simulation at start quantum-classical threshold · 2018. 11. 13. · vlsidesign 2001, vol. 13, nos....

VLSI DESIGN2001, Vol. 13, Nos. 1-4, pp. 155-161Reprints available directly from the publisherPhotocopying permitted by license only

(C) 2001 OPA (Overseas Publishers Association) N.V.Published by license under

the Gordon and Breach Science Publishers imprint,member of the Taylor & Francis Group.

Simulation at the Start of the New Millenium:Crossing the Quantum-Classical Threshold

D. K. FERRY

Department of Electrical Engineering and Center for Solid State Electronics Research,Arizona State University, Tempe, AZ 85287-5706, USA

It is clear that continued scaling of semiconductor devices will bring us to a regime withgate lengths less than 50nm within another decade. The questions that must beaddressed in simulation are difficult. Pushing to dimensional sizes such as this will probethe transition from classical to quantum transport, and there is no present approach tothis regime that has proved effective. Contrary to the classical case in which electrons arenegligibly small, the finite extent of the momentum space available to the electron setsize limitations on the minimum wave packet- this is of the order of a few nano-meters- and leads to the effective potential. The latter is an approach to find theequivalent classical potential, by which the actual potential is modified by quantumeffects. The use of the effective potential for analyzing the effect of quantization onsemiconductor devices will be discussed. The manner in which this leads to newformulations for quantum transport will be discussed.

Keywords: Effective potential; Quantum transport; Device simulation

1. INTRODUCTION

As the density of integrated circuits continues toincrease, there is a need to shrink the dimensionsof the devices of which they are comprised.Smaller circuit dimensions lead to more transistorson a single die. Advances in lithography havedriven device dimensions to the deep-submicronrange. Currently, 0.18 gm is the state-of-the-artprocess technology, but even smaller dimensionsare expected in the near future. Groups fromToshiba and Lucent Bell Labs have fabricatedn-channel MOSFETs with effective gate lengthsbelow 25nm, thus demonstrating that thesefeature sizes are feasible.

155

In this regime, the transport is expected to bedominated by quantum effects throughout theactive region, even though quantum transport forthese small (and inhomogeneous) devices is not wellestablished within a consistent conceptual frame-work [1]. Nevertheless, several approaches tosimulation ofsemiconductor devices have appearedin which the transport is handled quantummechanically [2]. An additional problem arises insmall structures, where one must begin to worryabout the effective size ofthe carriers themselves [3].In recent work, we have discussed the arguments forvarious sizes for electrons in semiconductor devices[4]. There are several reasons why this becomesimportant in small devices.

156 D.K. FERRY

FIGURE A conceptual device, with an applied source-drain bias. The source is at the left and the drain is at the right. The dashedareas are the "transition" regions, which must be considered as part of the active device.

Consider, for example, the conceptual device inFigure 1. The active channel length L is boundedby two contact transition regions, which in turnare coupled to the source and the drain contacts[5]. It is in these contact transition regions that thecarriers lose their coherence completely so that theactual contacts can be considered as being inequilibrium. The drain region is the distance overwhich the energetic channel carriers equilibrateand can be as much as 10nm [6]. On the otherhand, the transition near the source is smaller, butrepresents an absolute minimum of the deviceresistance which cannot be reduced [7].We may consider the importance of these

transition regions from the size of importantlengths and times within a Si MOSFET. Theseare listed in Table I. Nearly all of these lengths areof comparable size. This means that traditionaltransport based upon the Boltzmann equation isinappropriate for such devices. On the other hand,

TABLE Important parameters

Parameter Value in Si

Gate length 35 100 nmInelastic mfp 30- 80 nmCoherence length 20- 50 nmCoherence time 50-100 fsTransit time 0.1 ps

approaches such as the ensemble Monte Carlotechnique, in which no distribution function isassumed, provide a viable approach to this regime,if suitable quantum effects can be incorporated.As mentioned above, the effective size of the

electron wave packet is one such correction. Wewill see in the next section that this leads to aneffective potential which characterizes the initialquantization effects in the channel. Moreover,the use of such an effective potential dramati-cally changes the way in which we utilize quan-tum transport equations, such as the Wignerequation of motion [8]. This will be discussedin the subsequent section, and limitations inquantum approaches that are used today will bediscussed.

2. THE EFFECTIVE POTENTIAL

In classical mechanics, we typically consider theelectron to be represented by a delta function inposition. In quantum mechanics, however, theelectron is represented by a wave packet, whichhas non-zero extent. In some sense, this size is aresult of the uncertainty relation between posi-tion and momentum. One rational connection inthis regard is the representation in terms of

QUANTUM THRESHOLD CROSSING 157

Wannier functions. Wannier functions are highlylocalized functions which arise from summing theBloch functions over a complete band. That is, thereare N Wannier functions, whose extent is roughlyof atomic dimensions this becomes the equivalentof a delta function in space. The truly localizedWannier function, of atomic size, only arises if allpossible Bloch states are in the summation. If theband is only partially full, however, the summationmay be made over the occupied states, and theresulting Wannier function has an extended range.This approach was used, for example, to find theorbit of electrons bound to impurities. Here, weadapt this approach to determine the occupiedstates from the appropriate quantum distributionfunction, which then leads to a broadened Wannierfunction in real space. We assert that this broaden-ed function represents the electron wave packetfor this system [3]. We will see later that this isdirectly connected to the theory of distributions [9],which leads to a formal theory of the delta function[10].

In order to describe the packet in real space, wemust account for the contributions to the wavepacket from all occupied plane wave states. Even atroom temperature, the carriers in the inversion layerin a Si MOSFET are a two-dimensional gas, andall states up to the Fermi energy are occupied.Nevertheless, we may take a non-degenerate ap-proach for which the wave packet becomes aGaussian. The spatial extent of this packet can beestimated as [3]

2 AF6r 2Ar (1)kF 7r

This can be used to define the effective potentialthrough

Hv / d3rV(r)i f d3r’f(r-r’)6(r’-ri)f d3r’6(r’- ri)Veff(r’)

J dgrtVeff(rt)n(rt)’ (3)where the density is now represented in a classicalmanner by delta functions. Here, the effectivepotential is defined by the wave packet itself, with

Veff(*) J + (4)As mentioned, a particular form for the spreadfunction is (in one dimension)

f(x) x/a0eXp -a02 (5)

While the natural value to take for Si is the 3.6 nmfound earlier, this is for "free" carriers in a two-dimensional gas. The quantization in the inversionlayer is normal to the interface, and we have toworry about the bound states.Feynman and Kleinert have used an effective

potential of this form in an energy minimizationprocedure for bound carriers in a parabolic well.When the levels are not well defined (due tothermal broadening), then we may approximatethe spread by

For a Si inversion layer, we thus find that theeffective size of the wave packet is about 3.6 nm ata density of 5 x 1012 cm- 2.The general Hamiltonian term for the potential

energy in an inhomogeneous situation is givenby [11]

Hv f d3rV(r)n(r). (2)

h2a 8mknT" (6)This gives a value of 0.64 nm in Si inversion layers.We can estimate the validity of this approach

by comparing the role played by the effectivepotential in a quantum well such as that at theinterface in a MOSFET. The approach we takeis to compare the effective potential derived from

158 D.K. FERRY

2

1.5

0.5

(a)

Effective Potential

Schrodinger-Poisson

2 1012 4 1012 6 1012 8 1012 1013 1.2 1013Inversion Density (cm"2)

160

140

120

100

(b)

Effective

2 1012 4 1012 6 1012 8 1012 1013 1.2 1013Inversion Density (cm"2)

FIGURE 2 The average inversion depth (a) and quantizationenergy (b) for carriers in a Si MOSFET. A value of a0 0.5 nmhas been used.

a triangular quantum well with the fully self-consistent solution of a coupled Schr6dinger-Poisson solver for the inversion layer. In Figure 2,we show the average depth of the inversion densityand the quantized energy level of the well itself. Itcan be seen that the agreement is good.

3. APPLICATIONS IN QUANTUMTRANSPORT EQUATIONS

We turn at this point to the Wigner function,which is a Fourier transform, in the difference

coordinate, of the density matrix [2]. Generally,use of the Wigner function has been hampered bythe fact that it has non-compact support it takeson values at spatial regions where the wavefunction itself is undefined. This has led to a verycomplicated equation of motion, particularly inthe connection of the spatially varying poten-tial energy with the momentum of the Wignerfunction.The most usual form of the Wigner equation of

motion can easily be derived by techniquesreviewed by Moyal many years ago [12]. In thisderivation, a delta function is used to uncouple thepotential from the Wigner function in the com-mutator driving terms. In the approach builtaround the finite extent of the wave packet, weuse (5) as a generalized distribution with which torepresent the delta function. Classically, oneshould take the limit as a00 in order to re-cover the delta function. In quantum statisticalmechanics, however, we need not take the fulllimit. Rather, we need only limit it to the sizeof the fundamental wave packet. When this isdone, we can write the field driving term in theWigner equation of motion as

OVerf Ofw(x,p, t)Ox @

(7)

This is essentially the classical term that appears inthe Boltzmann equation with the exception thatthe actual potential is replaced by the effectivepotential (4). However, for strict validity of theapproximation used, we also require that a0 not betoo small; in fact, we need

a0p > (8)

Here, p is the "uncertainty" of the momentum,such as the thermal spread of this value. In thissense, a0 should be close to the thermal Debyewavelength. It is well known that the Wignerfunction is non-negative, but these negative regionsvanish if the function is averaged over phase spaceregions of a size given by the uncertainty relation.


Here, (8) insures that the smoothed Wignerfunction we use is averaged over these regions sothat the uncertainty has been incorporated. Insome sense, this approach is comparable to aWKB approximation, although this has been as-serted for the Wigner function previously.

Recognition of the wave packet size is funda-mental in the collision term as well. Complicationsare created by the fact that the collision takes afinite time to complete, and quantum transitionsare non-local. Indeed, it can be estimated thatoptical phonon collisions take 2-5 fs to complete[13]. While this is quite short, an electron travelingat 3 107 cm/s will cover 0.6-1 nm in this time.This distance is not negligible in a 35 nm MOSFETor in a resonant tunneling diode! Consider forexample the conceptual problem of Figure 3, inwhich a particle approaches a potential barrier andabsorbs a phonon near the barrier, as indicated inthe figure.

Classically, the transition indicated in Figure 3is forbidden. First, it is nonlocal in space.Secondly, absorbing a phonon leads to an increasein the momentum wave vector, but here the final

state has a lower value of momentum wave vectordue to the rapid increase in the potential energy.The horizontal lines are meant to be the spatialextent of the initial and final wave packets (whichare both moving). In quantum mechanics, theenergy that is conserved is the total energy, not justthe kinetic energy. In Figure 3, it is the intracolli-sional field effect [14] which rapidly changes themomenta of the two states with position (time).Quite beyond the estimates above, an electronapproaching the top of a 0.3eV barrier (in thecladding layer) is traveling at more than 108 cm/s,and will cover 3-5 nm (which may be the entirebarrier thickness) during the collision. If we usethe non-zero spatial resolution of the delta func-tions through (5), then the in-scattering term,for phonon emission, in the Wigner equation ofmotion can be written as

OfwOt coil

2 Ine l 2=j2Z 5 (Nq -t-- 1)q

x jf (x + x’, (9)

E

kV (x)

where fix’) is given by (5), and the energyresolution term

x + x’) + V(x + x’)+ e(Z:) V(x) (10)

provides the energy conservation of the system. Itshould be pointed out that the Lorentzian formused here is an approximation. Moreover, thisterm cannot be treated in isolation. In fact, it isthe product of the generalized distribution f(x)and this Lorentzian that provides the probabilityfunction to find the final state, which is definedby both a position and a wave vector. The broad-ening term is the same as found previously [14],and is given, within the above approximation, by

FIGURE 3 Conceptual picture of an electron absorbing aphonon near a potential barrier.

3hqTcoll 0Veffm* Ox (11)

160 D.K. FERRY

Space prohibits the full discussion (and derivation)of these terms, but this will be provided else-where 15].

4. CONNECTION WITH OTHERQUANTUM POTENTIALS

We can expand the effective potential when it is aslowly varying function of position. That is, wecan take the effective potential from the defininglines in (4) and use a Taylor series expansion as

The first term allows us to bring the potentialoutside the integral, while the second termvanishes due to the symmetry of the Gaussian.The third term becomes the leading correctionterm, which gives us

W(x) g(x) nt- oz202V-d- x2+ (13)

The major purpose of the effective potential is tofind a dependence of the density upon thispotential as exp(-/3V). Using this fact, we canreplace the potential in the second term of (13)with the density, and

c2 02 ln(n/no) +... (14)W(x)- V(x) Ox.

This particular form is often connected with aquantum potential appearing in hydrodynamicequations that are obtained from the equation ofmotion for the Wigner function [16] or the densitymatrix [17]. The second term has been termed the

quantum potential in this approach. We can nowreplace the density with the square root of thedensity, and if we ignore terms that are quadraticin the first derivatives and of higher order, (14) canbe written as

W(x)=V(x) 2c20)ln(v/n/n)+/ OX2

2OZ2 /;92=V(x)

Within a factor of 2, the second term is nowrecognized as the density gradient term first dis-cussed for quantum hydrodynamics by de Broglie[18] and Madelung [19], but known more as theBohm potential [20]. It has become popular touse this correction in current device simulationsto try to incorporate some quantum effects, re-ferring to the corrections as the "density gradientpotential".From the above, we can see that the various

forms for the quantum potential are really approx-imations to the full effective potential. As a result,use of the latter is to be preferred, since the integralsmoothing will reduce fluctuations while thederivative forms amplify fluctuations. Moreover,it is well known that the Bohm potential reproducesthe quantization energy of the ground state, andthis is found as well in the effective potential. Thismeans that the effective potential is already of anature to be used for mixed wave functions, where-as the density gradient approaches have severeproblems in this case, particularly near nodalpoints of the composite wave function [1]. Byjudicious choice of the extent of the wave packet,excellent fit can be obtained for the quantumparameters in e.g., the MOSFET inversion layer.

Acknowledgements

The author would like to express appreciationfor helpful discussions with H. L. Grubin, M.Fischetti and D. Vasileska. This work was sup-ported in part by the Office of Naval Research andthe Semiconductor Research Corporation.


References

[1] Ferry, D. K. and Barker, J. R. (1998). "Open Problems inQuantum Simulation in Ultra-Submicron Devices", VLSIDesign, 8, 165- 72.

[2] Ferry, D. K. and Grubin, H. L. (1995). "Modeling ofQuantum Transport in Semiconductor Devices", In: SolidState Physics, 49 (Academic Press, San Diego), 283-448.

[3] Ferry, D. K. and Grubin, H. L. (1998). "Electrons inSemiconductors: How Big are They"? Proc. IWCE-6,Osaka (IEEE Press, New York), pp. 84-7.

[4] Ferry, D. K. (2000). "The onset of quantization in ultra-submicron semiconductor devices", Superlatt. Mierostruc.,27, 61-6.

[5] Ferry, D. K. and Barker, J. R. (1999). "Issues in generalquantum transport with complex potentials", Appl. Phys.Lett., 74, 582-5.

[6] Gross, W. J., Vasileska, D. and Ferry, D. K. (2000)."Ultra-Small MOSFETs: The Importance of the FullCoulomb Interaction on Device Characteristics", VLSIDesign, these proceedings.

[7] Datta, S., Assad, F. and Lundstrom, M. S. (1998). "Thesilicon MOSFET from a transmission viewpoint", Super-latt. Microstruc., 23, 771-80.

[8] See, e.g., Grubin, H. L. (2000). "Wigner FunctionMethods in Modeling of Switching in Resonant TunnelingDiodes", VLSI Design, these proceedings.

[9] See, e.g., Carrier, G. F., Krook, M. and Pearson, C. E.(1966). Functions of a Complex Variable (McGraw-Hill,New York), p. 318ff.

[10] See, e.g., Erd61yi, A. (1961). "From Delta Functionsto Distributions", In: Modern Mathematics for theEngineer, Edited by Beckenbach, E. F. and Hestenes,M. R. (McGraw-Hill, New York), pp. 5-50.

[11] See, e.g., Kadanoff, L. P. and Baym, G. (1962). QuantumStatistical Mechanics (W. A. Benjamin, Reading, MA),Chapt. 6.

[12] Moyal, J. E. (1949). "Quantum Mechanics as a StatisticalTheory", Proc. Cambr. Phil. Soc., 45, 99-124.

[13] Bordone, P., Vasileska, D. and Ferry, D. K. (1996)."Collision-duration for optical-phonon emission in semi-conductors", Phys. Rev. B., 53, 3846.

[14] See, e.g., Barker, J. R. (1978). "High Field CollisionRates in Polar Semiconductors", Sol.-State Electron., 21,267; Barker, J. R. (1980). "Quantum Transport Theory",In: Physics of Nonlinear Transport in Semiconductors,Edited by Ferry, D. K. et al. (Plenum Press, New York),pp. 126-152.

[15] Ferry, D. K. and Grubin, H. L. (2001). "Quantum Trans-port Physics", Int. J. High Speed Electron. Sys., in press.

[16] Zhou, J.-R. and Ferry, D. K. (1992). "Simulation ofUltra-Small GaAs MESFETs Using Quantum MomentEquations", IEEE Trans. Electron Dev., 39, 473-8.

[17] Ferry, D. K. and Zhou, J.-R. (1993). "On the Form of theQuantum Potential for Use in Hydrodynamic Equationsfor Semiconductor Device Modeling", Phys. Rev. B, 48,7944- 9.

[18] de Broglie, L. (1926). "Sur la pssilbilit6 de relier lesphnomnes d’interf6rence et la diffraction t la th6orie desquanta de lumire", C.R. Acad. Sci. Paris, 183, 447;(1927), "La structure atomique de la matire et durayonnement et la m6canique ondulatoire", C.R. Acad.Sci. Paris, 184, 273.

[19] Madelung, E. (1926). "Quantentheorie in Hydrodyna-mischer Form", Z. Phys., 40, 322.

[20] Bohm, D. (1952). "A Suggested Interpretation of theQuantum Theory in Terms of ’Hidden’ Variables", Phys.Rev., 85, 166; 85, 180.

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