D ID-A65 464 TRANSIENT TRANSPORT IN BINARY AMD TERNARY tSENICONDUCTORS(U SCIENTIFIC RESEARCH ASSOCIATES INCGLASTONBURY CT H L GRUDIN ET AL. 2? FED 86

UNCLSSIFIED SR -R939-FARO -±848S5-ELFG 2/12NL

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TRANSIENT TRANSPORT IN BINARY AND TERNARY SEMICONDUCTORS

FINAL REPORT -R930009-F

H.L. GRUBINJ.P. KRESKOVSKY

M. MEYYAPPAN

B.J. MORRISON

FEBRUARY 1986

U.S. ARMY RESEARCH OFFICE D ICONTRACT: DAAG29-82-C-0023

Prepared by

SCIENTIFIC RESEARCH ASSOCIATES, INC.P.O. BOX 498

GLASTONBURY, CONNECTICUT 06033

L.J ~APPROVED FOR PUBLIC RELEASE; K~f5DISTRIBUTION UNLIMITED 4

-- e

lb1NTASSTTVp 28 February 1986 f4~ / iLSECURITY CLASSIFICATION OF THIS PAGE (When, Dera EnEv~I~~ ~~12

REPOT DMMENTTIONPAGEREAD ERSTRUCTIONSREPOT DCUMETATON PGE EFORE COMUPLETING FORKI. REPORT NUMBER I.GVT ACCESSION No. 3. RECIPIENT'S CATALOG NUMBER

Aeo Y*OSL.I N/A NIA14. TITLE (and S..btjqi.)' S. TYPE Of REPORT & PERIOD COVERED

Transient Transport in Binary and Ternary FnlRpr27/09/82 - 26/09/85Semiconductors6. PERFORMING ORG. REPORT NUMBER

7. AUTHOR(.) 6. CONTRACT OR GRANT NUMSER(e)

H.L. Grubin M. Meyyappan DA2-2C02J.P. Kreskovsky B.J. Morrison

9. PERFORMING ORGANIZATION NAME AND ADDRESS to. PROGRAM ELEMENT. PROJECT. TASK

Scientitic Research Associates, Inc. AREA 6 WORK UNIT NUMBERS -

P.O. Box 498Glastonbury, CT 06033

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U. S. Army Research Office 2 eray18Post Office Box 12211 13. NUMBER OFPAGES

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WS SUPPLEMENTARY NOTES

The view, opinions, and/or findings contained in this report arethose of the author(s) and should not be construed as an officialDepartment of the Army position, policy, or decision, unless so

ig. K EY WODg'"S4 C ou ore reverse aide If naceeaeay and Identify by block number)

Nonlinear transport, semiconductors, microelectronics, quantum transport,Boltzmann transport, drift and diffusion, gallium arsenide, aluminum galliumarsenide, indium gallium arsenide, transient transport.

20.~ ~~~~~~I AINAT(e~a eeu f~ necoe~ md Idenily 6r block nimmbor)--A description of recent studies of two and three terminal band structure

dependent compound semiconductors is given.

D JAN~ 43 EWtNFIOSIOSLT UNCLASSIFIED 28 Februaryv 1986

SECURITY CLASSIFICATIONE OF THIS PAGE (When Data Encered)

TRANSIENT TRANSPORT IN BINARY AND TERNARY SEMICONDUCTORS

H.L. Grubin, J.P. Kreskovsky, M. Meyyappan and B.J. Morrison

Scientific Research Associates, Inc.P.O. Box 498

Glastonbury, Connecticut 06033

The objectives of this study were several fold: 1) To examine space and timedependent transport in gallium arsenide, aluminum gallium arsenide, and indiumgallium arsenide; 2) To begin a reformulation of quantum transport for mixedstates. The technique used for studying transient transport was throughsolutions to the moments of the Boltzmann transport equation. The quantumtransport was primarily formulatory.

In examining the binary and ternary transport application of Vegards rule wasapplied to determining the effective masses and a variety of other band struc-ture parameters. The parameters for indium arsenide and aluminum arsenide wereobtained from previous calculations of Littlejohn, et.al., and modified forimplementation using the moments of the Boltzmann transport equation. Theinitial calculations for steady state were performed using parabolic bands. It

became immediately apparent that the parabolic band structure would be inade-quate for examining transport. Thus during the initial phases of the study, f 'dthe moments of the Boltzmann transport equation were modified to include theLpresence of generic energy versus momentum dispersion relationships. Allc2/ Fg-.'''calculations, however, were performed for the situation in which c + c .4j2 k2 /2m*, where cg is the separation between the conduction band minimiaand the top of the valence band. Nonparabolicity was used for the r, L, and Xportions of the conduction band for all semiconductors of interest.

In all calculations a displaced Maxwellian distribution function was used.For the case of parabolic bands, the derived moments of the Boltzmann transport

equation are direct and have been published in variety of places includingthose by the Principal Investigator. A new feature introduced under the AROstudy was the implementation of the displaced Maxwellian for nonparabolicbands. In this case, it was necessary to expand the formal displaced

Maxwellian, f(r,k,t) in powers of k. The expansion as performed in this study

consisted of four terms:

f(r=,t) f0 + fl + f2 + f3 (1)

higher order terms were ignored. Most, if not all, previous discussions ofnonparabolic transport have included only the f0 and f, terms. The departurein this study was the inclusion of the f2 and f3 terms. The first questioninvestigated was: How important is the inclusion of the f2 and f3 terms?

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To determine the importance of the f 2 and f 3 terms, the moments of theBoltzmann transport equation, which had already been obtained exactly forparabolic bands was rederived for parabolic bands using the four term expan-sion. It was determined that the form of the moment equations, as obtainedusing the complete parabolic displaced Maxwellian, and the four term expansion, L'were identical only when all four expansion terms were included. This resultindicated that all previous nonparabolic studies using the moment equations andonly the fo and f, terms were significantly incomplete. The summary in the :next few paragraphs discusses the significance of these latter contributions,as well as the general nonparabolic contributions which have not been discussedin the past.

To begin, it is worthwhile to restate the obvious, particularly, when nonpara-bolic transport is considered: The transient dynamics of carriers both underuniform and nonuniform field conditions, involves momentum transport ratherthan velocity transport. The moments of the Boltzmann transport equation de-scribe the time dependence of carrier momentum, and all overshoot phenomenawhich is generally ascribed to velocity, is rather momentum overshoot. Underuniform field conditions, the time rate of change of carriers due to inter-

*" valley transfer, is affected only through nonparabolic contributions to the* scattering intervals. Under nonuniform field conditions the divergence term,

div(n&k/m), which appears in the continuity equation, is unchanged in form forboth parabolic and nonparabolic bands. The significance physics, however, isaltered: For parabolic bands 4*k/m is the mean carrier velocity; for nonpara-bolic bands it is not.

With regard to momentum balance for uniform fields, the form of the equation isthe same for both parabolic and nonparabolic bands. The only alteration is inthe nonparabolic contribution to the momentum scattering integrals. However,for nonuniform fields there is a significant alteration in the pressure tensor.The alteration in the pressure tensor for nonparabolic bands arises from the f2 ". *.

contribution in the expansion of the distribution function. Complete cancel- .lation of the f2 contribution occurs for parabolic bands.

With regard to the third moment equation, the equation for energy balance, wefind nonparabolic contributions arising even for uniform fields. First it isnoted that there are nonparabolic contributions to the energy scattering in-tegrals. Of more significance, it is a fact that the time derivative of themean thermal energy of the electrons, a[3/2nkBTe]/at, becomesa[3/2nkBTeg(Te)]/3t. The effect in sample calculations of the contri-bution g(Te) is to increase the time to relaxation. Under nonuniform fieldconditions, the kinetic energy contribution undergoes similar alterations.There are additional changes, each of which can be traced to the inclusion ofthe f4 term in the expansion of the distribution function. Again, under para-bolic conditions, these additional terms undergo specific cancellation. Insummary, the presence of nonparabolic band structure significantly alters theform of the governing equations for transient transport within the framework ofthe moments of the Boltzmann transport equation. The question next is: How dothese changes alter our picture of velocity overshoot in semiconductor devices?

The typical way in which transient transport problems are attacked under uni-form field conditions is as follows: 1) A set of band structure parameters areestablished which yield steady state velocity versus field curves that

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are compared either to experiment, if available, or other calculations. 2)

Once an established set of parameters is obtained, the time dependent calcula-tions are performed. The results of these calculations whether through Monte

Carlo procedures, moment procedures, or iterative procedures, all show similarvelocity overshoot. The wrinkle in this procedure is as follows: For nonpara-bolic bands another set of band parameters is chosen from which a mean momentum

versus field relationship calculated. From an equation relating velocity tomomentum, the resulting steady state velocity versus field is obtained. Now,when the transient calculations are performed, the transients involve the ap-

proach of the mean momentum to steady state, from which the velocity transient Lis obtained. From a practical point of view, it will not surprise anyone thatthe time to steady state with and without nonparabolic contributions is dif-

example: When a negative differential mobility element is designed to operate

as an active device under 94GHz constraints, it is necessary to transfer car-

riers from the r to L and X valleys, and back. One criteria for sustainedoscillations is that the peak to valley velocity ratio be sufficiently high.This implies that a significant amount of electron transfer from the r valley

has occurred. For nonparabolic bands, insofar as velocity and moment are notlinearly related, there are situations where a large peak to valley ratio invelocity does not necessarily imply a similarly large peak to valley ratio inmomentum. This has emerged from calculations in which the compound semicon- _ductors were placed in a circuit containing reactive elements. Thus a newcriteria must be established for determining the upper frequency limit foractive device operation in which nonparabolic contributions are introduced.Consider next, the effects associated with nonparabolicity in semiconductors.

Calculations were performed to determine specific nonparabolic effects. Thesenonparabolic effects were observed for GaAs, InGaAs, and AIGaAs. For example:Steady state calculations were performed for GaAs subject to parabolic bands.The field dependence of velocity, carrier density, and electron temperature foreach of the three valleys were obtained. For the same set of band structureparameters, e.g., phonon frequencies, intervalley deformation potential cou- .-

pling coefficients, etc., the calculation was redone, but with a nonparabolicdispersion relationship. Under steady state conditions, the only alteration inthe equations is in the scattering integrals. The results of the calculationwere initially, partly unexpected. First, as anticipated the presence ofnonparabolic bands reduced the low field mobility of GaAs from its parabolicvalue. The unusual result was that the alteration in the scattering rates alsoeffected the rate of electron transfer. The result being that more carrierswere retained in the r valley than were expected from parabolic calculations.There was thus, an apparent 'improvement' in the 'calculated' velocity fieldcurve. There cannot, of course, be any 'improvement' in the velocity field

curve, and so the band structure parameters for the GaAs required readjustmentto agree with experiment. It is noted that very similar results were reported . 'by A. Sher, at the recent 1986 DARPA EHF Review. These characteristics werealso seen for ternary materials.

The above discussion with concerned specifically with the effects of nonpara-

bolic behavior. The InGaAs and AIGaAs studies were performed within thecontext of broader goals. First, it was anticipated that the band structure ofInGaAs was such that a larger fraction of carriers could be retained in the

-3-

r valley than was possible for GaAs, and that with its high mobility, it wouldoffer superior current drive capabilities for submicron applications than GaAs,but that its inferior high field saturated drift velocity would make it lessuseful than GaAs. Nonuniform field calculations for submicron GaAs and InGaAswere performed and demonstrate the superior current drive characteristics ofthe InGaAs. An additional question was asked with respect to GaAs versusInGaAs (note, in this discussion we are concerned with a 50% In concentration);namely, what are the upper frequency limits of GaAs versus InGaAs. Others haveaddressed this question, and have concluded that the upper frequency limit ofGaAs is higher than that of InGaAs. Indeed, a recently completed DARPA study 4performed by the Principal Investigator, using very general scaling concepts - .corroborates this result. The final conclusions, however, are not in. Forexample: Nonuniform field calculations were performed through solutions to theBoltzmann transport equation to simulate Gunn oscillations. These oscillationsinvolve the movement of dipole layers from a nucleation site to a collectingsite. Oscillations in GaAs have been measured at frequencies somewhat inexcess of IOOGHz. Comparative GaAs and InGaAs calculations were performed.Both of them require different levels of background doping to achieve highfrequency oscillations. Both semiconductors displayed oscillations in the1OOGHz regime. A comparative upper frequency limit has not been obtained yet.Further work is needed. It is noted that in the nonuniform field calculationsall transport was assumed to be parabolic.

The situation with AlGaAs was also considered. Here calculations were perform-

ed with varying compositions of AlGaAs. The experimental observation of asignificant dip in the mobility near a 50% composition of Al was not observed,

even when alloy scattering was included. It may be anticipated that suchquantities as the deformation potential coupling coefficient will requiresignificant nonlinear increases near this composition range to account for thedip in mobility. Major attention was, however, focused on a 30% composition ofAl. Within this range AlGaAs exhibited a marginal region of negative differen-tial mobility. The interest in the AlGaAs is clear from the point of view ofthe new heterostructure studies, and calculations were performed for AlGaAs/

GaAs HEMT devices.

The HEMT studies were performed using the semiconductor drift and diffusionequations and reveal the following results, some of which were initially sur-prising. The first result was that while a two dimensional electron gas formedat the AlGaAs/GaAs interface, it wasn't as had been expected confined to a nar-row, sub-100A width. Rather, there was significant charge injection within thesemi-insulating substrate, and within the heavily doped AlGaAs region. Thisresult was initially quite controversial. And it wasn't until very recently,that we understood the origin of the result to rest with the diffusivity as-sumption of the calculation. For example: In virtually all HEMT studies, bothanalytical and numerical, the Einstein relationship has been assumed. This

Ir relationship underestimates the high field diffusivity and as our studies in-dicated, result in narrow confinement. We have used an empirical relationshipthat more closely represents experimental observation and permits higher -

diffusivity. This latter result is responsible for the numerical results.

The above discussion involves solutions to both the semiconductor drift anddiffusion equations and the moments of the Boltzmann transport equation. Asis well known there are difficulties when one is considering transport at

- 4.-

heterostructure interfaces. To overcome these difficulties, we have beendirectly involved in formulating quantum transport for submicron devices.Initially, in 1981, through application of the Wigner distribution function,a series of moment equations were formulated for pure state. In the absence of "scattering, only the first two of those equations were independent. These twoequations were equivalent to Schrodinger's equation. During the course of theARO study, the moment equations were generalized to a mixed state. In general-izing these equations, the formulation was performed in terms of the densitymatrix rather than the Wigner function, although the two are related through atransformation. In this formulation it was immediately apparent that a separ-ation could be established isolating specific quantum contributions from theclassical contributions. Further, it was also immediately apparent that noassumptions on the properties of the potential were necessary, other than thefact that the potential seen by the carriers be continuous.

Each of the above studies: Boltzmann transport, drift and diffusion, andquantum transport have been presented at major technical meetings, summerschools, and government reviews. They are all being prepared for technicalpublication. Copies of publications arising from this study are included withthis report. A list of open presentations is given below.

PRESENTATIONS

Scaling and Band Structure Transport, 1983 APS March Meeting.

NrLarge Signal Numerical Calculations of Three Terminal Selectively DopedHeterostructure Field Effect AlGaAs/GaAs Transistors, 1985 WOCSEMMAD.

Role of Diffusivity in HEMT Calculations, 1986 WOCSEMMAD.

Moments of the Density Matrix Incorporating Dissipation, 1985 Quantum Transport

Workshop.

A Study of the Moments of the Density Matrix with Applications toUltrasubmicron Structures, 1986 APS March Meeting.

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594 Surface Science 132 (1983) 594 -6022Norih-1 lolland IPuhlishing (ompany. -

TIlE ROLE OF BOUNDARIES ON IIlGI! SPEED COMPOUND.SEMICONDUCTOR DEVICES

H.L. GRUBIN and J.P. KRESKOVSKY

Scientific Research Associates. Inc.. I'. 0. Bos 498. ;laitontpur'. Connst'ut t¢,)33. USA-.

Received 2 September 1982; accepted for publication II November 1982

The role of boundaries and interfaces on the electrical characteristics of long and submctl

scale compound semiconductor devices is discussed.

I. Introduction

Theoretical and experimental studies of compound semiconductor devicesover the past decade have demonstrated that conditions at the device boundariesare the most important determinant of the operating characteristics of the

device. Studies in long two-terminal gallium arsenide devices [I]. indium..phosphide [21, germanium [31, and cadmium sulphide [4] have demonstratedthat conditions at or near the contacts control the current-voltage relations

and the electrical characteristics of any resulting instabilities. For two-terminaldevices it is often possible to correlate the pre- and post-threshold device

behavior. For three-terminal devices, most interface studies have focused onLthe role of the layer just under the principle region of electron transport. In the

case of gallium arsenide field effect transistors, heterostructure interfaces havebeen incorporated with the object, e.g., of confining carriers to the activeregion in the case of a heavily doped active region, [5.6]. or of providing a sea

of carriers in a nominally undoped region.In near and submicron scale devices, the relative importance of the

boundaries increases as these regions occupy a sizeable fraction of the deviceactive region length. and the up- and downstream boundaries begin to con-municate with each other. Controlling these boundaries is likely to he the mostsignificant task of device physics studies in the immediate future. The motiva-

tion for this outlook is based upon the fact that transport on a near andsubmicron scale involves nonequilibriun effects on a picosecond time frame,and its effectiveness is based on transport by high mobility carriers. Thus,questions such as: H ow many carriers are injected into the I' valley of gallium.-'arsenide, and with what energy and velocity?" enter the picture prominantlv.

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The purpose of this paper is to highlight the effects of boundaries oncompound semiconductor devices, particularly gallium arsenide. In doing so,the discussion is separated into two parts: (I) the role of boundaries on long,low frequency ( < 20 GI iz) gallium arsenide devices, and (2) the role of theboundary on high frequency near and submicron scale devices.

For long devices in which a rich body of experimental work exists, a reviewof boundary effects is given in terms of a "pinned" cathode field and--

pinned" cathode current model. The basis for these models is tile notion thatcontact boundaries may be phenomenologically represented as either tunnelingor thermionic emission dominated regions, with a varying barrier height. Thedescription of long devices assumes that all picosecond scale transients haveoccurred, and that all band structure population statistics are adquatelyrepresented by steady state conditions.

For the shorter devices it is the short time transients and the spatial andtemporal transients within the bands that are calculated. The descriptionrequires solutions of the Boltzmann transport equation and resolution on thescale of a fraction of a bulk mean free path is needed. These problems arediscussed in section 3 where a review of such phenomena as velocity overshootis given. The ability to attain contact and boundary effects permitting therealization of the high overshoot speeds in devices is the core on which the,tudics of section 3 are based.

All of tile theoretical studies presented arise from solutions of differentialequations. For long devicesthe "drift and diffusion" equations are solved. Forshort devices, moments of the Boltzmann transport equation are solved. In-both cases true contacts are represented as boundary conditions to the dif-ferential equations. Thus, the studies illustrate the effects of the contactsand/or interfaces on device behavior. The results of these studies, when "successful, tend to highlight what is unknown about device material parame-ters. For the case of long devices, it is the cathode boundary condition that isunknown. I lere the sensitivity of the results to varying the cathode fieldidentifies tile significance of tle contacts on device behavior Ill. Similarsensitivity :;tudies are discussed in connection with solutions of the Boltzmanntransport equation, where a range of parameters is chosen to identify tleconditions for "injecting" and "blocking" contacts. A sensitivity analysis oftile electronic contribution to the thermal conductiviiv is also included as itseffects are dramatic.

2. Boundan conditions to negative differential conductivity devices

In this section a brief review of the influence of boundaries on the behaviorof N I)C devices is given. Fig. I displays typical boundary-dependent data fromthree different gallium arsenide two-terminal devices 171. The lower portion of

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each diagram displays current versus voltage characteristics, while the upperportion shows voltage versus distance at one bias point. Fig. )I showsmeasurements for a device in which the metal contacts are far removed fromthe active region of the device. "The current-voltage relation is relatively linearuntil a point where current oscillations occur. 'he field profile just prior to theoscillation is relatively uniform within the active region of the device, and isnear zero at the ends of the active region. Fig. lb represents a set ofmeasurements in which the metal contact abutted the active region of thedevice. The current-voltage characteristics remained linear to threshold which-again was manifested by a current oscillation. Notably different here is thelower average field prior to the instability and the enhanced voltage drop at thecathode. Fig. Ic displays results for another device with a metal contactabutting the active region. For this case there is a sublinear current voltagecharacteristic and no instability. The probed voltage versus distance shows alarge voltage drop at the vicinity of the cathode.

The electrical characteristics associated with fig. I have been described asrepresentative of "ohmic" (fig. Iia), -slightly blocking" (fig. lb), and "stronglyblocking" (fig. lc) contacts. One of the earliest models employed for explainingthese results assumed a "pinned" value of cathode electric field I1. Othermodels in which the cathode conductivity [8] or doping [9] have also beensuggested with varying degrees of success. The "pinned" cathode field modeldeveloped, partially as a consequence of the way the governing equations,describing current instabilities was written. Here the one-dimensional differen-tial equation for total current,

J(T)=q(NV- )1 ' T" (4)

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This is a second order partial differential equation requiring two bou nda rsconditions on 1'( X, I ) and one initial condition. In the above J( T) representscurrent density, N is carrier density, V and D are field dependent velocity anddiffusion.

Solutions to eq. (2) have been used successfully to simulate device resultssimilar to those of fig. I. For gallium arsenide in which the threshold field for

or negative differential mobility is approximately 3.2 kV/cm = l I.qualitatively

similar results occur for solutions with cathode fields falling into anv, one ofthe following three groups: 0 E( ' 0, T) E- , E I I,< L-(" (I, T) < 4E" ,. 1 -- (X , T) > 4 F l-he simulations with pinned fields falling in eithergroup I, "2 or 3 vield electrical characteristics similar to those of figs. la, Ib and -

Ic, respectively. lhe crucial feat ure of this model is that tile cathlode field Is

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pinned, necessitating that any instabilities in current occur at a critical value ofcurrent density. The field profiles associated with cathode fields in the range0 < E( X = 0, T) < E-1. , and E 11 < E( X = 0. T) < 4 E l, are sketched in figs.2a and 2b respectively. For reference, a velocity field curve with velocity scaledto current as qNoV(F)=J,,( I-), and with a region of negative differential .mobility is also included. Fig. 2 is understood as follows: The second column ,of each section shows the electric field versus distance profiles. E( X) beginswith a value E, at the cathode and extends downstream to a value E,. Bycurrent continuity, the current everywhere within the device is given byJ = qNV( Eh). For J < J,,( E.), a region of charge dep;etion forms near thecathode for E. below and within the region of negative differential mobility(NDM). Increasing the current until I J,,(l-') introduces charge neutralityeverywhere for E, < EFl . However, because E is a double valued function ofV, for E, within the NI)M region, approximate charge neutrality exists nearthe cathode for J < J( ," ), and for regions sufficiently far downstream fromthe cathode. Charge neutrality breaks down between these two regions. Finally,for J > J,( E,) an accumulation layer forms near the cathode. For fig. 2a. theaccumulation layer is stable until the bulk field exceeds ETII. For fig. 2b. theaccumulation layer, followed downstrean by the depletion layer, is often

unstable and leads to cathode originated instabilities 1101.The situation corresponding to fig. Ic is often represented by very high

cathode fields. The field profiles are those appropriate to a wide region ofcharge depletion near the cathode. The profiles are electrically stable.

The characteristic feature of these nonuniform field profiles is that theirstructure is significantly affected by the field being a double valued function ofvelocity. The pinning of the cathode field is not necessarily common, however.to all semiconductor devices. For example, it was also applied to InP devices.where it worked for a significant number of cases. However, a broad class ofdevice behavior could not be accounted for through its use [2]. These device.showed anomalously high efficiency and significantly low DC current level".Spontaneous Gunn type oscillations did not occur. Rather, device oper:ationrequired a tuned circuit. The details of the oscillation were thought to dependcritically on the cathode boundary condition, which in this case was taken as afixed cathode conduction current [10.1 I].

The distinction between "pinned" cathode field and "'pinned- cathodeconduction current is placed in perspective in fig. 3 and in the followingequation

dE. L__

dT (3)

Eq. (3) is the equation for total current through the boundary to the device.J( E ) represents the current- field relation at the cathode 1121 which may beexpected to differ from tlhat of the semiconductor device. Two such tvpes ofI

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Fig. 3. Cathode conduction current curve for an approximnately "pinned" cathode field, curve A:and pinned cathode conduction current (at high field fields). curve B. Curves are obtained from eq. 1-(41. Also shown, for reference, is the neutral current versus field curve for GaAs. From ref. [101,with permission.

curves are represented by curve A and curve 13 of fig. 3. Curve A is closelyrelated to the pinned cathode field model while curve B is associated with thepinned cathode current model. The similarity in "form" of curves A and B tomoderate barrier height tunneling and thermionic emission dominated con-I.d -ics deliberate [ 10), and the equation used to arrive at these curves is shown

ex -. exp-

nk T )1 k ] (4)

w-hich was adapted from studies on the unalloyed metal/semiconductor con-tact 1131. Its use here presumes a similar description. For the unalloyed contact.it is the ideality factor and describes the contact as dominated by thermionicemlission (ur =u) or by tunneling (it >y ) J. is the reverse current flux andmay be related to the barrier height phenomenologically through the Richard-son equation 1101.

Detecting a particular cont(act effect on a device is a difficult procedure. Forlong devices current voltage characteristics as represented by fig. I are oftensignatures of a contact classification. For short devices proximity effectsintroduce an additional complication and current-voltage measurements areless valuable. One type of measurement which may serve to provide infortna-pinn about the boundar is a noise measurement.

J ere the situation to envision Is that If the field is pinned within the 4

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negative differential mobility region, Increasing the bias will result in anincrease in the length of the NI)M region. Any fluctuation will sustainenhanced amplification and the noise will increase. If increasing the bias

* resulted in carrier injection into the device, the field at the cathode is likely todecrease with increasing bias and the noise is expected to decrease. While these * ""-results should be folded in with the field dependence of velocity and diffusion,a simple analytical noise calculation assuming a three piece linear approxima-tion to represent GaAs has been performed 114].

In this calculation, the "impedance field method" 1151 is applied to calculat-ing noise due to thermal velocity fluctuations amplified within the device. Themean squared noise voltage per unit hand width 115] is computed,

N v4q2 ,IVZ12 NI) d(vol). (5)

K "Vo-l -

where vZ is the impedance field vector (15]. The calculation is performed for a10 pm long element with a doping of l0/cnQ. The element sustains the fieldprofile of fig. 2b where it is seen that the NDM region increases withincreasing bias. The calculations, which are discussed in detail elsewhere. [14]are expressed in terms of the noise figure f161:

NF=I+ (64k0 rlRi (6)

where R is the real part of the device impedance. The results of the calculationare displayed in fig. 4, where the noise figure is sketched as a function of biascurrent and transit angle. The results appear as a signature of the effects of thecathode boundary. First, at low values of transit angle 0 = wT(A), where T() Ais the transit time across the negative differential mobility region, the noisefigure increases with increasing bias. This corresponds to an increase in thelength of the negative differential mobility region and enhances amplificationof any fluctuation originating there. More interesting structure is present athigher frequencies and higher bias where the noise figure increases and then

lshows a singularity. On the other side of the singularity there is a "U"-shapedregion ending again at a singularity. The strong increase in noise figurerepresents the approach of JRI - 0. Here. at low frequencies, the real part ofthe impedance is positive, and becomes negative at frequencies somewherebetween r < 0 < 27r. In going from positive to negative values it passes throughzero, hence the singularity. The frequency range for small signal negativeresistance increases with increasing bias 110] reflecting the broadening of thenegative differential mobility regions - a broad "U"-shaped region appears.Both the increasing noise figure at low frequencies. and the " U"-shaped regionat high frequencies are characteristics of an increasing depletion layer width.Note that increasing the bias still furthcr will result in an electrical instability11o].

. . . . .. : :> . .a. n . k- .. tt, .. r . wZ.... . . : , -

602 I. .. (;rulhn. J. P. Kreskovsky / Ifigh speed cvapound s.,mntductor devices .. '...

0.48=j

40-

=0.47. 0475 = 1

Wi 30

,

O 20"z

0.44= J

..-.0.40=.

07T/2 7T 3r/227

TRANSIT ANGLE, 8

Fig. 4. Noise figure versus transit angle and normalized bias current] = J,/NoeVp , for a 10 jum

long element with a depletion layer profile. From ref. 1141, with permission.

3. Electron transport in near and submicron devices: the role of the boundaries

The discussion of the above sections dealt with devices whose lengths were

typically 10 tm long or longer. Transport for these devices is generallydiscussed though use of the drift and diffusion equation (eq. (1)). For near andsubmicron length devices electron transfer is generally not complete until asubstantial fraction of the device has been traversed. Consequently the driftand diffusion description is inadequate and solutions to the Boltzmann trans-port equation are required.

For two level transfer in GaAs the steady state velocity field curve forcarriers in the central and satellite valleys are shown in fig. 5, where the netvelocity V is

V = N1 V, + N, 2 . (7)

Ilere the subscript I denotes transport in the central valley (F" valley for GaAs)

and the subscript 2 denotes transport in the satellite valley. Most device design .'"

is concerned with controlling the time spent by the numbers of carriers in

'I - --.-

rr .r- r . -' , . - , .- ,-' 7 '

-- -= " .- " .'. "-' -.- ' ." -,7- .- -: .''-. " -. -. ,.,-- ,- =.- - -_ - j'.. V r , , '

II. L Gribn. J. 1. KreskovA / high speed 'ompouad semiconductor devices 603

in valleys I or 2; and circuits, interfaces or contacts are sought which will allow -

for this control. Solutions to thelBoltzmann transport equation which providethe required nonequilibrium transport behavior are obtained by a number of

6.0

C EN TRA L, (VALLEY

3.0 -

" 2.0 -E

NET VELOCITY

0.6 T'"LCI

0

In-

0.-

w0.41

OSATELLI TE~VALLEYVALLEY

,

0.I

0 10 20 30

FIELD (kV/cm)

Fig. 5. Steady state velocity versus field curves for F' and satellite valleys for galliumn arsenide.From ref. 1221, with permission.

different techniques, the most familiar of which is the Monte Carlo method

[171. Other methods include the Rees iterative technique [18] and balanceequation solutions [19], which are considered below.

The balance equations discussed below are the first three moments of theBoltzmann transport equation. They form a subset of the infinite hierarchy ofmoment equations, and as such do not form a closed set of equations. [achequation introduces a higher order moment not defined by the given set ofequations. The most common form of the balance equations is obtained byassuming a distribution function of the form.

ILf)I, N1 exp ( J (8)'-A )-c.T . 3 12 e p21 k ., _

6(4 //,1. Grbn J- P. Kresk.onski, 1Ii~h xpeed de~psm inc,,s v icesi

for each species i. More generally 120]

Ia a 2 _ _

Ijl+AA + IA + ~ (9)A 4 ak ~Uku, + A4,,,, a U, du, aU,,

where it Is Insisted that thle first term Ii thle expansion gives thle correct localvalues of djensity, velocity and energy. The coefficients in the above expansionare inversely proportional to density and are model dependent. Thle firstcorrection to thie local equilibrium halance equtions, the so-called hydrody-namic approximation y'ields

+ I(/~V Af (10)

*at ('T

a, 'It e,-ll lAk divaOj \dT11

+ diV(K -as ~(2grdi)+ (17'

where T-, is ani electron temperature,

P Ar I'. It' -- Anz 1 2 + N k T;k.- Ais at stress tensor arising from nontunifr oeoiditiuins, and K is thle

thermal conductivity. Thle stress term is dissipative inl that when it nonuniformveiocity distribution is impressed on aneeto tem teewl eratve

forces tending to smooth them out. At this point. howkever, these terms areregarded as phenomenological entries.

The boxed terms above represent conlt ribuittions from thle nounsphericalnature of the distribution function. These con tributnins has e not been includedinl evaluating the collision integrals. i.e., bu .. ,~Ht %%i ll be discuIssed Ii afuture paper. [hei collision integrals are di.CsSce In)IC 1211. [he Underlinedand boxed itrms are ignored Ii thc drift aind di flusiona prxiain

U nder u nif ri field conditions the mnean resps n~ )I ca triers to aI suddenchange tin electric field is, showkn inlhl 11", 1, 1 ewh1 peaik veloci tv Is atconsequence of carrier,, bet in retaiined inl tile V'' al' for t inle dIIi tionlupward,, of (0.2 ps before un idergoing trainsfer ito thre tippi 'alles s [he peak .-

veloci tv inl fig. 0 is ext remel s high It r epw sen t, sl .11 ppei Hotind on t ile carriervelciitv that ma11I he expcted at t his held aMld pa ,'s ide t(lie met i\at ioni frdesigning bouindairies th.1t pert Iichescreii*ttr'c' it

I he peatk 'slsi'.losesi.a eiiicI'' ('is' 11,C timeW .l hint la

problems to, be .addlressel i lcsieviiii .ipji'priiit dcirs bs ouiihlt' condi-

ionv ee Ilie 7 u wiirhrr'iec bc.5 11 i 1 11)1 11111" 1,h It,1iiird tot Carriei-s to

%

70.-

GaAsNo =101

300 KIF 27 ky/cm

8

E

540

-J

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

TIME (ps)

Fig. 6. Transient velocity versus time, for electrons subject to at sudden chinge in field ofmagnitude 27 kV/cn1. From ref. [221. with permission.

transfer into subsidiary portions of the conduction band, on a near andsubmicron scale. the velocity field characteristics are expected to be lengthdependent. See fig. 8. Thus, near and submicron devices wvill categorically besensitive to both boundar- effects and device length. neither of w'.hich will beindependent of the other.

Figs. 5 to 8 are signatures of high speed submicron transport. What are theconsequences for device behavior? Can a suitable set of contact or interfaceconditions be found to achieve high speeds associated with overshoot'? While adetailed study on a scale similar to that for long devices is not yet available,some results are known. We discuss these below for a coilection of 2 pin longdevices, essentially undoped. No = 5 X 10' 5/cm 3 each subj. ected to a bias of 2-V. These one-dimensional studies are in steady state. i.e.. a/dT= 0. and aresubjected to the following cathode boundary conditions:

y~log N, =-A.,13

X4

W. %

IF 17.6 kV/cm%

100.0 %

50.0

5.0

0:5

0.12 3 4 5 6 7 8 9 .

(V x f10 ) cm/Sec

Fig. 7. Peak carrier velocitv versus bias rise time for electrons subject to a final field of 17.6 -

kV/rni.

4L

L00

-J

> L

04to 20

FIELD (k/ c r )

I ig 8 Vxsit %el cit% of (-,irrier% at the terminu' of a device of specified lengthI. and Indicated %alue

of electric field ( arm ivr enter Aith icro veloiicat and are ujetto a suddel 'hinge In field I rom

rel 12 11, i, ip r iisi i * -

.2 . . .

-v 1

Hii. L rhn J.PI. Kreskovskv /High speed ColIi~l eJAd~ rdevie.% (07

oI o Ii (1a5) T/ X0

1.0 -

-0

z

00 0.

30

02

DISTANCE (microns)

Fi .()Ircinlpp lto fcrirN N na2pnln eie"t nfr o ia

30igo 0"C1" ujc oaba f2V N stettlcrirdniv ,i h are

deSti lecnrlvle .()Eeti ild vru itne F r(I, ac lto 2 X1K II sA .2andB K0

608 11-.. ru .. I'. Kr,'skms'k i- / high speed compound .,i',c,,duutor de( ces

For A positive (negative) local charge accumulation (depletion) occurs at thecathode boundary. 1B is generally greater than, or equal to, 300 K and is ameasure of the mean thermal energy of the /'-valley carriers. As in section 2,the results are placed into three categories, "ohmic", "slightly" and "strongly"blocking contacts. The "ohnic" results are shown in fig. 9.

For "ohnic" boundaries an appropriate set of constants, with ref. 1191. are:A = 0.2, u, = 12,000 cm 2/V .s, and T, = 300. For this case carriers enter the

device with speeds greater than that associated with the central valley mobility.Consequently there is an accumulation of carriers at the boundaries, resultingin low values of cathode field. The field starts off at nearly I kV/cm andapproximately 2500 A must be traversed before significant transfer occurs.Increased transfer results in a lowering of the mean carrier velocity, necessitat-ing an increase in mobile charge as the anode is approached. The average "velocity across this device obtained from the relation

V =J/qN,, (17)

is V, = 1.78 × 10 cm/s. with N, = 5 x 10' 5 /cm 3. The advantage of overshoot

is not fulfilled by this contact device length configuration. -Lowering the cathode mobility to a value below that associated with the F

valley results in a more rapid dispersal of carriers, and cathode adjacent charge

depletion, associated with slightly blocking contacts, occurs. This is seen in fig.10 for A = -0.11, !. =6000 c 2/V -s, and T, = 300 K. It is noticed that the

cathode field for this case is approximately 4 kV/cm, which is higher than that Lassociated with the "ohimic" contact condition of fig. There are, however,important similarities between figs. 9 and 10. In both cases the carriersadjacent to the cathode are, for all practical cases, F-valley electrons. Very littletransfer, which is determined by carrier energy (temperature), has occurred. Inaddition, sufficiently downstream from the cathode the carriers appear to be

ignoring the cathode condition and are dominated by the downstream voltage "" "

drop, whose spatial distribution is about the same for both. The averagevelocity for this case is [I,, = 1.75 x 10' cm/s, slightly below that of fig. 9.

A significant change occurs when the mean energy of the F-valley enteringcarriers is elevated. For the parameters of fig. 10, but with Tr, 1200 K, a

substantial amount of transfer occurs at the cathode, resulting in a lowering of

the current through the device. The cathode field is approximately 7 kV/cm.,higher than that associated with fig. 10, and the downstream field is lower (see

fig. II). The average velocity of the carriers in this case is lowered to 1.28 x 10'cm/s, even though the central valley carriers are traveling at higher speeds (see __

fig. 5).The presence of moderately high cathode fields is attractive if a sufficient

number of carriers can be retained in the central valley where they can sustain"

high transit velocities. While this case is discussed in more detail below, the

simple ruse of injecting excess carriers into a device with the contact conditions

L- % .

ii. 6. (rih,,i. J. P. Ari-v4(o, i / I I, i/ speed cantipound vepincoiducttnr devices 6k

N T

z0i.-V

(1 1.00

0

0 2DISTANCE (microns)

30

o20

4

2 -

0 2DISTANCE (microns)

Fig. 10. As in fig. 9. hut with IL O 60&) cin/V s. A 0. 11 and B 300 K.

of fig. I I does not always yield the sought after current levels. This isillustrated in fig. 12, where now A =0.2. Here, the excess charge serves tolower the cathode field, which does not change the downistreami characteristicsin any significant way. The average velocity for this case Is virt1ualy Unchanged

aL -, - . ... .. r ' . S . .% ---... - '-

610) II. I.,(rut .J 1'. Kre.% 4,sk ' / Ihgi .%ped 44 ,...wid,1 w ,,,444 4 ,,r ,t' q %/C 4.'/..%r

Z NT

J 0

0.5 -

0 2

o "L

DISTANCE (nuc rons ) .- -

0

DISTNCE(m.c"ns

U

2

0 2"

DISTANCE (microns) " ' .-

-__

Fig. 11I. As in f'ig. 9. but with u, 6000 cm2/V -s, A = 0. 11 and B =1200 K.

when compared to fig. I1I. In this case V =1.30 X 10' cm/s. " 11 -'.

The situation of a strongly blocking contact is illustrated in fig. 13 forA = 9.0,ju = 1000 cm 2/V - s, and T = 3000 K. For this case virtually all carriersare swept away from the cathode, with the satellite valley carriers accountingfor most of the transport in the transition region. The cathode field is- .extremely high, approaching 60 kV/cm. The average velocity is approximately "

• .'

zero.The above results focus attention on the role of the upstream boundary in

the distribution of charge near the interface. Te results are more general than'those obtained using the stgy btate field dependent velocity relation espe-

cially in identifying the fact that electron transfer, en in the case of pariall iblocktng contacts, may not occur until some point downstream from thethoseobtaned singthe teadystat fied deendet velcityrelaionespe

cialy inidetifyng te fct tat eecton tansfr, ven n th cae"ofpartal/

blocing ontctsmaynot ccu untl sme pin ownsrea-fro th

HI. LG? (;pu/It, J. P'. Ar~~tv ,. Ilh (" 1npowidc %(,i ~'I# II(...Om/r h~ti . 611I

IN<

I-

z

<0 .

00 2

DISTANCE (microns)

20

IE

1-

10

5

3

0 2

COSTANCE (microns)

Fig. 12. As in fig. 9J. but with =604K! cin-/V s, A 0.2 and B =12MX K.

6,12 I1.. ,

P A,, '1,1j d

. , ,, --

,m ,,h, rt d '

"u'"-

LZ

0

I-1 0

IJ 110

0 2-

46-

-j

02

60

0 20

DS- (-co sF A g t ,s 93

Fig. loA ng cacuig. . Tihe reso f0o0 his is -9.0ct and carir hav travrse K.

path of sufficient length to exceed the threshold energy for electron transfer. In 1this case electron transfer results in velocity saturation and currents ap-propriate to the steady state parameters. Thus, while a blocking contact mayhe expected to yield high entrance velocities for one species of carriers, electrontransfer prevents a large fraction of these carriers from enjoying the speedadvantage. The message here is that a forgiving contact is useful only when

..2 '

//.L . (ruhin. J.PI. A'r.Vko, 'A v / Ibh %jpced c,'mip, 'ud *wmuiiw f,)r direvw6 I 03

coupled to an appropriate device design. Within the framework of the dis-placed Maxwellian. improvements in device structure are synonymous withvalues of electron temperature that are below that required for electrontransfer.

0

0

ox 1014/0

00

10.0

00 2 4

DISTANCE tmicrons)

% Fig. ~~~~~1. 5a oigdsrbto o eie.R sIV n odtosa h ahd

bonayae10.0i. .N Eetrcfedditiuin

614 1. L (rubn. J. J'. Kreskotsk- H/ /igh speed cornp)unld seu'v hitor det n ' %

One way of reducing the electron temperature is simply to go to shorterdevice structures, and recent results for gallium arsenide at room temperaturesuggest that device structures not greater than 0.25 pm may be required. Astructure that has recently been discussed within the framework of near andsubmicron devices is the N-N--N structure shown in fig. 14. This structurehas four interfaces to contend with. The most significant aspect of thisstructure is that it introduces an abrupt change in field at the N-N interface,which tends to emphasize overshoot contributions. The detailed input to thiscalculation is as follows. The N region is doped to 5 x 1015/cm 3 with the N-region an order of magnitude less in doping. For this device the cathode ohmicconditions of fig. 9 were imposed, and an average field of 5 kV/cm wasimposed. We note that over a distance of approximately 1.0 jim there is verylittle electron transfer. Current is carried mainly by F-valley electrons whose .carrier velocity peaks near 6 X 107 cm/s, showing substantial overshoot. Theelectron temperature for this calculation is shown in fig. 15, and is reasonablylow. V., for this case is 1.3 X 107 cm/s.

The results of this calculation are very encouraging. However, they are notsolely a consequence of structural device changes. The retention of F-valleyelectrons in the fig. 15 calculation is a consequence of using an electronicthermal conductivity appropriate to 5 X 1015/cm 3 through the Wiedemann-Franz ratio. A reduction in electronic thermal conductivitiy results in steepergradients in electron temperature. Fig. 16 shows results for a value of thermal ' -

conductivity in which substantial transfer occurs within 0.5 14m of entry intothe N - region. The average current has dropped to 6.5 x 106 cm/s. The firstset of results which virtually eliminate transfer must be regarded as an upperbound on current, whereas the second set is a lower bound. Actual deviceresults are likely to fall between the two. One important measurement likely toprovide significant information is whether a 1 m, 10' 4/cm 3 element will showsmall signal gain.

Returning to the boundary conditions, there are several points to be made.First, there is a clear indication as to the procedures necessary for generatingan entire set of "ohmic" and blocking contacts. It is not likely, however, thatthe boundary prescriptions of figs. 9 to 13 are unique; other sets of conditionscan be envisioned to yield similar results under DC conditions. Distinguishingbetween different boundary condition effects will come from time dependentstudies. The question then is: "How model dependent are the results?"

The governing equations are derived starting from the condition of adisplaced Maxwellian, which assumes strong electron-electron interaction. 5Boundary scattering is likely to substantially alter this interaction - near theboundary 1251. Results that appear to be model independent are those associ-ated with the entrance velocity. If central valley carriers enter with speedsgreater (less) than that dictated by equilibrium band structure considerations.carrier accumulation (depletion) will occur. L

kF

* .

. . ..-... "- ".o .,. -- . ,. - .o° o • " ..-.. - "."-.o° , * .. • .. ° -'. - ° ° ., ,. - . . . .. - .- . .- . .

C

1- 05

aN

0

00

05

00 2

DISTANCE (microns)

20

10

20

0 2

DITNE (mcos

47

I g (1 a t olL o u a i n h 'fl a a lU c o i , ( l c w l t lil r t r . f l-

-. 0

4 1.0

0

0 20DISTANCE (microns)

7--

o' °,.. -

UI. L. (rijhin. J. P' A're.4kovs.k v Ihgxh vwe.d c'omp,,w iut widu(~h tr i It'ri v% 617 " ''. .*1.

JJ J " . ,

11 12 - 13 IN -

J I 3.. N J.

T

J21 . 22 J23 1 2N .

Fig. 17. Representation of spatial dependence of current flow through two level device as a serial

chain of parallel elements. The first subscript in each element represents current through thecentral valley I or satellite valley 2.

Another point worth raising, particularly with regard to the absence ofovershoot in figs. 9 to 13, is the fact that for at least half of these devices thespace distributions were in steady state equilibrium with results similar to thoseof the drift and diffusion equations. While going to shorter device lengths doesnot necessarily prevent this from occurring [261, the length dependence has notbeen explored.

The results of figs. 9 to 13 also suggest that transport in a multivalleyedsystem may be controlled by one of the valleys. In the above examples, controlis by the central valley; satellite valley boundary conditions are relativelybenign. To see the reasons for this it is necessary to turn to fig. 17, whichrepresents current flow through the nonlinear elements as a linear chain ofparallel resistors. In this figure

J,,= N11VI,, i = 1, 2,..., N,

J2 = N2 V2 , i = 1, 2... N.

The voltage across element i is FAX. Thus if, e.g., the F valley in element Isustains a net carrier density below the background, it will have a higherresistance and voltage drop than an element with a density closer to back-ground. This, of course, is consistent with the results of fig. 11 and suggeststhat the portion of the conduction band with boundary conditions moststrongly departing from the uniform field conditions will be the dominantboundary condition.

The contact boundary conditions discussed in figs. 9 to 13 address only asmall part of the problem. It is certainly unrealistic to assume that undesirabledepartures in doping will be absent. A simulation of a nominally 10 7/cm 3

doped device, with a 10% decrease in doping over a distance of 1000 A, isshown in fig. 18. The distortion in electric field, for the set of boundaryconditions listed in the caption, is such as to prevent any real overshoot fromoccurring.

The above results which reflect the influence of space charge on transport indevices should be compared to uniform field calculations to indicate the goalsthat perhaps should be taken for some device structures. Fig. 19 displaysvelocity versus distance curves for electrons subjected to a sudden value of

.e

tobm J tt 4" A O

9 -

81.

hN E

0

06 0

0.[4

,.00

0.98 -N0 t

0 2 .2 50 1 !DITNC ICO S

I g 1 a.Eeti il ess lsm c 1)(.rird niv emdsm~ o n l id0 sL LS ok t ieasta.tet ttA %ilvcirir(itrb tin Ir m tl1(J %d c m ~ in

/IL Gmbm J I. A"Pcs A mn A iI, Ih.L 1: %pe (I f pI/ ia'Ild I IJrlmlI htw 14 1,E 6 11

T1 300 K<

i0 F -L-in <100>cm

% ko (0.065,0,0)

8

- 6

E

4

2

F 200

0 0 500 1000 10

DISTANCE (R)

- - Fig. 19. Velocity versus distance for electron% Aith a finite entrance energy sujctled to a Siddent

change in electric field. From ref. 1271. w4ith perimision.

electric field. In this case tile energy5 of thle entering carriers Is just below thatrequired for electron transfer. It is seen thiat moderate valuies of' electric fieldare necessary to sustain high transit veloci ties. InI fig. 20 the dependence oif'ransit velocity onl entrance enlergN Is calcu licd. It is %een that there is a1

window for high transit velocities, and it is tis veloci ty level that Is sought.

0i

- F, 7 7

021) H. L.. (;r1b,,, J. P'. KresA o.s v /High speed compoun wi mt .. ,?i icitr dei jv V

VELOCITY VERSUS DISTANCE VERSUS ENTRY ENERGY

N h F 1 0 -- in <100>

.N. Ky kz = 0 ,

T 300 K

.

e

i. E d0- C!~7

k.ox" Eo(eV)

o 0 0b 0.02 0. 041- Ac 0.03 0.07d 0.04 0.11e 0.05 0.16

f 0.06 0.21

0 g 0.07 0.27h 0.08 0.34

0 500 1000 1500

DISTANCE (a)

Fig. 20. Velocity versus distance for electrons with varying entrance energies, subject to a suddenchange in electric field. From ref. [27], with permission.

4. Summary

The experimental situation is such that, with the exception of long com-pound semiconductor devices, there are very few data on the role of boundariesand contacts to submicron devices. The reason for the paucity of data lies inthe fact that most submicron devices are three terminal device designs and thethird terminal tends to mask the role of the contact boundaries. This isextremely unfortunate since it is likely that two terminal device measurementswill indicate what can be achieved in controlling the entrance dynamics of thecarriers. To date, most two terminal device measurements on simple devicestructures have concentrated on the role of transport within the device, andraise the question of whether " ballistic" motion is possible 1281. Based on the

V. V. -:1 1. V V 9r~ ~ 9-

11.1~~~ ~~~ 61di .'KLAn% l 2

II. I.. (Gi 'u . J./' Ar,'U i.UA-v / I /lgh vped compound %e'ucondutor tht ae' (21 -

history of vacuum tube dynanmics 129j it should be recalled that, if transport isballistic, the electrical characteristics will be controlled by the contacts.

The situtation in submicron devices is further complicated by communica-tion between the up- and down-stream contacts. Thus it may be expected thatthe influence of a blocking contact on the electrical characteristics of long andsubmicron devices will be different. For submicron devices simple currentvoltage measurements may be rendered useless as a diagnostic tool. This iscertainly not the case in long devices.

The role of numerical simulations in these boundary and device studies hasbeen to act as surrogates for measurements that are not feasible. In one case,obtaining cathode boundary fields from measurements was not possible. Thusfor long devices the sensitivitiy of the numerical results to numerical changes in - -"

the boundary conditions, when coupled to experiments, provided the key to therole of contacts on device behavior [I]. For submicron devices, the difficultiesof direct correlation of experiment with specific transport phenomena areapparent and simulation through parametric studies will provide a key to therole of boundaries. But the description of transport on a submicron scale is stillinadequate and the descriptive role of boundaries is correspondingly weak. Forexample, most space charge dependent problems still treat the background as a"jellium" distribution. The discrete nature of impurities is ignored, as arestructural variations in the contacts. The extent to which this affects suchmeasurements as current-versus-voltage is yet to be determined. Notwithstand-ing these uncertainties, a considerable amount of information can be obtained Lby extrapolation from the ideal cases which can provide bounds on the limitsof transport through both the boundary and active region of the device.

Acknowledgments

The author acknowledges critical discussions with M.P. Shaw, G.J. lafrateand D.K. Ferry. This work was supported by the Office of Naval Research andthe Army Research Office.

References

Ill M.P. Shaw, P.R. Solomon and H.L. Grubin. IBM J. Res. Develop 13 (1969) 587.121 D.J. Colliver. K.W. Gray, [).J. Jones, H.D. Rees. G. Gibbons and P.M. White, in: Pro:. 4th

Intern. Symp. on GaAs and Related Compounds. Boulder, CO, 1972, Inst. Phys. Conf. Ser. 17(Inst, Phys., London, 1973) p. 286.

131 W.G. Guion and t).K. Ferry, Appl. Phys. L~etters II (1967) 337: J. Appl. Phw,. 42 (1971)254)2.•." "

141 K.W. Boer. !1.J. Ilans'h and V. Kummel. Z. Ph. ik 155 (1969) 170:K.W. Hoer and G. I)ohler. Phvs. Rev. 186 (1969) 793:

see also M.P. Shaw. P.R. Solomon and It.l. (;rubin, Solid State ('ommun. 7 (1969) 1619

.,~ -- .. -~ _______________________________________________

622 /LI I. (I iuhin. .t 1'. A rc A Av ./b %I~/, "' (w~It/ 11 '1411d %4 -11I tm' 11,JI 1, 1 e1p

15 Sees. eg. D- Ho(kon(;hot. _11,'XI\ndle. P' liatict aild 1.1' I Lall,11%. 11II) tan I~I lcironI)ececs ED1-27 ( 1980)) 114 1.

161 I). lDelagebeaudeuif andI N'. I.ln. ILIA.. Ir.:ns. IeCI:':: lDcs1c I: )-28 ( 19)X I) 7904.171 P.R. Soloio::n. Mv.PI. Sh::aw. I ITI. (, ;iibin anid RI1). KaiiLI. II E I rally I-lec t:.i: Device, 1[I )-22

(1975) 127.18 W..C Gr ev, Flctron. I~itr 7: 971 711:

see also: K.W. Bcer and G. lDohler. I'livs. Rev. Igo6 ( 19o9) 793-191 E.MNI onwell. IFE 'Iran,,. Electron D~evices FD)- 17 (1970) 262.

11 M I. Shaw. P.R. Solomo~: n and I1.LL. ( ;rub::: I lbe G.unn II :Jsu in I:ffect (Acainic Press. NewYork, 1979).

Jll 1I.1). Rees, in: Metal-Semniconductor Conutacts. Inst. Ph.. Conf. Ser. 22 ( nst. Ilhys.. London.1974).

II12J It. Krociner, IEEE Trans.. Eilectron D)evices FI)- 15 (1968) 819.131 V.L. Rideout. Solid State Hlectron. 18 ( 1975) 54 1.

1 141 Il.L. 6rubin. R.F. Greene and 6.J. lafrate. ito he published.1151 W. Shockley,. iA. Ciopeland and R.Pl. Jamecs. Quantunm Theorv oif Atom:s. Miolecules and the

Solid State (Academic P~ress, New York. 1966).

1161 H W. Thim. Electron. Letters 7 (1971) 106.1171 T Kurosawa. J. Phys. Soc. Japan Suppl. 21 (1966) 424.1181 111. ). Rees, Solid State ('ommun. 26A ( 1969) 416: J. Phys. (C5 (1972) 64.

1 191 R. Bosch and 11. -Thim., IEEE Trans. ElAectron D~evices IiI-21 (1974) 16.120) See. e.g.. A. Sommerfeld. Thermodynamics and Statistical Mechanics (Academic Press. New

York. 1956) section 43.[211 H.L. Gruhin, D.K. Ferry. G.J. lafrate and J.R. Barker. in: VLSI E-lectronics. Vol. 3

(Academic Press. New York. 1982) p. 198.

1221 ILL. (irubin and D.K. Ferry. J. Vacuum Sci. Technol. 19 (1981) 540.

1231 ILL. Gruhin. D.K. Ferry and JR. Barker. in: Proc. 1979 1IEL)M (1979) p. 394.1241 B. Faugemberg. M. Pernisek and F. Constant, in: Proc. 1982 Workshop on thle Physics of

Submicron Structures, to be published.

1251 R.F. Greene. H.L. Gruhin and G.J. lafrate. in: P1roc. 1982 Workshop on the Physics ofSubmicron Structures, ito be published.

1261 H.L. (irubin. CiJ. lafrate and I).K. Ferry. in: Proc. 1981 IEI)M ( 1981) p. 633.

1271 G.J. lafrate. R. Malek. K. Hess and J. Tang. to he published.1281 M.S. Shur and L.F. Eastman. IEEE Trans. Electron D~evices EI)-26 (197()) 1977.1291 See. e.g.. [IF. Ivev. Advan. Electron. Electron. Phys. 6 (1959) 1 38.

I.

- - -. - .o.-.

PHYSICS AND MODELING CONSIDERATIONS FOR VLSI DEVICES

H.L. Grubin & J.P. Kreskovsky

Scientific Research Associates, Inc.

Glastonbury, Connecticut 06033

G.J. lafrate

U.S. Army Electronics Technology & Devices Laboratory

Fort Monmouth, New Jersey 07703

D.K. Ferry

Colorado State University

Fort Collins, Colorado 80521

R.F. Greene

Naval Research Laboratory

Washington, D.C. 20375

ABSTRACT

Recent results concerning spatial and temporal transport insubmicron devices identify significant aspects of the role of

boundary conditions, scaling for suggesting new materials, and struc-

tural device changes. These results are discussed as a means of

achieving high speed and high frequency devices.

INTRODUCTION

There are several recent results concerning spatial and temporal

transport in submicron devices that are likely to have an impact

on the design of future high frequency sources. These results, which

emerge from Monte Carlo, and momentum moment equation solutions to the

"Wigner-Boltzmann" quantum transport equation1

63

S

. . .. . .. . . . . . . . . .

. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . .

64 H. L. GRUBIN ET AL.

3f 3f fw p.- w 2 h 2 (1) "-''

__ (.--- ( il x)f (X'p) 1, )col I?-t m x h 2 (x ("

and the Boltzmann transport equation (BTE)

f P-af - a-

+ + p - ()o (2)

identify crucial aspects of boundary conditions, the role of scalingin choosing suitable materials, and the significance of alterationsin otherwise simple device structures for achieving high speeds.These topics are reviewed below. (It is noted: In equation (1) theposition gradient in the brackets operates only on the potentialenergy, f is a single coordinate and momentum distribution function

W

fwp 2 fdy* (x + --) T (x - )eip y /h

and T(x) represents the state of the system in the coordinate repre-sentation).

SCATTERING MODIFICATIONS TO BALLISTIC TRANSPORT

Ballistic transport implies carrier transport unimpeded by interactions (electrostatic, or otherwise) with other carriers, or withscattering events. The extent to which scattering centers are sensedby transiting electrons is therefore of significance. The first setof calculation shown, Figure 1 (Ref. 2), represents scattering eventsin GaAs for a collection of electrons entering a uniform field regionwith an initial energy of approximately 0.30ev. Intervalley I-L enerseparation for this Monte Carlo calculation is 0.33ev. It is seenthat, with the exception of very high fields, approximately 50% of thcarriers are unscattered over the first 500°A.

The velocity versus distance curves for this calculation are displayed in Figure 2 (Ref 2), and it is seen that high speeds over use-ful distances can be achieved even in regions where a high number of

scattering events has occured. The optimum conditions for this ap-pear to require moderate fields, generally near the threshold fieldfor electron transfer, and moderate injection energies. The depend-ence on the latter is displayed in figure 3 (Ref.2), where the lowestachievable velocities over a distance of 15000A are for entry elec-trons with zero initial velocity. It is also seen that an optimum

-. -I

PHYSICS AND MODELING CONSIDERATIONS FOR VLSI DEVICES 65

,oo - - I . . .I - "

90 T=300 K

70 -- "

40

50 =50 \ g: O

0

30 50.

F =200"\' -.- ,

'00 500 '000 n500

DISTANCE (A)

Figure 1. Percentage of unscattered electrons versus distance.

Entrance electrons have a finite energy. From reference2, with permission.

entry energy exists; for energies near the F-L separation, scatter-ing quickly reduces the net drift velocity. The optimum conditions

identified by these calculations imply that voltage control near

the entrance boundary must be near 5Omv.

The examples of Figures I through 3 are for uniform fields, and

carriers subject to sudden changes in field. Studies in which spatial

gradients accompany nonuniform fields are more recent, several of which

are considered at this Workshop. One aspect is considered below.

SPATIAL AND TEMPORAL TRANSIENTS

A relatively direct way of handling spatial and temporal trans-

ients is through moments of the transport equations (1) and (2).

Little has been done with the moments to the Wigner - Boltzmann equa-

tion and the following deals exclusively with BTE moments. The

moments of the BTE form an infinite hierarchy of moment equations.Each equation introduces a higher order moment not defined by the

Ni

66 H. L. GRUBIN ET AL.

8J

" v.'.'4

0 G)6

r = 2A00

.

0 50 *W 10

Figure 2. Velocity versus distance for electrons with a finiteentrance energy subject to a sudden change in electricfield. From reference 2, with permission.

given set of balance equations. The most common form of the balanceequations is obtained by assuming a distribution function of theform

2f f i exp - fm(u-v) 2

3/2 2 k T.

for each species. More generally (Ref. 4)2 3 -

f =(1,Xk + AkkUl+Aklm a I k ,1 um o.-(5)

where it is insisted that the first term in the expansion gives the Ucorrect local values of density, velocity and energy. The coeffic-ients in the above expansion are inversely proportional to densityand are model dependent. The first correction to the local equilib-

rium balance equations, the so-called hydrodynamic approximation yiel(I

4

~ ~~. . . . .. . . . . . :: ._ :

PHYSICS AND MODELING CONSIDERATIONS FOR VLSI DEVICES 67 .

VIIITY Aw4SUS DISTANCI V1[WSUS EPT41P I rLTl. . .*..

T 300- K

0 0 0

bI 0 002 0 04C 0 03 0 07

d 0 04 0O11l0 05 0 16

f0 06 0 2

T 0 007 0 27h 0 08 0 34

o 500 1000 (A00

Figure 3. Velocity versus distance for electrons with varying en-trance energies subject to a sudden change in electricfield. From reference 2, with permission.

I n.nan + (v . (n) \at /Coil (6)

at1

4-4

i + v -en.F -grad n k T. 7

at0

+1 div (K gra T) at)Cl

whreP 2 3 TwhreiPg frmnvn iir nif~nv + -n k.} a is a stress tensor

ariingfro nouniormvelocity distri~utions, and K i5 the thermalconductivity. The stress term is dissipative in that when a nonuni-form velocity distribution is impressed on an electron stream therewill be reactive forces tending to smooth them out. Detailed informa-tion regarding suitable values for these terms are not available, andthey are regarded as phenomenological entries; they and their modifi-cation by the collision terms are to be studied.

68 H. L. GRUBIN FT Al

7

0Z i5 3

0 ~No '5x 10,/cm0

to

z

14 35x 10 /cm

0.1

0 2DISTANCE, (microns)

10

E

U.

Ix

-j

0 20 DISTANCE, (microns)

Figure 4. a. Doping distribution for an N -N -N advice. b. Sel:consistent electric field for a bias of 1 volt. (From Ro4, with permission).

PHYSICS AND MODELING CONSIDERATIONS FOR VLSI DEVICES 69

Tile boxed terms above represent contributions arising from tile

nonspherical nature of the distribution function. The underlined

terms are ignored in the drift and diffusion approximation. Figure "

4 shows the result of a solution to the balance equations for two

level transfer for a two-terminal device whose structure is receiving

considerable attention as an illustration of velocity overshoot. The

structure considered here consists of a low doped region LUlpm "*-sandwiched between two regions of hipher donine concentration. The

structure is subject to a bias of 1 volt. Four interfaces are involved

but most attention is focused on the two interior faces where signifi-

cant gradients in field occur, all associated with differences be-tween the background and mobile carrier density. The carrier con-

centration is displayed in figure 4c, for the case of injecting cathode

contacts (Ref. 4). It is seen that (a) significant charge injection

is present in the low doped region, and (b) electron transfer occurs

near the downstream section of the low doped region and is significant

in the downstream N region. It is noted that the field dependence of

carrier density in the r valley, Nr(F) is not the same as that asso-ciated with uniform field steady state values. Spatial gradients re- -

sult in a spatial lag.

Also shown in figure 4C is the drift velocity of carriers in the

F-valley, and it is apparent that the carriers acquire speeds consid-

erably in excess of the steady state peak velocity of electrons in

GaAs. These velocities are not, however, much different in value than

those associated with F valley electrons under uniform field condi-

tions. The velocity shown in figure 4c is not, however the same as

the carrier velocity computed under uniform field conditions. The

velocities computed here include temperature as well as momentum

gradients, and are properly defined as the current flux/carrier

density. Thus, increased velocity here is often associated with de-

creased carrier density, through current continuity. The key to all

device design is that high velocities must be accompanied by only mar-

ginal transfer out of the high mobility section of the conduction

band. This will require device lengths shorter than the active region

of the device shown in this figure. (Modifications to this statement Farising from variations in material parameters are considered in

Ref. 4).

The calculations of figure 4 show the situation where a high

carrier velocity occurs near the downstream edge of the low doped ..V

region. Often high velocities are required near the upstream C-

boundary as indicated by figures I through 3. Figure 5 (Ref. 6)

shows a spatially dependent result for carriers subjected to a pre-

specified value of electric field. The calculation is for silicon and

the highest carrier velocities occur near the 0.4 1im boundary. An

accompanying plot of carrier density (Ref. 6, figure 9) shows the

lowest level of carrier density in the region of highest velocity.

b.. ,. - -.

70 H. L. GRUBIN ET

1.5 1 ,

0 -

<0

0 I.0- -. 100(:L -,oI. - - -O

N,NT\ "

0

R0.5- f. 4., with -e5sso .rr' ~ ~\ .--.--. / -.-

0 I

I 2 r. .

DISTANCE (MICRONS) ""-

Figure 4C. Fractional population and central value velocity. (Fri 22.j

Ref. 4., with permission.) '_-

Coupling the above results to the need for moderate fields forhigh speed operation a useful multiterminal device very likely willrequire the presence of a local moderate field region that "kicks"electrons into high speed regions. The simplest high field in-jecting region could come from local charge depletion or "notches"(Fig. 6). Note that in Figure 6 the gate is treated generically.It could be a metal Schottky contact, a p AlGaAs layer, etc.

SCALING

In the absence of, or in concert with structural variations indevices it is also necessary to examine changes in material parametThe change in material parameters should have as its goal high mobility and high characteristic velocities. Thornber (Ref. 8) providedgeneral set of guidelines for choosing material parameters throughan alteration of the scattering rates, one of which is duscussed be]

The collision term in the Boltzmann transport equation is.--p

. . . ..°

"-.

PHYSICS AND MODELING CONSIDERATIONS FOR VLSI DEVICES 71

,.6r

'.,

2 kV/cm 2V/cm -

0.4 0.8DISTANCE (MICRONS)

E

2 1.5

0-J LO

W

Ia

I I I I I0.4 08

DISTANCE (MICRONS)

Figure 5. Variation of carrier velocity in silicon for electrons

subject to the illustrated field profile. From Ref. 6with permission

(:)coll J d3p, f(p')W(plp) - f(p)W(p,p')} (9)

where W(p'p) is the total scattering rate from pto p.Thornbersuggested several scaling changes in the BTE, to alter the driftvelocity.

72 H. L. GRUBIN ET AL.

HIGH ENERGY INJECTION SITE

GATE

SOURCE GaAs DRAIN

GATE

Figure 6. Schematic of a high energy injection region in a gated

submicron structure..I

v= J3pf/m 3pf. (10)

The one illustrated below involves uniformly altering the scatteringrate by a constant F, and results in the following relation i

v (x,t,F) v(Fx,rt, F/r) (11)

Where the right-hand side is the unscaled velocity. In this scalingmobility is altered but saturated drift velocity is not.

A dramatic consequence of the effects of scaling is illustratedbelow, but it is necessary to note that the possibility of relevantmaterial scaling over a broad range of electric field values is re-mote. Rather, it is more likely to be appliable over a restrictedrange of field values. This is illustrated for GaAs. For GaAs, overa field range of approximately 3 - 15 kv/cm, intervalley F-L couplingis a dominant scattering mechanism. Figure 7 shows the velocity fieldcharacteristic for F-L-X ordering with three different values of theF-L deformation potential. All these valleys are taken as parabolicwith N =0, while all other material parameters are those of Little-

0john, et al. (Ref.9). (We note that similar values for material con-stants yield differences between Monte Carlo and balance equation cal-culations.) In Figure 7 the fields at points 1, 2 and 3 occur at valu,5.6 kv/cm, 8.2 kv/cm and 12 kv/cm, respectively. The dominantF valley total momentum scattering rates are 4.78x10 1 2 /sec., 7.32x,0 1 2

sec., and l0.0x10 1 2 /sec., respectively. The ratios F/F are1.17xlO- 12 kv-s/cm, 1.12xlO- 1 2kv-s/cm and 1.20xlO- 12 kv-s/cm, respect-ively in general agreement with the rule of Equation 11.

Figure 8 illustrates the consequences of uniform scaling on velocity overshoot. The solid curve displays ovei.hoot for a galliumarsenide element subject to a field of 27kv/cm (Ref. 10). The dashed

................................................- .

i • ~

PHYSICS AND MODELING CONSIDERATIONS FOR VLSI DEVICES 73

o 050

ELECTRIC FIELD C klovollso/cm I

Figure 7. Steady state velocity versus field for F-L-X ordering as

obtained from balance equations. Different curves are for

different values of the deformation potential coupling

constant for intervalley r-L scattering.

(dotted) curve is a sketch of a scaled curve for F--2(F=0.5) and afield of 54kv/cm (13.5kv/cm). For r-2, high overshoot velocities

occur over a short period of time and require high bias levels. Byreducing F to values below 1.0 high overshoot velocities are retained

for a longer period of time and at lower values of field than for GaAs.This result is highly significant insofar as it suggests higher transitvelocities for extended temporal scales, and indicates a direction of

material selection for high frequency sources.

GENERAL CONCLUSION .

It has been known since 1969 that the compound semiconductors,GaAs in particular, are theoretically capable of providing high fre-

quency, near 100GHZ oscillations. Overshoot has also been known sincethat time (Ref. II). The difficulty in attaining high frequency three-

terminal and sometimes two-terminal operation has over the past decadebeen attributed to inadequate contact regions (Ref. 12), material

preparation, etc. Many of the high frequency problems were thought

to be reduced by going to small dimensions, where in addition toshorter transit lengths the benefits of overshoot would emerge. While

it is still too early to state how these benefits can be implemented

in practice, it is clear that special contacts or injection rep.ions

* >:. -. -..- "..]

.. . . . ..1

74 H. L. GFIUBIN ET At..

SCALED

CURVE ,. '%.

SCALEDS-,. CURVE

8 -F-54k No= 10 .

300"K .r-.

E F15F=27rVV/cm Fm .3.5 kcmF - 13. ....

0

4-

0 0j 02 Q3 0.4 0.5 0.6 0.7

TIME (ps)

Figure 8. Transient velocity characteristics for scaled and unscaledscattering rates. Solid curve is from reference 10.

are required. It appears, therefore, that the 1969 questions as towhy devices are not operating closer to theoretical limits is stillvalid today. The dual solution approaches of the last decade alsoappear equally valid today, where the pursuit of novel device struc-tures emphasizing the boundary role is continuing, while simultaneous:new material directions are sought.

ACKNOWL EDGEMENT

This work was supported by the Office of Navel Research, and theArmy Research Office, to whom we are grateful.

I..

PHYSICS AND MODELING CONSIDERATIONS FOR VLSI DEVICES 75

REFEIRENCES

l G.J. lafrate, H.L. Crubin and D.K. Ferry, J. de Physique C7,

307 (1981).

2. G.J. lafrate, R. Malik, K. Hess and J. Tang. To be published.

3. See, e.g., A. Sommerfeld, Thermodynamics and Statistical Mechanics,

Sec. 43. Adacemic Press, NY (1956).

4. H.L. Grubin and J.P. Kreskovsky, Surface Science 39 (1983).

5. 1I.L. Grubin, G.J. Iafrate and D.K. Ferry, J. de Physique C7,

201 (1981).

6. E. Constant and B. Boittiaux, J. de Physique C7,73 (1981).

7. R.K. Cook and J. Frey, IEEE Trans. Electron Devices, Ed-29 (1982).

8. K. Thornber, J. Appl. Phys. 51, 2127 (1980).

I 9. M.A. Littlejohn, J.R. Hauser and T.H. Glisson, J. Appl. Phys. 48,4587 (1978).

10. H.L. Grubin and D.K. Ferry, J. Vac. Sci. & Tech. 19, 540 (1981).

11. P.N. Butcher and C.J. Hearn, Electron. Lett. 4, 459 (1968).

12. M.P. Shaw, H.L. Grubin and P.R. Solomon. The Gunn-Hilsum Effect,

Academic Press, NY 1979.

I....*.13 -

71 .7 W 7 1 l 17 1 WL t. . -Y . ., "- . p ._ • .

-.'olume 87A, number 4 PHYSICS LETTERS 4 January 1982 '..r

THE WIGNER DISTRIBUTION FUNCTION

G.J. IAFRATEUS Army Electronics Technology and Devices Laboratory (ERADCOM, Fort Monmouth, NJ 07703, USA

H.L. GRUBIN.Scientific Research Associates, Glastonbury, CT 06033, USA

and . -

D.K. FERRYColorado State University, Ft. Collins, CO 80523, USA

Received 17 August 1981Revised manuscript received 9 October 1981 -

in this letter, the relationship between the characteristic function for two arbitrary noncommuting observables and ageneralized Wigner distribution function is established. This distribution function is shown to have no simple interpretationin the sense of probability theory but, in lieu of its special properties, can be used directly for calculating the expectationvalues of observables.

Whereas classical transport physics is based on the ables in a manner quite analogous to that of classicalconcept of a probability distribution function which theory, i.e. by integrating the product of the obsery-is defined over the phase space of the system, in the able and the Wigner distribution function over allluanturn formulation of transport physics, the con- phase space.cept of a phase space distribution function is not pos- This letter, in part, reviews the salient features ofsible inasmuch as the noncommutation of the posi- the Wigner distribution function. Although the Wignertion and momentum operators (the Hteisenberg uncer- function is generally defined in terms of all the gener-tainty principle) precludes the precise specification of alized coordinates and momenta of the system ina point in phase space. However, within the matrix question asformulation of quantum mechanics, it is possible toconstruct a "probability" density matrix which is of- .. = dy1 ... d..nten interpreted as the analog of the classical distribu- PW(Xl 2... ,P-..P) -Jtion function.

There is yet another approach to the formulation X 4f*(x I + Yl, ...Xn + Yn) (i)of quantum transport, based on the construction ofthe Wigner distribution function 1ll. As we shall X 4(x 1 - 4yl, ""Xn -Yn)

show, this distribution function has no simple inter- X expti(ply, + ... +pnnpy)hJ

pretation in the sense of probability theory [21 but,in lieu of its special properties, can be used directly we will discuss the properties of the Wigner functionfor calculating expectation values 13-51 of observ- in terms of a single coordinate and momentum. In

this case, we let

Partially supported by the Army Research Office.

145

V.

- L

A

Volume 87A. number 4 PHYSICS LETTERS 4 January 1982

function to the classical distribution function is appar-

Pw(x,p)=' f dy P'*(x + y)'IP(x - ly)eiPy/, ent by examining the equation of time evolution for-- (2) Pw(x,p). Upon assuming that %'(x) in eq. (2) satisfies

the Schr6dinger equation for a system with hamilto-where *l(x) refers to the state of the system in the co- nian H = p 2/2m + V(x), it can be readily shown thatordinate representation. Pw(X, p) satisfies the equation

The distribution function of eq. (2) has interesting 6Pw lat + (p/m)aPw/aX + 0 •Pw 0 (5)properties in that the integration of this function over ' 0-(.)

all momenta leads to the probability density in real wherespace, conversely, the integration of this function overall coordinates leads to the probability density in too- 2 "P 2 (-l)rnentum space. In mathematical terms, h n=0 (2n + 1)!

a2n+l V() 2n+lPw(x,p ) :,.fPw (x, p) clp = 41"*(x) IP(x) (3a) X Vw) (6)

ax2n+j op2n+l1'

and It is evident that in the limit h 0 0, 0 -Pw in eq. (6)becomes

fPw(x,p) dx =*(p)O(p) , (3b) 0 "Pw = -(v/ax)(aPw/ap) (7)

so that eq. (5) reduces to the classical coflisionlesswhere Boltzmann equation.

The Wigner distribution function defined in eq. (2)3 (p) (2,, h)-1/2 e-ip-/'(x) dx .is derivable [6] from the Fourier inversion of the ex-_f " pectation value (with respect to state *r(x) of the op-

erator ei(tI+Oi) (here, i and / satisfy the commuta-It foUows immediately from eq. (3) that, for an tion relation [.fi- ih). As such,

observable W(x,p) which is either a function of mo-mentum operator alone or of position operator alo~ie, Pw(X, p) = CW(T, O)e-i(TP+°xdr dO , (8a)or any additive combination therein, the expectation 4'"2

value of the observable is given by where .

(IV) ff WPw(x, p) dx dp , (4) Cw(r, 0) f %P(x)i(7P+°X)iN(x) dx , (8b)

which is analogous to the classical expression for the and the interval of integration is (_o% oo) unless other-average value. Herein lies the interesting aspect of the wise specified. In order to show that the right-handWigner distribution function; the result of eq. (4) side of eq. (8a) is indeed the Wigner distribution func-suggests that it is possible to transfer many of the re- tion as defined in eq. (2), note, from the Baker-suts of classical transport theory into quantum trans- Ilausdorff theorem 17J, that eit"P+ ° x) can be written a!port theory by simply replacing the classical distribu- r =+Oi) eiTfi12eiOieiru12tion function by the Wigner distribution function. (9)However, unlike the density matrix, the Wigner dis- in wiuch case Cw(-r, 0) of eq. (8b) becomestribution function itself cannot be viewed as the quan-turn analog of the classical distribution function since Cw(rO) [r-

it is generally not positive definite and nonunique ' f l'.,[,-

[Pw(x, p) of eq. (2) is not the only bilinear expres- -- (10)

sion 11,3-5] in I that satisfies eq. (3)]. which further reduces toFurther resemblance of the Wigner distribution

146

S 41

Volume 87A. number 4 PHYSICS LETTERS 4 January 1982

observables destroys the convenient probability inter- .Cw~r, 0) =f '"(x - rh)ei~xqI(x + rh) dx . (11) pretation of the characteristic function implicit in

eq. (15).

In order to demonstrate this point, assume the char-Then, by inserting Cw(T, 0) of eq. (11) into the right- acteristic function for two noncommuting observables,hand side of eq. (8a), integrating over the variable 0 i and B, to beby using the relation CAB(I2)= e(16)

ei (x *- x'") dO = 2irS(x' - x"), Observables and B are assumed to have eigenvaluef' spectra

and letting T = -y/h, the desired result is obtained. A IA') =A'IA') , BIB') = B'IB') , (17) - -

The method outlined above to arrive at the Wigner and are chosen so that [A1, [A, bi I = [B,[A, 1 il = 0. .distribution function is based on the notion of a char- This assumption is imposed so that the identityacteristic function. The characteristic function of anobservable, A, with respect to state IxP) (here, the eiQJtA+t2B) = e tt 1 " eit2h e-tl E2 IA.h 1/2 (18)Dirac notation is utilized for purposes of generality) may be used.is defined as Inserting eq. (18) into eq. (16) while obtaining theCA (Z)= (, leitA I q) (12) matrix elements of eitiA in the A'-representation and

ei28 in the B'-representation results in

where is a real parameter. Assuming A to possess aneigenvalue spectrum given by A A') A'IA'), CA(,) CAB( I,2) = e-tEt2 14,hIl/2 f dA'f dB'can be evaluated in the A'representation as (19)

X ei (ti A '+t2B)(4IA')G4' IB')(B' I,') .

CA()=fdAf dA"(*IIA')(A'IeitAi A")CA"Is>). In eq. (19), it is assumed that [ABI is a c-number in-(13) dependent of the eigenvalues A' and B'. We define

Since (A'IeitA IA") = ei EA'6(A' - A") in theA'-rep- F(A', B'), the generalized Wigner distribution func-resentation, CA (Q) in eq. (13) reduces to tion * , to be

CA( ) f dA'eiA' lA12 , (14) F(A',B')= (41 IA')(A'IB')(B'I P) , (20)

so thatwhere I''A = A 12 ' 1 -)12 aP(A'), the probabilitydistribution function for measuring A' while in state F(A', B')= I fdl f do2 th ac c t,h]u2n.-nb.Ai te.!).1 10. llence, the characteristic function for A is the (,)

Fourier transform of the probability dstrihu1t iOnfunction P(A'). Subsequent inversion of eq. (14) X CAB( I t 2 )e -i(tA '+t2 B'). (21)

above leads to It is evident from C(". (20.21 that

The Wigner distribution function was derived bytaking the Fourier transform of the characteristic * The form of the generalized distribution function derivedfunction for ettrp * Ox). In view of the connection be- is sensitive to the manner in which expli((i + taB)l istween the probability distribution function and the expanded. lor example, if instead of eq. (18), we used thecharacteristic function for a given observable, this ap- forms expit 2B) ep(tiA]4)eXp(tj tl Il. h1/2), exp(iqA/2)

seems to ba nX expOtE2 ) exp(it i4/2), or exp(2tC282) exp(iti1A) exptitproach be natural way of obtaining dis- X B12), all of which are equivalent, we would indeed ob-tribution function for momentum and position. Un- tam a different form of the generalized distribution func-fortunately, the noncommutative nature of the two ion, yet one which obeys the sum rules of eq. (22).

147

• - . .

Volume 87A, number 4 PHYSICS LETTRS 4 January 1982

fF(A', B'>dA' _= I(Bfd1 12eqs. (20, 21) differs from the Wigner function due to2sr(2a) the presence of the phase factor etI2 li .1/2 in the

X C~(0, itBintegrand of eq. (21). ForA =X and h = p.PW(X'p)in eq. (23) reduces to the Wigner function of eq. (2),

an d whereas F(x,p) defined from eq. (20) becomes

Thus, eq. (21I) establishes the relationship between the where 0(p) is defined in eq. (3b). It is evident thatcharacteristic function for two arbitrary noncomrnut- there is a family of functions which are bilinear in %Ping observables and the generalized Wigner distribution yet satisfy the sum rules of eqs. (3a, b).function. The generalized distribution function has There are some interesting questions to be resolvedthe essential properties of the conventional Wigner concerning the uniqueness and positive definitenessfunction in that an integration of the generalized of Wigner-type quantum distribution functions. Never.function over the eigenvalue spectrum of one obserV- theless, these distribution functions serve a usefulable leads to the probability density in the canonically purpose for calculating quantum mechanical observ-conjugate observable [eq. (22)]. ahles in transport [51 studies and numerous solid-

There is no simple probability interpretation of state [8,91 problems.F(A', B') in eqs. (20, 21 ) because of the necessaryoverlap between the states of the noncorruuting oh- Referencesservables. However, if A and B are made to commuteso that IA') and IB') are a common set of eigenvec- III E.P. Wigner, Phys. Rev. 40 (1932) 749.tors, then F(A', B') reduces to the probability distri- [21 R.F. O'Connell and E.P. Wigner, l'bys. Lett. 83A (1981)

bution function for A andB.15 131 G.J. lafrate, IL.L. Grubin and D.K. F err-y, Bull. Am. Phys.Finally, it is noted that the conventional Wigner Soc. 26(1981)458.

distribution function for observables Aand - is yb. 141 IL. Grubin, D.K. Ferry, G.J. lafrate and J.R. Barker, in:

tamned from VLSI electronics: microstructure science, ed. N.G.EinsprUch (Academic Press, New York) to be published.

AAIA' ' fdt f, [5) G.J. bfratc, II.L. Grubin and D.K. Ferry, J. de Pliys. Coil.P.A.(A 7T) 2 d 1 j~ C7 (198 1) to be publishcd.

(2n 2 161 J.E. * .loyal, Proc. Cambridge Philos. Soc. 45 (1949) 99.

> ~ -xi~ 1A '+C2 B') (2) 171 A. N1 -ssiah, Quantum mechanics, Vol. I (lnterscience,

with CAs(Ql t2) defined in eq. (16), whereas the alter- 181 G,. Niklasson, Phys. Rev. 1310 (1974) 3052.

native distribution function, F(A', B'), introduced in [91 1:. l3roscns, L.Iz. Lemnmens and i.T. Devreese, I'hys. Stat.Sol. (b)81 (1977) 551.

148

-. ° °. -- -.

7. T

CONSIDERATIONS ON THE FINITE, VERY-SMALL SEMICONDUCTOR

DEVICE AND SUPERLATTICE ARRAYS

David K. Ferry, Robert 0. Grondin, andRobert K. Reich

Colorado State University, Fort Collins, CO

Harold L. GrubinScientific Research Associates, Glastonbury, CT

Gerald J. LafrateElectronics Technology and Device LaboratoryFort Monmouth, NJ

ABSTRACT

The size of expected future very-small devices will result in

their being strongly coupled to the environment in which they arelocated. In this paper, we examine several of the limitations on the

device physics that can be expected to arise in these structures,including the role of the environment and its effect on ballistictransport. In addition, we look at submicron arrays of devices in

a lateral surface superlattice (LSSL) and examine transport in such

an LSSL through a Wigner function and Monte Carlo approach.

INTRODUCTION

Many people have examined the limit to which semiconductor de- I .

vices can be scaled downward 1- 3. While small devices in the range0.1 - 0.3 pm gate length have been made , problems such as intercon-nections, electromigration and thermomigration of metallization, and

power density within tie device strongly affect the packing densityand device size that can be achieved 5. Yet, no one has evaluated theoperation and performance of very-small semiconductor devices, i.e.those that can be conceived in the sub-0.1 pm size range, and theinteractions within arrays of such devices. One reason for this isthat the transport within such a device cannot be treated in isolation.Because of the size of such a very-small device, it is coupled strongly

267

268 D. K. FERRY ET AL.

to the environment in which it is located. The basic transport equa-tions cannot be separated from their casual boundary conditions andboth of these factors must be modified to account for the influenceof the device environment 6 . In previous work, some of us have pre-sented a formalism to address this 4'7, but this formalism remains -".

untried due to its inherent complexity. In the present work, we wish

to examine qualitatively several of the limitations on the devicephysics that can be expected to arise in the very-small semiconductordevice, particularly with respect to the finiteness of source andsink regions, contact regions and barriers, and the influence of these

latter quantities on the ballistic transport that may exist in thedevice 8 , 9 . In particular, we examine these effects through the useof discrete area-preserving phase-space maps which display the char-acteristics of transport in a generic device. In addition, we treatthe cooperative interactions that can arise in densely packed arraysof devices.

BALLISTIC TRANSPORT

Potential barriers within the device can play a significant rolein the quantum ballistic transport of carriers through it. Suchbarriers are found in very-small devices, for example, to confinecarriers to the active region3 , and are an intimate part of devicessuch as the planar-doped barrier transistor1 0 , tunneling barrier de-

vices 1, real-space transfer devices 12

, or superlattice avalanchephotodiodes 13. When a barrier is present at the contact region, caremust be taken to adequately handle turning-point reflection of the

electrons from this barrier. Even when the electron has sufficientenergy to pass over the barrier, there is a well-known quantum mechan-ical reflection at the barrier interface. If the potential barrier issmooth, i.e. introduces a transition over many wavelengths, the re-

flection and wave function matching can be handled by well-knownapproximation techniques such as the WKB approximation 14 , in whichthe potential barriers represent turning points for a near-classicalpath. In the very-small device, however, the barriers are expectedto be sharp on the scale of the electron wavelength and care must beexercised in matching wavefunctions and determining reflections. Thereflection problem is further complicated in real-space transfer de-vices12 due to the different band structure on either side of thebarrier. Here, additional terms arise due to the spatial variation

of the effective mass.

If the potential barriers are slowly varying on the scale of thewave packet, the trajectories are largely those of the classicalmotion. Even if this is not the case, as we expect for the very-small

device, nearly semiclassical trajectories can be expected if the var-

iation of the action is limited to a few low-order derivatives 15.

In semi-classical systems, the phase space of the classical motionforms a natural framework in which to examine problems such as these.

-• - Q 7".- V

SEMICONDUCTOR DEVICE AND SUPERLATTICE ARRAYS 269

While classical one-dimensional transport appears to be basically -simple, there exists recent work that suggests that even this simpleproblem contains a number of unexpected subtleties16 . In this paper,we begin to investigate the role of barriers in submicron devices

through the use of such finite, area-preserving phase-space mappingsof classical dynamics. The use of such mappings reveals a variety

of complicated structure 16 , 1 7, and has had recent success in explain-ing the cause of excess noise in Josephson junction parametric amp-

lifiers 18' 19. The results that we obtain indicate that if thesemappings are applicable to the VLSI scale, then present concepts of ",submicron transport may require substantial generalization.

The types of sublet4 es to which we are referring are best ill-

ustrated by Fig. 1. There, we are using an area-preserving mappingof particle position (horizontal) and momentum (vertical) within adevice active region bounded by two Gaussiam potentials V(x) (over-

" laid in the figure). In addition, an electric field has been applied.

p

, , ."

Figure 1. The phase-spacing mapping of an initial uniform distri-

bution of electrons in a generic very-small semiconductorIdevice. The contact regions are represented by a twin- .

gaussian potential V(x). The horizontal direction isposition while the vertical direction is momentum.

S* . ¢''..:'_ ",,t' . ." .\ .* . ...' ." •

1 270 D.K. FERRY ET AL. "'

The canonical mappings are area-preserving since we are consideringa collisionless (conservative) system and looking at the ballistictransport. The classical differential equations of motion are

aV(x)

(1) r =

x = + p/m ,

and the discreet classical area-preserving maps are

Pn+l Pn -T x n+1(2)

Xn+l =x + T Pn/m

In Fig. i, the potential has been scaled so that the two potentialshave a weak overlap. For small values of the total energy, resonantorbits occupy the central part of the figure. These orbits are in the

region of the classically integrable motion. For larger values ofthe energy, however, the orbits are such that the particles are sweptout of the well by the field. An energy dissipating collision candrop a high energy particle into the resonant region, thus trappingit within the structure. We expect these particles to contribute adiffusive component of current. Large angle scattering, however, canmove a particle to a back-flowing orbit, which effectively causes areflection of particles from the device input.

A number of interesting factors arise. The potential is generic,in the sense that it is similar in form not only to the devices men-tioned above, but also to the sinusoidal force term in the Josephsonjunction devices18,19. In the latter case, interaction between devicescan lead to a parametric pumping of the potential which yields period-doubling bifurcations and chaotic behavior. The system (2) is nottruly generic though, as the set (x,p) are not proper action-anglevariables. We have examined the behavior through potential pumping,with dissipation, through replacing the first equation of (2) with

(x n+l)

p = P T Typ + TF sin(Qt , (3)n+ x n 0- n

where Vo is the set of Gaussian potentials. The factor Y is an ef-

fective damping factor. In Fig. 2, we plot the (Fo,Q) plane results.The curve is a separatrix below which a stable device results. Abovethe curve, the device is unstable. No period-doubling bifurcationswithin a single device are found, contrary to18.

• .'.<.v, -v .., .-. .,. . ...,.,,v .-. .v -,. -. ;-,'.'.'", .'' L, .. -'.-. .. -".-" ....,, ." . . , .• ,..". ."• '.. ,

SEMICONDUCTOR DEVICE AND SUPERLATTICE ARRAYS 271 Z "

1.10

1.05 1

1.00

F0

.95 j .,

.90

b 5

.80

.75

.70F2 .5 .8 .7 .8

Figure 2. The separatrix in (Fom) space for parametric pumping ofthe device of Fig. 1. For values of (Foq) above the

curve, electrons are ejected from the device, while forvalues below the curve the device is stable.

DEVICE ARRAYS

When device sizes begin to shrink toward the O.lim or less region,the line-to-line capacitance in dense device arrays begins to dominatethe total node capacitance4. This parasitic capacitance leads to a

direct device-device interaction outside of the normal circuit orarchitectural design. In conventional descriptions of LSI circuits,each device is assumed to behave in the same manner within the total

system as it does when it is isolated. In the dense arrays discussed

here, this will no longer be the case. :

The possible device-device coupling mechanisms are numerous andinclude such effects as the capacitive coupling mentioned above, but 'Ialso include such effects as wave function penetration or tunneling

and charge spill-over. Formally, however, one may describe these

effects on system and device behavior by assuming the simplest formof coupling. Arrays of devices, interacting in this manner, form a i "

," -."

.. . . . . .

- -- r . .

272 D. K. FERRY El

lateral surface superlattice 20 ,2 1. Lateral superlattices, in whthe superstructure lies in a surface or heterostructure layer, oconsiderable advantages for obtaining superlattice effects in pl.technology. While a surface MOS structure is formally similar t,array of CCD devices 20 ,2 2, superlattices can also be fabricatedthe use of electron and ion beam lithography and selective area ,taxial growth 21 . If the coupling is capacitive, then the limitato a spacing less than the de Broglie wavelength is removed4 . W, --

have examined transport in such lateral surface superlattices (L: r".through a Wigner function approach. Before proceeding, however,is worth noting that the circuit theory view of LSSLs4 is generi-that of cellular automata 2 3. Many of the image processing applic 4tions proposed for LSSLs24 arise from the "games" aspect of cell%automata 2 5.

Superlattice structures give rise to sinusoidal energy mini-bands with relatively narrow widths. The shape of such bands re!in interesting electrical transport properties. Lateral surfacosuperlattices have cosinusoidal bands in two dimensions, with thEthird dimension quantized with discrete energy levels.

We have calculated the transport properties for such LSSLs Ia Wigner function formulation. A complete integral equation is ctained for the Wigner function and must be solved to obtain the tport coefficients6, 2 7 . However, the form of the solutions can .--found by taking the average velocity and energy from the first arsecond moments of an equivalent Wigner representation of the trarport equation2 8 . A constant electric field is applied in the plaof the sinusoidal bands. For simplicity, a constant relaxation t 4T can be used. Solving these two equations simultaneously leadsthe velocity and energy as functions of the field 2 9

(2a/fiL) 2 eF T"<v> E<E> <E >)

z

<E> = + <EL> + I + ± (2rzz

where L = 2i/D, D is the LSSL spacing, c is one-half the band-widand <Eo> and <Ej> are the equilibrium and transverse energies, respectively.

The velocity as a function of field has the same basic analyexpression as that obtained by Lebwohl and Tsu 30 except for the e:factor in front of the expression for the field. The differencein effect caused by the different equilibrium distribution chosenIn the latter paper, the authors assumed the initial equilibrium

. .

.. . . . . . . . . . . . . . . . . .. . . . . . . .

(-.--i .... .. i. ,?-.j. ,;iQ ....-. _ ,...x ~ i Li i. -. .- i-.. . . ..- V.. .-.. . .- i... .-.......- _-.-.-.- . . -- ' ---.---.- -- -.-

['-- W TO - .- I Twr

. SEMICONDUCTOR DEVICE AND SUPERLATTICE ARRAYS 273

tribution is very close to a zero temperature Fermi-Dirac function,while here the distribution is a finite temperature Wigner distributionand includes the physics of the energy band shape. Note that the

velocity as a function of the electric field shows the negativedifferential mobility predicted earlier and also exhibits the generalshape expected of the velocity-field curve.

To more exactly illustrate the mobility and negative differential

conductivity, we have carried out a Monte Carlo calculation. Thescattering processes are calculated for the model of lafrate et al.21,and thus are for the system of GaAs/GaAlAs. The scattering rates foracoustic and polar optical phonons have been obtained using a two

dimensional density-of-states for cosinusoidal energy bands. A Van

lHove singularity occurring in the density-of-states produces a sing- --larity in the scattering rate, which was removed by including the K

sel f-energy corrections due to the phonons in the vicinity of thesingularity. The widely-spaced discrete energy levels in the third ,dimension allows scattering and transport in that direction to be *i1 ,negl icted~i '

InI this surface superlattice, as in others, the conduction band-: '

splits into subbands. Here the lowest energy subband was nearly flat.Therefore, transport dominantly occurs in the next higher minibands.The satellite valleys and next subband are at energies of 0.2eV and0.3eV, respectively, above the subband considered. Their contributionto the Lransport of electrons is insignificant since there are no i .intermediate energies through which the electrons can scatter to aid

. ~)poptilation of upper bands.

r The overall transport properties of this system are calculated

bv an ensemble Monte Carlo technique. The results of the simulation* are the velocity-field curves and are shown in Fig. 3. The lower

curv re.sults for a field applied along one of the (10) basis vectorsof the ;qua-e lattice array of cylinders while in the top curve thefield i; applied along a (11) direction. At low fields, both curves,;how - linea,'ir region as expected for most structures. At approxi-matelv 8-10 kV/cm the curves begin to bend over to the peak near13 kV/cm. As the field is further increased, the velocity begins todecreas,;e and for this model continues to decrease to zero as thefield te(ds to infinity.

In summary, transport and scattering in a generic surface super-

lattice structure exhibits a negative differential mobility arising -from Bloch oscillations 31. This surface superlattice negative dif-ferential mobility is expected to be useful at much higher frequenciesthan that due to the conventional Gunn effect. Alternatively, relatedinstabilities may be an ultimate limit on very large scale integra-

tion,

o. • o -. • • .• . -. . .

.. , :2 ._ .&-. ,-2 i~i.-, -.j -. .-. . . .--.- -." 2.. ,..--.. i i • . .. ."- ?. i . . ..,i .....-... 5 , ,. , ., .. .

274 D. K. FERRY ET AL. "-'

14

12 , .

E

0 6

w

>

4 (11) Direction(10) Direction

0 5 10O 1 5 20 25 30

FIELD (ky/cmn)

Figure 3. The drift velocity (as a function of electric field) fora LSSL at 300 K. The GaAs/GaAlAs model of ref. 21 hasbeen used.

ACKNOWLED)GEMENT

This work was Fupported by the Office of Naval Research, theArmy Research Office, and the North Atlantic Treaty Organization.

REFERENCES u

I. B. Hoeneisen and C. A. Mead, Sol. State Electronics, 15, 819,891 (1972).

2. J. T. Wailmark, in Inst. of Physics Conf. Series, 25, 133 (1974)~3. J. R. Barker and D. K. Ferry, Sol. State Electronics, 23, 519,

531 (1980).4. These are reviewed in D. K. Ferry, Adv. in Electronics and

Electron Physics, 58, 312 (1982).5. R. W. Keyes, IEEE Trans. Electron De., ED-26, 271 (1979), and

references conta ined the re in

been sed.

SEMICONDUCTOR DEVICE AND SUPERLATTICE ARRAYS 275

6. M. P. Shaw, H. L. Grubin, and P. R. Solomon, "The Gunn HilsumEffect", (Academic Press, New York, 1979).

7. J. R. Barker and D. K. Ferry, in "Proc. IEEE Conf. on Cybernetics,Science, and Society", 1979, pp. 762-5 (IEEE Press CH1424-I |1979). /•-.'

8. J. R. Barker, D. K. Ferry, and H. L. Grubin, IEEE Electron Dev•Letters, EDL-1, 209, (1980), and references contained therein.

9. D. K. Ferry, J. Zimmermann, and P. Lugli, IEEE Electron Dev.Letters, EDL-2, 228, (1981). See also the papers in IEEE

Trans. Electron Devices, August (1981).10. R. J. Malik, M. A. Hollis, L. F. Eastman, D. W. Woodard, C. E. C.

Wood, and T. R. AuCoin, in Proc. Eighth Biennial Cornell

Electr. Engr. Conf., Ithaca, N.Y., (1981).11. P. A. Blakely, J. R. East, and G. I. Haddad, "Tunneling Trans-

istors for Millimeter-Wave Operation", submitted for pub-

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H. Morkoc, and B. G. Streetman, Appl. Phys. Letters, 38,36, (1981).

13. F. Capasso, W. T. Tsang, A. L. Hutchinson, and G.F. Williams, "' -

Applied Physics Letters, 40, 38, (1982).14. See, e.g., L. I. Schiff, "Quantum Mechanics", Third Ed.,

(McGraw-Hill, New York, 1968), pp. 269-79.15. L. S. Schulman, "Techniques and Applications of Path Integration",

(Wiley-Interscience, New York, 1981).16. See, e.g., "Topics in Nonlinear Dynamics", Ed. by S. Jorna,

A.I.P. Conf. Proc. 46 (American Institute of Physics, NewYork, 1978).

17. V. I. Arnold and A. Avez, "Ergodic Problems of ClassicalMechanics", (Benjamin, New York, 1968).

18. B. A. Huberman, Applied Physics Letters, 37, 750, (1980).19. N. F. Pedersen and A. Davidson, Applied Physics Letters, 39,

830, (1981).20. R. T. Bate, Bull. Am. Phys. Soc. 22, 407 (1977).21. G. J. lafrate, D. K. Ferry, and R. K. Reich, Surf. Sci. 113,

485 (1982).22. D. K. Ferry, Phys. Stat. Sol. (b) 106, 63 (1981).23. A. W. Burks, "Essays on Cellular Automata" (Univ. Illinois Press,

Urbana, 1970).24. See J. R. Barker, these proceedings.

25. M. Gardner, Scientific American 224, 112 (Feb. 1971).26. I. B. Levinson, Sov. Phys. JETP 30, 362 (1970).27. D. K. Ferry, H. L. Grubin, and G. J. lafrate, "Nonlinear Trans-

port in Semiconductors", (Academic Press, in preparation).28. G. J. lafrate, H. L. Grubin, and D. K. Ferry, J. Physique 42,

(Suppl. 10) C7-307 (1981).29. R. K. Reich and D. K. Ferry, Phys. Letters A, in press.30. P. A. Lebwohl and R. Tsu, J. Appl. Phys. 41, 2664 (1970).31. R. K. Reich, R. 0. Grondin, D. K. Ferry, and G. J. lafrate,

Phys. Letters A, in press.

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