Three-dimensional simulation of nanocrystal Flash memoriesG. Iannaccone and P. Coli Citation: Applied Physics Letters 78, 2046 (2001); doi: 10.1063/1.1361097 View online: http://dx.doi.org/10.1063/1.1361097 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/78/14?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Numerical model of a single nanocrystal devoted to the study of disordered nanocrystal floating gates of newflash memories J. Appl. Phys. 109, 094506 (2011); 10.1063/1.3580511 Semianalytical model of tunneling in nanocrystal-based memories J. Appl. Phys. 100, 074316 (2006); 10.1063/1.2356917 Nonlinear I - V characteristics of nanotransistors in the Landauer-Büttiker formalism J. Appl. Phys. 98, 084308 (2005); 10.1063/1.2113413 Effects of crystallographic orientations on the charging time in silicon nanocrystal flash memories Appl. Phys. Lett. 82, 2685 (2003); 10.1063/1.1566479 Three-dimensional self-consistent simulation of the charging time response in silicon nanocrystal flash memories J. Appl. Phys. 92, 6182 (2002); 10.1063/1.1509105

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Three-dimensional simulation of nanocrystal Flash memoriesG. Iannacconea) and P. ColiDipartimento di Ingegneria dell’Informazione, Universita` degli Studi di Pisa, Via Diotisalvi 2,I-56122 Pisa, Italy

~Received 10 November 2000; accepted for publication 9 February 2001!

We have developed a code for the detailed simulation of nanocrystal Flash memories, which consistof metal–oxide–semiconductor field-effect transistors~MOSFETs! with an array of semiconductornanocrystals embedded in the gate dielectric. Information is encoded in the MOSFET thresholdvoltage, which depends on the amount of charge stored in the nanocrystal layer. Nanocrystals arecharged through direct tunneling of electrons from the channel. Such memories are promising interms of shorter write–erase times, larger cyclability, and lower power consumption with respect toconventional nonvolatile memories. We show results obtained from the self-consistent solution ofthe Poisson–Schro¨dinger equation on a three-dimensional grid, focusing on the charging processand on the effect of charge stored in the nanocrystals on the threshold voltage. ©2001 AmericanInstitute of Physics.@DOI: 10.1063/1.1361097#

A nanocrystal Flash memory is basically a metal–oxide–semiconductor field-effect transistor~MOSFET! inwhich the gate dielectric is replaced by a gate stack consist-ing of a thin tunneling oxide, a layer of semiconductornanocrystals embedded in silicon oxide, and a thicker oxide.Such memory devices have been recently proposed1,2

and extensively studied in industrial and academiclaboratories.3–7

The memory is programmed by applying to the gate apositive voltage of a few volts that lowers the thin oxideconduction band and enhances tunneling of electrons fromthe substrate to the nanocrystals. Electrons get trapped in thenanocrystals, since further tunneling to the gate is inhibitedby the thicker top oxide. However, due to already observedCoulomb blockade effects at room temperature,3 only a welldefined number of electrons~depending on the applied gatevoltage! can occupy each nanocrystal, so that charging of thenanocrystals is a self-limited process.

When an electron is added to each nanocrystal theMOSFET threshold voltage increases in steps, so that bothsingle and multibit storage is possible. The informationstored in the memory is then simply read by measuring thesaturation current corresponding to a gate voltage signifi-cantly smaller than that used for programming. The memoryis erased by applying a negative gate voltage that ejects elec-trons from the nanocrystals into the channel.

While a conventional Flash electrically erasable pro-grammable read-only memory~EEPROM! has a tunnelingoxide typically thicker than 7 nm and requires programmingvoltages larger than 10 V, a nanocrystal memory has a verythin tunneling oxide~2–4 nm!, and exhibits short program-ming times with direct tunneling and much smaller write–erase voltages~'3 V!. It is therefore extremely promising interms of endurance to write–erase cycles and powerconsumption,4 while it usually has a poor retention timecompared to the 10-year requirement typical of commercialFlash EEPROMs.

Nanocrystal memories have interesting applications asquasi-nonvolatile memories, where cyclability and powerconsumption are more important than 10-year data retention,or as dynamic random access memories with very long re-fresh time. In addition, Coulomb blockade effects should en-able easy implementation of multibit storage schemes.

Several materials have been investigated for the nanoc-rystals and the dielectric layer, and are listed in Table I. Atpresent, silicon rich oxide deposited by low pressure chemi-cal vapor deposition~LPCVD! on SiO2 ~Ref. 4! and im-planted Si or Ge in SiO2 ~Ref. 5! are the most promisingfrom the point of view of compatibility with current comple-mentary MOS ~CMOS! technology, retention time, andthreshold voltage shift per electron.

Since the device structure is inherently three-dimensional~3D! and electrons in the dot are strongly con-fined ~the nanocrystal diameter is usually in the range of3–10 nm!, an equivalent electrical circuit would not be ad-equate for obtaining accurate results. For this reason, wehave developed a code for the self-consistent solution ofPoisson and Schro¨dinger equations on a 3D grid, based ondensity functional theory with local density approximation.8

While nanocrystals are of course randomly distributed inthe layer, we have considered a simplified situation in whichdisorder is removed, that is, nanocrystals occupy a perfecttwo-dimensional lattice in the dielectric layer. We considerthe device structure shown in Fig. 1: a MOSFET with a gatestack consisting of a 3 nm thick bottom tunneling oxide, a

a!Electronic mail: g.iannaccone@iet.unipi.it

TABLE I. A few prototypes of nanocrystal memories presented in the lit-erature, with measuredDVT per electron and retention time.

Dot/insulatormaterial

DVT per electron~V! Retention time Reference

Si in SiO2 0.36 Not shown 1InAs/AlGaAs 0.25 1 h at 100 K 6Si or Ge in SiO2 0.3 '1 h 5SiGe/SiO2 0.4 1 day 7SRO/SiO2 '1 Not shown 4Si/Si3N4–SiO2 0.48 '3 h 3

APPLIED PHYSICS LETTERS VOLUME 78, NUMBER 14 2 APRIL 2001

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rectangular two-dimensional array of silicon nanocrystalsembedded in SiO2, and a 6 nmthick top oxide layer. Withoutloss of generality, we assume cubic dots with 5 nm edgesand average donor doping of 1019cm23 ~here we will notconsider the effect of discrete dopants!. The substrate has anacceptor doping of 1018cm23 and the gate is metallic.

The regular arrangement of the dots allows us to simu-late only the region that represents the elementary cell of thelattice structure, and then to apply periodic boundary condi-tions on the potential. The domain in which we actuallysolve the Schro¨dinger–Poisson equation is therefore the oneshown in Fig. 2. The surface density of nanocrystalssnc isthe inverse of the area of the domain considered on the hori-zontal plane: we consider an area of 10310 nm2, corre-sponding tosnc51012cm22.

Let us point out that we are making two implicit assump-tions: that all nanocrystals are in the same charge conditionsand that edge effects acting on nanocrystals located close tothe contacts are not relevant. While the latter can be readilyaccepted if we assume a sufficiently large number of nanoc-rystals under the gate~e.g., at least 535!, the former is onlyjustified because here we focus on stationary properties ofthe memories: it would not be acceptable if we were inter-ested in the time-dependent process of nanocrystal charging.The potential profile in our domain is determined by thePoisson equation

¹@e¹f~r !#52r~r !52q@p~r !2n~r !

1ND1~r !2NA

2~r !#, ~1!

wheref is the scalar potential,e the dielectric constant,pand n the hole and electron densities, respectively, andND

1

andNA2 the concentrations of ionized donors and acceptors,

respectively, that depend onf as indicated in Ref. 9. Whileelectrons and hole concentrations in the substrate are com-puted with the semiclassical approximation,9 electrons in thenanocrystal are strongly confined, and therefore their densityis computed by solving the Schro¨dinger equation with den-sity functional theory

2\2

2¹S 1

m¹C D1Ec~r !C1Vxc~r !C5EC, ~2!

where Ec is the conduction band in the nanocrystal@Ec

5Ec(f50)2qf#, \ is the reduced Planck constant, andVxc is the exchange-correlation potential in the local densityapproximation8

Vxc~r !52q2

4p2e0e r@3p3n~r !#1/3. ~3!

Nanocrystals are randomly oriented, therefore we prefer notto use the effective mass tensor for silicon, but only a singleeffective mass for the density of statesm50.32m0 , wherem0 is the electron mass at rest. Each eigenfunctionC i ofenergyEi has a degeneracy of 12, due to spin degeneracyand to the presence of six equivalent minima in the conduc-tion band. If the dot is occupied byN electrons, andl andmare two integer such asN5123 l 1m ~with m,12! the elec-tron density reads

n~r !512(i 51

l

uC i~r !u21muC l 11~r !u2 ~4!

under the assumption that the electron density in the dot isnot appreciably different from that in the ground state.

The Poisson–Schro¨dinger equation is self-consistentlysolved for several values ofN and of the gate voltageVG .The maximum number of electrons that can occupy a dotNmax is a function ofVG : if m(N,VG) is the chemical po-tential of the dot withN electrons and applied gate voltageVG , andEF is the Fermi energy in the substrate,Nmax mustsatisfy the conditions

m~Nmax,VG!,EF ; m~Nmax11,VG!.EF . ~5!

The chemical potential is computed with Slater’s transitionrule10 asm(N)5EN21/2, whereEN21/2 is the highest occu-pied eigenvalue of the system withN21/2 electrons, com-puted with density functional theory.

FIG. 3. Electron density on thex–z plane for an applied gate voltage of 1.2V and one electron in the dot.

FIG. 1. Device structure considered in the simulation. A regular two-dimensional array of cubic Si nanocrystals is embedded in the gate oxide ofa MOSFET.

FIG. 2. View of the simulation domain: The nanocrystal is a silicon cubewith 5 nm edges, and the thickness of the top and bottom oxide is 6 and 3nm, respectively.

2047Appl. Phys. Lett., Vol. 78, No. 14, 2 April 2001 G. Iannaccone and P. Coli

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By keeping fixed the number of electrons and computingthe electron density in the channel as a function ofVG , onecan directly obtain the value of the threshold voltage for anynumber of electrons in the nanocrystal.

The electron density in the nanocrystal on thex–z planefor an applied voltage of 1.2 V and an occupancy of oneelectron is plotted in Fig. 3. As can be seen, the electrondensity has the shape of the first eigenfunction, with only onepeak at the center.

In Fig. 4 we plot the sheet electron density in the channelas a function of the gate voltageVG for a number of addi-tional electrons in the nanocrystal ranging from 0 to 5~thinsolid lines!. From those curves the threshold voltage can beobtained as the intercept on the horizontal axis of the lineapproximating each curve in the quasilinear region. As canbe seen, the threshold voltage with no electrons in the nano-crystal isVT050.5 V and theVT shift is approximately 0.4 Vper each stored electron.

In Fig. 5 we plot the chemical potentialm of the dot as afunction ofVG for a number of electronsN ranging from 1 to7. The Fermi level in the channel is zero, therefore the inter-cept of all the curves with the horizontal axis gives the gatevoltage at which an additional electron can enter the nano-crystal.

We can see from Figs. 4 and 5 that several options forwrite and read voltages are available. For example, if weassume that the logical ‘‘0’’ corresponds to zero electrons inthe nanocrystal, we could write a logical ‘‘1’’ by applying a

voltage pulse of 2 V, thereby introducing three electrons inthe nanocrystal~as can be seen from Fig. 5! and obtaining athreshold voltageVT151.7 V ~as obtained from Fig. 4!. Thenwe could choose a voltageVGR50.6 V to read the status ofthe memory, since it satisfiesVTO,VGR,VT1 and, in addi-tion, is small enough not to introduce any additional electronin the dot~see Fig. 5!.

We have developed a numerical model for the three-dimensional quantum-mechanical simulation of nanocrystalFlash memories, and we have shown results for a typicaldevice structure. The model developed is a useful tool forexploring the effects of process parameters and of geometryon the electrical properties of such memory devices.

For simplicity, we have considered a perfectly regulararray of nanocrystals. A disordered distribution of nanocrys-tals would have an effect very similar to that of the randomdistribution of dopants in conventional MOSFETs, causingstatistical dispersion ofVT and ofVT shifts. From this anal-ogy, we can say that the relative dispersion ofVT shifts in-creases when the channel area is reduced, but decreases withincreasing nanocrystal densitysnc, and can be therefore keptunder control by increasingsnc.11 However, a detailed in-vestigation on this subject is required.

A very important aspect for the industrial application ofnonvolatile memories based on nanocrystal layers is theevaluation of write, erase, and retention times. Promisingresults have been obtained in experiments3–7 and the issuehas been partially addressed from the theoretical point ofview in Ref. 12. An in-depth analysis of this aspect is alsoneeded to evaluate the maximum operating temperature as afunction of the required retention time. This issue is beyondthe scope of this letter and will be addressed in the nearfuture.

The authors gratefully acknowledge support from ISTProject No. 10828 NANOTCAD~NANOTechnology Com-puter Aided Design!.

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FIG. 4. Sheet electron density in the channel as a function of the gatevoltage for a number of electrons in a nanocrystal ranging from 0 to 5.

FIG. 5. Chemical potential of the nanocrystal as a function of the gatevoltage for a number of stored electrons ranging from 1 to 7.

2048 Appl. Phys. Lett., Vol. 78, No. 14, 2 April 2001 G. Iannaccone and P. Coli

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