Landau-Khalatnikov Theory for the Hysteresis Loops of a Ferroelectric Thin Film

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1 CHINESE JOURNAL OF PHYSICS VOL. 5, NO. 3 Jun 4 Landau-Khalatnikov Thory for th Hystrsis Loops of a Frrolctric Thin Film Lian Cui,, Zhiyou Han, Zhongyang Qiu, Shujuan Hao, Xinxin Li, JiXin Ch, Quan Xu, and Tianquan Lü 3 Dpartmnt of Physics, Daqing Normal Univrsity, Daqing 637, China Ordnanc Enginring Dpartmnt, Air Forc Aviation Univrsity, Changchun, 3, China 3 Cntr of Condnsd Mattr Scinc and Tchnology, Harbin Institut of Tchnology, Harbin 5, China (Rcivd July 4, 3; Rvisd Fbruary 9, 4) W us th Landau-Khalatnikov quation of motion to illuminat th hystrsis bhavior of a frrolctric thin film with surfac transition layrs. Our modl can rflct a mor ralistic situation compard to th prvious tratmnts of th sam problm using th socalld xtrapolation lngth. Th influnc of th surfac transition layr on th hystrsis loops undr a driving lctric fild is discussd in dtail. Th rsults show that th surfac transition layr can induc a transfiguration of th hystrsis loops in a frrolctric thin film. DOI:.6/CJP.5.9 PACS numbrs: Ac, p., 77.8.Fm I. INTRODUCTION Frrolctric thin films hav attractd incrasing attntion for applications in frrolctric mmoris in rcnt yars du to thir bistabl polarizations [ 3]. Thus th study of th hystrsis bhavior of frrolctric thin films has bcom a hotspot in both xprimnts and thoris. Exprimntally rsarchrs hav adoptd som tchniqus, such as pizolctric forc microscopy (PFM) and scanning prob microscopy (SPM), for studying th hystrsis action of frrolctric thin films, and obtaind crtain significant phnomna [4 7]. In ordr to dscrib th dynamical hystrsis bhavior, a numbr of thortical modls hav bn dvlopd. Th Kolmogorov-Avrami-Ishibashi (KAI) thory first intrprtd th polarization rvrsal proprtis of frrolctric thin films in th light of crystal growth thory [8]. Othr modls wr prsntd subsquntly to dscrib th hystrsis bhavior of frrolctric systms: a non-kai modl put forward by Tagantsv t al. [9], and a discrt modl basd on th Landau-typ fr nrgy [ 3]. Howvr, in ths modls, th surfac ffcts of th hystrsis bhavior in a frrolctric thin film wr not discussd. At prsnt, th Tilly-Zks modl of th Landau-Dvonshir fr nrgy xprssion by adding a Landau-Khalatnikov quation of motion has bn usd to study th phnomna of hystrsis bhavior in frrolctric thin films [8, 9, 4 6]. In this modl, th surfac ffct is includd in th fr nrgy rflctd by th so-calld xtrapolation lngth, which Elctronic addrss: cl_9834@6.com 9 c 4 THE PHYSICAL SOCIETY OF THE REPUBLIC OF CHINA

2 9 LANDAU-KHALATNIKOV THEORY FOR THE HYSTERESIS... VOL. 5 was proposd by Krstchmr and Bindr [7], that th polarization at th surfac can ithr b supprssd or nhancd. For xampl, Muslh t al. [8] xtndd th Tilly-Zks modl for a frrolctric thin film to study th surfac ffcts on proprtis of hystrsis bhavior, i.., a corciv fild. Th rsults illustratd that th corciv fild dcrass with dcrasing xtrapolation lngth, which supprsss th polarization at th surfacs, whil as th xtrapolation lngth dcrass, th corciv fild incrass, which augmnts th polarization at th surfacs. In this schm, th uncrtainty is that th choic of th xtrapolation lngth has a crtain dgr of arbitrarinss. And a suddn transition from th surfac to th intrior is assumd, which is not ralistic bcaus th transition of structur should b gradual. Subsquntly, to avoid th insufficincis brought by xtrapolation lngth, th surfac transition layr, rprsnting th gradual transition of structur, was first introducd in a frrolctric thin film by Lü and Cao [8]. Thy hav xpoundd th ffcts of th surfac transition layr on th static proprtis (polarization distribution, phas transition tmpratur, dilctric suscptibility, tc.) of a frrolctric thin film [9 ]. This modl rflcts a mor ralistic situation in comparison with that using th xtrapolation lngth. Howvr, as far as w ar awar, th dynamical proprtis of polarization rvrsal in a frrolctric thin film ar not availabl in th litratur, and hr w fill this gap. Th prsnt study of this papr is intndd to shd som light rgarding th influncs of th surfac transition layr on th hystrsis charactristics of a frrolctric thin film undr th action of a driving fild, making us of th Landau-Khalatnikov quation of motion. II. THE MODEL AND THEORY A frrolctric thin film with surfac transition layr locatd btwn two mtallic lctrods is considrd as th stacking of thin layrs, as shown in Fig.. Th proprtis of vry layr with finit thicknss z ar homognous in th plans paralll to th surfac, and th variation is only along th z-dirction. Th numbr of this layr stack is N. Thus, th whol thicknss of th frrolctric thin film is L = N z. If th origin is locatd at th lft surfac, w st z = thn an arbitrary layr locatd at a position z can b rcognizd by th indx i, i.. z = i z ( i N). To study this systm, a modifid Landau-typ fr nrgy for th systm nds to b stablishd. According to Rf. [9], w introduc a scond powr of polarization for ach layr in th Landau-typ fr nrgy, and assum its cofficint to b a function of position in ordr to rflct th contribution of surfac transition layrs. So th fr nrgy of th film can b xprssd by th Landau-typ fr nrgy xprssion [5, 6, 9] G = N i= [ A (T T c) Pi + Bψ (z) Pi + 4 CP i 4 + K (P i P i ) EP i ], ()

3 VOL. 5 LIAN CUI, ZHIYOU HAN, ZHONGYANG QIU, ET AL. 93 FIG. : Gomtric structur of th frrolctric thin film undr study. with th boundary condition [8] ( ) dp = whn z = and z = L, () dz whr P i is th polarization of th nth layr; th cofficints A, B, C, and K ar indpndnt of tmpratur T and position z; T c is th transition tmpratur of th bulk frrolctric; E is th applid fild. A nw trm (th scond trm) is addd in Eq. () in ordr to rflct th contribution of th surfac transition layr. Hr, B > dnots a dcras in th local polarization at th surfacs, and B < xprsss an incras in th local polarization at th surfacs. Th distribution function ψ(z) rflcts th surfac ffct. Th K (P i P i ) trm in Eq. () rprsnts th coupling ffct btwn th nighboring layrs. Th polarization gradint (dp/dz) can b includd in th finit diffrnc xpansion of th K (P i P i ) trm [5]. Th tim variation of th polarization during switching can b dscribd by th Landau-Khalatnikov quation as follows: γ P i t = G P i = A (T T c ) P i Bψ(z)P i CP 3 i + K (P i+ + P i P i ) + E, (3) whr γ is th cofficint of viscosity, which causs a dlay in domain motion. It is convnint to rscal th variabls into dimnsionlss forms. W st f i = P i /P with P = AT c /C, t = T/T c, δ = B/B with B = AT c, ζ = z/ξ with ξ = K/AT c, l = L/ξ, η = tat c /γ, k = K/AT c, σ = (ε AT c ), and = E/E with E = P /ε whr ε is th dilctric constant of th vacuum. So w can obtain th normalizd quation blow: f i η = ( t) f i δψ(ζ)f i f 3 i + k (f i+ + f i f i ) + σ. (4) Th corrsponding rscaling boundary condition is ( ) df = whn ζ = and ζ = l. (5) dζ

4 94 LANDAU-KHALATNIKOV THEORY FOR THE HYSTERESIS... VOL. 5 Anothr indx j is introducd to rprsnt th discrtizd tim η = j η. Th tim dpndnt polarization at ach layr is dnotd as f i,j. Through th forward diffrnc in tim for th lft-hand sid of Eq. (3), th diffrnc quation for f i,j can b writtn as: f i,j+ = f i,j + η [ ( t)f i,j δψ(ζ)f i,j f 3 i,j + k (f i+,j + f i,j f i,j ) + σ j ]. (6) According to th finit diffrnc for th drivativ (df/dζ) in th scond-ordr of smallnss, Eq. (5) bcoms { f, = f,j = f,j ζ =, (7) f N,j = f N,j = f N,j ζ = l. Th ovrall polarization of th whol film is dfind as f j = N N f i,j, i= (8) and th xtrnal lctric fild is dtrmind by ( ) πη j = m sin = m sin (πfη), (9) T whr m, T, and f ar th normalizd amplitud, priod, and frquncy, rspctivly. As a continuous structur, th proprtis of th surfac transition layr continuously chang to match th proprtis of imprfct surfac at on nd and prfct intrior at th othr nd. Thus, th dscription for th surfac transition layr is position dpndnt. Thn w can choos a simpl function (xponntial function, trigonomtric function, or any propr function) to dscrib th variation of th intractions. Sinc w do not hav any xprimntally masurd data availabl on th ffcts of th surfac transition layr, a simpl form is chosn for th distribution function ψ i (ζ). Th pculiar choic of ψ i (ζ) dos not affct th gnrality of th rsults and conclusions [9]: ψ i (ζ) = ( i ζ u λ ) i < u ζ, u ζ i N u ζ, ( i ζ (N ζ u ) λ ) N u ζ < i N, whr u and u ar th thicknss of th lft and right surfac transition layr, rspctivly. Paramtr λ (λ ) rflcts th variation intnsity of th fr nrgy dnsity of ach layr nar th lft (right) surfac. For simplicity, th two surfac transition layrs ar assumd symmtric, namly, u = u = u, λ = λ = λ. In this papr, w considr th rvrsal dynamics undr a driving fild. As statd, th polarization in th initial stat is th ngativ rmnant polarization, that is to say, th initial polarization is at zro fild; thn at th tim η = th fild is switchd to a positiv valu. ()

5 VOL. 5 LIAN CUI, ZHIYOU HAN, ZHONGYANG QIU, ET AL. 95 III. NUMERICAL RESULTS AND DISCUSSIONS Th hystrsis loops of a frrolctric thin film with diffrnt thicknss of surfac transition layrs for δ =. and δ =. ar shown in Fig.. In Fig. (a), w st δ =., i.., th surfac polarization is dprssd rlativ to th intrior of th film. It can b clarly sn that with incrasing th thicknss of th surfac transition layr, th avrag rmnant polarization and corciv fild of th film dcras. Th dcras of th avrag rmnant polarization is rmarkabl. A thickr surfac transition layr maks th frrolctricity of th whol film wakn. Th cas of δ =., i.., th surfac polarization is nhancd rlativ to th intrior of th film is givn in Fig.. Th avrag rmnant polarization and corciv fild of th film augmnt obviously as th thicknss of surfac transition layr incrass. In a word, th strongr surfac ffcts can mak th avrag rmnant polarization and corciv fild dcras for surfac polarization dprssd rlativ to th intrior or incras for surfac polarization nhancd rlativ to th intrior. Th abov rsults ar in good agrmnt with thos obtaind in Rfs. [8, 9]. It is noticd that in Fig., whn th thicknss of surfac transition layr is largr (u =.7 and u =.9), th shap of th hystrsis loops changs, that is to say, svr distortion occurs. Th rason for this phnomnon is that th polarization switching in th surfac transition layr and intrior of th film is not simultanous. Th surfac transition layr polarization is largr than that in th intrior of th film. And th polarization of th thin layrs includd in th surfac transition layr nhancs gradually (from u to and from u to L). Th lctric fild, which just inducs th polarization switching for th intrior of th film, cannot rsult in th rvrsal of th polarization of th whol film. Incrasing lctric fild is ndd to rvrs in turn th polarization of th thin layrs (from u to and from u to L) comprisd in th surfac transition layr. Thus, th thickr th surfac transition layr thicknss is, th mor distortd th hystrsis loop bcoms. This phnomnon, as far as w know, has not bn invstigatd. Th ffct of λ on th hystrsis loops of th film is illustratd in Fig. 3. Whn δ =., th avrag rmnant polarization rducs obviously with dcrasing λ, as shown in Fig. 3(a). Th small valu of λ implis a strong surfac ffct, according to Eq. (7). Th corciv fild of th film changs unconspicuously. For δ =., th avrag rmnant polarization and corciv fild of th film incrass vidntly as λ rducs. Morovr, whn th valu of λ is smallr (λ =.4), th varity of shap for th hystrsis loops appars. Th rason is similar to that of Fig.. In comparison with Fig., th influnc of th thicknss of th surfac transition layr on th avrag rmnant polarization and corciv fild of th film is mor obvious. Figur 4 shows th influnc of th thicknss of th film on th hystrsis loops for δ =. and δ =.. Figur 4(a) rvals that th avrag rmnant polarization and corciv fild rduc with dcrasing thicknss of th film. Th rsult is in good agrmnt with that obsrvd by Wang t al. [3], and also is consistnt with th thortical rsult obtaind in Rf. [9]. For δ =. in Fig. 4, both th avrag rmnant polarization and th corciv fild augmnt as th thicknss of th film rducs, which coincids wll with that rportd by Ong t al. [9]. According to Fig. (), th physical rason for th abov

6 96 LANDAU-KHALATNIKOV THEORY FOR THE HYSTERESIS... VOL. 5 P/P - =. k f=. z=. =. =4. l=5. m =. =.3 =. u=.5 u=. P/P - =. k f=. z=. =. =4. l=5. m =. =.3 =-. u=.7 u=.5 u=.9 - u=.5 (a) FIG. : Hystrsis loops for various valus of th surfac transition layr thicknss with (a) δ =., δ =.. =. k f=. z=. =. =4. l=5. m =. u=. =. =.3 =. k f=. z=. =. =4. l=5. m =. u=. =-. =. P/P - =.6 =. P/P - =.6 =.4 - (a) FIG. 3: Hystrsis loops for various valus of λ with (a) δ =., δ =.. phnomna is that with rducing th thicknss of th film, th rlativ thicknss of th surfac transition layr to th whol film bcoms largr. Th hystrsis loops of th film with diffrnt valus of lctric fild amplitud m for δ =. and δ =. ar illustratd in Fig. 5. From Fig. 5 (a), w can s that th corciv fild of th film incrass vidntly with incrasing m, which is consistnt with th thortical rsult obtaind by Omura t al. []. Th avrag rmnant polarization of th film dos not incras rmarkably. For th cas of δ =., th rsult is similar to Fig. 5 (a). A main point of diffrnc is that a distortion of th hystrsis loops of th film mrgs. Th rason is similar to that of Fig.. Th impact of th tmpratur on th hystrsis loops of th film for δ =. and δ =. is shown in Fig. 6. It is wll known that th systm of a frrolctric tnds

7 VOL. 5 LIAN CUI, ZHIYOU HAN, ZHONGYANG QIU, ET AL. 97 P/P - - =. k f=. z=. =. =4. m =. u=. =.3 =. l=3 l=5 l= (a) P/P - - =. k f=. z=. =. =4. m =. u=.8 =.5 =-. l=4 l=5 l= FIG. 4: Hystrsis loops for various valus of th frrolctric thin film thicknss with (a) δ =., δ =.. =. k f=. z=. =. =4. l=5. u=. =.5 =. m =.5 =. k f=. z=. =. =4. l=5. u=. =.5 =-. m =.5 P/P m =. P/P m =. - m =. - m =. - - (a) - - FIG. 5: Hystrsis loops for various valus of m with (a) δ =., δ =.. to b disordrd at highr tmpratur, so that th systm nds a lowr applid lctric fild to switch from on stat to anothr, and th avrag rmnant polarization is lowr at highr tmpratur. From Fig. 6 (a), w can s that th avrag rmnant polarization and th corciv fild of th film dcras with incrasing tmpratur. Th rsult is in good agrmnt with that obsrvd by Zhang t al. [7], and also is consistnt with th thortical rsult rportd in Rf. []. For δ =., th rsults ar similar to thos in Fig. 6(a). But, th svr transmutation of th hystrsis loops of th film appars. Th rason is that th rvrsal of th polarization of th diffrnt thin layrs in th surfac transition layr and intrior of th film is asynchronous. Th lowr tmpratur, which maks th systm tnd to b ordrd, can rsult in rvrsing th polarization of th whol film with highr lctric fild. Hnc, th distortion of th hystrsis loops is svr whn th tmpratur is lowr.

8 98 LANDAU-KHALATNIKOV THEORY FOR THE HYSTERESIS... VOL. 5 k m =. z=. f=. =. =4. l=5. u=. =.5 =. =.4 k m =. z=. f=. =. =4. l=5. u=.8 =.5 =-. P/P - =. =.8 P/P - =.8 =.4 = (a) FIG. 6: Hystrsis loops with diffrnt tmpraturs for (a) δ =., δ =.. IV. CONCLUSIONS Th hystrsis loops of a frrolctric thin film with surfac transition layrs hav bn discussd within th framwork of th Landau-Khalatnikov quation of motion. Th rsults w obtaind can b summarizd as follows: () du to th xistnc of a surfac transition layr, a distortion of th hystrsis loops will occur for surfac polarization nhancd rlativ to th intrior, lading to srious hystrsis phnomnon; () th avrag polarization and corciv fild of th film rduc (incras) with strongr surfac ffcts for surfac polarization supprssd (nhancd) rlativ to th intrior; (3) th corciv fild of th film incrass rmarkably with nhancing th lctric fild amplitud, but th avrag rmnant polarization dos not incras obviously; (4) an incrasd tmpratur maks th corciv fild and avrag rmnant polarization of th film rduc. Acknowldgmnt This work has bn supportd by th Youth Scinc Foundation of Hilongjiang Provinc undr grant No. QC3C7. Rfrncs [] A. Q. Jiang, Y. Y. Lin, and T. A. Tang, J. Appl. Phys., 45 (7). [] N. Sttr t al., J. Appl. Phys., 566 (6). doi:.63/ [3] F. Yang t al., Appl. Phys. Ltt. 9, 49 (7). [4] L. Pintili, I. Pasuk, R. Ngra, L. D. Filip, and I. Pintili, J. Appl. Phys., 646 (). doi:.63/

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