Brandbyge, Phys. Rev. B 72, 201101(R) (2005)
Lowest order expansion (LOE) of SCBA
Current formula [1]» All matrices evaluated at the Fermi energy» Non-interacting GFs
Effective, temperature dependent transmission function:
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Electron-phonon coupling matrix for phonon mode l
Sum over phonon modes
[1]: Paulsson, Frederiksen, Brandbyge, Phys. Rev. B 72, 201101(R) (2005)
Results from LOE
Close agreement between LOE and “Landauer+MD” and Boltzmann[1]
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[1]: PRL 94, 086802 (2005)
Results
Calculated vs. experimental[1] mobilities
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[1]: http://www.ioffe.rssi.ru/SVA/NSM/Semicond/index.html
Results
Calculated vs. experimental[1] mobilities
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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
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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.
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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
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