What can we learn from the temperature dependence of carrier mobility?

Talk given at CAM-IES Mixed Ionic Electronic Conductors, 2022

Transcript

1. What can we learn from the temperature dependence of carrier mobility? Alex Ganose
2. Thermoelectric materials convert waste heat into usable electrical energy Snyder and Toberer, Nat. Mater. 7, 105 (2008) Thermoelectric figure of merit: 𝛼 = Seebeck coefficient 𝜎 = conductivity 𝑇 = temperature 𝑘!"## = lattice thermal conductivity 𝑘$!$% = electrical thermal conductivity
3. Computational prediction of experimentally verified thermoelectric materials is rare — hampered by obtaining accurate transport properties Urban; Menon; Tian; Jain; Hippalgaonkar, J. Appl. Phys. 125, 180902 (2019) all experimentally verified computational thermoelectric predictions 2008 2012 2016 2020 Year 0.0 0.4 0.8 1.2 1.6 Exp. ZT n-LiZnSb p-SnS p-NbFeSb p-TmAgTe2 p-YCuTe2 n-Er12 Co5 Bi n-KAlSb4 p-Cd1.6 Cu3.4 In3 Te8 p-TaFeSb n-type p-type
4. group velocity (easy) lifetime (hard) energy (easy) Boltzmann Transport Equation determines carrier transport properties
5. Lifetimes determined by the matrix elements of scattering processes matrix element overlap
6. τ Pros: – Quick – Widely used Cons: – Fixed lifetime – Arbitrary results A hierarchy of methods exist for calculating electron lifetimes Madsen; Singh, Comput. Phys. Commun. 175, 67 (2006) Madsen; Carrete; Verstrae, Comput. Phys. Commun. 231, 140 (2018) ... constant lifetime
7. τ Pros: – Quick – Models acoustic phonon scattering Cons: – Still arbitrary – Fixed scattering process Yan; Gorai; Ortiz; Miller; Barnett; Mason; Stevanovic; Toberer, Environ. Sci. 8, 983 (2015) Xi; Pan; Li; Xu; Ni; Sun; Yang; Luo; Xi; Zhu; et al., J. Amer. Chem. Soc. 140, 10785 (2018) ... ∝ DOS–1 constant lifetime A hierarchy of methods exist for calculating electron lifetimes
8. τ Pros: – First principles matrix elements – High accuracy Cons: – Expensive – Limited to < ~15 atoms Giustino; Cohen; Louie, Phys. Rev. B 76, 165108 (2007) Verdi; Giustino, Phys. Rev. Lett. 115, 176401 (2015) ... DFPT ∝ DOS–1 constant lifetime A hierarchy of methods exist for calculating electron lifetimes
9. τ ... DFPT AMSET ∝ DOS–1 constant lifetime Explicitly calculate scattering rates while remaining computationally efficient Aim: accuracy of DFPT+Wannier at 1/500th computational cost A hierarchy of methods exist for calculating electron lifetimes
10. Multiple scattering mechanisms incorporated through computationally cheap matrix elements acoustic deformation potential (ad) deformation potential, elastic constant ionized impurity (ii) dielectric constant piezoelectric (pi) dielectric constant, piezoelectric coefficient polar optical phonon (po) dielectric constant, polar phonon frequency
11. 200 300 400 500 Temperature (K) 103 104 Mobility (cm2/Vs) ad ii po total 3.1 3.2 3.3 Energy (eV) 0.01 0.1 1 10 ønk (fs) ad ii po • All first principles inputs with no fitting or tuning parameters • k-dependent scattering rates • 10 min runtime on laptop 200 300 400 500 Temperature (K) 103 104 Mobility (cm2/Vs) ad ii po total 3.1 3.2 3.3 Energy (eV) 0.01 0.1 1 10 ønk (fs) ad ii po GaN ωpo ne = 3×1016 cm–3 AMSET predicts switch from impurity to polar phonon scattering in GaN Ganose, et al. Nat. Commun. 12, 2222 (2021)
12. AMSET obtains similar mobilities to DFPT & significantly improves upon CRT 200 400 600 800 Temperature (K) 103 104 Mobility (cm2/Vs) n-GaAs 1016 1018 Carrier concentration (cm°3) 102 103 Mobility (cm2/Vs) n-Si 150 300 450 Temperature (K) 0 200 400 600 800 Mobility (cm2/Vs) p-SnSe AMSET DFPT+Wannier CRT Ma; Chen; Li, Phys. Rev. B 97, 205207 (2018) Zhou; Bernardi, Phys. Rev. B 94, 201201 (2016) Poncé; Margine; Giustino, Phys. Rev. B 97, 121201 (2018) b–axis AMSET AMSET DFPT+Wannier CRT DFPT+Wannier CRT
13. AMSET shows close agreement to experiment for the mobility and Seebeck coefficient across many materials n-GaAs GaN ZnS ZnSe CdS CdSe CdTe p-CdTe SiC InP PbS SnO2 ZnO Si p-CuAlO2 SnSe p-GaAs 100 101 102 103 104 µ experiment (cm2/Vs) 100 101 102 103 104 µ calculation (cm2/Vs) GaP (a) n-GaAs GaN ZnS ZnSe CdS CdSe CdTe p-CdTe GaP SiC InP PbS SnO2 p-CuAlO2 SnSe °4.0 °3.2 °2.4 °1.6 °0.8 ¢ ln µ/¢ ln T experiment °4.0 °3.2 °2.4 °1.6 °0.8 ¢ ln µ/¢ ln T calculation (b) 0.1 1 m§ c (me ) 0.1 1 m§ c (me ) aAs N CdS InP Si As 104 /Vs) n-GaAs GaN ZnS ZnSe CdS CdSe CdTe p-CdTe GaP SiC InP PbS SnO2 p-CuAlO2 SnSe °4.0 °3.2 °2.4 °1.6 °0.8 ¢ ln µ/¢ ln T experiment °4.0 °3.2 °2.4 °1.6 °0.8 ¢ ln µ/¢ ln T calculation (b) 0.1 1 m§ c (me ) 0.1 1 m§ c (me ) n-GaAs GaN CdS CdTe Te InP PbS nO2 Si p-GaAs 103 104 (cm2/Vs) n-GaAs GaN ZnS ZnSe CdS CdSe CdTe p-CdTe GaP SiC InP PbS SnO2 p-CuAlO2 SnSe °4.0 °3.2 °2.4 °1.6 °0.8 ¢ ln µ/¢ ln T experiment °4.0 °3.2 °2.4 °1.6 °0.8 ¢ ln µ/¢ ln T calculation (b) °800 °400 0 400 Æ experiment (µV/K) °800 °400 0 400 Æ calculation (µV/K) n-GaAs GaN InP CdS PbS Si SnO2 SnSe ZnO p-GaAs (c) 0.1 1 m§ c (me ) 0.1 1 m§ c (me ) 1016 1017 1018 1019 1020 °3 00 ) 1016 1017 1018 n (cm°3) mobility Seebeck coefficient • All DFT calculations performed using PBE functional • Comparisons against single crystal measurements Ganose, et al. Nat. Commun. 12, 2222 (2021)
14. Docs: https://hackingmaterials.lbl.gov/amset/ Support: https://matsci.org/c/amset AMSET is an open source python package that you can run today installation pip install amset usage amset run --static-dielectric 10 ... Can be controlled through the command line or python interface
15. Carrier mobility is a key materials property that is relatively independent of carrier concentration how quickly an electron or hole moves when pulled by an electric field µ = σ/en mobility conductivity carrier concentration
16. Temperature dependence of carrier mobility has been used since the 1930’s to understand the quantum behaviour of materials First expressions for mobility in semiconductors include a T–3/2 dependence
17. 25 years later, these expressions do remarkably well predicted the transport in n-type Germanium Impressive, since Wilson’s was the first to treat: 1. Valence/conduction bands separated by a gap 2. Impurities as the source of free carriers 3. Lattice vibrations as the source of scattering
18. Since then, T-dependence of mobility has been used as experimental signature of the dominant scattering mechanism Acoustic/optical deformation µ~T–3/2 Polar optical µ~T–0.75 Alloy µ~T–1/2 Ionized impurity µ~T3/2
19. Since then, T-dependence of mobility has been used as experimental signature of the dominant scattering mechanism Acoustic/optical deformation µ~T–3/2 Polar optical µ~T–0.75 Alloy µ~T–1/2 Ionized impurity µ~T3/2 all derived from simplified models of transport in a single parabolic band yet these limitations are often forgotten in many studies
20. As computational tools have improved/become tractable, theory tells a different picture modern approaches allow transport properties to be predicted quantitatively
21. DFPT+Wannier shows excellent agreement across many materials classes – this is not an issue with theory / 1 341 . ( (3 2 . 2  0 (  . 1 (  . 2 .1 7/ 0 ( .1 7/ . 1 ( .1 7/ . 1 ( .1 7/ State-of-the-art approaches agree with experiment across different - scattering types - band structures - temperatures
22. Tin selenide is the poster boy for thermoelectric materials, long assumed to be limited by acoustic phonons SnSe DFPT+Wannier reproduces experimental mobility dependence of T–3/2 BUT polar optical phonons dominate scattering
23. Does T-dependence tell us anything about the dominant scattering?
24. Does the T-dependence of mobility give insight into the dominant scattering mechanism - Review of all DFPT+Wannier studies (60+ materials) - No correlation between T- dependence and scattering type
25. Does the T-dependence of mobility give insight into the dominant scattering mechanism - Review of all DFPT+Wannier studies (60+ materials) - No correlation between T- dependence and scattering type Not safe to assume a scattering type based on T-dependence alone
26. τ ... DFPT AMSET ∝ DOS–1 constant lifetime AMSET is a new framework for calculating transport properties Explicitly calculate scattering rates while remaining computationally efficient close to accuracy of DFPT+Wannier at 1/500th computational cost hackingmaterials.lbl.gov/amset/ Ganose, Park, Faghaninia, Woods-Robinson, Persson, Jain Nat. Commun. 12, 2222 (2021)
27. Using AMSET calculations, we find the primary factor controlling T-dependence are the phonon frequencies - Calculations performed on a single parabolic & isotropic band - Phonon frequencies have large impact Polar optical phonon scattering only
28. Using AMSET calculations, we find the primary factor controlling T-dependence are the phonon frequencies / 1 341 . ( (3 2 . 2  0 (  . 1 (  . 2 .1 7/ 0 ( .1 7/ . 1 ( .1 7/ . 1 ( .1 7/
29. Using AMSET calculations, we find the primary factor controlling T-dependence are the phonon frequencies / 1 341 . ( (3 2 . 2  0 (  . 1 (  . 2 .1 7/ 0 ( .1 7/ . 1 ( .1 7/ . 1 ( .1 7/ Phonon frequencies control T-dependence irrespective of scattering type
30. Band structure features also affect the T-dependence but to a lesser degree non-parabolicity anisotropy
31. We performed 24,000 mobility calculations using AMSET with machined learned materials parameters Dielectric tensors ~8,000 Polar freqs ~8,000 Deform potentials ~3,000 Elastic tensors ~11,000 Band structures 24,000 Mobility database 24,000
32. AMSET can separate mobility into contributions from different mechanisms, we calculate the T-dependence in each case average T-dependence for polar and acoustic phonon scattering are very similar
33. So why is this so important?
34. Example 1: µ~T–3/2 has been used to justify unphysical materials properties Material Fitted SnSe 24 eV PbTe 22 eV BiCuSeO 24 eV Calculated 10 eV 7 eV 4 eV
35. T-dependence primarily controlled by phonon frequencies and BS features T-dependence of mobility does not indicate the dominant scattering mechanism Polar optical phonon scattering much more common than previously thought
36. Funding U.S. Department of Energy, Basic Energy Sciences, Early Career Research Program Computing NERSC Anubhav Jain and you for your attention! Junsoo Park Acknowledgements
37. Scattering rate equations acoustic deformation potential (ad) piezoelectric (pi) polar optical phonon (po) ionized impurity (ii)
38. Example 2: the dominant scattering type can be used to optimize the optimal doping and temperature range for thermoelectrics - Different mechanisms have different optimal doping levels - Determining the wrong scattering type can frustrate optimisation of TE performance