Supplementary Information

Transcription

1 Supplementary Informaton SUPPLEMENTARY FIGURES Energy of electrons ev E C E F P cathode 1 2 Free carrer concentraton 1 25 m -3 E V anode y x Supplementary Fgure S1: Two-dmensonal potental profle. and bendng n the vcnty of head-to-head and talto-tal scdw, and electrodes wth dentcal work functon as ato 3. Polarzaton dvergence at head-to-head scdw nduces band bandng nto an extent when the bottom of the conducton band, E C, drops bellow the Ferm level E F, allowng the presence of screenng free electrons. Top of the valence band, E V, rses over the Ferm level at tal-to-tal scdw, screenng polarzaton charge by holes. Electrodes anode and cathode dctate flat potental profle whch results n formaton of transent regons between scdw and electrodes. These regons are exposed to hgh bult-n electrc felds whch nucleate wedge domans at junctons between head-to-head scdw and anode and tal-to-tal scdw and cathode. Delectrc gaps are created at the remanng two types of junctons. Note that ths smulaton s vald for deal defect-free materal. Densty m n n D /q N D p Dstance, x µm φ 1-1 Electrc potental, φ V Supplementary Fgure S2: Potental profle and charge-carrer and defect denstes. 1-D phase feld smulaton of charge carrer and defect denstes left axs and electrc potental φ rght axs across the head-to-head and tal-to-tal doman walls. Compensaton of polarzaton charge at head-to-head wall requres accumulaton of electrons, n, and depleton of oxygen vacances N D. The neglgble remanng densty of vacances s not onzed whch lowers the charge carrer densty, n D/q. The head-to-head wall accumulates holes p and almost fully onzed oxygen vacances N D. The electrc potental φ forms a zg-zag profle across the doman walls. The oxygen vacances almost fully replace screenng holes at the tal-to-tal doman wall after 1 1 hours wth ntal defect concentraton N D t= = 1 18 m 3. It makes the tal-to-tal walls sgnfcantly less conductve.

2 2 b a 18 A 1 µm c x A phase deg. ampltude a.u. max 1 µm z d 2 nm A x: 4 z: 4 µ m nm µm P y x Supplementary Fgure S3: Scannng probe mcrographs. The atomc force AFM and pezo-response PFM scans of [11]c sample surface near a head-to-head scdw were taken after removng the Pt electrodes. PFM ampltude, a, and phase, b, together wth change of slope at AFM topography, c, suggest that the straght boundary s 9 ferroelectrc/ferroelastc doman wall whle the curved boundary A s 18 doman wall as llustrated n d. cathode n Densty m-3 12 φ nd/q p ND -2 Electrc potental, φ V Dstance, y nm 2 Supplementary Fgure S4: Charge/defect denstes by the cathode. 1-D phase field smulaton of charge and defect denstes left axs and electrc potental along the head-to-head doman wall n the vcnty of the cathode. The electron concentraton n almost fully screens the polarzaton charge at the wall. The presence of the Pt electrode causes a drop of the electron concentraton n and potental φ n the regon 1 nm bellow the anode. Ths regon, on the other hand, accumulates oxygen vacances ND whch carry charge nd /q and holes p. 14 φb = 1 V Current A 1 φb = 2 V 1-4 φb = 3 V arrer thckness, d nm 1 Supplementary Fgure S5: Tunnelng current. Dependence of tunnelng current on barrer thckness for barrer heghts ϕb = 1, 2, 3 V. Voltage V =1 V.

3 3 1-2 Current A φ b = 1 V φ b = 2 V φ b = 3 V Voltage V Supplementary Fgure S6: Tunnelng current. Tunnelng current aganst voltage for barrer heghts ϕ b = 1, 2, 3 V when the barrer thckness s d=5 nm. 1 2 Current A d = 15 nm d = 5 nm d = 1 nm Schottky barrer heght, φ b V Supplementary Fgure S7: Tunnelng current. Tunnelng current aganst barrer heght for thcknesses d = 5, 1, 15 nm. Voltage V =1 V.

4 4 SUPPLEMENTARY TALES Parameter Value Unt Ref. α 1 T Jm/C 2 α 11 T Jm 5 /C 4 α Jm 5 /C 4 [36] α T Jm 9 /C 6 α Jm 9 /C 6 α Jm 9 /C 6 c J/m 3 c J/m 3 [36] c J/m 3 q Jm/C 2 q Jm/C 2 [36] q Jm/C 2 G Jm 3 /C 2 G Jm 3 /C 2 [36] G Jm 3 /C 2 Γ C 2 /Jms [36] ε [36] µ n, µ p.1 cm 2 /V s [4] β cm 2 /V s τ 1 ps [41] E C 3.6 ev E V 6.6 ev E D 4. ev [33] E F 3.98 ev 5.1 deally defect-free case ev N m 3 [33] N D t= m 3 deally defect-free case m 3 z 2 1 g 2 1 [33] Values of Vö moblty β and concentraton N D t= of undoped ato 3vary n lterature by many orders of magntude. We choose hgher estmate of β from [38, 42, 43] for 4 K, whch s counterbalanced wth very low estmate of N D t= compared to Refs. [22, 33]. Supplementary Table S1: Values of materal coeffcents for ato 3 used n the smulatons.

5 5 SUPPLEMENTARY NOTES Supplementary Note 1: The phase-feld model The phase-feld smulaton ncorporates couplng between ferroelectrc and wde-bandgap semconductor propertes ncludng moble defects. Model equatons are obtaned by Lagrange prncple from Helmholtz free energy densty [35]: f[{p, P,j, e j, D }] = f e bulk + f ela + f es + f grad + f ele, where P s the ferroelectrc part of polarzaton, P,j ts dervatves the subscrpt, represents the operator of spatal dervatves / x, D the electrc dsplacement and e j = 1/2u,j + u j, s the elastc stran where u s a dsplacement vector. The bulk free energy densty f e bulk [{P}] = α 1 P 2 + α e 11 P 4 + α e 12 P 2 Pj 2 + α 111 >j +α 112 P 4 Pj 2 + Pj 4 P 2 + α 123 >j s expressed for a zero stran as a sx-order polynomal expanson [36], where α, α e j, α jk are parameters ftted to the sngle crystal propertes Supplementary Table S1. The remanng contrbutons represent blnear forms of denstes of elastc energy f ela [{e j }] = 1/2c jkl e j e kl, where c jkl s the elastc stffness, electrostrcton energy f es [{P, e j }] = q jkl e j P k P l, where q jkl are the electrostrcton coeffcents, gradent energy f wall [{P,j }] = 1/2G jkl P,j P k,l, where G jkl are the gradent energy coeffcents, and electrostatc energy f ele [{P, D }] = 1/2ε ε D P 2, where ε and ε are permttvty of vacuum and relatve background permttvty, respectvely. The zero-stran coeffcents α e can be expressed n terms of usually ntroduced stress-free coeffcents α j as follows: α e 11 = α q11 q q q 12 2, 6 c 11 c 12 c c 12 α e 12 = α q11 + 2q q 11 q q c c 12 c 11 c 12 4c 44 y usng the Legendre transformaton to electrc enthalpy h[{p, P,j, u,j, φ, }] = f[{p, P,j, e j, D }] D E, where E = φ, s the electrc feld and φ the electrc potental, and usng Lagrange prncple, we can unformly express the set of feld equatons whch govern the knetcs of ferroelectrcs: h =, S3 e j,j h E 1 P h Γ t P,j,,j P 2 P 6. j S1 S2 = qp n + n D, S4 = h P. Equaton S3 defnes the mechancal equlbrum whle nerta s neglected. Equaton S4 represents Gauss s law of a delectrc ncludng a nonzero concentraton of free electrons n, holes p, and charge densty of onzed donors n D. Equaton S5 s the tme dependent Landau-Gnzburg-Devonshre equaton [37] whch governs the spatotemporal evoluton of spontaneous polarzaton wth knetcs gven by coeffcent Γ. Couplng between the ferroelectrc/ferroelastc system wth ts semconductor propertes s ntroduced by consderng a nonzero densty of free carrers electron-hole n the electrostatc equaton S4. The dstrbuton of free carrers s governed by contnuty equatons: S5 q n t + J n, = qr n, S6 q p t + J p, = qr p, S7

6 where electron and hole currents J n µ n qne + n, and J p and J p, respectvely, are governed by drft and dffuson as follows: J n = = µ p qpe p,. Here µ n and µ p are electron and hole mobltes, respectvely. In the frst step of calculaton we analyze only the statonary soluton n thermal equlbrum. In ths step we can ntroduce the computatonally convenent form of recombnaton rates R n and R p as follows: R n = n n /τ and R p = p p /τ, where τ s lfe-tme constant and n and p are electron and hole concentratons n thermal equlbrum: n = NF 1/2 E C E F qφ p = NF 1/2 E F E V + qφ Here F 1/2 s the Ferm-Drac ntegral. Densty of states s gven by the effectve mass approxmaton: 3 meff 2 N 2, 2π 2 where effectve mass m eff = cm e s assumed equal for electrons and holes. Results presented n the graphs correspond to c =.117,.e. N = 1 24 m 3 [33]. The charge densty of onzed donors s obtaned as n D = qzfφn D, where z s the donor valency,,. fφ = g exp ED E F qφ s the fracton of onzed donors wth the donor level E D and the ground state degeneracy of the donor mpurty level g [33]. The donor densty N D evolves through dffuson, N D t WD βn D + qzfφφ =, S8 N D where β s the donor moblty [38], and W D s the contrbuton to the free energy due to defects whch s assumed to be the usual free energy of mxng at small concentratons [39]. Values of the smulaton parameter are ntroduced n Tab. S1. The two-dmensonal smulatons Fg. 4, Supplementary Fg. S1 and squares n Supplementary Fg. S4 were performed wth zero defect concentraton N D =, on a smulaton doman of 2 6 µm 2 Fg. 4 and Supplementary Fg. S4 and 6 6 µm 2 Supplementary Fg. S1. The numercal soluton of equatons S3-S7 on the defned subdoman was performed by a fnte element method wth lnear trangular elements of sze 4 nm n the vcnty of doman walls and 4 nm nsde domans. The boundary condtons are set to potental φ = ϕ = n Fg. 4 and Supplementary Fg. S1, and φ = ϕ =.8 V n Supplementary Fg. S4, zero free-carrer flux, zero stress, and zero polarzaton gradent. Perodc boundary condtons n x-drecton were appled n case of smulaton shown n Supplementary Fg. S1. The smulatons start from ntal condtons that are defned as zero for all varables except polarzaton whch s P = 21, 1P for x < and P = 2 1, 1P for x >, P =.262 C/m 2 n the reference frame of Fg. 4. The ntal condton n case of Supplementary Fg. S1 s P = 21, 1P for x > 1.5 µm and P = 2 1, 1P for x < 1.5 µm. The smulatons reach thermal equlbrum n < 5 ns and gves solutons for the spatal dstrbuton of polarzaton P, mechancal dsplacement u, electrc potental φ, and concentratons of electrons n and holes p. The calculaton wth appled voltage uses the thermal equlbrum as an ntal condton and contnues wth recombnaton gven by R n = R p = np/τn + p + G where G = 1 2 s 1 s a small free carrer generaton. The boundary potental s appled as φ = V y/6 1 6 V. The one-dmensonal calculaton nvolves also equaton S8 whch gves drft and onzaton of donors and excludes the elastcty equaton S3 by puttng q j =, α e 11 = α 11 and α e 12 = α 12. Snce the defect drft s orders of magntude slower than the polarzaton changes, the smulaton s splt nto two steps. Frst, the dstrbuton of polarzaton P 1, potental φ, and free carrer denstes n, p are calculated and, second, the polarzaton s frozen and drft of donors N D s calculated. The thermal equlbrum s reached after 1 5 s. The result of ths smulaton s shown n Supplementary Fg. S2. 6

7 The one-dmensonal calculaton along the doman wall Supplementary Fg. S4 uses the polarzaton charge dstrbuton extracted from two-dmensonal smulaton and contnues wth calculaton of potental φ, free carrer denstes n, p, and donor densty N D. The results are shown n Supplementary Fg. S4 for head-to-head wall n the vcnty of the cathode. 7 Supplementary Note 2: Electron tunnelng To estmate the tunnelng current we assumed the trangular potental barrer and the Wentzel - Kramers - rlloun approxmaton of the transmsson probablty for the Fowler-Nordhem tunnelng. Tunnelng current s calculated as [44] J t = A 4πm eq h 3 Emax E mn T C ES f EdE, S9 where A s the effectve area of the tunnelng assumed 1 nm 2 µm, E mn s the bottom of the conducton band n metal, E max s the top of the potental barrer and h s the Plank constant. The supply functon S f E s for Ferm-Drac dstrbuton calculated as 1 + exp S f E = ln 1 + exp E E F1 E E F2 where E F1 s the Ferm level poston n the metal when the external voltage s appled and E F2 s the Ferm level at the head-to-head scdw. The transmsson probablty for the Wentzel-Kramers-rlloun approxmaton s 2me d T C E = exp 4 q ϕ E3/2 3 qv where d s the barrer thckness. The tunnelng current equaton S9 s calculated numercally and the results summarzed n Supplementary Fgs. S5-S7. The Fowler-Nordhem tunnelng wthout assstance of defects exceeds the measured currents for barrer thckness d 9 nm, and barrer heght ϕ b 2V. SUPPLEMENTARY REFERENCES [35] L, Y. L. and Chen, L. Q. Temperature-stran phase dagram for ato 3 thn flms. Appl. Phys. Lett. 88, [36] Hlnka, J., Ondrejkovc, P., and Marton, P. The pezoelectrc response of nanotwnned ato 3. Nanotechnology 2, [37] Semenovskaya, S. and Khachaturyan, A. G. Development of ferroelectrc mxed states n a random feld of statc defects. J. Appl. Phys. 83, [38] Yoo, H. I., Chang, M. W., Oh, T. S., Lee, C. E., and ecker, K. D. Electrocoloraton and oxygen vacancy moblty of ato 3. J. Appl. Phys. 12, [39] Porter, D. and Easterlng, K. Phase Transformatons n Metals and Alloys, Second Edton. Taylor & Francs, [4] Yoo, H. I., Song, C. R., and Lee, D. K. Electronc carrer mobltes of ato 3. J. Eur. Ceram. Soc. 24, [41] Smrl, A. L., et al. Pcosecond photorefractve effect n ato 3. Opt. Lett. 12, [42] engugu.l. Electrcal phenomena n barum-ttanate ceramcs. J. Phys. Chem. Solds 34, [43] El Kamel, F., Gonon, P., Ortega, L., Jomn, F., and Yangu,. Space charge lmted transent currents and oxygen vacancy moblty n amorphous ato 3 thn flms. J. Appl. Phys. 99, [44] Duke, C. Tunnelng n solds. Academc Press, 1969.