Supplementary Information to: "Low electronpolar. of high carrier mobility in methylammonium

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1 Elctroc Supplmtary Matral ES or Physcal Chmstry Chmcal Physcs. Ths joural s th Owr Socts 6 Supplmtary ormato to: "Low lctropolar optcal phoo scattrg at th udamt o hgh carrr moblty mthylammoum lad-odd provsts." A. Flpptt, A. Matto, C. Caddo, M.. Saba, ad P. Dlugas sttuto Oca d Matral, CNR-OM SLACS Caglar, Cttadlla Uvrstara, Mosrrato 94- CA, taly. Scuola trazoal d Stud Supror Avazat - Va Booma 65, Trst, taly.. Rlaxato tm: quatum mchacal modlg For our lctroc trasport calculatos w usd modl xprssos or th scattrg rlaxato tm, dpdt o carrr rgy ad tmpratur. Ths xprssos ar drvd rom a quatum mchacal tratmt wth som sutd approxmatos, amly parabolc bad bhavor but cludg asotropc masss or th lctroc bad structur ad smpl paramtrzatos or th phoo bads lar dsprso or acoustc phoos, xd rgs or polar optcal phoos. W cludd thr scattrg procsss: mpurty scattrg S, acoustc phoo scattrg APS, ad polar optcal

2 phoo scattrg POS. Th total scattrg rat s thus obtad accordg to th Matthsss rul, as: rl S APS POS th ollowg, w rst drv th gral quatum mchacal xprsso or th rlaxato tm cas o lastc scattrg, ad th th spcc ormulas or th thr procsss mtod abov. Fally, w dscrb our calculato or th dormato pottals o MAP, whch s a y paramtr or th modlg, ad at our owldg ot yt rportd ltratur.. Gral xprsso or lastc scattrg Bloch-Boltzma trasport starts rom th assumpto that th ma r path o th lctro s much largr tha th charactrstc lctro wavlgth. ths hypothss, v prsc o scattrg out o qulbrum th crystall momtum ca b stll assumd as good quatum umbr ad th lctro stll occups a Bloch stat ; th acto o scattrg s thus movg th carrr rom to assumg tra-bad scattrg oly. Th chag o dstrbuto ucto du to collsos s: t col - d W W whr th trms ar or scattrg to or out o, W th scattrg rat, ad Paul prcpl has b accoutd. At qulbrum th t collso rat vashs prcpl o dtald balac: W 3 W Also th Boltzma dstrbuto ca b wrtt, to th lar ordr th lctrc ld:

3 4 ; g v E Atr substtuto o Eq.3 ad 4 Eq. ad som mapulatos, t ca b show that th collso rat cas o lastc scattrg ca b xprssd through th dto o a rgy-dpdt rlaxato tm: 5 t col whr: 6 cos, W 3 * * * z y x d m m m m. mpurty scattrg. Followg th Broos-Hrrg dvlopmt 3D, th probablty that th lctro udrgos a momtum chag rom to du to th scattrg wth th Coulomb pottal o a ozd mpurty wth oc charg Z ad coctrato s: 7 4, W q Z Clarly th dlta ucto xprsss th rgy cosrvato; s th dlctrc costat, ad T K q B / th squard Dby scrg lgth. Substtutg Eq.7 to Eq.6 ad solvg th tgral assumg parabolc bad modlg, w obta th wll-ow Broos-Hrrg ormula: 8 /8 ~ 8 ~ log 4 ~, 3/ m q q m m Z T S

4 m~ * * * whr / 3 m m m x y z s th gomtrcally avragd bad mass..3 Acoustc phoo scattrg. Strctly spag, th scattrg wth acoustc phoos s a lastc procss; howvr, sc acoustc phoo rgs ar typcally small, w ca stll us Eq.6 to d a rlaxato tm. From th Frm gold rul: N, N N H N E E 9 W t, N, N whr, N ad, N ar lctro momtum ad phoo occupacy o tal ad al stats, rspctvly. W ca modl th log-wavlgth lmt o th acoustc wavs as a homogous stra: H t r r whr th lar chag o th bad dg wth a appld stra s calld dormato pottal D: D whr =x, y, z, ad th stra ca b xprssd as th lar-ordr drvatv o th spac dlatato: r u r r Th, substtutg Eqs.- to Eq.9 ad Eq.6, ad assumg parabolc bad modlg ad lar acoustc phoo dsprso, t ca b show that th rlaxato tm or th acoustc phoo scattrg th dormato pottal approxmato s:

5 ~ 3/ m K BT D 4 v ~ / m KBT 4 / AP s / 3 whr s th mass dsty, s th soud vlocty, m ~ th gomtrcally avragd bad mass. Emsso ad absorpto cotrbutos ar both cludd Eq.3. Th d trm to th rght s th pzolctrc scattrg,.. th cotrbuto du to th chag polarzato ducd by a lastc stra cas o polar matrals; s a admsoal paramtr calld lctromchacal couplg..4 Polar optcal phoo scattrg. Th scattrg wth optcal phoos s sstally lastc, thus th smpl xprsso o Eq.6 caot b appld. Nvrthlss, t was show by Rdly,3 that a approprat gralzato to th rlaxato tm ca stll b urshd. Howvr, th xact tratmt s too cumbrsom to b usd practcal calculatos, thus w rcur to a approxmato whch s stll suctly accurat or our mas. Followg Rdly, or a rlvat logtudal optcal phoo o rgy w ta: POS POS 3/ Z C 3/ Z C A B 4 5 whr: A N / sh / 6

6 B C N / cosh / 7 N sh N / / cosh 8 Z / * m 4 9 Eq.4 s qut accurat or, whl Eq.5 s xact th lmt, ad a rasoabl approxmato or. Wh mor tha a sgl phoo s mportat or th scattrg procsss, thy ca b tratd as dpdt scattrrs ad th summd accordg to Matthsss rul..5 Paramtrs. Th paramtrs trg th rlaxato tm xprssos or MAP ar larg porto wll stablshd ltratur: s =3.x 5 cm/s, =6 at room- =6.5, m ~. m, =8.9 cm -. Also, w ta =, sc at room-t whch s th tmpratur rgo o mor trst or our mas th pzolctrc rspos o MAP ca b assumd small or dscardabl. O th othr had, at our owldg, a valuato o th dormato pottals ltratur was mssg, thus w prormd accurat calculatos to stablsh th valus o D or lctros ad hols th orthorhombc structur.

7 . Dormato pottals o MAP. W calculatd th dormato pottals or MAP by t drcs,.. applyg % stras alog th thr axs o th orthorhombc structurs, ad valuatg th chags ducd at th bad xtrms. Th bad structur rsults ar rportd Fg.S. th calculatos udr stra th atoms ar pt xd to thr groud-stat crystal coordats, ordr to clud oly th chag du to a pur spac dlatato or cotracto. Also, th spurous rgd-sht cts o th bad structur du to lctrostatc pottal chag udr stra has b xtractd out Fg.S by carul ralgmt o th dpst Pb 5d sm-cor lvls. Fgur S: Bad structur o orthorhombc MAP th qulbrum structur blac ls ad udr a appld postv rd ls ad gatv gr ls stra o %. Pals a, b, c dsplay coducto bads or stras appld alog th a, b, c axs o th orthorhombc cll, rspctvly; pals d,, ar or stras appld alog a, b, c but or valc bads. Th valc bad top ad coducto bad bottoms at pot calculatd or th orthorhombc groud-stat structur dscrbd R.4 ad th structur udr postv ad gatv % stra, ar rportd Tab.S. Th groud-stat VBT s xd at zro, thus th CBB quals th bad gap. Th dormato pottal s calculatd as th chag th bad rgy xtrma or th appld stra, avragg ovr postv ad gatv stras; thus or coducto:

8 D j.. CBB CBB CBB CBB or j=a, b, c, ad aalogous quato or th VBT. Tabl S: Calculatd valc VBT ad coducto CBB bad xtrms or MAP th groud-stat orthorhombc structur ustrad ad udr appld % stra alog th axs o th orthorhombc cll. Th dormato pottals D avrag th bad shts or postv ad gatv stra. D s th dormato pottal avragd ovr th 3 axs. Th bad gaps or th stras alog th thr drctos ar also rportd. VBT V CBB V gap V ustrad a axs = % = +% D a b axs = % = +% D b c axs = % = +% D c D From Tab.S w s that th dormato pottals ar always gatv,.. a postv stra lattc strtchg sht dow both VBT ad CBB rgs alog ach drcto, whl a gatv stra lattc shrg sht VBT ad CBB to lowr rgs. Ths s rasoabl, sc th at-bodg Pb 6s, 6p stats ar stablzd by a cras o Pb- dstac, ad dstablzd by a Pb- shrg. Notc that th sg o th dormato pottal has o cts o th scattrg rat wth th acoustc phoos, whch dpds o D Eq.3. Also, sc w ar trstd th 3D-avragd trasport proprts, w cosdrd th 3D-avragd dormato pottals, also rportd th Tabl: -9.3 V ad -7.3 V or hols ad lctros, rspctvly, ar thos usd or our trasport

9 calculatos dscrbd th artcl. Ths valus ar typcal o ordary smcoductors,.g. or bul GaAs th coducto bad dormato s rportd btw -7 ad -9.5 V 5. Ths smlarty hghlghts oc mor th orgac charactr o MAP. M. Balas, ad R. F. Walls. Smcoductor Physcs ad Applcatos, Oxord Uvrsty Prss, Nw Yor,. B. K. Rdly, J. Phys.: Cods. Mattr 998,, D. R. Adrso, N. A. Zahlu, M. Babr, B. K. Rdly, ad C. R. Btt, Phys. Rv B,, 63, A. Flpptt ad A. Matto. Hybrd provsts or photovoltacs: sghts rom rst prcpls. Phys. Rv. B, 4, 89, X. Za, P. N. Butchr, ad M. J. Kary, Phys. Rv. B 994, 49, 75.