We revise the possibility of having an amplified surface potential in
ferroelectric field-effect transistors pointed out by [S. Salahuddin and S.
Datta, Nano Lett. 8, 405 (2008)]. We show that the negative-capacitance regime
that allows for such an amplification is actually bounded by the appearance of
multi-domain ferroelectricity. This imposes a severe limit to the maximum
step-up of the surface potential obtainable in the device. We indicate new
device design rules taking into account this scenario.Comment: 4 pages, 3 figure

American Physical Society

Cano, A.

Jiménez Jiménez, David

English

arXiv

A. Cano

D. Jiménez

Crossref

Applied Physics Letters

Multidomain ferroelectricity as a limiting factor for voltage amplification in ferroelectric field-effect transistors

We revise the possibility of having an amplified surface potential in ferroelectricfield-effect transistors pointed out by [S. Salahuddin and S. Datta, Nano Lett.8, 405 (2008)]. We show that the negative-capacitance regime that allows for such amplification is actually bounded by the appearance of multidomain ferroelectricity. This imposes a severe limit to the maximum step-up of the surface potential obtainable in the device. We indicate new device design rules taking into account this scenario

Cano, Andrés

Diposit Digital de Documents de la UAB

Multidomain ferroelectricity as a limiting factor for voltage amplification inferroelectric field-effect transistorsA. Cano and D. Jiménez  Citation: Applied Physics Letters 97, 133509 (2010); doi: 10.1063/1.3494533 View online: http://dx.doi.org/10.1063/1.3494533 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/97/13?ver=pdfcov Published by the AIP Publishing                      This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:158.109.223.71 On: Tue, 18 Feb 2014 13:22:13Multidomain ferroelectricity as a limiting factor for voltage amplificationin ferroelectric field-effect transistorsA. Cano1,a and D. Jiménez2,b1European Synchrotron Radiation Facility, 6 rue Jules Horowitz, BP 220, 38043 Grenoble, France2Departament d’Enginyeria Electrònica, Escola d’Enginyeria, Universitat Autònoma de Barcelona,08193 Bellaterra, SpainReceived 22 June 2010; accepted 8 September 2010; published online 30 September 2010We revise the possibility of having an amplified surface potential in ferroelectric field-effecttransistors pointed out by S. Salahuddin and S. Datta, Nano Lett. 8, 405 2008. We show thatthe negative-capacitance regime that allows for such amplification is actually bounded by theappearance of multidomain ferroelectricity. This imposes a severe limit to the maximum step-up ofthe surface potential obtainable in the device. We indicate new device design rules taking intoaccount this scenario. © 2010 American Institute of Physics. doi:10.1063/1.3494533The operation of field-effect transistors FETs generatesa heat whose dissipation imposes severe restrictions to theminiaturization of integrated circuits. The lowering of theFET operating voltage is therefore highly desirable, whichhas to be accompanied with a reduction in the threshold volt-age to maintain performances. This, however, implies theincrease in the stand-by power since the inverse of the so-called subthreshold slope appears limited to 60 mV/dec atroom temperature. At present, this is considered as an impor-tant roadblock for the transistor scaling down.1 RecentlySalahuddin and Datta2 have suggested that the 60 mV/declimit can be overcome in ferroelectric FETs as the sketchedin Fig. 1. These FETs has a long history as candidates fornondestructive readout memory elements.3 The idea ofSalahuddin and Datta is to exploit a negative capacitanceregime of the ferroelectric in which the surface potential ofthe semiconductor Vs is up-converted and therefore the so-called body factor of the transistor mVs /Vg−1, whereVg is the gate potential, becomes smaller than one.The physics behind this negative capacitance regime isassociated to depolarizing field effects, i.e., the electric fieldthat accompanies the polarization of the finite-size ferro-electric. As a result of this field, there is a shift in ferroelec-tric transition point and the ferroelectric can operate in itsotherwise unstable paraelectric state. What is more, in sucha state, the voltage drop V through the ferroelectric de-creases by increasing the gate voltage. This yields the desiredamplification, since the changes in the surface potential Vsare then larger than the ones in the gate voltage Vg=V+Vs. This possibility is explained in Fig. 2. Here we plot theload line Q=CsVs=CsVg−V, where Q and Cs are thecharge and the semiconductor capacitance, respectively, andthe QV characteristic of the ferroelectric. The slope ofthis later is always positive if the ferroelectric is well insideits paraelectric phase Fig. 2a and therefore the intersec-tion between this function and the load line shifts towardhigher voltages if the gate voltage is increased. The ferro-electric then behaves as the conventional oxide in the FET.However, when the ferroelectric QV characteristic ac-quires its S-shaped form, its slope is negative for V=0Figs. 2b and 2c. This happens below the nominal tran-sition temperature of the ferroelectric see below. If such aslope is more negative than −Cs Fig. 2b, then the inter-section with the load line shifts toward lower voltages as thegate voltage increases. This means that the surface potentialis enhanced as we have explained before, which can associ-ated with a negative-capacitance behavior of the ferroelec-tric. If the ferroelectric QV characteristic gets sufficientlyflat for low voltages then there appears three points of inter-section with the load line Fig. 2c, from which only themarked with dots correspond to stable states for the ferro-electric now in its ferroelectric state. The voltage amplifi-cation holds until these points shift again toward higher val-ues if the gate voltage is increased as shown in Fig. 2c.This eventually translates into hysteresis loops for gate volt-ages varying cyclically from positive to negative.According to these reasonings, the maximum amplifica-tion of the gate potential is expected to be limited by theeventual transition into the ferroelectric state. Salahuddinand Datta have tacitly assumed that this transition implies thesingle-domain state in which the spontaneous polarization isuniform through the ferroelectric. The actual situation, how-ever, can be far more subtle. As a result of the depolarizingfield the ferroelectric instability is generally equivalent to theappearance of multidomain ferroelectricity in which the po-larization varies in space, with transition temperature shiftsrather different from the assumed by Salahuddin and Datta.4The aim of this paper is to show that the negative capaci-tance regime then gets saturated, which puts severe limits tothe maximum amplification of the gate voltage that can beobtained in ferroelectric FETs.aElectronic mail: cano@esrf.fr.bElectronic mail: david.jimenez@uab.es.gatesource drainbodysemiconductorferroelectricVsVgn-region n-regionp-regionCFECs+xzFIG. 1. Color online Schematic cross section of a ferroelectric FET leftand equivalent circuit for the metal/ferroelectric/semiconductor stack right.APPLIED PHYSICS LETTERS 97, 133509 20100003-6951/2010/9713/133509/3/$30.00 © 2010 American Institute of Physics97, 133509-1 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:158.109.223.71 On: Tue, 18 Feb 2014 13:22:13To obtain the response of the FET to the applied voltagewe follow the Landau-type approach described in Ref. 4. Onone hand, the behavior of the ferroelectric is described by theequationsaP + bP3 − c2P = − zV , 1ax2 + y2 + 0z2V − zP = 0. 1bHere P is the distribution of polarization along the ferroelec-tric z-axis and V the electrostatic potential in the ferroelec-tric. Equation 1a, which can be derived from a Ginzburg–Landau–Devonshire free energy, represents the constitutiveequation for the ferroelectric whose electrostatics, as followsfrom Maxwell’s equations, is eventually determined by Eq.1b. The instability that gives rise to ferroelectricity is de-scribed by the vanishing of the coefficient a=aT−Tc0 asusual, where Tc0 is the nominal transition temperature in theabsence of depolarizing field while the rest of coefficientsare assumed to be positive constants. We thus assume asecond-order continuous phase transition, which a priori isthe most favorable scenario for the amplification of the FETgate voltage. On the other hand, the semiconductor is as-sumed to be undoped or lightly doped operating within itssubthreshold regime as in Ref. 2. Thus, its mobile carrierdensity can be neglected and we simply have the equation2Vs=0 for the electrostatic potential in the semiconductor.At the ferroelectric-semiconductor interface these quantitieshave to satisfy the electrostatic matching conditions V=Vsand 0zV− P=szVs, where s is the dielectric constant ofthe semiconductor.5 In addition, we have the boundary con-ditions V=Vg at the gate and Vs=0 in the semiconductorbeyond its depletion layer see Fig. 1. Vg is assumed to bebelow the FET threshold voltage in the following.As long as the ferroelectric stays in its paraelectric phasethe body factor of the FET is given bym = 1 +s1 + 0alwa . 2Here l and w represent the thickness of the ferroelectric andthe width of the semiconductor depletion layer, respectively.Noting that 0a1 in the vicinity of the ferroelectric insta-bility one obtains m	1+sal /w=1+Cs /CFE, where Cs=s /w and CFE=1 / al, which is the result reported bySalahuddin and Datta.2 The desired up-conversion of the sur-face potential is obtained if the body factor is m1. This ispossible if the bare polarization stiffness a gets negative andhence the ferroelectric can act as a negative capacitanceCFE0. If depolarizing fields were totally screened, thatwould mean the instability of the paraelectric phase withrespect the spontaneous polarization of the system. This po-larization, however, is accompanied with some depolarizingfield in the FET, which produces an increase in the polariza-tion stiffness. In consequence, the ferroelectric can remain inits paraelectric phase even if the bare stiffness is a0. Infact, the ferroelectric can be proven to be stable with respectto uniform distributions of polarization up to the pointa=−1 / 0+sl /w where the expression 2 for the bodyfactor would give zero. This point, however, generally doesnot correspond to the transition point in the ferroelectric FETas we show in the following.The actual phase transition point is determined by thepoint at which the Eqs. 1a and 1b have their first non-trivial solution P0 for Vg=0. This can be found byfollowing the general procedure outlined in Ref. 4. Forthe FET geometry and the typical values for the depletionwidth of lightly doped semiconductors w	0.1–1 m,such a solution corresponds to a polarization waveP0x ,z= p0 coskxxcoskzz, with kz= /2l and kx= c−1/4 /2l1/2, and appears atac  − cl. 3The parameters entering in this expression can be estimatedas  /0	1–100 and c	dat2 /0, where dat is the character-istic atomic distance c	10−9–10−11 J m3 /C2, see e.g., Ref.6. In the FET setups w, ldat, and consequently aca. So,in fact, much before the paraelectric phase can get unstablewith respect to the uniform polarization, the ferroelectrictransits into its multidomain ferroelectric phase at ac. Atthis point, the body factor in Eq. 2 takes the valuemmin  1 −scw, 4which we anticipate to be the minimum obtainable in theferroelectric FET. It is worth noting that mmin depends onmaterial parameters of both the semiconductor and the ferro-electric but not on the thickness of this later. The abovenumbers give mmin	0.99 only, though this could be furtherreduced to 	0.7 considering the state-of-the-art semiconduc-tor capacitance Cs=0.1 F /m2 which however requiresstrong doping and therefore is beyond our model.As we have mentioned, the expression 2 for the bodyfactor is valid as long as the ferroelectric stays in itsparaelectric phase. That is, for a	ac. To obtain the corre-ΔVQVgQ QΔV ΔVVg Vg(a) (b) (c)FIG. 2. Color online Sequence of possible states in the ferroelectric FET and geometrical determination of the bias point: a and b paraelectric states,and c hysteretic ferroelectric state. In b the negative capacitance regime with the step-up conversion of the surface potential is achieved.133509-2 A. Cano and D. Jiménez Appl. Phys. Lett. 97, 133509 2010 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:158.109.223.71 On: Tue, 18 Feb 2014 13:22:13sponding expression for aac it is important to take intoaccount that there is a nonzero background polarization inthe ferroelectric. Close to the transition point such a polar-ization is well described by the polarization wave P0 foundbefore and, to our purposes, higher harmonics can be ne-glected. Within this approximation the amplitude of the po-larization wave is p0 4 /3a−ac /b. Furthermore weexpress the total polarization as Ptot= P0x ,z+P, where Pis due to the applied gate voltage, and linearize the equationswith respect to this quantity Ptot3 P03+3P02P. We then ob-tain the body factorm  1 + slwac + 13 a − ac . 5As we see, the body factor in fact increases once the ferro-electric enters in its multidomain ferroelectric phase. Thishardening results from the cubic P3 term that eventually sta-bilizes the system. Nevertheless, the ferroelectric stays in thenegative capacitance regime for some range of temperaturesbelow the transition point. The precise computation of thisrange requires to go beyond the single harmonic approxima-tion to describe properly the evolution of the ferroelectricground state, which is beyond the scope of the present work.It is worth mentioning that the negative capacitance regimein the multidomain ferroelectric phase also manifests in theunusual hysteresis loops with negative slopes described inRef. 7 to rationalize experimental data on ferroelectric thinfilms.8 The behavior of the body factor is illustrated in Fig. 3as a function of the control parameter a. The maximum am-plification of the gate voltage corresponds to Eq. 4 at thetransition point ac. We note that, in order to obtain a signifi-cant gain, the semiconductor capacitance should be engi-neered to be Cs=  /c / per surface area. This design ruleis independent of the ferroelectric thickness. Such a thick-ness simply sets the amplification window that, for practicalpurposes, has to be tuned about room temperature. In a mul-tidomain scenario the gradient coefficient c plays a moreimportant role, giving the above design rule in sharp contrastthe reported in Ref. 2.In conclusion, we have shown that appearance of multi-domain ferroelectricity may substantially limit the maximumvoltage amplification expected in ferroelectric FETs.D.J. acknowledges support from Ministerio de Ciencia eInnovación by Project Explora TEC2008-01883-E/TEC andMinisterio de Educacion y Ciencia by Project TEC2009-09350.1International Technology Roadmap for Semiconductors, 2009 Edition.www.itrs.net.2S. Salahuddin and S. Datta, Nano Lett. 8, 405 2008.3See, e.g., M. Dawber, K. M. Rabe, and J. F. Scott, Rev. Mod. Phys. 77,1083 2005, and references therein.4See, e.g., A. M. Bratkovsky and A. P. Levanyuk, J. Comput. Theor.Nanosci. 6, 465 2009, and references therein.5Experimentally, suitable ferroelectric-semiconductor interfaces are diffi-cult to process and buffer layers are frequently needed to avoid, e.g.,interdiffussion problems. In addition, different species such as SiOx mayappear at the interface. This can be seen as an additional capacitance Cintin series with the semiconductor capacitance Cs. Its influence on the bodyfactor can be easily taken into account by replacing Cs=s /w in Eqs. 2,4, and 5 by there resulting effective capacitance CsCint / Cs+Cint.6W. Y. Shih, W.-H. Shih, and I. A. Aksay, Phys. Rev. B 50, 15575 1994;O.G. Vendik, S. P. Zubko, and L. T. Ter-Martirosayn, Appl. Phys. Lett. 73,37 1998; A. G. Zembilgotov, N. A. Pertsev, H. Kohlsted, and R. Waser,J. Appl. Phys. 91, 2247 2002.7A. M. Bratkovsky and A. P. Levanyuk, Appl. Phys. Lett. 89, 2531082006.8D. J. Kim, J. Y. Jo, Y. S. Kim, Y. J. Chang, J. S. Lee, J.-G. Yoon, T. K.Song, and T. W. Noh, Phys. Rev. Lett. 95, 237602 2005; Y. S. Kim, D.H. Kim, J. D. Kim, Y. J. Chang, T. W. Noh, J. H. Kong, K. Char, Y. D.Park, S. D. Bu, J.-G. Yoon, and J.-S. Chung, Appl. Phys. Lett. 86, 1029072005; Y. S. Kim, J. Y. Jo, D. J. Kim, Y. J. Chang, J. H. Lee, T. W. Noh,T. K. Song, J.-G. Yoon, J.-S. Chung, S. I. Baik, Y.-W. Kim, and C. U.Jung, ibid. 88, 072909 2006.mmin01ac log |a|ma*FIG. 3. Color online Schematic behavior of the ferroelectric FET bodyfactor m as a function of the control parameter a=aT−Tc0: blue coloronline ignoring nonuniform distributions of polarization red color onlinetaking into account multidomain ferroelectricity. Since ferroelectricity is ex-pected in a multidomain state, m remains finite with its minimum at thecorresponding transition point ac. Otherwise it could be downed to zero atthe paraelectric↔uniform ferroelectric transition point a.133509-3 A. Cano and D. Jiménez Appl. Phys. Lett. 97, 133509 2010 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP:158.109.223.71 On: Tue, 18 Feb 2014 13:22:13

Multidomain ferroelectricity as a limiting factor for voltage ampliﬁcation in ferroelectric ﬁeld-effect transistors

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Multi-domain ferroelectricity as a limiting factor for voltage amplification in ferroelectric field-effect transistors

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