computing the intrinsic mobility of electrons and holes in semiconductors · 2019. 12. 5. ·...

Computing the intrinsic mobility of electrons and holes in

semiconductors

Part I : concepts and resultsABIDEV 2019

(1) Guillaume Brunin(2) Henrique Miranda

Matteo GiantomassiGian-Marco Rignanese

Geo�roy Hautier

May 21, 2019

Institute of Condensed Matterand Nanosciences

Electronic lifetimes and phonon-limited mobility

Electron-phonon coupling dictates various phenomena : intrinsic mobility,indirect light absorption, superconductivity,...Electrons are scattered by phonons : mechanism dominating the electroniclifetimes and the mobility at high temperature

[Yu & Cardona, Fundamentals of Semiconductors]1


Electron mobility in the relaxation-time approximation of the linearized Boltz-mann transport formalism :

µe,–—(ÁF , T ) = ≠e

�ne

ÿ

nœCB

⁄dk

�BZvnk,–vnk,—·nk(ÁF , T )ˆf (Ánk, ÁF , T )

ˆÁ

Electronic lifetimes due to the scattering by phonons :

1·nk

= 2fi~

ÿ

m,‹

⁄

BZ

dq�

BZ

|gmn,‹(k, q)|2 ◊ [(n‹q + fmk+q)”(Ánk ≠ Ámk+q + Ê‹q)

+(n‹q + 1 ≠ fmk+q)”(Ánk ≠ Ámk+q ≠ Ê‹q)]

Perturbo

No need for Wannier functions or atomic orbitals !

2

Lifetimes in Silicon : agreement between ABINIT and EPW

Lifetime = 1/linewidth

3

Tetrahedron integration to replace Lorentzian broadening

Linewidth Ã ”(Ánk ≠ Ámk+q ± Ê‹q)Linewidth around the CBM of Si for a 60 ◊ 60 ◊ 60 k-point grid (300K)

More scattering channels for Á ≠ ÁCBM > ÊLO ∆ larger linewidthsA Lorentzian broadening requires convergence studiesTetrahedron method : no broadening parameter

4


Linewidth Ã ”(Ánk ≠ Ámk+q ± Ê‹q)Linewidth around the CBM of Si for a 60 ◊ 60 ◊ 60 k-point grid (300K)More scattering channels for Á ≠ ÁCBM > ÊLO ∆ larger linewidths

A Lorentzian broadening requires convergence studiesTetrahedron method : no broadening parameter

4


Linewidth Ã ”(Ánk ≠ Ámk+q ± Ê‹q)Linewidth around the CBM of Si for a 60 ◊ 60 ◊ 60 k-point grid (300K)More scattering channels for Á ≠ ÁCBM > ÊLO ∆ larger linewidthsA Lorentzian broadening requires convergence studies

Tetrahedron method : no broadening parameter

4


Linewidth Ã ”(Ánk ≠ Ámk+q ± Ê‹q)Linewidth around the CBM of Si for a 60 ◊ 60 ◊ 60 k-point grid (300K)More scattering channels for Á ≠ ÁCBM > ÊLO ∆ larger linewidthsA Lorentzian broadening requires convergence studiesTetrahedron method : no broadening parameter

4

Convergence of the linewidths in Silicon

Linewidth Ãs

BZ

dq�

BZ

|gmn,‹(k, q)|2...Linewidths on a 9 ◊ 9 ◊ 9 k-point grid, for increasing q-point gridsThe Tetrahedron integration converges fast (like for a DOS !)

5

Double-grid technique : 1 grid for matrix elements, 1 grid for energies

1·nk

= 2fi~

ÿ

m,‹

⁄

BZ

dq�

BZ



6

Double-grid technique : errors on the linewidths in Silicon

Linewidth Ãs

BZ

dq�

BZ

|gmn,‹(k, q)|2...Linewidths on a 9 ◊ 9 ◊ 9 k-point gridBlue : 9 ◊ 9 ◊ 9 q-point grid for matrix elements, increasing density forthe energy grid only

7

Polar materials : divergence of the matrix elements

g

LRmn,‹(k, q) = i

4fi�

e

2

4fiÁ0

ÿ

Ÿ

3~

2NMŸÊ‹q

4(1/2) ÿ

G ”=≠q

(q + G) · ZúŸ · eŸ‹(q)

(q + G) · ÁŒ · (q + G)

◊ È�mk+q|e i(q+G)·r|�nkÍ

128 S. Poncé et al. / Computer Physics Communications 209 (2016) 116–133

6

2

0

-2

-4

-6

4

80

60

20

40

500

400

300

200

100

0

Fig. 15. (Color online) (a) Electronic bandstructure of w-GaN along high-symmetrylines in the BZ. The red line highlights the band considered in the bottom panel.(b) Phonon dispersion interpolated from a 6 ⇥ 6 ⇥ 6 � -centered q-point grid. Thered line is the optical mode considered in the bottom figure. The filled dots areinelastic X-ray scattering data [145] and the filled triangles are Raman data [146].(c) Calculated electron–phononmatrix elements at k = � for the highest band andmode index, starting from 6 ⇥ 6 ⇥ 6 electron and phonon grids. A gauge-invariantelectron–phonon matrix element is obtained by averaging the square moduli overdegenerate states. The blue dashed line shows the standard interpolation, and thered line shows the polar interpolation using Eq. (13). The two sets of data arecompared with direct DFPT calculations at each wavevector (filled dots).

symmetry directions of the BZ (filled dots) at k = � , m = nbeing the highest valence band, and ⌫ the highest optical mode.The selected band and mode are indicated in red in the top andmiddle panels of Fig. 15. The square moduli of electron–phononmatrix elements are averaged over degenerate bands and modes.The dashed blue line shows the standard Wannier interpolation ofRef. [46], and has to be comparedwith the red line that implementsthe polar interpolation of Eq. (13) [60].

Fig. 15 clearly shows the importance of correctly treatingthe long-range interaction in the Wannier interpolation whenstudying polar semiconductors and insulators. This feature isactivated in EPW by using the lpolar input variable.

10.5. Superconductivity inMgB2

The last example presented here focuses on the superconduct-ing properties of magnesium diboride (MgB2). This tutorial can befound in the EPW/examples/mgb2 folder of EPW. An online ver-sion is available at epw.org.uk. This tutorial is based on theworkof Ref. [59] and includes some additional results.

Fig. 16. (Color online) (a) Electronic bandstructure and (b) DOS of bulk MgB2; (c)phonon bandstructure and (d) PDOS at the experimental lattice parameters.Source: The inelastic X-ray scattering experimental data (black dots) at 300 K aretaken from Ref. [149].

Fig. 17. (Color online) Calculated isotropic Eliashberg spectral function ↵2F(!) ofMgB2 (blue solid line) and integrated electron–phonon coupling strength � (blackdashed line).

MgB2 is the prototypical multi-gap phonon-mediated super-conductor, with a critical temperature of Tc = 39 K [150]. Theanisotropic gap arises from the � and ⇡ Fermi-surface sheets [10,151–155]. This superconductor has beenwidely investigated theo-retically [10,90,151,156–167] due to its high Tc. We re-investigatehere MgB2 using EPW to show that its fascinating properties cannow be computed easily and accurately.

The calculations were performed within DFT-LDA [63,68] usingthe Quantum ESPRESSO [62] code. The valence electronicwavefunctions were expanded in a plane-wave basis set using akinetic energy cutoff of 60 Ry and the charge density is integratedon a � -centered 24 ⇥ 24 ⇥ 24 k-point mesh. The experimentallattice parameters a = 5.826 and c/a = 1.142 bohr wereused throughout the calculations [150]. AMethfessel–Paxton first-order smearing [122] of 0.02 Ry was applied. The first-orderpotential perturbation and dynamical matrices were calculatedusing DFPT [42,44,45] on an irreducible 6 ⇥ 6 ⇥ 6 � -centered q-point mesh.

The electronic bandstructure is shown on the top panel ofFig. 16. The associated phonon bandstructure at the bottom ofFig. 16 slightly underestimates the inelastic X-ray scattering datameasured at 300 K [149].

The isotropic Eliashberg spectral function ↵2F(!) calculatedusing Eq. (12), and the associated cumulative electron–phononcoupling strength �(!), are shown in Fig. 17. The spectralfunction shows a dominant peak around 60 meV, and a secondaryone around 86 meV. The cumulative electron–phonon coupling

[Poncé, CPC 209, 116-133 (2016)]8

Convergence of the linewidths in GaP

Linewidth Ãs

BZ

dq�

BZ

|gmn,‹(k, q)|2...Linewidths on a 8 ◊ 8 ◊ 8 k-point grid, for increasing q-point gridsThe Tetrahedron integration converges slower than in SiliconThis is due to the polar divergence of electron-phonon matrix elements

9

Double-grid technique for polar materials : less e�cient...

Linewidth Ãs

BZ

dq�

BZ


10

Double-grid technique for polar materials : special treatment

1·nk

= 2fi~

ÿ

m,‹

⁄

BZ

dq�

BZ

| gmn,‹(k, q)¸ ˚˙ ˝gLR

+gSR

|2 ◊ [(n‹q + fmk+q)”(Ánk ≠ Ámk+q + Ê‹q)


11

Double-grid technique for polar materials : faster convergence

Linewidth Ãs

BZ

dq�

BZ


12

Double-grid technique for polar materials : faster convergence

Linewidth Ãs

BZ

dq�

BZ

|gmn,‹(k, q)|2...Linewidths on a 8 ◊ 8 ◊ 8 k-point gridBlue : 8 ◊ 8 ◊ 8 q-point grid for matrix elements, increasing density forthe energy and g

LR only

12


Electron mobility in the relaxation-time approximation of the linearized Boltz-mann transport formalism :

µe,–—(ÁF , T ) = ≠e

�ne

ÿ

nœCB

⁄dk


ˆÁ

Electronic lifetimes due to the scattering by phonons :

1·nk

= 2fi~

ÿ

m,‹

⁄

BZ

dq�

BZ



Perturbo

No need for Wannier functions or atomic orbitals !

13

Transport properties : do we really need the lifetimes everywhere ?

The mobility contains a derivative of the Fermi-Dirac occupation function :

µe,–—(ÁF , T ) = ≠e

�ne

ÿ

nœCB

⁄dk


ˆÁ

14

Lifetimes in CB pockets : do we really need to sample all q-points ?

Transitions from k to k + q : momentum conservationq-points are also limited by the energy conservation : Ánk ≠ Ámk+q = ±Ê‹qOnly a limited set of q-points contribute and need to be taken intoaccount !

15

Mobility of electrons in Silicon with ABINIT

New implementation in ABINIT :Selects the important k- and q-pointsComputes the lifetimes and velocities for these pointsPerforms the integration to obtain the phonon-limited mobility

16

Mobility of electrons in GaP with ABINIT

17

First part : conclusion

We can compute lifetimes with ABINIT, without using Wannier functionsor atomic orbitalsNote : this was already possible in previous versions of ABINIT

Use of the Tetrahedron method :correct behavior without having to deal with a broadening parameter

faster convergence w.r.t. q points

Double-grid technique :the ” functions require denser grid than the gmn,‹(k, q)

the diverging part of the matrix elements can be computed on a denser grid

Phonon-limited mobility : we compute only what is necessary·nk for few % of k points, gmn,‹(k, q) for few % of q points

Second part : by Henrique Miranda

19

Electron-phononself-energyusingplanewaves

gmn,⌫(k,q) = h mk+q|�⌫qVKS | nki

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⌃nk(!, "F , T ) =X

m,⌫

Z

BZ

dq

⌦BZ|gmn,⌫(k,q)|2

n⌫q + fmk+q

! � "mk+q + !⌫q + i�+

n⌫q + 1� fmk+q

! � "mk+q � !⌫q + i�

�

<latexit sha1_base64="OlVS14rD6XSQeP6qN/aQHqXm0U4=">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</latexit>

1


•  InterpolatetheDFPTpoten;als–  Removeandaddthelong-rangepart(Fröhlich)–  Reducememoryforpoten;als:boxcutmin,singleprecision,distribu;onoverperturba;onsandq-points



⌃nk(!, "F , T ) =X

m,⌫

Z

BZ

dq


n⌫q + fmk+q

! � "mk+q + !⌫q + i�+

n⌫q + 1� fmk+q

! � "mk+q � !⌫q + i�

�


1


•  InterpolatetheDFPTpoten;als–  Removeandaddthelong-rangepart(Fröhlich)–  Reducememoryforpoten;als:boxcutmin,singleprecision,distribu;onoverperturba;onsandq-points

•  Wavefunc;onsfromNSCFcalcula;on•  Computeandaccumulateon-the-fly(avoidIO)



⌃nk(!, "F , T ) =X

m,⌫

Z

BZ

dq


n⌫q + fmk+q

! � "mk+q + !⌫q + i�+

n⌫q + 1� fmk+q

! � "mk+q � !⌫q + i�

�



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Electron-phononself-energyworkflow

WFK(k=q)

DEN

BECS,DDB1 POT1,DDB1 POT2,DDB2

DDB

mrgddb

2

Electron-phononself-energyworkflow

WFK(k=q)

DEN


DVDB DDB

eph

SIGEPH

WFK(k>q)

mrgdv mrgddb

2

CheckpointandRestart

•  Longcalcula;ons->restartfeature!•  Addi;onalarraywith“done”/”notdone”wri1eninsidetheloopoverk-points

•  Ifthecalcula;oniscrashed,restartwith eph_restart 1!

•  Savecomputa;onal;me!

3

•  MobilityinthelinearizedBoltzmannequa;on:

BolzmanntransportintheSERTA

µ↵�("F , T ) =e

n("F , T )⌦

X

n

Zdk

⌦BZvnk,↵vnk,�⌧nk("F , T )

✓�@f("nk, "F , T )

@"

◆

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4


•  Life;mesfromimaginarypartofself-energy:


µ↵�("F , T ) =e

n("F , T )⌦

X

n

Zdk


✓�@f("nk, "F , T )

@"

◆


⌃nk(!, "F , T ) =X

m,⌫

Z

BZ

dq


n⌫q + fmk+q

! � "mk+q + !⌫q + i�+

n⌫q + 1� fmk+q

! � "mk+q � !⌫q + i�

�

<latexit sha1_base64="qn65Tn8puE9xC4BkLo4xNFHB84c=">AAAExniclVNdb9MwFM3WAqN8bfDIi8U0tGld1w4keEGahoT2xhDrNrF0lePepFZsJ42djsmLxM/jL/AH+B3cuAHWrBLCUtObe47vOffaCVLBtel2fywtN5p37t5bud968PDR4yera09PdJJnDPosEUl2FlANgivoG24EnKUZUBkIOA3i9yV+OoVM80Qdm6sUBpJGioecUYOp4dryd1/BJUukpGpk/TgorC+pGQehjYuiNQdOboCTOqjiwlpVFqgBUCL+lGaQai5Q03HrJBlPaiyJtbZLzTo1nCd+qOFTFdO/PqfF0Llq+1SkY1ovhuRgETkAc4t7qfLSYyIhomUT+QJzURynhY3QvWojo9h01SbBVo138AWnhQ9MBxBxZTUetsG3zzySdOYCNzutNrbcPt4i71q+ziWWdpWJz5UZYiHihxlldjQ7no/OXVmfXDs31xd7rZe+gNCcV0z1xzzZJuHcoKvuyA5xJ0KQ4NrGf058A1+Nu3E2EDmgWESxnQKtZDwam07J9iuxziKxHhb+h+DOfwgOWni5Rr9HN1xd73a6bpHbQa8K1r1qHQ1Xf/qjhOUSlGGCan3e66ZmYGlmOBOAZ5FrSCmLaQTnGCoqQQ+s81OQDcyMSJhk+FOGuOzNHZZKra8ktr1RXi5dx8rkQizIaAympm7CtwPLVZobUGwmHuaCmISU3zQZ8QyYEVcYUJZx9E/YmOL4DX755WB69THcDk72Or1Xnb1Pr9f3D6oRrXjPvRfeptfz3nj73qF35PU91tht9BsXjWHzsKmaefNyRl1eqvY88+ZW89svcGClJA==</latexit>

1

⌧nk= ⇡

X

m,⌫

Z

BZ

dq

⌦BZ|gmnk,⌫q|2 ⇥ [(n⌫q + fmk+q)�("nk � "mk+q + !⌫q)+

(n⌫q + 1� fmk+q)�("nk � "mk+q � !⌫q)]<latexit sha1_base64="yeoN4q1PKiMtC+t9dZKZBmk45WU=">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</latexit> 4

SpeedingupimaginarypartofSE

DoublegridEigenvaluesdensegrid:1.SKWinterpola;onbs_interp_kmult!2.NSCFcalcula;ontolwfr 1e-15!irdwfkfine!getwfkfine!

5



Tetrahedronintegra;oneph_intmeth 2!

5




QandK-pointfilteringkerange!

sigma_erange!

5



5

Tetrahedronmethod

zcutsmalltoreproducethelimit,largeenoughtoavoidnumericproblems

Doubleconvergenceγandq

⌃nk(!, "F , T ) =X

m,⌫

Z

BZ

dq


n⌫q + fmk+q

! � "mk+q + !⌫q + i�+

n⌫q + 1� fmk+q

! � "mk+q � !⌫q + i�

�

<latexit sha1_base64="qn65Tn8puE9xC4BkLo4xNFHB84c=">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</latexit>

6

Tetrahedronmethod

zcutsmalltoreproducethelimit,largeenoughtoavoidnumericproblems

Doubleconvergenceγandq

tetrahedronlinearinterpola;onoftheeigenvaluesandmatrixelements

Convergeqonly

1

⌧nk= ⇡

X

m,⌫

Z

BZ

dq

⌦BZ|gmnk,⌫q|2 ⇥ [(n⌫q + fmk+q)�("nk � "mk+q + !⌫q)+

(n⌫q + 1� fmk+q)�("nk � "mk+q � !⌫q)]<latexit sha1_base64="yeoN4q1PKiMtC+t9dZKZBmk45WU=">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</latexit>

6

Tetrahedronmethod101

•  TessellatetheBrillouinZoneusingtetrahedrons

•  Simpleanaly;calexpressionsdependontheenergiesatthesummits

⇡I(!) =

Z

BZFk�(! � "k)

<latexit sha1_base64="3N4V6Au2ju/H/F0BIgeVl8R2s1w=">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</latexit>

[1]A.H.MacDonald,S.H.Vosko,andP.T.Coleridge,JournalofPhysicsC:SolidStatePhysics12,2991(1979)[2]P.E.Blöchl,O.Jepsen,andO.K.Andersen,Phys.Rev.B49,16223(1994)Manyothers...AlsodicussionwithAtsushiTogo

⇡I(!) = ⌦tetra

NtetraX

i=1

gi(!)4X

s=1

Iis(!)Fis

<latexit sha1_base64="+CXaMHXu9+3cWS8m/H87mkUKH8Q=">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</latexit>

7

Oldtetrahedronimplementa;on1.  Createlistofalltetrahedra2.  Hashandsort3.  Reducetounique

(Doublememoryalloca;on)

memorytraceabinit--abimem-level3abimem.pyplot<file>GaAs72x72x72kandqgrid

8

Newtetrahedronimplementa;on1.  Generate1tetrahedronandhash2.  Ifnewstore,ifalreadyexistsaddmul;plicity3.  Repeatforalltetrahedra(Lowermemoryfootprint)

memorytraceabinit--abimem-level3abimem.pyplot<file>GaAs72x72x72kandqgrid

8



µ↵�("F , T ) =e

n("F , T )⌦

X

n

Zdk


✓�@f("nk, "F , T )

@"

◆


4

Groupvelocitymatrixelements

•  Before:DFPTrunonFBZforeach3direc;onsandreaddiagonalmatrixelementsfrom1WFfiles–  LotsofIOandwaste!

10

vnk,↵ = h nk|@k,↵H| nki<latexit sha1_base64="XHPNsxVLD184LAWqCazkSXiNrpA=">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</latexit>


•  Before:DFPTrunonFBZforeach3direc;onsandreaddiagonalmatrixelementsfrom1WFfiles–  LotsofIOandwaste!

•  Computeon-the-flyusingnc_ihr_comm(chiinGW)–  Onlyworksfornorm-conserving–  Notmostefficientforoff-diagonalmatrixelements

vnk,↵ = h nk|p̂↵ + i[VNL, r̂↵]| nki<latexit sha1_base64="av8P7d5F0aV0kCw8mllFXZRLOOE=">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</latexit><latexit sha1_base64="av8P7d5F0aV0kCw8mllFXZRLOOE=">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</latexit><latexit sha1_base64="av8P7d5F0aV0kCw8mllFXZRLOOE=">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</latexit><latexit sha1_base64="av8P7d5F0aV0kCw8mllFXZRLOOE=">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</latexit>

10


DFPTrou;nes:load_spin_hamiltonianload_spin_rf_hamiltoniangetgh1c_setupgetgh1c

11



DFPTrou;nes:

load_spin_hamiltonian

load_spin_rf_hamiltonian

getgh1c_setup

getgh1c

Objectdkkop_t:ddkop_setup_spin_kpoint

ddkop_apply

ddkop_get_velocity

Canbereusedindifferentcontexts:transport,chi,

interpola;on,etc…

WFsinmemory->recompu;ngbe1erthanIO

SameforEPHmatrixelements11


Transportcomputa;ondriver

µ↵�("F , T ) =e

n("F , T )⌦

X

n

Zdk


✓�@f("nk, "F , T )

@"

◆


12


µ↵�("F , T ) =e

n("F , T )⌦

X

n

Zdk


✓�@f("nk, "F , T )

@"

◆


µ↵�("F , T ) =e

n("F , T )⌦

ZK(!, "F , T )

✓�@f(!, "F , T )

@"

◆d!

<latexit sha1_base64="If4RKd4y8pHl8pp7dChC6k4JkVA=">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</latexit>

K(!, "F , T ) =X

n

Zdk

⌦BZvnk,↵vnk,�⌧nk("F , T )�(! � "nk)

<latexit sha1_base64="Zn64bznB2Q5DwEgFh/Y+rUd2Mgs=">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</latexit>

•  Canusethetetrahedronmethod!12


•  Newtransportcomputa;ondriverstar;ngfrom*_SIGEPH.ncfile

•  Computedautoma;callyaxersigmaphwheneph_task -4!

•  Computeconduc;vity,mobility,Seebeck.•  Writeto*_TRANSPORT.ncnetcdffile•  AnalyzeresultsusingAbipy

12

Transportcomputa;onworkflow

WFK(k=q)

DEN


DVDB DDB

SIGEPH

mrgdv mrgddb

13

Transportcomputa;onworkflow

WFK(k=q)

DEN


DVDB DDB

SIGEPH

WFK(k>q)

mrgdv mrgddb

TRANSPORT

WFKFINE(k>q)

KERANGE

13

Unittests

•  Testindividualrou;nesoutsideoftheirnormalscope95_drive/m_unittests.F90!

•  Twousecasessofar:– Newtetrahedronrou;nes(realpartofSE)–  Implementsymkptandlistkkrou;neswithbe1erscalingforourapplica;ons

•  Workinprogress…

14

Lessonslearned•  IOisbad!LotsofIOwillquicklybecomethebo1leneckofthecomputa;on

•  FLOPSarecheapRe-compu;ngisbe1erthanIO

15

Lessonslearned•  IOisbad!LotsofIOwillquicklybecomethebo1leneckofthecomputa;on


•  TreatintegralsnicelyWhenperformingdifficultBZintegralsprefertetrahedron(orbe1er)

15

Lessonslearned•  IOisbad!LotsofIOwillquicklybecomethe

bo1leneckofthecomputa;on


•  TreatintegralsnicelyWhenperformingdifficultBZ

integralsprefertetrahedron(orbe1er)

•  SavememoryAimto2GBpercore,distribute

memory,recompute,avoidunnecessaryalloca;ons

15

Lessonslearned•  IOisbad!LotsofIOwillquicklybecomethe

bo1leneckofthecomputa;on


•  TreatintegralsnicelyWhenperformingdifficultBZ

integralsprefertetrahedron(orbe1er)

•  SavememoryAimto2GBpercore,distribute

memory,recompute,avoidunnecessaryalloca;ons

•  CheckpointisgoodSavework,allowrestart

15

Furtherwork

1.  SpecialtreatmentofFröhlichmatrixelements2.  Improveconvergenceofmobilitywithk-points

usingquadra;ctetrahedronmethod3.  Fasterandscalablek-pointsmachinery:

symkptandlistkk!

16

Acknowledgments

Super

Post :

Funding :

Geo�roy Hautier

Henrique Miranda

F.R.S.-FNRS

Gian-Marco Rignanese

Matteo Giantomassi

Institute :

Institute of Condensed Matterand Nanosciences

18

Thankyou!

computing the intrinsic mobility of electrons and holes in semiconductors · 2019. 12. 5. ·...

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