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.olid.-State Electronics Vol. 32, No. 12. pp. 1185-1189. 1989 0038-1101/89 S3.00 + 0.00Printed in Great Britain. All rights reserved Copyright © M92 Pergamon Press plc

ELECTRON DIFFRACTION THROUGHAN APERTURE IN A POTENTIAL WALL

M. Cahay, J. P. Kreskovsky and H. L Grubin

Scientific Research Associates, Inc., P. O. Box 1058Glastonbury, CT 06033

Abstract

We study the diffraction of a two-dimensional Gaussian wavepacket through a rectangular aperture in a finitepotential wall (one slit experiment). For wavepacket with incident wavevector k. satisfying the diffractioncondition, ko = 2-x/w, w being the slit width, the near field (Fresnel-like) diffraction pattern behind the slit canbe clearly seen for small time duration ( < 0.2ps). At later time steps, the diffracted beam is fragmented intolobes (perpendicular to the direction of incidence of the wavepacket) as a result of the multiple reflections ofthe wavepacket inside the slit (assumed to be of finite thickness). At later time, no far-field Fraunhofferdiffraction pattern is observed in our numerical simulations.

Keywords

Electron diffraction; Schroedinger equation; ballistic transport; split gate.

Introduction

With the advent of sophisticated growth techniques such as Molecular Beam Epitax, and Metal OrganicChemical Vapor Deposition, there has been an increased theoretical interest in vanous quantum mechanicaltunneling problems, ncluding: (1) the resonant tunnelin. of electrons through double barrier heterostructures,a problem of primary iportance in asserting the switching time of resonant tunneling device (RTD) (Haugeand Stovneng, 1989; Goldberg and coworkers, 1967), (2) the electron propagation through narrow ballisticconstriction (defined by a split-gate) in the two-dimensional electron gas of a G.As-AlyGal.xMsheterostructure. The latter followed the recent experimental discovery (Van WeeA and-cogrkers, 1988) thatthe conductance of such constrictions increases in a sequence of steps of height 2e'/h (at sufficiently lowtemperature). More recently, the possibility of using narrow split-gate for transistor applications his beensuggested by Kriian and coworkers (1988) in their newly prop QUADFET.

Both RTD's and QUADFETs have potential high-speed device applications within the terahertz regimeTDaughasd beovnen , 1ey investgatpadhyay u ng p worke, 1989). On one hand, the fast switching resnse ofTDs hand Stoveeng 1989; Bestigatedy gpa y dimensional analysis of wavepacket propagation through

double barrier heterostructures (Hauge and Stovneng, 1989; Goldberg and coworkers, 1967). In practicehowever, electrons are injected from 3D contacts and tunneling through the quantum well of the RTD ischaracterized by two-dimensional dynamics. On the other hand, in view of their potentiality for high-speedapplications (such as in the QUA DFET), there is now a call for a transient analysis of quantum transportthrough narrow ballistic constrictions to supplement the two dimensional steady-state analysis completedrecently by several authors (Szafer and Stone, 1989; Krrczenow, 1989 and references therein).

Hereafter, we limit our numerical simulations to the diffraction of a gaussian wavepacket through a narrowaperture in a potential wall of finite height. The thickness d and length w of the slit are assumed to be 100A and200A respectively, which corresponds to an aspect ratio a = d/w equal to 0.5. The choice of such a narrow slitwas imposed to make the problem tractable numerically without having to use an excessive numberof gridpoints. However, the time evolution of the diffracted beam shows interesting features which could still bepresent in the electron diffraction through a more realistic split gate realizable with present-day technology(Thornton and coworkers, 1986; Zheng and coworkers, 1986). The use of electron diffraction through asplit-gate was recently proposed by Kriman and coworkers (1988) for interesting device applications.

The Numerical Approach

Recently, Ancilatto and coworkers (1989) have developed a method to solve the multi-dimensionalSchro in er equation based on the Chebychev polynomial expansion of the time evolution operator. Eachterm in this expansion is calculated using FFT computations. Tije subsequent effort to calculate thequantum-mechanical wavefunction 0(t) scales roughly as - MNa 19 N, where M is the number of terms in theChebychev expansion, d is the number of space dimensions and N is the total number of grid points. Onedrawback of this approach is that it requires the use of uniformly spaced grid points. Another drawback of theChebychev expansion of the time-evolution operator is that, being a truncated se!-,.., expansion, it diwa notconserve the unitarity of the time-evoltion onuator "nlt-' the Liuncated series converges. As a consequence,en.rgy and norn-, z of thL 4fai,-'.ction need not be conserved at any time step.

IX Mi. (CA11AY Cf ial.

in this paper, we used an alternative approach to solve the time-dependent Schroedinger equation based upon afinite-difference solution procedure to solve a set of coupled equations governing the real and imaginarycomponents of the wavefunction. The use of Crank-Nicolson differencing scheme insures conservation of thenorm of the wavefunction at all times. The resulting coupled difference equations are then solved using a blockalternating direction implicit (ADI) technique following the scalar ADI development of Douglas and Gunn(1964). Recently, a similar algorithm has been used by Barker (1989) to study the wavepacket propagationthrough Aharanov-Bohm rings. The technique can be used with non-uniform grid spacings, allows for anexplicit time-dependence of the potential energy profile and can readily be extended to include theyresence ofa spatially varying effective-mass, of an external magnetic field and to three dimensional configurations (Cahayand coworkers, 1989).

Numerical Examples

As an illustrative example, we consider the diffraction of a two-dimensional Gaussian wavepacket through anarrow constriction such as the one shown in the upper left frame of Fig. 1. The subsequent framies show thetime-evolution of the wavepacket impinging on a 100 A wide potential barrier of height 0.06eV containing a 200A wide slit where the potential is assumed to be zero. The slit is disposed symmetrically with respect to thecenter of the simulation domain, a square of dimensions 3,000A x 3,OOOA. Our numerical simulations wereperformed using a non-uniform grid spacing with 69 and 79 grid points in the x and y directions respectively. Inan actual split-gate, the slit can actually be varied from a few hundred to a few thousand angstroms whilesweeping t e gate voltage. The potential in the 2D electron gas is also different from the sharp potential wallassumed in the present work. We will comment on this point in our conclusion section. The initial wavepacketis assumed to be

O(x,y,t=o) = 1i/ Jo2] exp[ikox] exp[-{(X-Xo) 2 -(y-yo) 2 )/2o2 ] (1)

Where (xo,yo) are the coordinates of the wavepacket center and o is rqual to 100A. Finally, the electronwavevector ik. is 0.0094 A-i (with this value of ko, a free electron (m = 0.067mo) travels a length of 1,000A in0.6 ps). The average kinetic energy of an electron in the state (1) is about 10 me-V for the values of theparameters listed above and is therefore about 116th of the barrier heigjtt. A Fermi energy of about 10 meVcorresponds to a typical 2D electron gas sheet density of - 3 x 1011 cm- . In Fig. 1, the frames are taken atvarious physical time steps equal to a multiple of 0.05 ps (which is c.qual to 10 times the computational timestep). WE have plotted the logarithmic contour p lots (IoglO) of the probability density It(t)12. For k

the electron De Broglie wavelength(: = 21/k1o) is about 660A and therefore bigger than the slitwidth. Consequently, even though part of the wavepacket-(a 35%) is transmitted on the right-hand side of thewall at time tŽ0.25ps, no diffraction lobes are detectable in the transmitted waveform. For an electron with anincident energy of 45 meV (such an electron is far in the tail of the Fermi distribution in the previous example),ko 0.0,65A- , which is close to the diffraction condition, ko=2x/w. In that case, as can be seen in Fig. 2, thereis actually a diffraction of the electron wavepacket through the slit as indicated by the existence of three distinctlobes in the contour plots of the charge density profiles in the immediate vicinity (on the right) of the slit attime t equal 0.15 ps. However, for later time steps, the diffraction lobes actually smear into a main one which(for this specific case) precludes the actual observation of a Fraunlaoffer-like diffraction pattern far from theslit. In fact, the diffraction condition, ko = 2i,/w, is only met by a small fraction of the various Fouriercomponents of the wavepacket incident on the slit. As a result, the diffraction pattern cannot be as sharp as inthe case of an incident plane wave, which is the idealistic situation equivalent to the one used in optics to studylight diffraction through a narrow slit. More numerical simulations involving modification of the shape of thewavepacket, potential walls and slit dimensions and extension of the simulation domain need to be performedhowever before determining if an appropriate set of parameters can eventually lead to a Fraunhoffer-likediffraction pattern far from the slit. The overall shape of the wavepacket behind the slit changes drasticallywhile varying the direction of incident wavevector ko. This is illustrated in Fig. 3 where the wavepacket with akinetic energy equal to 10 meV was assumed to be incident at a 45° angle on the slit. This example stresses theimportance of collimating the electron beam in order to observe a diffraction pattern with different lobesbeyond the slit. This point was already stressed by Kriman and coworkers (1988) in their steady state analysis ofthe QUADFET.

The width of the potential barrier below the split gate is of primary importance in determining the diffractionpattern behind the split-gate. As a result of the finite width, the electron wavepacket suffer multiple reflectionsbetween the podt,,e.2 -'ells defined by the slit. The resulting spread in transit times introduces additionalstructure in the diffracted electron bezm. For instance, the time frame t = 0.35 ps in Fig. 2 above clearly showsthe presence of successive maxima in the transmitted part of the wavepacket in the direction perpendicular tothe potential wall. Finally, additional lobes can be clearly seen in the reflected part of the wavepacket, i.e.. onthe left side of the slit. Such "Back diffraction" would probably also exist while considering realisticsplit-gate and it could be eliminated by appropriately collimating the incident electron beam.

Cncliusion-

Preliminary results describing the time-evolution and diffraction of a wavepacket through a narrow slit in afinite potential wall have been reported. In order to save computational time, the size of the slit was chosen tobe smaller than the actual split-gate realized with present day technology (Thornton and coworkers (1986); VanWecs and coworkers (1986). Several intere'ting features in the time evolution of the wavepacket have howeverbeen pointed out: (1) A Fresnel-like diffraction pattern is seen at early time once the diffraction requirement is

Electron diffraction through an aperture 1187

satisfied, i.e., ko=2u/w (k. being the wavepacket incident wavenumber and w being the slit width); (2) Due tothe finite thickness of the slit (in our simulation, the aspect ratio of the slit is equal to 0.5), the main diffractedlobe is found to be fragmented in the dii ection perpendicular to the slit aperture as a result of the multiplereflections through the slit of finite length; (3) No Fraunhoffer-like diffraction pattern is seen far from the sliteven when the slit width is comparable to the electron De Broglie wavelength.

One should emphasize that the Fresnel-like diffraction pattern observed in our simulations is an artifact of theuse of infinitely sharp coih.,;rs ir the potential energy profile. A more realistic simulation would requireself-consistent calculations of the potential energy profile below a split-gate such as reported by Laux andcoworkers (1988). Even with appropriate collimaUon of a nearly monochromatic incident electron beam, therounding of the corners below the split-gate on the scale of the wavelength of the incident electron could be thedominant source of suppression of the diffraction pattern at low temperatures.

L tO Z 0.0 Yps

I- IO0

0 oI~500~ I100A

0 0,500A 3 0 1,500A 3,0OA

S000X 3,000o

t 0.15 Ps t z0.25ps

I ,,500A

00

0 '0,

0 1,5004 3,0004 0 1,5004 3,0001

3,000ot 0.35 ps

Figure 1. Time evolution cf a Gaussianwavepacket moving through a slit (inthe direction indicated by the arrowko). The incident kinetic energy of

the wavepacket is 10 meV. No

1,5001 diffraction pattern with differentlobes is seen beyond the slit. Thisfigure should be compared with Figure 2

wavelength is approximately equal to

the slit width.

0 0

0 1,500A 3,OOOA

1188 NI. C'AMA~Y el, al.

3 ,oooA 3,oooA=0.05 psy t =0.15 Ps

1,5001 1,500A o

0 1,500A 5,000A 0 1, 500A 3,OOOA

0

3,oooA 3,OOOA

t 0.25 ps t 0.35 Ps

0 0 0o oo 1,5001 3,0001 0 I,SOOA 3,0OOO

Figure 2. Same as Figure 1 for a wavepacket with an incident energy equalto 45 meV. Note the formation of diffraction lobes after crossing the slitwhich can clearly be seen at time t - 0.15ps. At time t - 0.35ps, the maintransmitted lobe of the diffracted wavepacket is clearly segmented intovarious fragments. These are due to multiple reflections of the wavepacketinside the slit which has finite length. As a result, different parts of thewavepacket are delayed further in time leading to successive transmitted lobes.Spurious effect due to reflection at the boundaries of the simulation domainare only seen for t - 0.Sps.

3,0001 3,0004

t 0.05 ps t = pS

1,5001 1,5001

0 00 1,500 3,000 0 1,500o s3,000

Figure 3. Same a3 Figure 1 while shooting at an angle of 45*

on the silt The incident kinetic energy is 10 -eV.

Electron diffraction through an aperture 1189

References

Ancilatto, F., Seloni, A., Xu, L F., and E. Tosatti, (1989). EbY5ý 39,2993.Bandyopadhyay, S., Bernstein G. H., and W. Porod (1989). Proc. First Conference on Nanostruure Physi,.s

a abrieation_ Ed. by M. A. Reed and W. P. Kirk, Academic Press, Boston.Barker, J. R. (1989). Proc. First Conference on Nanostructure Physics and Fabrication. Ed. by M. A. Reed

and W. P. Kirk, Academic Press, Boston; Also, Proc. of San Miniato Workshop on QuantumTransport (1987), San Miniato, Italy.

Cahay, M., Kreskovsky, J. P, and H. L Grubin (1989). Unpublished.Douglas, J., and J. E. Gunn (1964). Numer. Math. 6.Goldberg, A., Shey, H. M. and J. C. Swartz, (1967). Am. J. Phys. 35, 177.Hauge, E. M. and J. A. Stovneng, (1989). Preprint.iL ayama, Y., Saku, T. and Y. Horikoshi (1989).Phb. Rev. B. 39,5535.Kirczenow, G., (1989). Phys. Rev. Lett. 62,2993 and references therein.Kriman, A. M., Bernztein, G. H., P-mIknccs% 13. S., and D. K. Ferry (1988). Proc. 4th Intl. Conf. on

Superlattices. Microstructures and Microelectronics Trieste, Italy.Laux, S. F., Frank, D. J., and F. Stern (1988). Su face Science. 196, 101.Szafer, A. and A. D. Stone (1989). Phy5 Rev. Leters 62,300.Thornton, T. J., Pepper, M., Ahmed, H., Andrews, D., and G. J. Davies (1986) R56, 1198.Van Wees, B. J., Van Houten, H., Beenakker, C. W. J., Williamson, J. G., Kouwenhoven, L P., Van der Marel,

D., and C. T. Foxon (1988). Phys. Rev, Letters. 60,848.Zheng, H. Z., Wei, H. P., Tsui, D. C., and G. Weimann (1986). Phys. Rev. B. 34,5635.

Acknowledgement

We would like to thank T. R. Govindan, F. J. de Jong and W. R. Briley for helpful discussions. This work wassupported by the Office of Naval Research and by the Air Force Office of Scientific Research.

Numerical Solution of the Eguation of Motion of theCoordinate Representation Density Matrix

T. R. Govindan, H. L. Grubin and F. J. deJongScientific Research Associates, Inc.Glastonbury. Connecticut 06033-1058

ABSTRACTEquation (2) is the definition of u(x,x') and

Mathematically well posed problems have been equation (3) is equation (1) written in terms of u

formulated and numerical schemes have been and p. The characteristic directions for the systemimplemented for solving the equation of motion of of equations (2) and (3) are

the density matrix. Equivalent first order systemsof equations have been derived for the equation of (x + x'}/2-constant (4)

motion of the density matrix and these systems aresolved rather than the second order equation of - (x - x')/2-constant (5)motion. The advantage of the approach is thatsuitable forms of boundary conditions can be In terms of the characteristic directions q and f,

inferred from the characteristic directions of the equations (2) and (3) can be written asfirst order system. Further. the first order systemcan be solved rapidly by a 'method ofcharacteristics' based numerical scheme that has U(X,X') + -- - 0 (6)been developed. Typical solutions. which arc 2m 8q

discussed, require about 20 seconds of Cray-XMP CPUtime and demonstrate the accuracy and efficiency of ap aU ithe solution procedure. - + - + - [v(x)-v(x')]p=O (7)

at af AINTRODUCTION

Frensley Ill was the first to suggest that solutions Suitable boundary conditions for equations (6) andto the equation of motion of the density matrix in (7) are the specification of p and u along the

the coordinate representation could be an important boundary x'-0 and the specification of u along thetool in understanding the physics of quantum well boundary x-L, where L is the length of the device."and quantum barrier devices. In [21 we described Along the boundary x-0, p is specified as thetypical results obtained from a solution algorithm complex conjugate of p(x,0), since p is hermitian,we have developed for the equation of motion of the and u Is computed from the outgoing characteristicdensity matrix. In this paper we describe essential equation (7). Along the boundary x-L, p is computedfeatures of the solution algorithm, from the outgoing characteristic equation (6).

SOLUTION PROCEDURE An alternate system of first order equations can beformulated in terms of the current matrix.

The equation of motion of the density matrix is [ll: iA ap

a I 2 21 j(xx.) + - 0 (8)-- p(x,x',t)----]P(X,x',t) 2m 8f

at 2m aX TX 8x' 2 Jp aj i

1)-- + - + -[v(x)-v(x')]p=o (9)

-- [v(x,t)-v(x',t)]p(x,x',t) 8t 8q AA

Equations (8) and (9) have the same characteristic

where p is the density matrix, m is the effective directions V and f as equations (2) and (3).mass, and v is the potential energy. We focus Suitable boundary conditions for equations (8) and

attention here on obtaining time-independent (9) are the specification of p and j along the

solutions of equation (1). boundary x'-0 and the specification of p along theboundary x-L. Along the boundary x-O, J is

For convenience of solution and determining suitable specified as the complex conjugate of J (x,0) and p

forms of boundary conditions, equation (1) is is computed from the outgoing characteristic

vritten in the form of a coupled first order system equation (8). Along the boundary x-L, J is computed

of equations: from the outgoing characteristic equation (9). Bothsets of the first order system of equations.

in (ep Of "1 equations (6)-(7) and equations (8)-(9), are useful

U (x,x*) + - t- + -I -"o (2) in applications since they allow different forms of

2z X boundary conditions. Both sets of equations can besolved by the same numerical procedure.

The overall solution procedure consists of solving

a, a e 1 I the first order system of equations as an initial

- + -- - + - [v(x)-v(x')]p-O (3) boundary-value problem sa rting from conditions

'8t tax ;x7 J A along the line x'-O and marching to the line x'-Lusing the method of characteristics. A character-istic net for the equation of motion of the densitymatrix can be constructed a-priori from grid points

*This study was supported by AFOSR, ARO and ONR of a uniform square grid. A discrete form ofequations (6) and (7) can be written on this grid as[Figure 1].

ij p(i,j)-pG(-lj-l) '

[U]av+- 0 (10)2m

la) u(i+1,)-1)-U(i'J) I I

Ltav -- '."

i ./" I -+ -- [V(x-v(X'lav[Plav 0

.where [.J•v represents an average over the grid ai. I1cell. Depending upon the form of averaging chosen,"equations (10) and (11) form a system of 2x2 blocktridiagonal or block diagonal algebraic equationsthat can be easily solved at x'-J from known valuesat x'-J-l. Thus, the solution procedure can bemarched from boundary conditions at x'-O, In stepsalong x', to x'-L. A similar procedure can beutilized for equations (8) and (9). Self-consistency is included in the analysis by iteratingthe solution of the density matrix equation with the ....solution of Poisson's equation to convergence.

G.mZ

RESULTS Vlx)lev

The equilibrium distribution of potential anddensity for a double barrier structure of [3] were Gacomputed. Computations were carried out on auniform grid of 300 x 300 points. Maxwelliandistribution of density was assumed at the boundary.Figure 2 shows the self-consistent potentialdistribution in the struc-ure. Figure 2 also showsthe density distribution with local accumul-tirn of _.w.........................charge in the quantum well due to tunneling. Figure M -= -W -7 . H • M3 is a plot of rhe density matrix on the (x.x')plane, showing Maxwellian distribution near theboundaries, depletion of charge in the vicinity of 0." Ithe barriers, and accumulation in the quantum well. X p )/10 /cmFor brevity, only one typical result has been WIpresented here. The solution procedure has beenused to compute distributions in a variety ofstructures. G.%

CONCLUSIONS P 0 1

The ability to compute solutions of the equation of p =0 I p I0'motion of the density matrix provides a 0 0

computational tool to examine fundamental issues 0.0 ! -, a , . .....

associated with mesoscopic structures. Ofparticular relevance is that the time independentcomputations took about 20 CPU seconds on a CrayX-MP and thereby provide a new efficient means of Figure 2. Potential and Density vs. Distanceexploring the physics of quantum structures.

REFERENCES

Ill U. R. Frensley, J. Vac. Sci. Technol. B5I. 1261(1985).

[2] T. f. Covindan, H. L. Grubin, and F. J. deJong, OENSIT,

Proceedings of the Workshop on ComputationalElectronics, Beckmann Institute, 21-22 May 1990.

[3] N. C. Kluksdahl, A. M. KrIman, D. K. Ferry, andC. Ringhofer, Phys. Rev. B39. 7720 (1989).

Figure 3. Density Matrix on the (x.x') Plane

SPACE-CHARGE EFFECTS IN COMPOSITIONALAND EFFECTIVE-MASS SUPERLATTICES

M. CahayI, M. A. OsmanI, H. L. GrubinI

Scientific Research Associates, Inc.

Glastonbury, CT 06033

M. McLennan2

School of Electrical EngineeringPurdue University

West Lafayette, Indiana 47907

I. INTRODUCTION

In the Effective-Mass Superlattice (EMSL) proposedby Sasaki [1,2], the effective-mass of electrons (orholes) is changed periodically and the conduction(for electrons) or valence band (for holes) issupposedly aligned. This eliminates the potentialdiscontinuities between the respective superlatticelayers which are present in doping and composiAonalsuperlattices. Previous work [3] neglecting space-charge effects has shown the threshold voltage fornegative differential resistance (NDR) in EMSL to bemuch lower (=10mV) and the current density

iSupported by ONR.2 Supported as a fellow by the SemiconductorResearch Corporation.

NANOSTRICTIRF PHYSICS 495 Copyright © 1989 by Acadmic Press, Inc.AND FABRICATION All rights of reproduction in arry fom reserved.

ISBN O-17-SSSoOO-X

(104-105 A/cm2 ) much higher than in resonanttunneling diodes (RTD). Hereafter, we reportpreliminary self-consistent calculations ofspace-charge effects in EMSL and compositionalsuperlattices under zero bias condition.

II. THEORY

We use the technique described "n ref. [4] tocalculate quantum-mechanically the electron chargedensity profile in various types of superlattices(see fig. 1). Under the assumption of ballistictransport, the Schroedinger equation throughout theentire structure is solved using the scattering-matrix approach [5] for each electron impinging fromthe contacts, with the usual boundLry conJitions forplane-wave solutions, requiring everywherecontinuity of the wave function and its firstderivative divided by the electron effective mass.Once the Schroedinger equation is solved forelectrons impinging from the contacts, the electrondensity is then calculated for the two streams ofelectrons incident from the left and right contacts.The calculation for electron density andelectrostatic potential are then performediteratively, until the electrostatic potentialconverges to a final solution.

4- SUPERLATTICE go

I a a a a a I I aI I I I I I I I II I I I I I a I I

--------- _J I ------------ L I

Buffer- _pacerm' ' ', ImI L pacer, Buffer1-. -... 1 'm2 m :Mi 2 : ml m2arn ml am1 i - I-I I • II I I r - I ' '

(ml) 1 (" 2 ) , I I (MI)

Fig. 1. Spatial variation of the conductionband edges of the device studied in the cext. Thedashed (full) curve describes a compositional (EMSL)superlattice respectively. Only the buffer regionsare assumed to be doped (10 1 8 cm- 3 ).

496

III. NUMERICAL EXAMPLES

The technique outlined abeve was applied tocalculate self-consistently the electron density and

conduction band energy profile of two types of

superlattices whose configuration are shown infigure 1. The first superlattice consists of five

regions of In 0 . 7 2 Ga 0 . 2 8 As 0 . 8 6 Po. 1 4 with width41A and effective mass ml=0.039m0 separated byInP regions of width 29.34A and effective mass

m2 =0.073mo. The current-voltage characteristicof this EMSL was calculated in ref. [3] neglectingspace-charge effects. As shown in ref. [1], thisspecial choice of layer thicknesses leads to equalenergy gaps between the EMSL energy subbands.

This EMSL is sandwiched between two undoped In?

spaces. layers (50A,) contacting two highly doped(10 1 8 cm- 3 ) InP buffer regions (500A). The

second superlattice is a compositional superlatticeobtained by replacing the In 0 . 7 2 Ga0 . 2 8 As 0 . 8 6 P0 . 1 4 /InPregions in the EMSL described above byAl 0 . 3 Ga0 . 7 As/GaAs respectively (ml=0.0919m0 andm2 =0.067m0 ). The conduction band discontinuitybetween Al 0 . 3 Ga 0 . 7 As/GaAs is taken to be 0.209 eV(see fig. 1).

Figure 2a) shows the self-consistent chargedensity profile in the two different superlatticesunder zero bias condition. Also shown forcomparison is the charge density profile obtainedwithout iteration, i.e., using the conduction band

energy profile such as shown in figure 1. As a

result of the doping gradients, internal contactpotentials are important (while different) in bothtypes of superlattices as can be seen in figure

2b). This leads to an upward shift of theconduction band energy profile in both superlatticesresulting in a substantial decrease of the chargedensity, when calculated self-consistently, fromtheir non self-consistent values. As shown in fig.2a), for identical doping concentration in thebuffer regions, the charge deasity is about one

order of magnitude higher in tha EMSL than in thecompositional superlattice.

497

0) 09

T 3.500 K SELF-CONSISTENT

, - - - NOT SELF-CONSISTENT

Io0l EMSL -.

zn -n-n

L.3

COMPOSITIONAL -

or 106

105

0 0035 0071 0.107 0142

POSITION (cm)(x iO4)

b)03

T 30-0 K

o S02za) COMPOSITIONAL

z2

0-1.)

zo

n- n- n' -EMSL

00.035 0071 0.107 0.142-4

POSITION (cm) l 10 )

Fig. 2. Self-consistent and non self-consistent(a) charge density profiles, (b) conduction bandenergy profile in an EMSL and compositionalsuperlattice (see text and fig. 1) with identicaldoping concentration in the buffer regions. Thedashed-dotted curves show the self-ccnsistentresults for a structure in which the compositionalsuperlattice has been replr:ed by undoped GaAs(n+-n-n+).

498

IV. CONCLUSIONS

Our preliminary (zero bias) self-consistentcalculations have shown that, in superlattices ofsimilar dimensions, space-charge effects are moreimportant in EMSL than in compositional super-lattices. The resulting increased overallcapacitance of EMSL should therefore have adetrimental effect on their potentiality forhigh-speed device applications [3]. Indeed,inclusion of space-charge effects in RTD has beenshown to reduce (up to 50%) their calculated peakcurrent and peak-to-valley ratio and to shift to ahigher voltage their NDR [4] (to an extentincreasing with the importance of space charge).A careful investigation of the superiority of EMSLover RTD for achieving low-power and ultrafastbistable switches must therefore await a detailedself-consistent calculation of their current-voltagecharacteristic.

REFERENCES

1. A. Sasaki, Phys. Rev. B 30, 7016 (1984).2. A. Sasaki, Surf. Sci. 174, 624 (1986).3. A. Aishima and Y. Fukushima, J. Appl. Phys. 61

(1), 249 (1988); Electronic Letters 24, 65(1988).

4. M. Cahay, M. McLennan, S. Datta and M. S.Lundstrom, Appl. Phys. Lett. 51 (10), 612 (1987).

5. M. Cahay, M. McLennan and S. Datta, Phys. Rev.B37, 10125 (1988).

499

FROM Kluwer Academic Publishers(199 1 )COMPUTATIONAL ELECTRONICSSemiconductor Transport andDevice Simulation

DENSITY MATRIX COORDINATE REPRESENTATION NUMERICALSTUDIES OF QUANTUM WELL AND BARRIER DEVICES

T. R. Govindan. I-. L Grubin and F. J. de JongScientific Research Assocates Inc.

Glastonbury, Connecticut 06033-6058

INTRODUCTION

Frensley [1] was the first to suggest that solutions to the equation of motion of thedsty matrix in the coordinate representation, p(r,xet) as an important tool in

A;tad- of the physics of quantum well and quantum barrier devices. Amathematicaffy well posed problem has been implemented here for solving thisequation.

In one dimension, the equation of motion of the density matrix is:

-(i/A) IV(xt)-V(x,,t) Ip (x,x,,t )

We have solved the above eqainusing two sparate solution procedures. Thefirst procedure for which otIm indepndn results =r di susse& Involvesthe mnethod of characteristics in which we obtain equivalent first: Order systems.For example, in one mae, the coupled first order system is obtained after definingthe current inatrix

(2) J(x,x'l)+(i/2a) ([p/ax-ap/ax']-o

which transforms the density matrix equation into:

(3) *P/*t+UJ/ax+ajfax,+(i/,) (V(x)-V(x,) ]p-o

Eq a )in and (3) represent an equivalent first order =ouWe system forotiig real and laiaparts of the denasity and auret matrices. Vith

ti .requit• the toual 7=0 densit r-.ai dthe ue lnt g the bouda.esex-.O, O<X<I, whereLi the lengthof te6devi=6 the spcfcto fciurrent alongOOx_% <I,e4and the spedification Ofdensitaloaim iO<ze'<L Self-constenqis included in the aalysls.

RESULTSEquillbaium zero bias calculation=: The first set of calculations rereentcqullbruin self-consistent calculations with Marwelian boundare In

glllbrim. at thc boundaries p (x~zo' - A exp-[(x-x')/:2)1' where 1' 11 z'IýmkTreIdisplays the diagonal elements of density and potential for an N÷IK+, withe indicated densities. The point to note about this stmmure isthat the barrier height cannot be obtained from the Wcal s

c•alculton the barie heigt is aproximatel 10mev less tha that of Fiur 1and ogges6Ts tat rsimle etimateis cofsstn wthe b rInmeso put struturebas aorelieytobeT k dinacurte.o Ftgue s 3 potndt isplay the notelldu tcaltats for ag otoube brirstructure ofwe as J ;whog rvobane soluions ugtedne

omulanion. ac e agreei ent for this zero bias calculation Ts remar w bhe Notetbatrfriter iscetrulylctued withdoingu to he s pauctu lyr hre is a~ reno of locals

d sugaccumulation. The eiera of the carreis near and about the res t

216

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-300 -450 0 150 300 -300 -150 0 150 3w

Fi&. L Potential and Density vs. Distance. F, 2. Potential and Density v. Distan.

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0.23. •0.23.

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O.60 N 0.44

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F,& 3. Potential and Density v Distance FS 4. Potential and Density vs. Distanoe.

217

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Fig•. "7. Potential and Density vs. Distance.I Fig. 8. Potential and Density vs. Distam

218

The calculations of Figures 1 through 4, assume Maxwellian boundary conditionson both the up-and downstream regions. If the distribution of quantum states isdifferent for the two boundaries, it is expected that the corresponding distributionof carriers will be difference. Several calculations of this type were performedunder the assumption of Maxwellian distributions at different temperatures. Nophysical significance is attributed to this temperature difference; rather thecalculations are displayed to shown the significance of the boundary conditions.Figure 5 shows the result in which *-he anode boundary temperature is higher thanthat of the upstream contact. Figure 6 corresponds to the case when thetemperature is lower. Note that in one case there is local accumulation at thedownstream region. In the second case there is local depletion.

Finite bias calculations: In a companion paper, [5]. the moments of the Wigerfunction are used to examine ftrasport in a double barrier beterostructure. Thebarrier height is 21mev, and the barrier widths are a narrow 5 A. The bias acrossthe device [5] in the study is 8inev. The same structure in the structure was alsoexamined usng the density matrix. The equlMirimn distribution of carrier isdisplayed in Figure 7 and the none•ilibrifim distribution is displayed in Figure &Note that in Figure 8, current inition with an entering velocity of 4x10' cmls wasassumed. The results are in qialtative agreement with that of [5] particularlywith respect to the almost linear variation of potential across the structure,althoug we not the high carrier density at the downstream boundary. Asindicated earlier this value is determined by the spectra of energies at thedownstream contact.

Time dependent calculations: A non self-consistent time dependent problem wassolved different algorithm. The structure was a 650 long Ga elementwith thic battlers arted byaS50L.regon. At timet=0, a.u.ormdenityofcarier ws ceatd.While this inta distribution is not physical, wepint out that within the first lOts of the cakualation, the carriers are removed

frmthe barriers and settle within the well and outside of the barriers.Subsequent time development shows that as the carriers outside of the doublebarrie move toward the contacts there is a reduction of charge within the well.Upon reflection at the contacts the charge in the well increases.

CONCLUSIONS

The above calculation demonstrates the ability of the density matrix to examinefindamental issues associated with mesoscopic structures. Of particularrelevance is that the time indendent calculations took less than 10 CPU secondsand thereby provide a new efficient means of exploring the physics of quantumstructures.

REFERENCES

[1] W. R. Frensley, J. Vac. ScL Technol, BI 1261 (1985).

[2] E. P. Wigner, Phys, Rev. 40, 749 (1932).

[3] I. L Grubin and J. P. Kreskovsky, Solid State Electronics, 37, 1071 (1989).

[4] N. C luksdahl, A. M. Kriman, D. K Ferryand C Ringhofer, Phys. Rev. B39,7720(1989)

[5] D.L Wollard, K A. Stroscio, M. A. Littlejohn, R. J. Trew, and ILGrubin, Proceedings of the Workshop on Computational Electronics,Beckmann Institute 21-22 May 1990.

"This study was supported by AFOSR and ONTR

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Sernicond. Sci. Technol. 7 (1992) B360-B363. Printed in the UK

Temperature description oftransport in single- andmultiple-barrier structures

H L Grubint, T R Govindant, B J Morrisont, D K Ferryt andM A Stroscio§tScientific Research Associates, Inc. Glastonbury, CT06033, USAtArizona State University, Tempe, AZ 85287-6206, USA§Army Research Office, Research Triangle Park, NC 27709-2211, USA

Abstract. Barrier calculations based upon solutions of the Liouville equation inthe coordinate representation reveal a complicated spatial dependence of thequantum distribution function near and within the barriers. Within the frameworkof classical transport this spatial dependence suggests equilibrium electrontemperature values that differ from the ambient. The prospect of quantum heatingand cooling under equilibrium conditions is examined and dispelled in favour ofan interpretation that includes density-gradient contributions.

1. Introduction position, as is the consequent electron temperature. Forquantum structures, in which quantum distribution

Calculations based upon solutions of the Liouville functions are required, e is generally spatially dependentequation in a density matrix formulation yield a com- [1], and on the basis of equation (1) suggests a spatiallyplicated spatial distribution with a mean kinetic energy dependent equilibrium carrier temperature. Because ofin equilibrium that differs significantly from the classical the significance of carrier temperature in interpretingresult. In particular, where classical physics teaches that hot-carrier phenomena, the spatial dependence of thethe energy per particle is kD T/2 per degree of freedom for mean energy per particle, and the origin of this de-a Boltzmann distribution, quantum physics, as pointed pendence is discussed through solutions to the Liouvilleout by Wigner [1], teaches otherwise. The origin of this equationdifference lies in the presence of quantum mechanical ihapop/at = [H. p] (2)forces arising from gradients in density (Ancona andlafrate, [2]), and are suggestive of a spatially dependent which in the coordinate representation is a differential

local temperature in both equilibrium and non- equation for p(x, x', t)

equilibrium cases, although spatial dependent carrier ap/at + (h/2miXV 2 - V' 2)p

temperature in equilibrium introduces interpretive dif-ficulties. To avoid this difficulty one either abandons the -(l/ih)[V(x, t) - V(x', t)]p = 0. (3)spatial-dependent temperature description, or retains it To expose the essential features of this discussion, wefor non-equilibrium studies and seeks another descrip- assume Boltzmann statistics, spatial variations onlytion for equilibrium. But in either case, it is necessary to along the x direction, and free particle behaviour alongdemonstrate its origin. This is provided below for the y and z directions. Transforming equation (3) toequilibrium conditions. centre of mass and non-local coordinates, r=(x +x')/2,

C=(x-x')/2, p=-p(r+ C,r- -, we find

p,+(h/2mi)p,.--(1/ih)[V(r+C, t) - V(r-C, t)]p=0. (4)In the above equation subscripts denote differentiation.

Classical physics indicates that the mean kinetic energy The potential V in equations (3) and (4) is the sum of allin equilibrium for carriers obeying Boltzmann statistics is heterostructure contributions, Vo(x), and contributionsfrom Poisson's equation:

(E> (=)3 J d3 p(p2/2m)f(x, p) =pkBT (1) 8/8X[C(X)8V,/x] = -e 2ep(x, x)-po(x)] (5)

where the subscript 'sc' denotes self-consistent; po(x) isthe background 'jellium' doping distribution. The dia-

where f(x, p) is the classical distribution function, and c, gonal components of solutions to equation (4) (along thethe mean kinetic energy per particle, is independent of diagonal r=x and C=0) provide the density, while the

Transport in barrier structures

expectation value of energy <E> is obtained from the consistent with the condition of global charge neutrality.diagonal components of the kinetic energy density matrix Figure l(b) also displays two additional plots. The curve[3] reaching the lowest value of density within the barrier is

E(r + C, r - =-h/8mp¢. (6) obtained from the classical Boltzmann relation between,density and potential energy; the curve reaching inter-

An approximate form of the expectation values of the mediate values of density within the barrier is obtaineddensity and the energy density [1,4,5] for one degree of from equatior (7). Note that away from the barrier thefreedom is: density from equation (7) approaches a value that is less

p(x) =p(x, x)=p exp[-(V + Q/3)/kB7] (7) than the classical value, a result that is a consequence of achange in curvature of the potential as the boundaries are

E(x) = [(k, T/2) + (h2/24mkB T))V..]p(x). (8) approached. Neither approximate solution can be regar-

In equation (7), Po is a reference density, and Q(x) is the ded as an adequate representation of the complete

Bohm quantum potential (see, e.g., [6]): solution, although the quantum-corrected solutionpossesses the general features of a density that is higher

Q(x)= -(h 2 /2mXp112 )_Jp 1 12. (9) (than classical) within the barrier and lower (than clas-

The second term of equation (8) is referred to as the sical) adjacent to the barrier. Figure l(c) displays the

Wigner contribution. In equilibrium the spatial de- potential distribution. The lowering of the potential

pendence of the energy per particle, e, as given by adjacent to the barrier (--20 meV) is a consequence of the

equation (8) is second order in h. To this order, if the excess charge and self-consistency. Figure 1(d) displays

potential appearing in equation (8) is represented by the the quantum potential; note that its value is greater than

Boltzmann relation between density and potential -300meV in the centre of the barrier. The energy

energy, p(x)=p3 :p[ -V(x)/kBT], it is seen that the density matrix represents the curvature of the densityspatial dependence of L is a direct consequence of the matrix in the non-local direction. As seen in figure l(a),spatial derivatives of dEnsity. dinthis contexthen n of t the curvature is steeper where there is excess charge andthe quantum correction to e is the same as the origin of changes sign within the barrier. The mean kinetic energythe quantum potential, per particle, e, obtained from the density matrix isdisplayed in figure l(el, along with the Wigner con-

tribution as obtained from equation (8). It is apparentthat the main origin of the structure leading to the

3. Calculations Wigner contribution is the quantum potential. Thenegative value of e within the barrier and the positive

The spatial dependence of z and the origin of the excess value of energy adjacent to the barrier suggestquantum contributions to transport arise from gradients that the Wigner contribution is not a correction, butin the carrier density. These features are illustrated represents a dominant effect, and that temperature con-through solutions to the Liouville equation for two cepts (which must include negative values) are not likelyequilibrium solutions using Maxwellian boundary con- to be germane within the context of equilibriumdition as discussed in [7]. Two cases are considered. For transport.the first calculation a single barrier characterized by a The spatial dependence of the mean kinetic energypotential per particle is also of significance in multiple barrier

V(x)=-300(meV)exp[-(x/l2.5 A)2] (10) structures. This is examined for a double barrierstructure with

is placed within a uniform, 1500 A long, structure dopedto 1018 cm- 3 . The two-dimensional density matrix, Vo(x) = (300meV){exp[-(x-75A)/I2-5A] 2

p(x, x') as obtained from the Liouville and Poisson +exp[-(x+754)/12.5 A] 2} (A))equations is displayed in figure l(a). In equilibrium thedensity matrix is real and symmetric, p(x,x')=p(x',x), The barriers are centrally placed within a 1500Aand the solution is completely represented by one-half of n n-nn structure with adjacent 10' 8cm-3 n+ regions,the matrix on either side of the diagonal, x = x', as and a centrally placed 500 A, 10" cm-n3 region. The two-displayed in figure I(a). The charge density p(x) = p(x, x) dimensional density matrix is displayed in figure 2(a),is displayed as a line plot in figure l(b), where since most obtained from the Liouville and Poisson equations.of the structure in the solution is contained within a There is excess charge between the barriers, a modestrange of 250 A, about the centre, only 500 A of the results increase in curvature between the barriers and a changeare displayed. Figure l(b) displays a significant reduction in sign of the curvature within the barriers. The line plotof charge within the barrier, as well as charge accumula- of density is shown in figure 2(b) over a reduced range oftion on either side of the barrier. While the reduction of 600 A. The density as obtained from equation (7) displayscharge within the barrier is a consequence of the presence a significantly lower charge densitv within the barrier butof the barrier, the excess charge adjacent to the barrier is order of magnitude agreement within the quantum well.a consequence of both self-consistency in the calculation The classical solution for density is completely unac-and wavefunction (or density matrix) continuity across ceptable. The potential distribution, shown in figure 2(c),the harrier The spatial dependence of the charee i; reaches flat-hand beyond 400 A on either side of the

H L Grubin et al

E " r (CENTER OF MASS DIR.)

•_ (NONLOCAL DIR.)

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-250 -5so -50 50 0so 250 -250 -450 -50 50 ISO 250DISTANCE (ANGSTROMS) ALONG THE DIAGONAL DISTANCE (ANGSTROMS) ALONG THE DIAGONAL

Figure 1. (a) Density matrix for a single-barrier structure. The physical dimension of the structure is 1500 A, requiring thatthe density matrix, which is calculated over a square matrix, is of side 1500A/.,/2. The centre of mass and non-localdirections are indicated; (b) diagonal component of the density matrix (- - -), from equation (7) (---), classical relation(-----); (c) potential energy V(x); (d) quantum potential; (e) energy per particle from density matrix --- -). fromequation (8) (---).

origin; its increase arises from self-consistency and the energy per particle, as seen in figure 2(e). As in the case ofreduction of charge in the low-doped region compared the single barrier of figure 1 the calculations suggest thatwith the bounding charge density. The quantum potent- the Wigner contribution is not a correction but repres-ial displayed in figure 2(d) is positive within the quantum ents a dominant effect, and that temperature concepts arewell, and emphasizes the reduction in charge density not germane in the context of equilibrium transport.compared with the classical value; it is negative withinthe barriers, as in the case of the single-barrier structure,and positive outside of the barriers. The positive value 4. Conclusionsoutside of the barriers is a consequence of wavefunctionand density matrix continuity within the classically The calculations of figures 1 and 2 reveal significantaccessible region. Note again that the structure of the spatial variations in energy associated with density gra-quantum potential is apparently the main origin of the dients. Mathematically, these energy variations, whichstructure leading to the Wigner contribution to the are a consequence of wavefunction continuity as repre-

Transport in barrier structures

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Figure 2. (a) Density matrix for a double-barrier structure; (b) diagonal component of the density matrix (---,fromequation (7) ( ---- ), classical relation (- -); (c) potential energy V(x); (d) quantum potential; (e) energy per particlefrom density matrix from equation (8)

sented by 'curvature' in the density matrix, are suggestive Referencesof quantum heating and/or cooling. Physically theseenergy variations represent the influence of localquantum mechanical density dependent forces on the [1] Wipner E P 1932 Phys. Rev. 40 749carriers. While their magnitudes indicate that they must [2] Ancona M A and iafrate G J 1989 Phys..Rev. B39 9536

[3) Grubin H L, Govindan T R and Stroscio M A to bebe accountf.-l for in all quantum mechanical treatments publishedof transport in mesoscopic structures, an interpretation [4] Grubin H L and Kreskovsky I P 1989 Solid-Statein terms of heating or cooling in equilibrium is Electron. 32 1071problematic. (5] Wollard D L, Stroscio M A, Littlejohn M A, Trew R J

and Grubin H L 1991 Proc. Workshop onComputational Electronics, (Deventer: Kluwer) p 59Acknowledgement (6] Philippidis C, Bohm D and Kaye R D 1982 Nuo0o

Cimento B71 75This work was supported by ARO and ONR (1L1 , F7] Govindan T R, Grubin 1- L and deJong F J 1991"TR(G, D)KF) and AFOSR (HLG, TRG and BJM). NASECODE Conference

Semicond. Sci. Technol. 7 (1992) B434-13438. Printed in the UK

Density-matrix and quantum-moment studies of single- andmultiple-barrier structures

H L Grubint, T R Govindant, B J Morrisont andM A StrosciottScientific Research Associates, Inc., Glastonbury, CT 06033, USAtArmy Research Office, Research Triangle Park, NC 27709-2211, USA

Abstract. The time-dependent LUouville equation for the density matrix in thecoordinate representation, incorporating scattering effects through a quasi-Fermilevel, and Poisson's equation have been solved numerically for a single- anddouble-barrier structure using algorithms based on 'characteristics', and showsignificant charge accumulation on the emitter side of the barrier, as well assignificant charge screening in multiple-barrier st-uctures.

1. Introduction zero terms are retained

Barrier calculations based upon solutions to the V(r + C) - V(r - V) .2_1V1 + 4Vx/3 (4)

Liouville equt-.in with a density matrix formulation To second order in h, a solution to equation (2) is [ljdisplay significant charge screening. In particular, the p + C, r - Pe exp[ - #(V + Q13)Liouville equation which in the coordinate represen-tation and in the absence of scattering is the differential + (g/.))2(1 + A2 V./6)] (5)equation where fl = l/kBT, A2 = fPh2/2m and

np/at + (h/2miXV2 -V'2 )p-(l/ih)[V(x,t)- V(x', t)]p=0 Q = (A2 #(VQ 1•22XV., -/ v)2 /2). (6)

With Q regarded as a quantum correction, inserting theyields solutions for p(x,x',t) that contain significant Boltzmann relation between density and potentialquantum departures from the classical solution when (fV= -In p), Q is transformed tothere are strong gradients in the carrier density. Weillustrate these features for Boltzmann statistics, spatial Q(x) = -(h2 !2mlptI2)/plI 2 (7)variations only along the x direction, and free particle which is a generalization of the Bohm quantum potentialbehaviour along the y and z directions. We also trans- (e.g., see [2]) for a multiparticle system and teaches thatform equation (1) to centre of mass and non-local screening is significant when there are strong gradients incoordinates, r = (x + xj/2, carrier density.C = (x - x/2, p =- p(r + C,r - 0 While the above discussion ignores scattering, when

( xrelaxation effects similar to. Fokker-Planck dissipation

p, + (h/2mi)pK - (I/ih)[V(r + 4,t) - V(r - , t)]p = 0. are introduced the diagonal component of density is

(2) approximately given by

In the above equation subscripts denote differentiation, p(x) = p(x, x) = poexp[ - fl(V(x) -4(x) + Q(x)/3)] (8)and the potential V includes all heterostructure con-tributions, V0(x) as well as contributions from Poisson's with a velocity flux density proportional to -pal,/ax,equation: and suggests that there are situations in which the effect

S e2 [jp(x, x) - p,(x)] (3) of scattering on the density can be qualitatively represen-(d/x)(x)Vx] = -e[ ,)-p0ted by introducing a scattering operator, part of which

The subscript 'sc' denotes self-consistent, and po(x) is the consists of an algebraic contribution, 4(x). As a rationale,background jellium' doping distribution Note: along the consider an n*n-n* structure in which the heavilydiagonal r = x and C = 0. doped regions are long enough for the carriers to relax to

Significant quantum effects which are revealed background. Under bias assume a quasi-Fermi levelthrou-h numerical solutions are also revealed through exists whose value is such that the density is constant atapproximate solutions when the potential in equation (2) the upstream and downstream boundaries. This as-is expanded in a Taylor series and only the first two non- stimption is implicit in most devicc alalvwcs 1c'.. Ic[ 11.

Density -matrix and quantum -moment studie!:

with the conbined effects of an applied bias and scattering Note the calculations indicate that global charge neu-else'where in the structure dettermining the remaining trality occurs (including the double-bat ncr calculations):charge distribution. i.e. fdx(p - p,) = 0. It is important to note that the

In the 'ibsence of a detailed description of scattering quantitative value of the charge adjacent to the barrier isin structures, how may one expect the density to vary? depeadent upon the value of ý(x). For example if ý(x)This is explc~ed through the assumption that an alge- were suddenly reduced prior to the barrier, thie densitybraic scattering potential, ý(x), qualitatively represents prior to the barrier would also be reduced. Global chargeýthe effects of scattering on the density within the c.nservation would require that local accumulationstructure, nd approaches the quasi-Fermi level, E~, regions would form elsewhere.within the boundary regions. ý(x) is taken as constant For the double-barrier calculations the low dopedand equal to the value of E, at the emitter boundary E#(), region extends over a distance of 600 A. Two sets ofuntil the quantum structure is reached. At the end of the simulations were performed. !n the first case the bias onquantum structure tý(x) is set equal to the value of E, at the collector ranged from 0 _< V1 'ppiie < - 300 meV.the collector boundary, Er~c). The intermediate values Here, the intermediate value of cý(x) was equal towithin the quantum region are indicated below. In terms E#4) to the beginning of the second barrier and variedof equation (2), incorporating t~(x) amounts to replacing linearly to Er~c) at the end of the second barrier. For theV(x) with V(x) - ýOx). second case the bias on the collector ranged fromn

- 100 meV <V.P,,i,, _ - 500 meV and the linear varia-tion of I(x) started at the beginning of the emitter barrier

2. Calculations and ended at the end of the collector barrier. Figures 2(a-d) represent the low-bias range, while figures 2(e-h) are

The Liouville. equation was solved for 1500A n~nn* for the high-bias range. Figures 2(a, b and c) displaystructures under bias. Displaced Maxwellian conditions density, potential and Q + V (within select regions of theare assu~aed and a flat band occurs at the boundaries. In structure). Figure 2(d) displays the real part of the densityone case a single barrier 50 A wide and 300 meV high was matrix for an applied voltage of - 300 meV. The potent-centrally loc? ted (figure I calculations); in the second case ial variation shows the presence of the formation of atwo barriers each 50A wide and 300meV high were notch region at the emitter side of the first barrier, acentrally placed and separated !y 50A (figure 2 cal- smaller fraction of potential falling across the emitterculations). For the single-barrier calculations ý(x) was barrier, compared to the collector barrier, as the bias ischosen to ensure that the density at the emitter and increased. Indeed, further increases in bias for this varia-collector regions was equal to the background of tion of ý(x) result in small variations in potential drop1018 cm <'. T~h( values of the algebraic scattering potent- across the emitter barrier region [5]. The density distri-ial across the barrier varied linearly from the value E,(e) bution shows the build up of charge in the quantum well,to E,(c). The small field that forms in the heavily doped and an increased accumulation on the emitter side cf theregýions under bias was ignored and the current through barrier. At higher bias levels the charge in the well beginsthe device was not accurately represented. Higher current to screen the emitter barrier from the collector and thevalues that would normally flow were computed from a potential drop across the emitter region is smialler thansimple circuit equation and yielded density distribution,, the corresponding case of the single barrier. The termsimilar to that obtained whe.- the current contribution Q + V within the quantum well is displayed because itswas ignored. The real part of the two-dimensional value is approximately equal to the value of the quasi-density matrix M~x, x') as obtained from the Liouville and bound state energy and moves down as the bias isPoisson equations is displayed for a bias range increased. Successive increases in bias, reduce the move-0 -< V.pplid 1< - 300 meV. p(x, x) is symmetric and ment of the emitter barrier and delay or inhibit thecompletely represented by one half of the matrix on movement of the quasi-bound state to values below theeither side of the diagonal, x = x'. The density matrix E#(), particularly when then- regions are very narrow.shows a build-up of charge on the emitter side of the The reduction of charge in the well occurs with thebarrier as well L;a broad depletion region on the variation in cý(x) as indicated for the second range of biascollector side. Linear plots of the density and potential levels in figures 2(e-h). (Figure 2(h) is the real part of theare displayed in figure 1(e) and 1(f) respectively. As the density matrix for a value of applied potential equal tobias increases there is a lowering of the barrier, and a - 300meV and is directly compared to figure (d)). Thebuild-up of a 'notch' potential at the emitter side of the result is a distribution of charge and potential in thebarrier, signifying the development of a region of charge vicinity of the emitter barrier more closely related to thataccumulation; the collector side of the bai-rier shows across the single-barrier structure. There is a morenearly linear variation in potential suggestive of a broad equitable potential drop across the double barriers [5],region of charge depletion. Both accumulation and de- and a skewing of the Q + V terms. A comparison of thepletion regions for single-barrier structures have been potential distribution and charge distribution atdiscussed hy Eaves et al (4]. Incrensed charge accumula- 1

',,,pi, = -300meV indicates that a larger potentialtion on the emitter side of the barrier tends to rediu-, the drop falls across the double barrier in the absence ofrelative change in potcntial on the emitter side of the charge In, tilc well, which as a consequetnce will lead toI_-. - '' ..- _ -- , . ", it,. 4 - #1 , -. 1-,.. rT.r in r,,rrrn v'i.i..C .,ld iiiO t thitl t~V.tprr ,.,,

H L Grub'in et at

r (CENTER OF MASS DIR.)to

_ ~ (NNLOCAL DIR.)0+

0 -F

2 '2

I W (d)

inE

0.

(e X (A)t N

-- -- -- - 0.4

N -0.

- -nm - - ~ - m I

I- I ' IC

I-Z 11__ -

-750 -500 -250 0 250 500 750 -750 -500 -250 0 250 500 750ODISTANCE (ANGSTROMS) AL0146 OHC WAIAAL~JDa DISTANCE (ANGSTROQd61 ALONG THE DIAGONAL

Figure 1. (a-d) Density matrix for a single-barrier structure. The physical dimension of thestructure is 1500 A. requiring that the density matrix, which is calculated over square matrix.is of side 1500 A/.,/2. The centre of mass and non-local directions are indicated. (a)V.POI(.d=O.OmeV, (b) V.,,1I1d = - 100meV, (c) V...11.d=~ 300meV. (d)VDplied = - 500 meV. (e) Diagonal component of density matrix: (- -- -) 0.0meV;(-)-100 meV; (- - -) - 300 meV; (-) -500 meV. (f) Potential energy -- - .0meV;( --- ) -100meV; (---) -300meV; (-) -500meV.

in the measured current voltage characteristics (see, e.g., tering results in a reduction of ý(x) prior to the collector[4]) is a measure of the variations in the charge distribu- barrier, for the next and higher range of applied biastion as a function of scattering within the structure, levels, then the build up of charge in the well of theIndeed if we assume that the range of scattering potential resonant tunnelling diode, and the subsequent redistri-where 4(x) is constant to the collector barrier is represen- bution of charge to a region upstream of the emittertative of the scattering dynamics until (he peak current is barrier at higher bias levels emerges from thisreached in resonant tunnelling strtictiircs, and that rcat- description.

Dewrit-nl uiorix and quantum Imoment Studies

(e))

07

-=; N

- 0t1*

1 1 1 I I - - - - 03

Iltl ttf0.2t N0.

-a - - - - - - 0.

(c)-0.3-0.2

0. _0ý

(0-- - -~ - 0.3 g

0 0

0 0.a n-- ý. - -.C- ---- - - -0.2 0.

- - - -. 4 -0.6

-750) -500 -250 0 250 500 M5 -750 -500 -250 0 250 500 750

DISTANCE (ANGSTROMS) ALONG THE DIAGONAL DISTANCE (ANGSrROMS) ALONG THE DIAGONAL

E

X (4,004,,5-

Figure 2. Double-barrier structure under bias: (a and d) density, (b and f) potential energy(c and g) 0 + V at select portions of the structure. (d and h) density matrix at - 300 meV.Differences in result depend upon algebraic scattering potential (see text). (a), (b) and (c)

(-)0.0Ome'!; ( 100 meV; (- )-300 meV. (e), (f) and (g) (--)0.0 meV;(--300 meV; (----)-500 meV.

3. Summary consequent change in the distribution of density and incurrent. In the absence of a detailed description of

An appropriate description of transport requires, at least scattcring it fc a enitcUc hog hat the boundaries, that the t-irriers approach their incorporation of anl algebraic scattering potential. Whileanticipated back-pround vajllucs I his is achieved lhroughd the results presented here are dependent upon the specificthe introduction of scaittcruing~ event,,. The presence of represen tatlonl of ~(~)they, aIre representative of thle fact

e&ItrjH vents ;Ilt(-rý 1110 1 IoII'ilko VoII3tionl , with A 1lw fleW, 111;o ; NIII)IItC nwhiretr of ir:iwl'noyl :oý nrovislel In

H L Grubin et al

standard kinetic and potential energy concepts, albeit References

quantum mechanical, is not likely to be adequate to

describe transport in multiple-barrier systems. [1] Grubin H L, Govindan T R and Stroscio M A to bepublished

[2] Philippidis C, Bohm D and Kaye R D 1982 NuovoCimento 71B 75

[3] Shockley W 1950 Electrons and Holes in Semiconductors

Acknowledgment (Princeton, NJ: Van Nostrand Reinhold)[4] Eaves L, Sheard F W and Toombs G A 1990 Physics of

Quantum Electron Devices ed F Capasso (Berlin:This work was supported by ARO and ONR (HLG, Springer)TRG) and AFOSR (HLG, TRG and BJM). [5] Ricco B and Azbel M Y 1984 Phys. Rev. B 29 1970

1 Introduction to the Physics ofGallium Arsenide Devices

HAROLD L. GRUBINScientific Research Associates, Inc., Glastonbury, Connecticut

1.1 INTRODUCTION

It is arguable that the history of gallium arsenide semiconductor devices,from the early 1960s to the present time, falls intc, three groups. First, therewas the experimental work of Gunn [1], demonstrating the generation ofsustained oscillations upon application of a sufficiently large dc bias. Thiswork opened up the possibility of fabricating bulk microwave and mil-limeter-wave devices, and hastened additional and intense studies of theproperties of compound semiconductor devices. Second, there was the studyof Ruch [21, whose results suggested that the transient, or nonsteady-state,aspects of semiconductor transport would improve the speed of devices byalmost an order of magnitude. This, of course, is the argument behind muchof the move toward submicron and ultrasubmicron structures. The third era,the one we are presently in, involves the incorporation of gallium arsenideinto material-engineered highly complex structures, some of which haveprovided remarkable millimeter wave characteristics, such as the pseudo-morphic HEMT [3]. Much of this book is concerned with this third era, andthus this chapter will only briefly touch upon it. Rather, this section willpresent a road map of the consequences of using compound semicon-ductors for device applications, using gallium arsenide as the paradigmexample.

The band structure of gallium arsenide is familiar to most and is displayedin Fig. 1. [4]. It is a direct bandgap material. The minimum in theconduction band is at F with relevant subsidiary conduction band minima atL and X. The curvature at F is such that the effective mass of the F-valley islower than that of the next two adjacent subsidiary L- and X-valleys. Forthe valence band, the two valleys of significance are those associated withthe light and heavy holes. We will concentrate on transport contributionsfrom these five valleys.

In equilibrium, the relative population of electrons in the valleys isdependent on the density of available states and the energy separation, for

2 INTRODUCTION TO THE PHYSICS OF GALLIUM ARSENIDE DEVICES

L6

-4

-6-8

'I_- OL6

-10 )(

L A r7 A X U.K 2:

WAVE VECTOR k

Figure 1.1 Band structure of the semiconductor gallium arsenide [4]. (Reprinted,with permission, from Physical Review [Section] B: Solid State.)

example:

~ o~m~ 3 1 2 expAnor= nL kT (1)

where nr and nL denote the equilibrium density of the F- and L-valleycarriers, respectively, mr and mL are their effective masses, and A is theF-L energy separation. Thus, in equilibrium virtually all of the electrons ofinterest are in the F-valley. For the holes, the valleys are degenerate.

Gallium arsenide is a compound semiconductor. At low values of electricfield, apart from carrier-carrier scattering, there are three important scatter-ing mechanisms: polar optical phonon scattering, acoustic phonon scatter-ing, and impurity scattering. For F-valley electrons, the contribution to themomentum scattering rate from polar optical phonons is approximately twoorders of magnitude larger than that of the acoustic phonon. Since, with

regard to mobility, scattering rates are additive, the polar optical phonon isthe dominant scatterer. Ideal room-temperature electron mobilities are inthe range of 8000 to 9000 cm 2/Vs . For the subsidiary valleys, the effectivemasses of the carriers are much larger than that of the F-valley, and the

It relative contribution of the acoustic phonon increases. Nevertheless, thepolar phonon dominates the transport. For holes, the situation is mixed,

INTRODUCTION 3

with the dominant scattering being polar and nonpolar deformation poten-tial coupling. For momentum scattering, the nonpolar deformation potentialscattering dominates.

At high values of electric field and for electrons, nonpolar phonons enterthe picture, intervalley transfer from F to L takes place, and the situationbecomes complex. For example, the spatially uniform, field-dependentvelocity characteristics of gallium arsenide, ignoring electron-hole interac-tion, display a region of negative differential mobility, as shown in Fig. 1.2[51, where at values of field in excess of 3 kV/cm the mean carrier velocitybegins to decrease with increasing electric field. This is an unusual situationand it is perhaps important to recognize that the mean electron velocity of agiven species of carrier, assuming a parabolic band, is not decreasing withincreasing electric field. Rather, the numbers of high-mobility electrons aredecreasing, due to transfer to the subsidiary larger effective mass valleys.

The situation with holes is different. Here, the dominant transport isthrough the heavy hole. Interband hole scattering is always present even atvery low fields, however the relative population is fixed through the ratio ofthe effective masses, and the existence of a dc negative conductance forholes, on the basis of available data, is ruled out. The field dependence ofthe mean hole velocity, ignoring interaction with the electrons, is displayedin Fig. 1.3 [6], and there are two important features of note. First, there isthe extremely low mobility of the holes at low-field values. Second, there isthe saturated drift velocity, which is expected to be higher than that ofelectrons at high fields. We note there is no hard data on the high-fieldcarrier velocity of holes in gallium arsenide.

Calculations of the type displayed in Figs. 1.2 and 1.3 have beendescribed by many workers and are routinely incorporated into simulationcodes. Of more recent interest, because of mixed conduction heterostruc-ture devices, are the modifications that may be expected when electron-hole

2

EU

> 0.8

0.6I I ! I I I

2 4 6 8 10 20 40 60 80 100

FIELD (kv/Cm)

Figure 1.2 Field-dependent electron mean velocity for gallium arsenide [5].


108 -- - ,'•- . . . . .-l

GaAs. 1

S, i /

+ 0 6 ! /- 'q-, -i

107 ¶,€2.10!

, i E (Vlcm)

* Figure 1.3 Field-dependent hole mean velocity for gallium arsenide [6]. (Re-printed, with permission, from Journai of Applied Physics.)

II

2 !

-1

01

E ..

> • 11 -'--- -3-

O i WITHOUT e-h P =5x101cm-*!! wITH e-h n c 0r&m 3

ii0 IFigure 1.4 Field-dependent electron mean velocity for gallium indium arsenide

assuming an interaction with heavy holes [8]. (Reprinted, with permission, from

Applied Physics Letters.)

,t2

T 00K

!,f

THE ROLE OF BAND STRUCTURE ON DEVICE OPERATION 5

scattering occurs. However, because of the limited number of studies withGaAs and because of similarities with other compound semiconductors,results of InGaAs studies are presented. Additionally, because of ex-perimental work in InGaAs [7], the role of carrier-carrier scattering hasbeen most extensively studied for that material. Monte Carlo calculationsincorporating electron-hole scattering are displayed in Fig. 1.4 [8]. Theresults require some detailed discussion and are considered later. Here, wesimply note that the presence of holes leads to reduction in the low-fieldmobility but an increased peak carrier velocity. These intriguing results arealso anticipated for GaAs.

The question of interest is how may we expect the role of the complicatedcompound semiconductor band structure to affect the performance ofdevices. This is considered next.

1.2 THE ROLE OF BAND STRUCTURE ON THE OPERATIONOF ELECTRON DEVICES

In examining the role of band structure on the operation of electron devices,there are several items of immediate interest: the effective mass, thelow-field mobility, and the direct bandgap energy of the binary III-Vmaterials (see Table 1.1 [9]). Additionally, the energy separation of theconduction band minima to the subsidiary valleys is listed in Table 1.2 [10].Note that of the seven binary materials listed, five are direct bandgapmaterials, and two, GaP and AlAs, are indirect materials. The indirect

TABLE 1.1 Critical Parameters of Select CompoundSemiconductors

ElectronLow-field Direct EnergyMobility Bandgap

Compound Effective Mass' (cm 2/V• s) (eV)

GaAs 0.063b 9,200 1.424GaPc 0.25'/0.91' 160 2.78GaSb 0.042 3,750 0.75InAs 0.0219 33,000 0.354InP 0.079 5,370 1.344InSb 0.0136 77,000 0.230Alsd 0.71b 300 2.98

Source: Ref. 9. Reprinted with permission of Springer-Verlag."Multiples of free electron mass at the conduction band minima.bDOS.cThe minima in the conduction ba.Id are at A-axis near zone boundary.d1The minimum in the conduction band is at X.


TABLE 1.2 Intervalley Energy Separation

F-L r-X L-XCompound (eV) (eV) (eV)

GaAs 0.34 0.48 0.14

GaP -0.27 -0.39 -0.37GaSb 0.08 0.37 0.23InAs 1.27 1.60 0.33InP 0.63 0.73 0.10InSb 0.41 0.97 0.56AlAs -0.15 -0.79 -0.64

Source: Ref. 10. Reprinted with permission from OxfordUniversity Press.

bandgap materials have the highest effective masses of the group and alsothe lowest mobility. Of these materials, GaAs, InAs, InP, and InSb possessregions of negative differential mobility. GaSb, GaP, and AlAs do not. It isperhaps not surprising that the first four mentioned materials possess aregion of negative differential mobility, nor that the last two materials donot. In the latter case, the minima in energy is associated with a largeeffective mass, high density-of-states energy level. The situation with GaSbis peculiar. But here, while the effective mass of the F-valley is the smallestof the three, its closeness in energy to that of the subsidiary L-valley is suchthat at low values of field conduction, contributions arise from both the F-and L-valley, effectively suppressing the contributions of intervalley transferto negative differential conductivity.

"The presence of a region of bulk negative differential mobility has, as amajor consequence, the possibility of electrical instabilities. These in-stabilities manifest themselves either as large-signal dipole-dominated oscil-lations, often referred to as the Gunn effect, or as circuit-controlled 'oscillations, where the semiconductor behaves electrically as a van der Poloscillator. The binary semiconductors GaAs, InP, and InAs have exhibitedelectrical instabilities associated with bulk negative differential mobility.While InSb has also sustained electrical instabilities, the interpretation ofthe instability is complicated by the small direct bandgap and the possibilityof avalanching at low bias levels.

, !An additional feature of importance is the intrinsic carrier concentrationsof some of these materials, as shown in Table 1.3 [9]. It is clear that theintrinsic concentration of InAs and InSb make them unsuitable for aunipolar source. Indeed, all transport calculations using these latter materi-als must necessarily include multispecies transport.

In choosing materials for electron devices, particularly as power sources,a figure of merit has been the peak to saturated drift Velocity ratio. Fromi? this point of view, indium phosphide is an attractive candidate, but this mustbe weighed with the fact that the low-field mobility of InP is less than that of

mmll ', I

THE ROLE OF BAND STRUCTURE ON DEVICE OPERATION 7

TABLE 1.3 Intrinsic Concentration

Compound n(/cM2)

GaAs 2.1 x 106

InAs 1.3 x 1015

InSb 2.0 x 10"6InP 1.2 x 108

Source: Ref. 9. Reprinted with permission of Springer-Verlag.

gallium arsenide. A recent study comparing these features suggests that theF-valley mobility is the dominant material parameter of submicron struc-tures, whereas the high-field saturated drift velocity is the dominant materialparameter of micron-length structures [11].t Additionally, if a choice for twoterminal sources is to be made between, for example, InP and GaAs, otherissues emerge. For instance, the scattering rates in InP indicate a shorterenergy relaxation time than that of GaAs. The consequence of this arehigher-frequency operation for InP. Thus, at least with respect to thesematerials, the peak to valley ratio of the materials is only one factor in thedesign of an electrical source.

140 -

120 163' K''

100ý----298 K

- 80 -

t: Oscillatory region-60 -Lowest current

. /for oscillations

0 0 1 2 3 4 5 6 7 8 9

V (drain-source) (V)

Figure 1.5 Temperature-dependent pulsed data for a GaAs FET, with a 3.0 tjmgate length, a source to drain separation of 8.5 tim, and an epitaxial thickness of3000 ± 500 A. Nominal background doping is 1017 /cm 3 [12].

'(Editor's Note) See, however, Chap. 2, page 82, last paragraph and reference [12]cited within for an additional view on this issue.

8 IN'TRODUCIION TO( TIlE PHYSICS OF GALLIUM ARSENIDE DEVICES

There is less to say about the effects of negative differential mobility onthe operation of avalanche diodes. Here, the effects of negative differentialmobility conductivity are present but are overshadowed by the effects ofavalanching. For example, recent simulation studies show the presence ofdomains in IMPATf's, whose presence is a direct consequence of negativedifferential conductivity. These domains can complicate the actual transittime of dipole layers associated with the avalanche generation, but thenegative differential mobility is a marginal issue. Such is not the case withthree-terminal devices.

For three-terminal device observations of bias-dependent white light inGaAs FETs, as from either the drain side of the gate contact and the gateside of the drain contact, are consistent with numerical calculations showingthe presence of local high-field dipole layers near the gate and draincontacts. In addition, for a range of bias, some devices display a currentdropback consistent with bias-dependent formation of high-field domainsand concurrent current oscillations. This last result is shown in Fig. 1.5 [121.Remaining questions of interest focus on the manner in which transport inthese devices is examined. We begin with the equilibrium description oftransport.

1.3 EQUILIBRIUM DESCRIPTION OF TRANSPORT

The steady-state equilibrium description of transport has traditionally pro-vided most details of device behavior. Nevertheless, the description ignoresacceleration. It assumes that the carrier velocity is determined by the localelectric field and that the total current is governed by a balance of a driftcomponent and a diffusive component. Typically, the continuity equation issolved simultaneously with the current equation, which for electrons is ofthe form

J, -e n,, - Dn - ) (2)

and for holes:

Jp = +e(pvp - Dp (3)

Here, n and p denote electron and hole concentration, respectively, vvelocity, and D diffusivity. The usual derivations of Eqs. (2) and (3) proceedfrom a linearization of the Boltzmann transport equation. The assumption isthen made that the equation is valid for high-field nonlinear transport.8/

EQUILIBRIUM DESCRIPTIONS OF TRANSPORT 9

Typically, the field-dependent velocity assumed in these equations is of thetype displayed in Fig. 1.2.

While the use of the field-dependent velocity in these equations isuniversal, the type of diffusivity coefficient used in these studies is almost asnumerous as the numbers of workers involved in numerical studies. How-ever, a number of important issues are at stake in the description of thediffusivity. For example, if the Einstein relation

D pckT (4)e

is used, then, under equilibrium and/or zero current conditions, the depen-dence of carrier density on conduction and valence band energy is given byeither the equilibrium Boltzmann or Fermi distribution. However, undernonequilibrium conditions (and near-zero current conditions), the Einsteinrelation inadequately describes diffusive transport [13]. To correct for thelatter deficiency, the field-dependent diffusivity often used in calculations isof a form similar to that shown below [14]:

AkT 2D e sat (5)

where at high values of electric field, the diffusivity only gradually de-creases. While the diffusivity coefficient of Eq. (5) more adequately repre-sents high-field phenomena, because its field dependence is conceptuallyconsistent only with the assumption of nonequilibrium conditions it isconceptually inconsistent with equilibrium conditions, and will lead toincorrect built-in potentials [15].

While the drift and diffusion equations (DDE) clearly offer conceptualdifficulties with respect to consistency of physics, they nevertheless offerconsiderable insight into the physics of device operation and are usefulproviding their limitations are kept in mind. For example, instabilities inlong GaAs structures are known to depend critically on conditions at thecontacts. A study in 1969 [16] demonstrated that by experimentally creatingdifferent conditions at the boundaries to the active region of GaAs, a widerange of different electrical instabilities could be obtained. Correspondingnumerical studies were performed through solutions to the above drift anddiffusion equations, in which a value for the electric field was specified at thecathode (and anode) boundary. It was found that the boundary-dependentelectrical behavior could be broken into three categories, as summarized inFig. 1.6 [16]. The key conclusion of the study was that the electricalbehavior of compound semiconductors devices was dependent in a detailedway on contact conditions. This same critical result has reappeared numer-ous times in a variety of different types of structures.

10 INTRODUCIION -O THE PHYSICS OF GALLIUM ARSENIDE DFiVICI:S

2.5

2.0

E B1.5

2 E

C V

0.5

0 5 10 15I I

Average electric field (V/cm x 10-3)

(a)

I ,2.5qI

2.0

UE

1.5

C= E

1 505 10 15

Average electric field (V/cm x 10-3)

Figure 1.6 (a) The v(E) curve and the computer simulated current density j as afunction of average ele.tric field (E) (A through C) for various cathode boundary

fields. The boundary field is zero for curve A, 24 kv/cm for curve C, and is indicatedby the arrow for curves B, and B2 . The sample is 10-2 CM long and has n = 10 cm-3

and i. = 6860cm 2/V- s. The right- and left-hand ordinates are related by j- nev,v2 =0.86 x 107 cm/s. (b) Experimental j- (E) curves (J-) and (-) (dashed) and

theoretical curves B and C (solid). The only significance in ' fact that the low-field

slopes differ is that the theoretical curve is for a mobility of 6860cm 2/Vs, whereasthe experimental curve is for a mobility of 4000 cm 2/V- s [16].

NONEQUILIBRIUM DESCRIPTIONS OF TRANSPORT 11

1.4 NONEQUILIBRIUM DESCRIPTIONS OF TRANSPORT

The situation of most interest lies in nonequilibrium transport. The mostcritical area of interest is the incorporation of acceleration into the govern-ing equations.

In examining nonequilibrium transport, several approaches have beenused. One is the Monte Carlo method, where the trajectory of a particle isfollowed through its acceleration and subsequent' scattering events. In thediscussion below, results of Monte Carlo calculations will be presented, butwe first concentrate on nonequilibrium phenomena as described by themoments of the Boltzmann transport equation. These equations, in theirsimplest form for parabolic bands, a position-dependent conductionband, and a position-dependent effective mass, take the form shownbelow [171:

Carrier balance:

an + Ikd 2 f d 3dk (6)+ -t m (2 ;T) 3 4t Coa l

Momentum balance:

d (nh 2kk)- nhkd + V m nVrE + qnv X B - rnkT

+ n k + 3 nkT) _Vm + (27r)2 f -{f k d3k (7)2m 2 M (2o7T)h3ddk c7i

Energy balance:

a r/(h2 kd'kd 3 )] (Izkd'kd 5d- n( + - kT~i + V . nv 2 + -k Tat L 2m 2 1J r 2m 2

2 farf hi2 k'k 3=-n.v VrEc + (21) J-t Icoi m d k (8)

In the above, hkd is the mean momentum of the carriers, T is the carriertemperature, and, for electrons, Ec is the position-dependent conductionband energy. B is an applied magnetic field. The terms on the right siderepresent scattering and/or electron-hole interaction, as through avalanch-ing. For example, the right side of Eq. (6) represents intervailky scattering.If avalanching occurs, generation is expressed through an erergy-dependentionization coefficient [18]. If a carrier temperature model is assumed, then


9 carrier generation is given by

[ na(T) (9)

where a(T) is the ionization coefficient. In the absence of a first-principle

determination of a(T), the following relation can be assumed as a starting

point:

a(T) ct*(F)v(F) (10)

where a*(F) and v(F) are the equilibrium ionization rates and field-dependent velocities, respectively and the relation between T and F isdetermined from the equilibrium solution. While the Eq. (10) relation isuncertain, it has the conceptual advantage of relating ionization to energy,rather than field.

But, perhaps the most significant feature of these equations is thepresence of acceleration, both spatial and temporal in the momentumbalance equation. These acceleration terms are absent from the drift anddiffusion equations. Additionally, under equilibrium conditions, and hence,

zero current (i.e., nhkd/m = 0), the electron temperature model teachesthat for any spatially nonuniform structure, such as a p-n junction, theelectron temperature is everywhere constant and equal to the ambient.Thus, co. xeptual problems arising from the form of the diffusion contribu-tion to the drift and diffusion equations do not enter here. Note that ageneralized drift and diffusion current term is obtained when the left side ofEq. (7) is set to zero.

Equations of the type shown above provide a considerable amount ofinformation with regard to transport. For example, with a F-L-X orienta-tion in GaAs the distribution of carriers as a function of field is shown inFig. 1.7 [5]. Here the relative distribution of carriers in each of the valleys isdetermined by the distribution of temperature in each of the valleys, whichin turn is driven by the electric field, as shown in Fig. 1.8 [5]. Note that forfields below 4 ky/cm, the carriers reside in the F-valley. At fields above6 kV/cm, the L-valley population exceeds that of the F-valley. It should beemphasized, however, that because of the very low subsidiary valley mobili-ty, most of the current, for fields up to 50 kV/cm, is carried by the F-valleycarriers.

The interest in nonequilibrium equations lies not in the steady-stateuniform field distribution, but in transients and nonuniform fields. Thetransient distribution of carrier density and velocity for electrons subject to asudden change in electric field is shown in Fig. 1.9 [51, where the high peakvelocity can be noted.

The high peak velocity in Fig. 1.9a is primarily associated with F-valleytransport. Indeed, the F-valley velocity at 2.8 ps is near 4 X 10' cm/s. The


1.0 -

r"•-)z 0.5

CnI,-

zwa

wN

o0.10--j

z0P_ 0.05C-,,IA-

l I I

1 2 4 6 8 10 20 40 6080100FIELD (kv/cm)

Figure 1.7 Steady-state uniform field carrier distribution for F-L-X orientation inGaAs [5).

5I0t5 -!

E 5 -

-J_w

o 104

wJ 5-J

10 I I1 5 10 15 20

r- TEMPERATURE /300 K

Figure 1.8 Temperature dependence of electric field for a F-L-X orientation inGaAs. Electric field is the independent variable [5].


I - . (o)j

> 2 I .. ...s

5 -

.E 4 1- . -

I '

0 2 4 6 8 10T/(.28 ps)

1.21.0 .(b)

0.8 r

-,0.6 -"0.4-- N

' L0.2 - -

00 2 4 6 8 10

T/(.28 ps)

Figure 1.9 Transient overshoot for a field of 9.6 kV/cm IS]. F-L-X orientation,and an applied field of (a) r and mean velocity, (b) F- and L-valley carrier density.

mean velocity

V (nrvr + nrVL + nxVx)/NO (11)

where n denotes net population of the F-, L-, and X-levels and No is thetotal carrier density, is also shown in Fig. 1.9. We note that the significantdrop in mean velocity is a consequence of electron transfer from the centralto the satellite valleys (see Fig. 1.9b). Also note a transient decrease inF-valley velocity. This is a consequence of the difference between the energy(longer) and momentum (shorter) relaxation times in GaAs. The time-independent spatially nonuniform situation also displays overshoot effects.

The situation when, in one dimension, space charge effects are intro-duced is displayed in Figs. 1.10 and 1.11 [5], where for a gallium arsenidedevice of different lengths we show the field and space charge distribution ofthe F-valley electrons and the current-voltage characteristics. The feature tonote is that as the device length decreases the current drive increases. Notethat in all cases the field is nonuniform and increases toward the anode.Electron transfer exists for all four structures, with the greatest amount oftransfer occurring for the longest device. Additionally, since high-field

Itma • u n


1.6 .25/.m 1. 50Am

1.21.

0.8 0.80.4 0".

GoA$ 0 0 0.5 1.0 0 0 .05 1.0

N =5X1015/Cm 0 4.0 2.0 tm 0o0 'I 3.63 f 0 32F. z5xI0 v/Cm IOm 3.2 -

0 2.8 2.8Wz2.4 2.4 h

2.0 2.0

1.6 1.6 -I

1.2 1.2

0.8 0.80.4 0.4

0 0.5 1.0 0 0.5 1.0 ,

DISTANCE/L

Figure 1.10 Electric field and F-valley carrier distribution for GaAs (with atwo-level model) two-terminal devices of different lengths and a mean field of5 kV/cm [5].

0.5

0.40

"0.25pm 0.50

~,0.3 10z

2.00

z 0.2

0.1

0 II I I I

0 I 2 3 4 5 6F (kV/cm)0

Figure 1.11 Current-voltage characteristics, as a function of lengths, for the struc-tures of Fig. 1.10 [5].


EMITTER

0400mf

0 02ý-m E r

5 x 10 16 N

0.6 jim

Figure 1.12 Dimensions and doping level of the simu-lated PBT [20]. 0.15/4m

values are synonymous with carrier accumulation, we see that electrontransfer here is accompanied by an accumulation of L-valley carriers.Saturation in the current density occurs at high bias, and even for theshortest device there is electron transfer at the anode side of the device.

The role of nonequilibrium transport on two-dimensional simulations isdiscussed for the vertical three-terminal GaAs permeable base transistor[19], one cell of which is displayed in Fig. 1.12 [20]. One importantadvantage of this structure is the parallel placement of the source and draincontacts* and the absence of any substrate through which current can flowand reduce the transfer characteristics of the device. The simulations wereperformed for a 1 jim long source to drain region and a 200 A gate. Also,for comparison, results of the drift and diffusion equation simulations wereincluded.

The computed I-V characteristics of the device shown in Fig. 1.12 arepresented in Fig. 1.13 [20]. The moments of the Boltzmann transportequation results are extrapolated to the origin, as indicated by the longbroken lines. The shorter dashed curves show the results for the DDE. Thecomparison shows that the predicted current levels are significantly higherfor the moment equation solutions, a result consistent with FET calculationsperformed by Cook and Frey [211, who used a highly simplified momentum-energy transport model. The present calculation results also indicate aregion of negative differential forward conductivity at VBE = 0.6 V. Theorigin of this phenomena is a consequence of electron transfer!t Thepresence of a dc negative forward conductance is also a feature of PBTmeasurements but is clearly absent from DDE simulations [20]."*(Editor's Note) They are also called emitter and collector in PBT terminology.'(Editor's Note) See also Chap. 2, Section 2.2.4 for additional views on the issue of

negative differential conductance.


16 MBTE VBE=O.--- DDE

1.4

1.2VE= 0.4

1.0-VE =0.2

-0.6-

0.4 - V =0.2/

0.2

00 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6

VCE (vols)

Figure 1.13 Collector current versus collector-emitter voltage for different valuesof base-emitter voltage [20].

A comparison between the total carrier density distribution along thecenter of the channel for drift and diffusion and MBTE solutions is shown inFig. 1.14 [201 for VCE = 1.0 V and VBE = 0.4 V. The moment equationprediction for the F-valley carrier density is also shown.

As seen in Fig. 1.14 for the DDE simulations, the carrier density reachesa maximum between base contacts and displays a significant dipole layer.Here, with the velocity in saturation and the cross-sectional area at aminimum, the carrier density must increase to maintain current continuity.In the moment equation simulation, the constraints of current continuity aremore complex. First, a decrease in the cross-sectional areas is, as in theDDE, accompanied by an increase in field along the channel. The fieldincrease under both equilibrium and nonequilibrium conditions is qualita-tively similar, as may be observed from Fig. 1.15 [20], which shows thepotential distribution along the center of the PBT channel. However,consequent changes in electron temperature, both increasing and decreas-ing, lag behind the equilibrium state. This leads to velocity overshoot (Fig.1.16 [20]) and a delay in electron transfer. As a result, for nearly the firsthalf of. the device, transport is almost exclusively F-valley transport. Theimplication is that if the F-valley carrier velocity increases with increasingfield, then the product of density and cross-sectional area normal to current


1.4

12 VCE 1.0 NT

108

Uj0.

I-ISI \

Lr

0.2 04

N'CENTRAL VALLEY

0 I \

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

Figure 1. 14 Carrier density versus distance along center of channel for the PBT(20]1.

flow must decrease to maintain current continuity. Since the velocityincreases faster than the area decreases, the carrier density decreases.

At moderate bias levels, typical PET calculations show a decreasing fieldas the gate region is passed. This also occurs in the PBT. Now, as thecross-sectional area increases, the V-valley carriers exhibit a decrease invelocity. It must be noted, however, that for the parameters of the calcula-

1.0 - - - -.

-DDE/

0.8- /f~VB 00.4

,, 0.6 / -E 00.40.

0.2 - -- DD

0I0.1 0.2 05 06 0.7 0.8 0.9 0

DISTANCE

Figure 1.15 Potential versus distance along center of channel for the P BT [20].

NONEQUILIBRIUM ELECTRON-HOLE TRANSPORT 19

0.8

-3.0

0.6

2.0

E•0.4-

-1.0

0.2

0 1 0

0 0.2 0.4 0.6 0.8 1.0

DISTANCE (microns)

Figure 1.16 F-valley velocity along the center of the PBT channel [201.

tions the L-valley carriers make a negligible contribution to current. Thus, adecrease in carrier velocity results in a net increase in carrier concentration.However, initially, the decrease in field is not accompanied by a correspond-ing temperature decrease (as experienced in the uniform field calculations).Thus, the high V-valley temperature results in transfer to the L-valley,giving rise to the second minimum in the F-valley carrier density shown inFig. 1.14. Further toward the drain, the field decreases. However, relaxationis incomplete and the field at the collector is not equal to the field at theemitter. Also note that the moment equation potential distribution gives riseto a slightly higher field upstream of the base, and a lower field, over alonger distance, downstream of the base compared with the drift anddiffusion result. More significantly, the electron temperature at the collectorexceeds that at the emitter. It is noted that as the field relaxes, the electronstransfer back to the central valley.

1.5 NONEQUILIBRIUM ELECTRON-HOLE TRANSPORT

Additional nonequilibrium studies were mentioned at the beginning of thischapter. This concerns nonequilibrium electron-hole transport, which forspecificity was discussed for InGaAs. The details are considered below.


In this recent study, ensemble Monte Carlo studies were performed inwhich electrons were injected into p-type InGaAs [8]. In one case, theacceptor doping was 10' cm-3 , and in the second case 5 x 1017 cm-3 . Thecalculations were at 300 K and the ratio of the injected electrons to themajority holes was taken to be 1:5. (Note: The ensemble Monte Carloavoids any assumptions on the magnitudes of the energy and momentumexchange in an electron-hole [e-h] scattering process and the evolution ofthe electron and hole distribution functions.) The electrons and holes wereassumed to be in equilibrium with the lattice when the electric field wasswitched on, and the band model consisted of three nonparabolic valleys forthe conduction band and two parabolic light- and heavy-hole bands. Therole of the light holes was suppressed in this study. The model includes theelastic acoustic phonon, impurity scattering using Ridley's model, alloyscattering, deformation potential, and intervalley and intravalley phononscattering process. The e-h and screened carrier-phonon scattering arecalculated from the expressiens given in Osman and Ferry [22], using aself-consistent screening model. There is only one LO phonon mode in this

50

(b)40.

30

o WITHOUT e-h T =30 0 0 K U)z0 20 ,

* WITH e-h cr2-I 17 -3

(a) = 10 A P=SxIQcm

-ULi 17 -3E n =10 cm

00

•_o J

ZI _J</

>-> 501- P O -3 r

2 r n 2x016cm-3 Mm 40LLJ n= xI cs•

>8

O0 4 8 0 1 2 o030:

ELECTRIC FIELD (kv/cm)(20O

IO d,6n =2xlO0 cm"

0 2 4 6 8 10 12ELECTRIC FIELD (kv/cm)

Figure 1.17 (a) Electron drift velocity in p-type InGaAs, (b-c) percentage ofupper-valley electrons in p-type InGaAs [(; (Reprinted, with permission, fromApplied Physics Letters.)

_____ _____ _ _I

NONEQUILIBRIUM ELECTRON-HOLE TRANSPORT 21

calculation. The electron transport parameters for InD.53Ga0.47 As are thesame as those reported in Ahmed et al. [23]. However, for hole effectivemasses and deformation potential constants, appropriate to GaAs wereused. The interaction between the L-valley electrons and the heavy holeswas ignored.

The drift velocities of the electrons injected into the p-type InGaAs as afunction of the applied electric fields are plotted in Figs. 1.4 and 1.17a [8],for doping levels of 5 x 1017 and 1017 cm-3 respectively. The curves connect-ing the open circles in these figures correspond to situations in which theinteraction with the mobile holes is ignored and only the interaction with theionized acceptor impurities is taken into account.

When the interaction between the minority electrons and the mobilemajority holes is taken into account, the electron velocities are lower forfields below 4 kV/cm compared with majority electrons, as can be seen fromthese figures. At these low fields, the electron transfer to the upper valleys isnegligible, as shown in Fig. 1.17b [81, and the energy loss through e-hinteraction is not significant because the rate at which the electrons gainenergy from the electric field is small, as can be seen in Fig. 1.18 [8].However, the e-h scattering, which has the same angular distribution as theimpurity scattering, has the same effect on the electron mobility as doublingthe doping level. Consequently, the mobility of the electrons is reduced,leading to lower velocities. As the electric field is increased, the fraction ofelectrons with enough energy to transfer to the upper valleys increases.

300P = 10 17 CMa- 3

16 -3 "O - - -- --n 2 x10 cm. " .--

>200 ,// 6 kV/cm

>-

C,

ZUJ 100

S~2 kV/cm i

00 I 2 3 4 5

TIME (ps)

Figure 1.18 Electron energy in p-type InGaAs as a function of field,181.

THIS

PAGE

IS

MISSING

IN

ORIGINAL

DOCUMENT

TRANSPORT IN ULTRASUBMICRON DEVICES 23

For instance, from tiF. equation of motion of the density matrix, for asystem including mobile carriers and scattering centers, the first threemoment equations have the following form [271:

1 (0(( b0))) + 1- d-O ((()) (([H,, P(0)1)) (12)

1,md (dx) Ii00p())) + I_ d ((p(2))) ((p(O))) + (([H., P(o)])) (13)

m Ox \dx(/(2))+ 1 d a 3 d (P(OV ))+ i

M a•(((3X -- 2 •((f1 +x W (([H., p(2)])) (14)

where the (( ) denote the quantum ensemble averages, and using Diracnotation, the operators of interest are of the form

p(0) = 1x0 >< x0 (15)

P() = (½1)(PIxo x >< x0 + IX0 >< x0IP) (16)

P(2) = ( ½ )2(plxo >< x1 + 2PIxo >< xjP + Ixo >< x0oP 2) (17)

p(3) = (½)3(p 3Ix° >< Xo1 + 3Pjxo xo1p 2 + 3P 2Ixo >< xojP

+ x0 >< x 0IP 3) (18)

where P is the momentum operator. Note that the terms involving H,incorporate dissipation. In a diagonal representation, the ensemble averageof the first three operators breaks down into the following form:

(ep(0())= p•Pn1(x 0 ) = n(xo) (19)

((P(')))= p•,nimv, n(xo)mud (20)

((()) p2 _ U+h2 -

2

- Y p.n. a. In ni4 ~dx,

t2 2

lxnd pjn, - In n, (21).4 dx

where po is the diagonal element of the density matrix, and

-f` = E p,5njm(v, - Vd)2 (22)

It is clear that with the exception of the third operator, which contains aterm involving Planck's constant, the equations are of a classical form. Thus,in the simplest approximation, there appears to be a close similarity between


the classical moment equations and that obtained quantum mechanically.The difficulty is, of course, in solving these equations.

There is, howevcr, an interesting situation to consideýr: that in whichPit= 1/N the system. In this case, the first two moment equations, includingdiss-pation in momentum, reduce to

dn- + div(nv) = 0 (23)at

dnv + 2±+2nv ( dV ) n(24)8t dx dv + -x + -M24

where

Q = d x2 (25)

The quantity V represents an imposed barrier and the self-consistent energyassociated with Poisson's equation. The potential Q [24] is density-depen-dent and tends to become significant near strong barriers, where thecurvature of vI- will either enhance or diminish the imposed barrier.Tunneling and resonance arise from Q.

In multiple dimensions, these equations are subject to the constraint

I mv . dx = nh (26)

or in a gauge that includes a vector potential, the constraint

(mv + eA) dx =nh (27)C

yecz

n+ I /777-d• a" n+

Contact -AIGaAs ContactI ,'e -,' ,' _ _ ',D , ,GaAs

IA> IS> x=O x=L

Figure 1.20 Configuration ý, itable for the Aharonov-Bohm constraint [291. (Re-printed, with permission, from Applied Physicsv Letters.)

REFERENCES 25

Presently, device systems are being constructed which are influenced by theconstraint of Eq. (27), often called the Aharonov-Bohm condition [28]. Inparticular, structures are being constructed in which the path of an incidentbeam of electrons is split and then recombined. The split path lengths of theoriginal beam are different, and under coherent reconstruction in which Eq.(27) is satisfied, conduction oscillations are anticipated. A structure original-ly proposed to deal with this is displayed in Fig. 1.20 [29].

ACKNOWLEDGMLNTS

The author acknowledges numerous conversations with G. I. lafrate, J. P. Kreskov-sky, M. A. Osman, M. Me)yappan, and D. K. Ferry. Portions of this work weresupported by ONR, AFOSR, and ARO.

REFERENCES

1. J. B. Gunn, Solid State Commun. 1, 88 (1963).

2. J. G. Ruch- IEEE Trans. Electron Devices ED-19, 652 (1972).

3. A. A. Ketterson, W. T. Masselink, J. G. Gedymin, J. Klein, C.-K. Peng, W. F.Kopp, H. Mork9c, and K. R. Gleason, IEEE Trans. Electron Devices ED-33,564 (1986).

4. J. R. Cheilkowsky and M. L. Cohen, Phys. Rev. B: Solid State [14] 556 (1976).5. H. L. Grubin, Lecture Notes in "The Physics of Sub-Micron Semiconductor

Devices," (H. L. Grubin, D. K. Ferry, and C. Jacoboni, eds.). Plenum Press,1988.

6. V. L. Dalal, A. B. Dreeben, and A. Triano, J. Appl. Phys. 42, 2864 (1971).7. R. J. Degani, R. F. Leheny, R. E. Nahory, and J. P. Heritage, Appl. Phys. Lett.

39, 569 (1981).8. M. A. Osman and H. L. Grubin, Appl. Phys. Lett. 51, 1812 (1987)9. 0. Madelung and M. Schulz, eds., "Numerical Data and Functional Relation-

ships in Science and Technology/Landolt-Bomstein," Vo.. 22. Springer-VXr!ag,Berlin, 1987.

10. B. K. Ridley, "Quantum Processes in Semiconductors." Oxford Univ. Press(Clarendon), London and New York, 1982.

11. H. L. Grubin and J. P. Kreskovsky, Physica 134B + C, 67 (1985).12. H. L. Grubin, D. K. Ferry, and K. R. Gleason, Solid-State Electron. 23 157

(1980).13. P. N. Butcher, Rep. Prog. Phys. 30, 97 (1967).14. K. Yamaguchi, S. Asai, and H. Kodera, IEEE Trans Electron Devices ED-23,

1283 (1976).

15. M. Meyyappan, J. P. Kreskovsky, and H. L. Grubin, to be published.16. M. P. Shaw, P. R. Solomon, and H. L. Grubin, IBM J. Res. Dev: 13, 587

(1%9).


17. E. M. Azoff, Solid State Electron. 30, 913 (1987).18. R. K. Froelich, Avion. Lab. Rep. AFWAL-TR-82-1107, 1982.19. C. 0. Bozler and G. D. Alley, IEEE Trans. Electron Devices ED-27, 6 (1980).20. J. P. Kreskovsky, M. Meyyappan, and H. L. Grubin, in "Proceeding of NUMOS

I" (J. J. H. Miller, ed.). Boole Press, 1987.21. R. F. Cook and J. Frey, COMPEL 1, 2 (1982).22. M. A. Osman and D. K. Ferry, J. Appl. Phys. 61, 5330 (1987).23. S. R. Ahmed, B. R. Nag, and M. D. Roy, Solid-State Electron. 28, 1193 (1985). t

24. C. Philippidis, D. Bohm, and R. D. Kaye, Nuovo Cimento Soc. Ital. Fis. B [11]71B, 75 (1982).

25. G. J. lafrate, H. L. Grubin, and D. K. Ferry, J. Phys. (Orsay, Fr.) 42, C7-307(1981).

26. E. Wigner, Phys. Rev. 40, 749 (1932).27. H. L. Grubin, to be published.

28. Y. Aharonov and D. Bohm, Phys. Rev. 115, 485 (1959).29. S. Datta, M. R. Mellcoh, S. Bandyopadyay, and M. S. Lundstrom, Appl. Phys.

Lett. 48, 487 (1986).

II

I,

II •• •

THE ROLE OF LONGITUDINAL-OPTICALPHONONS IN NANOSCALE STRUCTURES

Michael A. Stroscio, Gerald J. IafrateU. S. Army Research Office

P. 0. Box 12211Research Triangle Park, North Carolina

K. W. Kim, M. A. LittlejohnDepartment of Electrical and Computer

EngineeringNorth Carolina State UniversityRaleigh, North Carolina 27695

Harold L. GrubinScientific Research Associates, Inc.

Glastonbury, Connecticut 06033

V. V. Mitin, R. MickeviciusDepartment of Electrical and Computer

EngineeringWayne State UniversityDetroit, Michigan 48202

As device dimensions in nanoscale structures and mesoscopic systemsare reduced, the characteristics and interactions of dimensionally-confinedlongitudinal-optical (LO) phonons deviate substantially from those of bulk polarsemiconductors. This account emphasizes LO-phonon effects arising in threeseparate systems: Short-period AlAs-GaAs superlattices; rectangular GaAsquantum wires embedded in AlAs; and metal-semiconductor interfaces such asthose in de Broglie wave quantum-interference devices.

I. INTRODUCTION

In nanoscale and mesoscopic systems, the effects of confinement oncarriers have been studied extensively. However, to properly model carrierenergy loss in nanoscale and mesoscopic systems, it is essential that calculationsof carrier scattering by longitudinal-optical phonons take into account the factthat confinement also changes the strength and spatial properties of longitudinal-

Nanostructures and 379 Copyright © 1992 by Academic Press. Inc.Mesoscopic All rights of reproduction in any form reserved.Systems ISBN 0-I1 2-409660-3

optical phonons. As a first example of phonon-confinement effects, we shallemphasize the dominant role of interface-optical phonons on electron scatteringin heterostructures with confinement lengths less than about 40 Angstroms.

Electron interactions with longitudinal-optical (LO) phonon modes inheterostructures are strongly affected by the changes in the Frohlich Hamiltoniancaused by phonon confinement and localization, as well as by the changes in theelectronic wave function due to the confining potential. The presence ofheterointerfaces produces large changes in the dielectric constant and gives riseto the confinement of optical phonons in each layer (i.e., confines modes) and thelocalization in the vicinity of interfaces (i.e., interface modes or surface-optical(SO) modes). These confined and interface modes arise because the abruptchanges in dielectric constant near heterointerfaces make it impossible to satisfythe normal bulk dispersion relation which normally restricts LO phonon frequen-cies to those where the dielectric constant vanishes. There have been suggestionsthat interactions by the interface modes can be significant and that the scatteringrate due to the confined modes can be considerably reduced in some structurescompared to the bulk LO-phonon scattering rate.l- 3 Therefore, an appropriatetreatment of the optical-phonon modes in quantized systems is essential forunderstanding electron transport in heterostructures. Recently, both macroscopicand microscopic approaches to electron-optical-phonon interactions inheterostructures 1-11 have been applied in theoretical treatments. Enhancedelectron-SO-phonon (i.e., electron-interface-phonon) scattering in polar semi-conductors with confining dimensions less than about 50 Angstroms has beenindicated recently 12,13 on the basis of scalar-potential modes. Physically, thisenhanced scattering rate is due to the interface phonon potential which increasesapproximately exponentially near heterointerfaces, where there is an abruptchange in the dielectric constant. The appropriateness of using a scalar potentialto model the electron-SO-phonon interaction has been questioned.14 Herein, wemodel the potential to calculate the ratio of electron-SO-phonon scattering toelectron-confined-LO-phonon scattering in a short-period GaAs/AlAssuperlattices. 15 The results provide a criterion which, in conjunction withexperiments, can be used to examine the validity of the scalar-potential model.

II. MODEL AND PROCEDURE

Figure 1 shows a schematic drawing of the GaAs/AlAs short-periodsuperlatfic% considered in this study. Each GaAs layer has width, a, and eachAlAs layer has width, b. We will consider the case where a uniform electroncurrent, J, is incident parallel to the superlattice heterojunctions. The ratio ofelectron-SO-phonon scattering to electron-LO-phonon scattering in the superlatticeof Fig. 1 is calculated through the Fermi golden rule by treating the electron-Lo-phonon and electron-SO-phonon interaction Hamiltonians 4,6 as perturbation

380

AlAs GaAs

b- a

1i2

FIG 1. Short-period GaAs/AlAs superlattice with a uniform electron curentparallel to the heterojunction interfaces.

Hamiltonians and by taking the electronic wavefunction in the z-direction as thatgiven by Cho and Prucnal for the maximum edge of the fi-st subband.16

To obtain an order-of-magnitude estimate of the ratio of electron-SO-phonon scattering to electron-LO-phonon scattering in theGaAs/AlAs superlatticeof Fig. 1, we take the ratio of the Fermi golden rule transition probability forelectron scattering from the symmetric AlAs-like interface mode to the corre-sponding transition rate for electron scattering from the lowest order confinedphonon mode.

In Fig. 2, this ratio is plotted as a function of the parallel phononwavevector times the quantum well width, qa, for a GaAs/AlAs short-periodsuperlattice with a=b. For qa < 7t, the ratio is larger than unity. A principal factorcontributing to the rapid increase for small values of aq (=X) is the exponentialincrease in the strength of the SO-phonon mode near the heterojunction. Inagreement with our results, a surge in the strength of the S+ mode with adecreasing well-width, a, is observed in a recent low-temperature experiment,17

which clearly demonstrates the dominance of the SO-phonon modes over theLO-phonon modes. These results are based on the implicit assumption that theuniform current, J, is used to prepare the electronic state given by Ref. 16. Thisassumption is reasonable if the electrons have incident energies significantly lessthan the GaAs/AlAs conduction band discontinuity and if the superlattice barrierwidths are roughly 50 Angstroms or less. Furthermore, the frequency of thesymmetric AlAs-like mode fora GaAs/AlAs heterojunction varies from about45

381

20-

U-

10

S..

0

.2

00

0 1 2 3 4 5

X (=q a)

FIG. 2. Approximate ratio of electron-SO-phonon (S+ mode) scattering to elec-tron-LO-phonon (m=1 confined mode) scattering.

meV for qa > 6 to 50 meV near qa=O and the frequency of the lowest-orderconfined mode is approximately 36.2 meV. Thus, the scaling illustrated in Fig.2 is expected to hold for a < 50 Angstroms and for incident electron energies inthe range from about 50 meV to very roughly 150 meV. In this energy range, thephonon momentum, q, will be approximately equal to the incident electr-"imomentum. Thus, we conclude that enhanced SO-phonon scattering will oM rin the regime of low-voltage (also, low-temperature in general) operationcharacteristic of mesoscale devices currently being considered by the electrondevice community. 18-20

As a second example, of the nature of Lo-phonon confinement effectsin polar semiconductor nanostructures, we consider Golden rule scattering ratesfor emission and absorption in rectangular GaAs quantum wires embedded inAlAs. In these calculations, the carriers are taken to be free in the x-directionand bound in the ground states of finitely deep quantum wells in the y- and z-directions. For a 40-Angstrom-by-40-Angstrom GaAs quantum wire of infinitelength in the x-direction, we find strong one-dimensional density-of-states effects

382

in the rates for surface-optical (SO) phonon emission. The emission peaks for theSO-phonons, also known as interface-optical phonons, occur near the AlAs-likeand GaAs-like LO-phonon frequencies. The confined and interface phononmodes used in obtaining these results are those considered previously by theauthors.6 ,9,1 2.2 1 Fig. 3 depicts the Golden rule scattering rates for the quantumwire. The strong SO-phonon emission rates near the AlAs-like and GaAs-likeLO-phonon frequencies are more than an order of magnitude larger than thecorresponding bulk LO phonon emission rates.9,2' However, if the lateraldimensions of the quantum wire are varied sinusoidally in amplitude by only tenpercent, the narrow one-dimensional density-of-states emission peaks haveaveraged values close to those of bulk carrier-LO-phonon interaction rates. 21

In this case, we learn that carrier-LO-phonon scattering rates may deviate

1015 d.4QA

80, f (a0e) • .

1013

! /so (,4,)LO (abs)

I SO(0be)

1 12Ip I I

0 20 40 so so 100

Electron Energy E. (meV)

FIG.3. Scattering rates at 300 K as a function of electron energy in thelongitudinal direction. The size of quantum wire is 40 Angstroms by 40 Angstroms.The SO-phonon scattering rate is denoted as SO, f forthe free-standing quantumwire. and as SO for the GaAs/AlAs quantum wire, respectively, while LOrepresents the rate of electron scattering with confined LO-phonon modes.Phonon emission (absorption) process is represented as emi (abs), respectively.

383

substantially from the corresponding wires; however, in the more realistic caseof variable quantum wire dimensions, one-dimensional effects are suppressed.

As a third and final example of how optical phonon properties innanoscale structures may deviate substantially from those of bulk opticalphonons, we consider a metal-semiconductor interface for the case where thesemiconductor thickness is restricted to nanoscale dimensions. In this analysis,we find that the nature of the interface phonon modes at the metal-semiconductorinterface are modified substantially for two reasons: (1) at the metal-semicon-ductor interface, only those phonon modes with Frohlich potentials whichapproach zero at the interface are allowed; and (2) the magnitude of the Frohlichpotential for the surviving interface phonon mode decreases exponentiallv withthe product of the phonon wavevector and thickness of the semiconductor slab.As demonstrated previously, 4.6 the components of the optical phonon polariza-tion vectors for dimensionally-confined modes have opposite parities for polar-ization components normal to the semiconductor interface and for polarizationcomponents parallel to the interface. Since the average tangential componentsof the optical-phonon electric field must vanish at the metal interface and sincethe phonon polarization vector is proportional to the phonon electric field, itfollows that only optical phonon modes with even polarization vectors normal tothe metal-semiconductor interface will survive. Furthermore, since the laplacianof the Frohlich potential for a given phonon mode is proportional to thedivergence of the polarization vector for that mode, it follows that only opticalphonon modes with odd Frohlich potentials satisfy the metal-semiconductorinterface. In particular, of the well-known symmetric and antisymmetricinterface modes,2 1 only the antisymmetric mode will survive. In the limit of nodispersion, the antisymmetric interface mode has a Frohlich potential whichscales as one divided by the square root of cosh(qd)*sinh(qd) where q is themagnitude of the phonon wavevector parallel to the metal-semiconductorinterface and d is the thickness of the semiconductor slab. For small values of qd,the Frohlich potential of the antisymmetric interface mode then decreasesexponentially with qd. Hence, we conclude that the carrier-optical-phononscattering rate decreases to zero at the metal-semiconductor interface and that thestrength of the Frohlich potential for the antisymmetric interface mode in thesemiconductor slab decreases exponentially with qd. (It is important to note thatthese results depend upon having a perfect metal-semiconductor interface sinceeven a thin oxide at the metal-semiconductor interface is sufficient to isolate thesemiconductor optical-phonon modes from the metal.) The impact of theseobservations on the physics of nanoscale structures such as the de Broglie waveinterference device is clear: carrier-optical-phonon scattering will be reducednear metal-semiconductor interfaces. Additionally, Schottky barrier heights atmetal-semiconductor interfaces in nanoscale structures may deviate from thoseof metal-semiconductor structures with semi-infinite metal and semiconductor

384

regions; the extent of this deviation will depend on the relative importance ofphonon scattering and other factors such as interface states which determine theequilibrium Schottky barrier height.

MI. CONCLUSION

In conclusion, the characteristics and interactions of dimensionally-confined optical phonon modes deviate substantially from those of bulk polarsemiconductors. In particular, theory and experiment point to greatly enhancedcarrier-LO-phonon scattering rates when dimensional constraints optimizeinterface mode strengths. For thecase of enhanced carrier-LO-phonon emissionrates in recta'gular quantum wires, our analysis indicates the one-dimensionaldensity-of-states effects may be greatly reduced as a result of minor fluctuationsin the laterial dimensions of the quantum wire. Finally, we conclude thatnecessary restrictions in phonon symmetries at metal-semiconductor interfaceswill modify carrier-LO-phonon scattering rates in nanostructures where chargetransport occurs in thin polar semiconductor regions in intimate contact withmetals.

ACKNOWLEDGEMENT

The authors acknowledge invaluable assistance from Ruby Dement forthe preparation of the manuscript.

Thanks are also due to Dr. J. W. Mink, Professor T. Ando, and ProfessorN. Mori for helpful discussions. Special thanks goes to Professor K. T. Tsen whomade his preliminary experimental data available to the authors. This work is,in part, supported by the Office of Naval Research under grant N00014-90-J- 1835and the U. S. Army Research Office under grant DAAL03-89-D-0003-05.

REFERENCES

I. R. Lassnig, Phys. Rev. B 30,7132 (1984).2. F. A. Riddoch and B. K. Ridley, Physica B+C 134B, 342 (1985).3. N. Sawaki, T. Phys. C' Solid SLat, r! iys. Y,4955 (13103).4. J. J. Licari and R. Evrard, Phys. Rev. B 15, 2254 (1977).5. L. Wendler, Phys. Status Solidi B 129, 513 (1985).6. N. Mori and T. Ando, Phys. Rev. B 40,6175 (1989).7. R. Chen, D. L. Lin, and T. F. George, Phys. Rev. B 41, 1435 (1990).8. B. K. Ridley, Phys. Rev. B 39,5282 (1989).9. M. A. Stroscio, Phys. Rcv. B 40,6428 (1989).10. K. Huang and B. Zhu, Phys. Rev. B 38, 13377 (1988).11. S. Rudin and T. L. Reinecke, Phys. Rev. B 41, 7713 (1990).

385

12. M. S. Stroscio, K. W. Kim, M. A. Littlejohn, and H. Chuang, Phys. Rev. B42, 1488 (1990); K. W. Kim, M. A. Stroscio, A. B hatt, R. Mickevicius and

V. Mitin, J. Appl. Phys. (to be published).13. S. Teitsworth and P. Turley, private communication.14. B. K. Ridley and M. Babiker, private communication.15. M. A. Stroscio, Gerald J. lafrate, K. W. Kim, M. A. Littlejohn, H. Goronkin,

and G. N. Maracas, Appl. Phys. Lett., in press (1991).16. H.-S. Cho and P. R. Prucnal, Phys. Rev. B 36,3237 (1987).17. K. T. Tsen and H. Morkoc, Private communication.18. K. Sols, M. Macucci, U. Ravaioli, and K. Hess, Appl. Phys. Lett. 54, 350

(1989); F. Sols and M. Macucci, Phys. Rev. B 41 11887 (1990).19. H. Sakaki, Jpn. 1. Appl. Phys. 28, L314 (1989).20. G. J. lafrate, in GalliumArsenide Technology, edited by D. K. Ferry (Sams,

Indianapolis, 1985), Chapter 12.21. K. W. Kim and M. A. Stroscio, J. Appl. Phys. 68,6289 (1990).

386

PHILOSOPHICAL MAGAZINE LETrERS, 1992, VOL. 65, No. 4, 173-176

Dramatic reduction in the longitudinal-optical phonon emission ratein polar-semiconductor quantum wires

By MICHAEL A. STROSciOt, K. W. KIM$, GERALD J. IAFRATEt,

MrrRA DurrA§ and HAROLD L. GRUBINII

t U.S. Army Research Office, P.O. Box 12211,Research Triangle Park, North Carolina 27709, U.S.A.I Department of Electrical and Computer Engineering,

North Carolina State University, Raleigh, North Carolina 27695, U.S.A.§ U.S. Army Electronics Technology and Devices Laboratory,

Ft Monmouth, New Jersey 07703, U.S.A.-II Scientific Research Associates, Inc., 50 Nye Road,

Glastonbu - Connecticut 06033, U.S.A.

[Received 8 November 1991 and accepted 17 December 1991]

ABSTRACT

Novel quantum-effect polar-semiconductor structures underlie technologiesportending dramatic enhancements in the capability to process information ordersof magnitude faster than is possible currently. In many embodiments of thesequantum-effect structures, charges are transported in quasi-one-dimensionalquantum wires which must support the transport of charges at high mobilities.However, it has recently been demonstrated that the longitudinal-optical (LO)phonons established at quantum-wire interfaces lead to dramatic enhancements incarrier-phonon interactions and concomitant degradation in carrier mobility. Thisletter demonstrates that phonon modes may be tailored through thejudicious use ofmetal-semiconductor interfaces in such a way as to dramatically reduce unwantedemission of interface LO phonons and, consequently, to lead to the achievement ofhigh quantum-wire mobility.

Modern fabrication techniques for making semiconductor structures withnanometre-scale dimensional features have been essential in leading to fundamentaldiscoveries (Esaki 1974, Stein, von Klitzing and Weimann 1983) as well as instimulating concepts for future information-processing systems based on the exploit-ation of quantum effects occurring in such structures (Capasso and Datta 1990). Asoriginally proposed by Sakaki (1980), the predicted high mobilities of quasi-one-dimensional wire-like regions of a semiconducting material underlie many proposedquantum-wire system concepts (Luryi and Capasso 1985, Sakaki 1989) and haveresulted in pioneering efforts leading to the actual fabrication of quantum wires (Watt,Sotomayor Torres, Arnot and Beaumont 1990). Furthermore, quantum-wire struc-tures have been essential to seminal studies underlying quantum-wire laser concepts(Tsuchiya et al. 1989) and quantum-coupled electronic systems (Reed et al. 1988).Recently, however, theoretical studies of the interaction between longitudinal-optical(LO) phonons and carriers in a polar-semiconductor quantum wire (Stroscio 1989)have revealed the presence of discrete LO phonon modes similar to those identified forpolar-semiconductor quantum wells (Kliewer and Fuchs 1966, Licari and Evrard 1977,Mori and Ando 1989, Kim and Stroscio 1990). As for the case of quantum wells,interface LO phonons are established at the semiconductor-semiconductor bound-aries of quantum wires (Kim et al. 1991). In addition, it has been shown that, for carrier

0950-0839/92 S3-00 V 1992 Taylor & Francis Ltd.

174 M. A. Stroscio et a!.

energies in excess of the interface LO-phonon energy, the inelastic scattering caused bycarrier-interface-phonon interactions dominates over other scattering mechanismswhen confinement occurs on a scale for about 40 A or less (Stroscio et al. 1991). In thisletter, it is demonstrated that establishing a metal-semiconductor interface at thelateral boundaries of polar-semiconductor quantum wires introduces a set of boundaryconditions that dramatically reduces or eliminates unwanted carrier energy loss causedby the interactions with interface LO-phonon modes.

Motivated by the recently demonstrated technology for the epitaxial growth ofmetals in intimate contact with polar semiconductors (Harbison et al. 1988, 1989,Guivarc'h et al. 1989), the dielectric continuum model of interface-phonon modes hasbeen applied to determine the carrier-interface-phonon interaction Hamiltonian neara semi-infinite metal-polar-semiconductor interface (Stroscio et al. 1992). Detailedmicroscopic calculations of interface modes in polar semiconductors (Rilcher, Molinariand Lugli 1991) indicate that the dielectric continuum model provides an accurateformalism for modelling carrier-interface-phonon interactions. As summarized byMori and Ando (1989), the components of the optical polarization vectors for confinedand interface phonons have opposite parities for polarization components normal tothe semiconductor interface and lu, polarization components parallel to the interface.Since the average tangential components of the optical-phonon electric field mustvanish at the metal interface (it may survive at a semiconductor-semiconductorinterface) and since the phonon polarization vector is proportional to the phononelectric field, it follows that only optical-phonon modes with even polarization vectorsnormal to the metal-semiconductor interface will survive. Furthermore, since theLaplacian of the Fr6hlich potential for a given phonon mode is proportional to thedivergence of the polarization vector for that mode, it follows that only optical-phononmodes with odd Fr6hlich potentials satisfy the metal-semiconductor interfaceconditions. Hence, only those confined and interface modes of odd potential satisfy thecorrect boundary conditions at the metal-semiconductor interface. In particular, of thewell known symmetric and antisymmetric interface modes, only the antisymmetricmodes will survive; then straightforwardly from the formulation of Kim and Stroscio(1990), it follows that

Sy( hexp 2 L-2 + 1/2

q~~~~~~ ~ KO000[IC~ct~d2:l2((.O)]"

1 . sinh (qz) 0<z<d

x (2q)112 exp (iq p) (aq + at_, ) sinh (qd/2)" 2

where HA denotes the antisymmetric electron-interface-phonon interaction Hamil-tonian in a polar-semiconductor slab of dielectric function KI(CO), bounded for z<Owith a semi-infinite metal and for z>d/2 with a semi-infinite region of dielectricfunction K2(w). In this result, q is the component of the interface-phonon wave-vectorparallel to the semiconductor heterojunctions, p is the pair of spatial coordinatesparallel to the planes of the semiconductor heterojunctions, z is the coordinateorthogonal to p, L is the normalization length, aq and at q are the usual phononannihilation and creation operators, ard eO is the permittivity of vacuum. In thedispersion regions where q does not 1 _pend strongly on a, HA scales with qd/2 as[cosh (qd/2) sinh (qd/2)] -̀ '2 or equivalently as 2exp(-qd/2); thus, in the dispersionlessregion (away from q=0), the strength of the electron-interface-phonon interactiondecreases exponentially as d/2 becomes large. The impact of these observations on the

Longitudinal-optical phonon emission rate 175

physics of nanostructures is clear: carrier--optical-phonon scattering will be reducednear metal-semiconductor interfaces.

For rectangular polar-semiconductor quantum wires embedded in a metal, the onlyconfined and interface-LO-phonon modes satisfying the metal-semiconductor bound-ary conditions are the modes with odd potentials at all boundaries of the quantum wire.For the case of confined modes, the electron-LO-phonon Hamiltonian is precisely thatgiven previously by Stroscio (1989) since all the modes derived earlier are odd at thequantum-wire interfaces; thus all results of Stroscio (1989) hold for the case of arectangular polar-semiconductor quantum wire embedded in a metal. For the case ofinterface modes, there are no allowed solutions since it is impossible to satisfy themetal-semiconductor boundary conditions simultaneously at two of the quantum-wire metal-semiconductor interfaces. While the demonstration holds rigorously onlyfor quantum wires with rectangular cross-sections, it is clear that the same conclusionwill be substantially correct for a quantum wire of arbitrary cross-section since theinterface modes with long-range Coulomb character must be reduced severely byinvoking metal-semiconductor boundary conditions at two opposing locations on thequantum-wire cross-section.

The conclusion that carrier-interface-phonon interactions will be absent inquantum wires surrounded by metal is critical in maintaining high-mobility transportin quasi-one-dimensional quantum-wire structures. Indeed, the recent theoretical andexperimental findings (Tsen et al. 1991) indicate that the strength of the carrier-interface-phonon scales approximately inversely with the thickness of the confiningregion. However, it is precisely the regime of the smallest dimensional scales which hasportended the greatest advances in the novel application and discovery of quantumeffects in semiconductor structures. As demonstrated in this letter, this highlyunsatisfactory situation for polar-semiconductor quantum wires can be circumventedby eliminating interface-optical-phonon modes through the judicious use of metal-semiconductor interfaces.

ACKNOWLEDGMENTSThe authors would like to thank Professor M. A. Littlejohn and Dr J. Mink for

helpful discussions. This work is, in part, supported by the Office of Naval Researchunder Grant No. N00014-90-J-1835 and the Army Research Office under Grant No.DAAL03-89-D-0003-05.

REFERENCESCAPASso, F., and DATAr, S., 1990, Phys. Today, 74, 75.ESAKI, L., 1974, Rev. mod. Phys., 46, 237.GUIVARc'H, A., CAULET, J., GUENAis, B., BALLINI, Y., GtERIN, R., PouDouLEC, A., and REGRENY,

A., 1989, J. Cryst. Growth, 95, 427.HARBISON, J. P., SANDS, T., TABATABAIE, N., CHAN, W. K., FLOREZ, L. T., and KERAMIDAS, V. G.,

1988, Appl. Phys. Lett., 53, 1717; 1989, J. Cryst. Growth, 95, 425.KIM, K. W., and STRosCIo, M. A., 1990, J. app!. Phys., 68, 6289.KIM, K. W., and STRoscio, M. A., BHATT, A., Mici, Evicius, R., and MITIN, V. V, 1991, J. appl.

Phys., 70, 319.KLIEWER, K. L., and FucHs, R., 1966, Phys. Rev., 150, 573.LICARI, 1. J., and EVRARD, R., 1977, Phys. Rev. B, 15, 2254.LURYI, S., and CAPASSO, F., 1985, AppL. Phys. Lett., 47, 1347.MORI, N., and ANDO, T., 1989, Phys. Rev. B, 40, 6175.REED, M. A., RANDALL, I. N., AGGRAWAL, R. J., MATYi, R. J., MOORE, T. M., and WETSEL, A. E.,

1988, Phys. Rev. Lett., 60, 535.

176 Longitudinal-optical phonon emission rate

ROcum.,, H., MOLINAkI, E., and LUGLI, P., 1991, Phys. Rev. B, 44, 3463.SAKAKj, H., 1980, Jap. J. appl. Phys., 19, L735; 1989, Jap. J. appl. Phys., 28, L314.STEIN, D., VON KLITzING, K., and WEIMANN, G., 1983, Phys. Rev. Lett., 51, 130.STROscIo, M. A., 1989, Phys. Rev. B, 40, 6428.STRoscio, M. A., IAFRATE, G. J., KIM, K. W., LITTLEJOHN, M. A., GORONKIN, H., and

MARACAS, G. N, 1991, Appl. Phys. Lett., 59, 1093.STROscto, M. A., IAFRATE, G. J., Kim, K. W., LiTrLEJOr•N, M. A., GRUBIN, H. L, MITIN, V. V., and

M cKEicius, R., 1992, Nanostructure and Mesoscopic Systems, edited by M. A. Reed andP. Kirk (Boston: Academic Press) (to be published).

TSEN, K. T., SMITH, D. S., TSEN, S.-C. Y., KtaAR, N. S., and MORKOC, H., 1991, J. appl. Phys., 70,418.

TSUCHIYA, M, GAINES, J. M., YAN, R. H., SrMES, R. J., HOLTZ, P. O., COLDREN, L. A., and PEMoFI,P. M., 1989, Phys. Rev. Lett, 62, 466.

WATt', M., SOTOMAYOR TORRES, C. M., ARNOT, H. E. G, and BEAUMONT, S. P., 1990, Semicond.Sci. Technol., 5, 285.

FIOM Kluwer Academic Publishers(1991)COMPUTATIONAL ELECTRONICSSemiconductor Transport andDevice Simulation

A NEW NONPARABOLIC HYDRODYNAMIC

MODEL WITH QUANTUM CORRECTIONS

D. L. Woolard(). M. A. Stroscio(b) M. A. Littlejohn('.b).

R. J. Trew(') and H. L. Grubin()

()Electrical and Computer Engineering DepartmentNorth Carolina State University

Raleigh. North Carolina 27695-7911

(O)U.S. Army Research OfficeResearch Triangle Park. North Carolina 27709

(')Scientific Pesearch Associates Inc.

Glastonbury. Connecticut. 06033-6058

Abstract -This paper presents a new hydrodynamic traosport model with non-parabolic conduction bands and quantum correction trmu. For the first timesolutions for the full quantum balance equations, applied to an ultrasmall .1ec-tron device, -re presented.

1. Introduction

Requirements for faster electronics have piodiced smaller devices influenced byquantum interference effects which can not be modeled by dassical theory. Thesedevices operate under nonequilibrium conditions where the average carrier energyreaches many times its equilibrium value. To address these problems several ap-proaches [1) have been considered. This paper describes a preliminary investigationinto solutions for electron density, average velocity and average energy for an ultra-small electronic structure under the conditions of nccdassical electron transport.using an approach previously described as quantum hydrodynamics (2).

2. The Nonparabolic Model

First. we present a brief summary of z unique form of the hydrodynamic transportModel applicable to nonparabolic conduction bands. A compl'te description of thismodel will be formalized elsewhere [31. Our hydrodynamic model was developedby studying moments of 'the Boltzmann transport eruation i41. This process was

achieved using the moment operators 'i = 1.- 1 = u(k) = k and *3;;(-F'") d s

im(k)u(k) - u(k) = E,(k). Here. the Kane dispersion relation. j'k = E,(1 +

oEi). has been used to define u(k) and m(k) = m 1 + ,,k'.. These moment

operators were chosen because they lead to a form which can be manipulatedmore easily and one in which simplifying approximations can be seen more dearly.Performing the moment process and simplying to first order yields immediately the

60

results: On = -V, -(nv) (1)at

Dy F 1-- = -v- V,v + " V, - [Pvj (2)

aw 1-- -- V, w+-F.v- -V, -(v -[Pw]+q) (3)

t• n

where m**. [Pv]. [Pw] and q have integral definitions [3] which depend on f.At this point additional assumptions or relations are necessary to dose the mathe-matical system . A familiar approach is to use a displaced Maxwellin distribution

function [7). However, to resolve the dilemma of two distinctly different effec-tive pressure tensors PT and Pw. (for parabolic bands and displaced Maxwellian.[PvJ = [Pw] = nkBT[I), we use stationary Monte Carlo calculations and physicalintuition to suggest the following distribution function

[n.(M)] ,-.W(TU)I-u- V1,f= eXP 2kBT. (4)

as a constitutive relation to dose the moment equations. Here T. has been chosento replace T because 1kgT., approximates the effective thermal energy well for thestationary transport case, and ,n,(T,) has been introduced because we expect anonconstant effective mass strongly dependent on T.. Using Equation (4) in thedefinitions for the transport parameters, and limiting the analysis to first and second

orders in T.. we arrive at the the supplemental relations; w = IkT.-+!-!-- )-!V.r.

,t--(T.) = ,n'(1 + 3akBT.), [Py] = .nkBT.[I]. [Pw] = -koT.[J[ and= 5a(-.) 2(kpT.) 2nui. These relations can be used to dose Equations (1) -

(3) expressing them in terms of electron density n, average velocity v and averageenergy w.

3. The Quantum Correction Terms

Since we desire a general model suitable for studying ultrasmall devices, an ap-proach is illustrated to develop some quantum corrections for our semi-classicalnonparaboric model equations. Grubin and Kveskovsky [2] have previously pre-sented a detailed derivation which will achieve ths goal. However. for our initialinvestigations we use a slightly more restricted version of their quantum hydrody-namic equations. Specifically. we neglect any spatial or nonparabolic effects on theelectron effective mass and ignore any deficiency of the general quantum distributionfunction to agree with Fermi statistics (low temperature effects). We begin withthe general moment equations of Stroscio [S]. The collisionless one-dimensionalforms of the first three moments are:

On _ 1• 0(ip S(5)Di mo ax

0 p4 _ 0 J(p),Uj l 1 0a+ 1-(a-2,U" j) - - (n((p- p)4)) (6)

61

_(p p)) 1 0 ) Pd 8

a - ' 1 (.((p _ p.)3)) _ p a (n((p - p) 2 ))

2 0 Ppd P+d((P - Pd)2 ) 8nM* (n((P -p))))•-+- (7)- F 8 •nm* 8z

where n is the electro n density, p is the single electron momentum, pd is theclassical average momentum and Ue/ is the total effective electric potential. If wefollow Ref. 2 and use the momentum displaced nonequilibrium Wigner distributionfunction.

(I + RI -.. (P - P4 2 a ( ) (8)

of Ancona and lafrate 161. we obtain n((p - pd)') = -(1 - I'll OfI ) 1. ) and

-((p-pd)3) = 0 where a = •./ = ji and -y . These results can

be used with Equations (5) - (7) to yield the collisionless quantum hydrodynamicequations:

8n . (9)S8z

Dv 8w 18 Q 1 a(-kar) (10)-=. (U.1 1 + T -

OW OW a___ Q 1 8(nhkBT) + L 12&r ()

3 n De 12WmenO z Be~

where W 1 fkT '+ lm'w - V" and Q = Compar-ing the above quantum equations to the semiclassical nonparabolic hydrodynamicEquations (1) - (3), approximating T. by T and using the supplemental relationsof Section 2 we arrive at the quantum corrected collisionless nonparabolic hydro-dynamic transport model in one dimension,

8n -- a- () (12)

8V Di IlB Q 208m n 2 ]

81 -v 7 - -Mo;-8z(U-1 1 + - - fw W- w - (13)

aw a 8Q, 8 _ Q. 2(8 .•- = -•-u- (U.-, + )- 5-,,In(w+,)(w-w,.- - '1-2w.,. (14)

81 B a B 3n 2a 8" 2with nonparabolic correction terms /(n, v, wo) = 1 + 2aw(w, ma)(w - w.- iV 2 ).

P(n, v, w) = w(w, wo) = +A19 p (n, v, wo) =l(fz1)2(w, - w.-__ .__ -•:3

211v') and quantum corrections w,, A °a1 ) and Q = -2L

Q is the standard form of the quantum potential and w. is a purely quantummechanical contribution as first noted by Wigner.

4. Results and Conclusions

An initial study has been made of a double-barrier heterostructure. A1..Ga1 _.,A&barriers with zc = 0.3, doped to 10"4cm- 3 with source and drain regions of length

62

200 A and doped to 10 16cm- 3 . This structure is shown in Figure 2 with the sourceand drain regions excluded. In the self-consistent simulations, the heterostructurewas treated using the Anderson rule; Ue11 = -q'Pqjf -x(z), where x(x) (AE, =0.697z,) is a position dependent electron affinity and the applied potential *1,ifwas determined from solving Poisson's equation. The dissipation mechanisms weretreated in a classical manner (7] and we assumed Tr, = 300K. The profile resultof w. and w., parameters intrinsic to the energy balance equation, are displayedin Figure 1 for *.pi•¢d = 8mV. Figure 2 shows the corresponding profile resultfor Q along with the conduction band E, (E, = U.1 1 ). This study is the firstto use the three quantum hydrodynamic (QHD) equations to study transport inheterostructure devices. The key additional feature of the QHD equations is theincorporation of the effects of density gradients through the quantum potential.Insofar as d classical solution does not exist in the presence of barriers. Q must atleast cancel the effects of the barriers and permit carrier transport. Indeed 2 doesapproximately balance the barrier potential a result consistent with [21.

i wq

2A~ 4W4

-10.0 ! ! I I I

X A (X 102)Figure 1. Average energy (w) and quantum correction (wt) at F.Wj.,d= 8MV.

S-2.0.

X ()(X 102)Figure 2. Conduction band (E.) and quantum potential ()at =BmV.

1. Hot Carriers in Semiconductors. in Solid-State Electronisc 32. 12, 1989.2. H.L. Grubin and J.P. Kreskovsky Solid-State Electronics 32. 12. p. 1071. 1989.3. D.L. Woolard. M.A. Littlejohn and R.J. Trew to be published4. E.M. Azoff IEEE Tronu. Electrn Devices. 36. 4. p. 609. 1989.5. M.A. Strosio, Supermlattices and Microstructures. 2. 1, p. 83. 1986.6. M.G. Ancona and G.J. lafrat, Physical Review B, 39, 10. p. 9536. 1989.7. D.L. Woolard et. al. Solid-State Electronics 32. 12, p. 1347, 1989.

FROM Kluwer Academic Publishers(1991;)COMPUTATIONAL ELECTRONICSSemiconductor Transport andDevice Simulation

CO ARATV NUMERICAL SIMULATIONS OFA GaAsSUBMICRON FET USING THE MOMENTS OF THE BOLTZMANN

TRANSPORT AND MONTE CARLO METHODS

I. P. KreskovkY. Q. A. Andrews, B. L. Morrison and H. L. GrubinScientific Research Associates, Inc

P.O. Bo1 1058Glastonbury, Connecticut 06033

Abstract

Numerical simulations of a submicron GaAs FMr have been performed usingboth Monte Carlo (MC) methods and the Momen of the Belaun Tnnsport

quation (MBTE). The I-V characteristics as well as details of the Internaldisuttion of carriers and potential were obtained. The MC calculations showno regions or negative forward conductance,. However, the MBTE results showthat negative forward conductance can be present or absent depending on the,value of the thermal conductivity.

introduction

Recentl, both MC and M1TE or r3d m approa to dch ec siulaton

o eoe pebut more fundamental methods metioned

While attention has fomed on the Iplemntati of MC and MBTB simulation

hav notbeen made. Thei s presetTs th reauls of t tvesia.

Whil th resltsare reh An T! i natre the dmon provide si. WMti

direction for future work.

The device conidered is a simple GaAs F•ETwith a 0.6 m•ro source-draincowgand a id e i aflf d 02 m o lin The device depth s taken as

to1 t/he use a t4mmd of t' .1lUnilia eta itbd -.- ý

T he employed fufows dseltha outlnaed _by oc -y andWhile attnion Thas focuseparnthe momentum and poMsae advn t In& anderJ t in com isofan olied field aeung a non-parabol band

structure. At to end of the fre fgh , the partes ae scaereL

Ahle thatie aeulsp carge drliisaryibuin isaomptedfro theydprve disributionduseciong a o fuou r-el aloritmadkis.seutoni ovdudtn h

potential. This process is repeated unti attsicalsteadysteisa eved. TheMC procedure considers tasfer the between r. X and Lvalleys.

The METE aproc solves continuity, momentumand e ner equations for rand Lvalley electrons. Thesg equations are expressed as

(1) *n/t - - an 1 V/x - n 1 /r 1 +n2/a

142

(2) anlVl /at - - anlV1Vl/ax - I/r anjTlko/aX +

nle/m, al/ax + p ,2v/ax 2 - nlv/T3

(3) an 1 Tl/at - - 8nlVlTI/aX - 2/3 nIT1 aV/lx +

2/3KO 8/ax (xcaTl/ax) - nlTl/i 5 + n 2 T2 /r 6 +

=IVl2/3ko (n 2 /r 2 - nl/rl) + 2nlrlVl 2/3k 0 T3

forr (subscript ,- 1)iley carriers Asimilar system withbscipt 2 applies tothe Lv eyarrers. The equ"ions are expressed here i- on•e- a. form.The extension the two-dimenioriul cse coidered here is straight forward. InEq. (2), the on...tu _ eqution, t coefien of h fouh term on the RHS,

sibution fncton.n general, we assume this *ent to be zero or y

smalLI o-eoadsgiiattesoeo h - hrceitc at low fedwill be snificantly in enr and below that anticipated from the low field mobilityof the maeialin 4ustion. e coeffcient• , apearing in the third term on the

RHS f th enegr quation, is the thermal condtityothelcrngsWhile its precise value is unknown, it is important and its role in the beharvior ofthe solu iosisinvestigated herm If itIs ero, there wilbeso heat transfer byconduction while If It approaches infinity the energy equation reduces to

(4) V'T - 0

Thec quantities indicated byri E~qs. (1-3) are relaxaton times determined froma por valuation, of theollso i ane are taken as energdepen�den the present study [2F.

Equatios (1.a) and the L•alli counter parts ae to Polssone equationthe r -smgystem ui soved using U -•n pmbsed on the eIM

Lineazed xc, t and u 4forsovingthe drift and

paralleling r e dse -tk em rains a0d

Fiffsioe 2 tions the a-V gori fthmI glyeficvien ad asig dee grledrmte tof -

discussion its application to the present system may be found in [5].

Results

Simulations of the FErishownin O& 1 weu e performed for ate bias levels of 0,0.3 and 0.5 volts with the drain bias varyng from 0 to 2.0 volts Both the MC andthe MBE siultIds used an lx6 c grid meoi t sh square cells with

sandlt tV =(an o•,alvegeof pereceMl) wereused h thenncalaatio reanre approimautely 2.5Minute Of Cray XW P U time prbias point. The MC simnulailons re~qured

aprxnaatel6 mintes bias Results forMCsimulamtio using 900parices hoe no sin~t tcrces but reurdsgiiatygreateraCu

Figure 2 shows the IN characteristics of the device as determined from the twos lation approach At a gte bias of 0.1 volts it is observed that both the MCand Ilh rxesults show goo agreement. Atthis bins level, the r valleZcrondo not become too hot (mxinu loa tmertreo 12 n

tas ia maximumao0 polmael Z* 2* valley and 38*9Lale. As thebias iincrased volt.3 we see that the agreeAment is good up to

am &Ox --Imtel-y 0.6 volts Y&, Above this value of danbias the hMET result:hibbitsnegaive forward conductance NO such effect is observed for the MCresult. t d - 2.0 volts, the maximu r valley electron temperature from thee

Wreac=e= 2400* K and the population of carriers in the r and Lvalleys is 2D% and 80*6 respectively at the drain-. the point of maximum transfer.

143

Thetred f wde diapeement between MC and MUME pedictions is see to

OWasthe gatebias ismcrased. AtV = 0.5 volts we obscrve that negativecoducacappears in the MBTh si aon at about V - 0.5 volts. H• ther valley temperatures reaches a madmum of 2M" K ancf t the drain thepopulation of carriers in the r valley is only 10% of the total. At Vds - L1 volts,the maximum r vally temperature is only 18K0"., and the r vallcy population atthe drain is 2596. It should be noted the O on of carrier in X t Valley fromthe MC results was negliAle at all bias Levls.

The reason for the negative conductance is straight forward. Once the bias levelsbecome high enough that sufficient energy is supplied to the carriers to causethem to transfer, the possibility exists that the carriers will transfer at a rate whichis greater than the r valy velocity increases Since the current is predomnantlythrough r valley transport, if this occurs the current will decrease. Tisiprecisce what is hapyenig here and further suggests two possible sources of thedis~epncy. Frst, the scatteng rate constants. r•, which are functions of theelectron temperature, could have an inappropriate high tmeauevrainHowever, under uniform field conditions the rests obtained firn both MC and

MBTEsimuatins yeldreasonable velocity field curves which are in generalagreement wit exjerimental data. This suggests that the problem lies elsewhere.and is associated with spatial nonuniformities in the so on. Ekminatimon of the

cotnut an moen0aitum ecluations does not reveal any osbesucso hS(As discussed earfier the coefficient of the o term,. affect

the low portion of the I-V curve, but has little effect at high bib)* In theequation, the only term which can be regarded with susp-conishba

term. This prmtdus to examine the result if the thermalcoaduct ityweeresulte.isishown for V - O5volts asthe curve

labeled in WPig. 2 and shows that indeed such a reoves the netine•wIn fat we have been able to obtain results which match the MC

reult byvamg tvalue ofx. However, the required Vale of i ybia

thed eve la ofbeled'Wv•"x the C rsl, M=yetatV -2.Oavalue 6fx 100 times ecurve valuegaveararentthatwas stillobelow the MC relt. One other possilft is that the scattering rates should bebased on a historically averaged temperature, rather than the local temperature asdone at present Clearly, more research needs to be done he•.

FnWly, we compare the distribution of potential and carrier denity within theas determined from the two procedures. in F' 3 and 4. FXg. 3 shows the

potent- sibution atV -O.3andV - 1.0. =etheI-Vcurvesam ingood agreement, Mad in FiC3 we o terve the potential distributions aralmoidenticaL The MC result shows a .di ht amount of noise which is to beexpected since the space charg must be derwed from the discrete earticledistributiom However, the similarity in the results indicat siumiar in the spaced• distribution. F 4 then shows a comparison of the MCpatd

=lowe hal) th te det" coutom obtained finm the bWrEsimulation. ere the gate lies; along the centerline of the figure. The similarit Inthe depletion regeo is evident here.

ConcusionsAa tive study of a submicron GaAs PET hres been perfore usng MC

In te I-V c at low values ogte bla,. As the gate bi wasincreased, the tTE result exhi'bited negatie forward conductance whereas onlypositiv forward condctanc was observed in the MC results Theprset ofnegative c e was found to be dependt on the m tude the thermalconductivty coefciecnt in the r valleyenergy equation, with negativeonductance disappearng altogether as he trmal approachedfty. At bias levels where good agreement was present in the pnrdicted I-V

144

result, good agreement was also observed in the potential and carrier d.....within the device. Overall the results indicate that both MC and NMT~approaches are viable for submicron device simulationi, however, further reae~his needed to provide proper closure of the NMB1 approach, specifical fltjhregard to the role of the heat flux terms in the energy equation, and thevaluation of scattering rates based on historic rather than local quantitic&

Acknowledgements

Th~is research was supported in part by AFOSR and byteNational Cenwer foComputational Electronics. Research sponsored by the Air Force Otffio ofScientific Research (AFSC) under contract F4962GM048011:3. The UnitedStates government is authorized to reproduce and distribute reprints forgovernment purposes not withstanding any copyright notation hereon.

References

L R. W. Hockney and J. W. Eastwood. Qgoe maon1-PlieAdam Huiger,(18.

2. IL L Orubin, 9)192S~rMy G. I. lafrate and J. R. Barker, VLSI Electronics,3, (1982).

3. W. R. Briley and I. McDonald, .CoIp. (9O,4. J. P. manyad LL GL~mPiz =J L (

An FWMfl TERMALCOUI

20Sa W*WMJf THERMAL COW.

T -

Figure L ScbIDmmic o(GaAZ FET%101isa

z A

so,

0 0.4 0.0 . L6 LO Z

CO. Vis LO.

Fundamental Research on theNumerical Modelling of Semiconductor

Devices and Processes

Papers from NUMOS I, the First InternationalWorkshop on the Numerical Modelling of Semiconductors

11 th - 12th December 1986, Los Angeles, USA.

Edited ByJJ.H. Miller

Numerical Analysis Group, Trinity College, Dublin

BoOL[BOOLE PRESS

THE MOMENTS OF THE BOLTZMANN TRANSPORTEQUATION AS APPLIED TO THE GALLIUMARSENIDE PERMEABLE BASE TRANSISTOR

J.P. KRESKOVSKY, M. MEYYAPPAN and H.L. GRUBINSienific Research Associates. Inc.. Glastonbury, CT 060373. USA

ABSTRACT: Solutions to the first three moments of the Boltzmann transport equation and Poisson'sequation are obtained for a permeable base transistor (PBT) using linearized, block implicit (LBI) andADI techniques. Two level electron transfer is considered. The results of the simulations are compared toresults obtained from the drift and diffusion equations. The comparison indicates that nonequilibriumtransport and velocity overshoot are important in the PBT. The predicted I-V characteristics of the deviceshow substantially higher current levels and' a higher cutoff frequency are obtained with the momentequations.

INTRODUCTION

Numerical and experimental studies of the PBT were first performed by Bozler andAlley [1). Their results suggested that a PBT could obtain cutoff frequencies as highas 300 GHz. This result has not yet been obtained. In some of their drift anddiffusion computations Bozler and Alley [1] used a two piece velocity-field curve inan attempt to account for nonequilibrium effects. The velocity varied linearly withfield up to a value of 4 kV/cm after which it was held constant at 2 x 10 cm/sec.This approach will yield higher current levels, but it offers no improvements ineither the qualitative or quantitative consequences of nonequilibrium transport.Nonequilibrium effects, in the presence of nonuniform fields, result in carrier andvelocity distributions that are significantly different than those offered by excess

saturation velocity coupled to the drift and diffusion equations.Nonequilibrium effects in a PBT have been investigated using Monte Carlo

methods by Hwang et al. 121. Their results showed significant differences comparedto the drift and diffusion results; the peak velocity along the center of the channelexceeded 4 x 107 cm/sec and the density distribution was also significantly different.The Monte Carlo results also indicated a cutoff frequency approximately 60% higherthan the corresponding drift and diffusion prediction.

In the present note, the performance of the PBT structure shown in Fig. 1 isanalyzed using both the drift and diffusion and the moments of the Boltzmanntransport equation, (MBTE). The base penetration of the present device structure isonly one half of the channel width. This in itself results in a higher fr. compared todevices with equal base penetration and channel width 131. However, further andsubstantial increases in fr are observed from the MBTE predictions as well as highercurrent levels.

99

(N) 1J. P. Kreskovsky el al.: The moments of the Bol~zmann Iraport equation

00s6-

Figure 1: Dimensions and doping level of the simulated PBT.

ANALYSIS AND NUMERICAL METHOD

The equations to be solved are obtained by taking the first three moments of theBoltzmnann transport equation. The resulting equations represent conservation ofmass, momentum, and energy, and may be expressed for the central valley carriersas

an, a nV I (1)

at ax, I 212()

dn,V', _ dn,V'V'j ko dn,T, !dcr1 + en , aip n 2at mX M x mI VB x, m , ax, 1 1. (2)

all T, _ n,V",T, 5 dV' 2 o," aV' 2 dadTIdx,at ax, 3''ax, 3k. dx1 3k0 ax,

- n, T,Fr + n,T.,F + m,(n1 -3-n,',+nr2 3o 3k(3

A similar system is obtained for the satellite valley carriers. These equations arecoupled to Poisson's equation

In eqs. (1 )-(4) n is carrier density. V' the velocity vector, T the carrier temperature.d/ the electrostatic potential, ko is Boltzmann's constant, m is the effective electronmass. rare scattering rate constants, cri is a stress tensor arising from thenonspherical nature of the distribution function, and K is a thermal conductivity. Thescattering rates are determined a priori from evaluation of the collision integrals [4].

Equations (1 )-(3). their satellite valley counterparts, and Poisson's equation forma coupled system of nine nonlinear PDE's in two space dimensions of the form

J. P. Kreskovsky et al.: Tire ,oments of the Itoltzmann transport equation 101

dr =DW)(4>) + S(,) (5)

where 4, represents the vector of unknowns, H and S represent nonlinear functionsof -4. and D(4O) represents a nonlinear, partial differential operator. To solve thissystem Poisson's equation is decoupled from the remaining eight equations bydiffercncing the electric field appearing in the momentum equation at the explicittime level. The remaining equations are solved by direct application of linearizedblock implicit (LBI) techniques 151. The equations are first time differenced

AH(O) D)[D(,), + S(,)"]. (6). lt

The nonlinear operators are then linearized using Taylor series, as for example:

D(>") = D(O") + - a4>OM+ O(At-). (7)

Equation (6) may then be expressed as

(A + AtL)A,"' AtiD(4") + S(4")i (8)

where

d (9a)

L jO (9b)

Equation (8) is then split following the ADI procedure of Douglas and Gunn 161.

(A + AtL):W= At[D(4i") + S(4O')! (10a)

(A + AtL,)d4>** = A-4.* (lOb)

and

A4A" = **+o(A 2 ) ( + Op)

As a result of the ADI splitting, the number of operations required to solve theremaining coupled equations varies linearly with the total number of grid points, anddue to the linearization process. the solution can be advanced in time withoutintroducing nonlinear iteration. After solution of the continuity, momentum, andenergy equations. Poisson's equation is solved using a scalar ADI procedure tocomplete a time step. Steady solutions, such as those to be presented here. areobtained from the lona time asymptotic transient solution. Since transient accuracyis not of interest under such conditions, the time step can be spatially scaled to speedconvergence to steady state.

The solutions to the drift and diffusion equations reported here were obtainedusing a related procedure, described in detail in 171.

102 I. P. Krr'sko'ikv el atl: The momennS of $he l'oliznwnn t'awlonl cqualion

COMPUTED) RESULTS

The computed I-V characteristics of the device shown in Fig. I arc presented in Fig.2. The results for the MBTE calculations were extrapolatcd to the origin, asindicated by the long broken lines. The shorter dashed curves show the results forthe DDE 131. The comparison shows that the predicted current levels are significant-ly higher for the MBTE solutions, a result consistent with FET calculationsperformed by the present authors, as well as by Cook and Frey 18) who used a highlysimplified momentum-energy transport model. The present MBTE results alsoindicate a region of negative differential forward conductivity at V,,. = 0.6 V. Theorigin of this phenomena is believed to be a consequence of electron transfer. Thepresence of a dc negative forward conductance is also a feature of PBT measure-ments 191. but is clearly absent from DDE simulations.

A comparison between the total carrier density distribution along the center of thechannel for drift and diffusion and MBTE solutions is shown in Fig. 3 forC= 1.0 V and V8, = 0.4 V. The MBTE prediction for the gamma valley carrier

density is also shown. The drift andidiffusion result was obtained using an equilib-rium velocity-field curve, and yields results qualitatively similar to that of Bozier andAlley.

As seen in Fig. 3 for the DDE simulations, the carrier density reaches a maximumbetween base contacts. Here, with the velocity in saturation and the cross sectionalarea at a minimum, the carrier density must increase to maintain current continuity.In the MBTE simulation the constraints of current continuity are more complex.First a decrease in the cross sectional areas is, as in the DDE, accompanied by anincrease in field along the channel. The field increase under both equilibrium andnonequilibrium conditions is qualitatively similar, as may be observed from Fig. 4which shows the potential distribution along the center of the PBT channel.

V -. Aa

/V 5 0..

0.- -0

0V OA00.

/7/

0 o. • 0.4, 0.4 0.*1 1.0 *. t S.C 4

Figure 2: Collector -urrcnt vs. collector emitter voltage for different values of ba~sc emiut oac-re..

J.P• Kreskov$sky et al.: The mromCnts of the 1foltzmana trafnsport e quaftiOn 103

0.* Doc

0 €1 3 0") 0.a 0.a 0 O .0 0.* 3.0

Figure 3: Carrier density vs. distance along center of channel fbr the PBT.

However, consequent changes in electron temperature, both increasing and decreas-

ing. lag behind the equilibrium state. This leads to velocity overshoot and a delay in

electron transfer. As a result, for peprly the first half of the device transport is

almost exclusively gamma valley transport. The implication is that if the gamma

valley carrier velocity increases with increasing field, then the product of density and

cross sectional area normal to current flow must decrease to maintain current

continuity. Since the velocity increases faster than the area decrease the carrier

density decreases.At moderate bias levels typical FET calculations show a decreasing field as the

gate region is passed. This also occurs in the PBT. Now, as the cross sectional area

increases the gamma valley carriers exhibit a decrease in velocity. [t must be noted.

however, that for the parameters of the calculations the L valley carriers make a

negligible contribution to current. Thus a decrease in carrier velocity results in a net

Os- W, .- &0 / /CIE

of

Figure 4: Potential vs. distance along center of channel for the PBT.

104 i.P. Kreskornsky et a/.: 1/ic motnrnvt of the Bioltzmmann tranLsport equation

Joe

so

- S t

&- 0. o

Figure 5: Cutoff frequency vs. base emitner voltage.

increase in carrier concentration. However initially, the decrease in field is notaccompanied by a corresponding temperature decrease (as experienced in theuniform field calculations). Thus. the high gamma valley temperature results intransfer to the L valley giving, rise to the second minimum in the gamnma valleycarrier density ohown in Fig. 3. Further toward the drain, the fild decreases.However. relaxation is incomplete and the f eld at the collector is not equal to thefield at the emitter. Also note, tti,: MBTE potential distribution ,Vves rise to aslightly higher field up.rtrea.- of the base, and a lower field, over a longer distance.downstream of the base compared to the DDE result. More significantly the electrontemperature at the collector exceeds that at the emitter. It is noted that as the fieldrelaxes. the electrons transfer back to the central valley.

All of the results presented thus far are qualitatively similar to the Monte Carloresults of Hwang et al. [2j. However. the results for cutoff frequ :-ncy. sho~vn ia Fig.5. are qualitatively different. At V,.,, = 0).5 V. jT is seen to incr .ase with V..,: fromapproximately 100 GHz to 130 GHz. value 2.5 ,o 3 times greater than that obtainedusing, the DDE. However. for higher values of V,.,,. fr is shown to decrease withincreasing V,,. For V,-, = 1.5 V. an fr in excess of 200 GHz is pred~icted forVsE + 0.2 V; while at V,,, = 0.6 V. tfr has dropped below 40 GHz. a result that isqualitatively consistent with the presence of the negative forward conductance. andone that would find no explanation in the DDE cal",-'. tions.

ACKNOWLEDGEMENTS

The authors think R.A. Murphy, C.O. Bozler and M.A- Hollis, for continuing'interest and for continuous discussions of these calculations. This wvork vab,a •-on-sored h% AFOSR and ONR.

J. P. Kreskovsky ei al.: The momnlets of the Bolhztunn frunsport equation WS

REFERENCES

III C.O. BozIcr and G.D. Alley. IEEE Tr:'s. Electron Dcv. ED-27(6) (1980).121 C.-G. Hwang. D.H. Navon and T.-W. Tang. IEEE Elcetron Dcv. Lett. EDL-6(3) (1985).131 M. Mcyyappan. J.P. Krcskovsky and H.L. Grubn, SRA rcport. in preparation.14) H.L. Grubin. D.K. Fcrry. G.J. lafrate and J.R. Barker. VLSI Elcctronics.3 (1982).151 W.R. Briley and H. McDcmald. J. Comp. Phys. (1980)).161 J. Douglas and J.E. Gunn. Num. Math. 6 (1964).17, J.P. Krcskovsky and H.L. Grubin. J. Comp. Phys. (1987).181 R.F. (;ook and J. Frey. COMPEL 1(2) (19N2).191 See. e.g.. C.O. Bczlcr. M.A. Hollis. K.B. Nichols. S. Rabc, A. Vcra and C.L. Chen. IEEE Eklctron.

Dev. Lett.'EDL-6(9) (19K-5).

Electron velocity overshoot and valley repopulation effects in diamondM. A. Osman, H. L Grubin, and J. P. KreskovskyScientfic Research Associates, Inc. P. 0. Box 1058, Glastonbury. Connecticut 06033

(Received 12 December 1988; accepted for publication 22 February 1989)

Using ensemble Monte Carlo procedures, electron velocity overshoot and transient valleyrepopulation in diamond have been investigated as a function of field strength and orientation.It is found that the response of electrons to the sudden application of a 50 kV/cm electric field

along (100> results in a velocity transient with a maximum of 1.9X 101 cm/s after 200 fs and a

steady state value of 1.30X 107 cm/s. For a field along (110), the corresponding maximum andsteady state values are 2.0X 107 and 1.4X 107 cm/s, respectively. The calculated temporal andspatial duration of velocity overshoot in diamond is longer than that of silicon but shorter thanthat of gallium arsenide.

The superior thermal conducting characteristics of longitudinal-acoustic and transverse-optical phonon modessemiconducting diamond, coupled to hole and electron mo- assist intervalleyfscattering and have temperatures of 1500bilities and saturated drift velocities that are superior to sili- and 1720 K, respectively.' These energies of the intervalleycon, make it an ideal candidate for high-speed, high-power, phonons are considerably larger than the corresponding val-

and radiation-resistant device applications." 3 To exploit ues in SO A single value, 8 10W eV/cm, was used for all ofthis, much effort is under way in the growth and doping of the intervalley deformation potentials, and a value of 8.7 eVsynthetic diamond, while interest in natural diamond re- was used for the intravallev acoustic deformation potential.2

mains high.-7 The successful development of a semicon- Initially, the electrons were assumed to be in therml equi-ducting diamond device technology requires the determina- librium corresponding to a Maxwellian distribution at roomtion of a number of key transport parameters, such asmobilities, transient velocities, etc. The mobility and steadystate drift velocity of electrons and holes in natural diamondhave been investigated by Nava et al.2 and Reggiani et al.3 In 2.0 -this letter we examine two effects occurring in diamond for -POthe first time: ( 1) velocity overshoot and (2) transient valley Erepopulation (as have been investigated in Si and Ge'). The -.o<0- ...................calculations reported below are for electrons subjected to 020sudden changes of field with values 20 and 50 kV/cm. Addi- .tionally, for fields of 50 kV/cm, the orientation dependenceis studied. It is found that velocity overshoot occurs on tem-poral scales of the order of 400 fs, which is longer than that ofsilicon and the same order as that of gallium arsenide. The Fspatial duration of the overshoot is of the order of 400 A, less (a)

than that of gallium arsenide9 but larger than that of sili- 00 (Q4 0.8

con. " Thus nonequilibrium effects are expected to be impor- TIME (pstant at feature sizes below 400 A.

In the ensemble Monte Carlo procedure used in this 2.0 T-'--0Kstudy of diamond, the kinetics of electrons are determinedDIAMONDrelative to the minimum in k space at which the valley is -FII<100

located. Electrons located in valleys on opposite sides of thesame axis, e~g., the (100) and (100) valleys, are not distin- 0guishable, and symmetry allows specific calculations to beperformed for three ellipsoidal valleys (m, = 1.4, m, =1.0

-= 0.36) located along the (100), (010), and (001) crystal- 2lographic orientations. The Monte Carlo procedure takes >

into account scattering by elastic intravalley acoustic phon-ons, g-type intervalley phonon scattering between two paral-lel valleys, andf-type intervalley phonon scattering betweenfour perpendicular valleys. Following an f-type scattering, 0b)the final valley is selected from the possible perpendicular 0 50 1OOvalleys using a random number. The interaction between the DISTAN CE (nm)electrons and ionized impurities is ignored in this investiga- FIG. 1. Dtift velocity of electrons in diamond as a function of (a) time and

tion. The longitudinal-optical phonon modes assist interval- (b) positio at 300K The orientations ofthe elteric field axe shown forley g scattering and have a temperature of 1900 K, while each curve.

1902 App. Phys. Lett. 54 (19). 8 May 1989 0003-6951 1891191902-03S01.00 (z) 1989 American Instituto of Physics 1902

temperature. The electric field was then switched on and the 150time evolution of the electron ensemble was examined forfields of 20 and 50 kV/cm applied along the (100) direction. - - - -To illustrate the orientation dependence of valley repopula-tion and velocity overshoot, the calculations were repeated 50kVln

for a 50 kV/cm electric field along the (110) direction. 1ooThe time evolution of the electron velocity during the

first picosecond after switching on the electric field is shown 2 /cmin Fig. i. For a 50 kV/cm electric field along the (100) 2-

direction, the transient velocity reaches a maximum of a:1.9x l0' cm/s after 200 fs and then decreases to a steady z

state value of 1.30X 10' cm/s. When a 20 kV/cm field is

applied along (100), the velocity increases more graduallyto a maximum of 1.4X 107 cm/s after 300 fs, and then de- F11<11O>creases to a steady value of L.0X 10' cm/s. Changing the FII<IO0>orientation nf the electric field at 50 kV/cm to (110), in-creases the maximum transient velocity and steady state ve- 0 02 0.4 0.6 0.8locity to 2.00X 107 and 1.4x 110 cm/s, respectively. The TIME (ps)steady state results are consistent with the earlier experimen- FIG. 2. Mean energy as a function of timne for electron in diamond. Differ-tal work reported by Nava et al., which showed higher drift ent curves refer to different magnitudes and orientation of the electric field.

velocities in natural diamond for fields along (110) com-pared to that along (100).2 In all of the above situations, the results in an initial rapid rise in average energy, as well as amaximum velocity is about 50% higher than the corre- higher average energy at longer times for the electrons in thesponding steady state velocity. The magnitudes of the maxi- (001) valleys relative to those in the (100) valleys (see Fig. 3mum and the steady state velocities are larger than those for both 20 and 50 kV/cm fields).reported for Si at 50 kV/cm.8 Additionally, the time dura- The (001) and (010) valleys are equivalent when thetion of velocity overshoot is longer in diamond than in Si. field is along (100) directions. Hence the rapid heating ofFor example, for a 20 kV/cm field along (100), the time electrons in these valleys results in a significant fraction ofduration of overshoot is approximately 250 fs in Si? com- the electrons undergoingf-type intervalley phonon scatter-pared to 400 fs in diamond as can be seen from Fig. 1 (a). The ing to the (100) valleys, whose transport is governed by thelongerovershoot duration makes itpossible to design diamond larger longitudinal effective mass. " This reduces the popu-devices with larger dimensions than Si devices, while main- lation of the electrons in the (001) valleys while increasingtaining shorter transit times across the device. Figure 1(b) thatof (100) valleys, ascan beseen in Fig. 4. Forexample, inshowsthat the distance over which overshoot effects are pro- response to a 50 kV/cm field along (100) [Fig. 4(a) ], thenounced is approximately 400 A for diamond. This over- population of the electrons in the (100) valleys rapidly in-shoot distance is longer than the 300 A in silicon at compara-ble field values.'°

Figure 2 displays the time evolution of the average ener-gy of the carriers subject to fields of 20 and 50 kV/cm. For 150 50kV/cmII<Il)•the 5M kV/cm field, and during the first 200 fs, the average 0 /lc0energy of the carriers rises rapidly and is nearly independentof orientation. However, at longer times the average energyfor the field along (110) is slightly higher than that along the . .50 kV/cmlllOO(100) direction. In response to a 20 kV/cm field along(100>, the average energy increases gradually during the >100first 400 fs and reaches a value of 130 mV which is below the -

threshold for the emission of g-type intervalley phonons. To "illustrate the valley repopulation effects and how it in- Wfluences the velocity overshoot, the mean energy and the W - .number of electrons in individual valleys were determined. 50 . "In Fig. 3, the average energy of the electrons in the (100) . "".

(dashed lines) and (001) valleys (solid lines) is plotted for20 and 50 kV/cm electric fields. Note that, regardless of field E10oorientation, e.g. along (100) or (110), the component of the - EO,electric field along the principal axis of the (001) valleys is 0zero, while it is finite along the minor axes where transport is 0 0.2 04 0.6 Q8determined by the smaller transverse effective mass. Conse- TIME (p$)

quently, the electric field heats the electrons in the (001) FIG. 3. Mean energy as a function oftime for electron in diamond in (001)

valleys at a rate faster than that of the (100) valleys. This valleys (solid lines) and (100) valleys (dashed lines) at 300 K.

1903 Appl. Phys. Lett., Vol- 54, No. 19.8 May 1989 Osman. Gnruin. and Kreskovsky 1903

50- noo -. 50kV/cm the (100) orientation, and the population of the electrons in" '-o.. both (001) and (100) valleys changes by less than 5% from

--------- °the initial thermal equilibrium values. The steady state ener-40- .. gy of the electrons in the (100) valleys remains well below

thef/type phonon emission threshold (see Fig. 3) when a 2030 kV/cm is applied along the (100) axis, so that only a smaller

"r_ (fraction of electrons undergoes intervalley scattering to0 ((010) and (001) valleys. Here the population of the elec-

0 0.2 Q4 0.6 0.8 trons in the (100) valley gradually increases to 50% after ITIME (ps)

0z4°0 5ps, while that of the (001) valleys decreases to 25% of the4 -- "50-V/c"" ... total electron population [Fig. 4(c)].

W "In conclusion, we have shown that the electrons in dia-_ 30mond exhibit orientation-dependent velocity overshoot and"LL FII<IIO> (b)

0 20 ,b) that the temporal and spatial duration of the overshoot isS0 02 0.4 0.6 0.8 longer compared to Si. The maximum velocities were about

TIME (ps) 50% larger than the steady state velocities. The magnitudes

20 kV/cm ... of the maximum velocities following a 50 kV/cm field were.50- .. . 2.00X 107 cm/s for (110) field orientation and 1.9X 10O

,IIci00o> - cm/s for (100) orientation. The valley repopulation is40 strongly affected by the orientation and the magnitude of the

-----. electric fields. Finally, the influence of the ionized impurity30 scattering on the above conclusions remains to be investigat-

ed. Because the known donor impurities, such as activated

201 lithium, require more than 100 meV to ionize and release0 0.2 0.4 0.6 0.8 free electrons, the concentration of free electrons and ionized

TIME (ps) donors is strongly dependent on the electric field. This can

FIG. 4. Fraction of electrons in (100) valleys (dashed lines) and (001) influence the transient response of electrons to applied elec-valleys (solid lines) as a function time after switching on the field at 300 K:(a) F= 50 kV/cm 11 (100), (b) F= 50 kV/cm, 11 (110), and (c) E = tic field due to the finite time required to release the elc-kV/cm 11(100). trons from the donor impurities.

This work was supported by the Strategic Defense Ini-

creases from its equilibrium value to 50% of the total after tiative Organization (managed by the Defense Nuclear

300 fs and then gradually decreases to 45% after 700 fs. The Agency, RDT&E RMSS code: B 76610 SF SB 0073 RAEV

physics behind this repopulation is displayed in Fig. 3, which 3230 A) and by the Office of Naval Research.

shows that after 200 fs the average energy of the electrons inthe (100) valleys exceeds the threshold for intervalleyphonon emission. This leads to a larger fraction off-typeintervalley scattering, a reduction of the (100) valley popu- 'V. K. Baxhenov, 1. M. Viokulin, and A. G. Gonor, Soy. Phys Semicond.

lation, and an increase in (001) valley population. 19, 829 (1985).When the orientation of the field is changed to (110), 'F. Nava, C. CanalL C. Jacoboni, L Reggiani, and S.F. Kozlov, Solid State

the (100) and (010) valleys are equivalent; and transport in Commun. 33,475 (1980).these valleys is through a combination of longitudinal and 'L Reggiani, S. Bosi, C Canali, F. Nava, and S. F. Kozlov. Phys. Rev. B

23, 3050 t1981).transverse effective masses. Several new features enter: First, '13. Singh, 0. R. Mesker, A. W. Levine, and Y. Arie, Appl. Phys. .ett. 52Zthe energy difference between the (001) and (100) valleys is 1658 (1988).smaller than for the field along ( 100); second, because of the 'V. S. Vavilov, M. A. Gukasyan. M. I. Guseva, T. A. Karatygina, and E. A.

contribution of the transverse effective mass, the electrons in Konorova, Sov. Phys. Semicond. 8, 471 (1974).Ks. L Moazd, R. Nguyen, J. . Zeidler. IEEE Electron Device Lett.

the (100) valleys have a higher average energy and heat up EDL-9, 350 (1988).at a faster rate compared to the situation when the field is 'M. W. Geis, D. D. Rathnu., D. J. Ehrlich, R. A. Murphy, and W. T.

along the (100) axis as can be seen in Fig. 3. Consequently, Lindley, IEEE Electron Device Lett EDL-8, 341 (1987).

the difference between thef-type intervalley scattering from 'C_ Jacoboni and L Reggiani, Rev. Mod. Phys. 55, 645 (1983).'r. J. Malone and J. Frey, J. AppL Phys. 48, 781 (1977).

(100) to (001), and the intervalley scattering rate from 'D K. Ferry and H. L Grubin, Microelectron. J. 12,5 (1981).(001) to (100) are smaller for the (110) orientation than for "K. Seeger, Semiconductor Physics (Springer, Viemna, 1973), Chap 7.

1904 Appl. Phys. Lett., Vol. 54, No. 19.8 May 1989 Oswan. Gnjbi, and Kreskovsky 1904

Monte Carlo investigation of carrier-carrier interactionand ultrafast cooling of hot photoexcited carriers in GaAs

M.A. Osman, H.L. Grubin, and J.P. KreskovskyScientific Research Associates, Inc.

P.O. Box 1058, Glastonbury, Connecticut 06033-6058

D.K. FerryCenter for Solid State Electronics Research

Arizona State University, Tempe, Arizona 85281

Abstract

The role of the electron-electron (e-e), hole-hole (h-h), and electron-hole (e-h)interaction on ultrafast cooling of carriers in GaAs is examined for excess excitationenergies of 40, 200, and 300 meV using an Ensemble Monte Carlo (EMC) approach. It isfound that when the initial energy of the carrier is below the phonon emission thresholdcarrier-carrier (c-c) interactions stimulate either the optical phonon emission orabsorption process depending on whether the initial energy of the carrier is above orbelow the thermal energy, respectively. The e-h interaction role is strong whenexcitation energy is below the LO phonon emission threshold.

Introduction

In recent years, the time-resolved femtosecond spectroscopy and optical time offlight measurement techniques have become an attractive tool for investigating theultrafast dynamics of photoexcited carriers in semiconductorsI- 3 . Considerableexperimental information about the carrier-carrier and carrier-phonon interactionsprocesses has been obtained. These processes show a strong dependence on carrierconcentration, energy, electric field, temperature, and material properties. Forexample, at high electric fields the minority electrons in p-type InGaAs exhibit highersaturation velocities than majority carrier electrons 4 , while in p-type Si theyexhibit mobilities that are much lower than majority carrier electrons5 . Similarly anincrease in the electron energy-loss-rates in the presence of a cold hole plasma6 , andan absolute negative mobility for the minority electrons and holes at low temperaturesin GaAs/AlGaAs quantum well structures have been observed2 . In all these situations,the e-h interaction has been suggested as the possible responsible mechanism. Morerecently it has also been suggested that electrons excited by excess energy of 0.9 eVrelax initially by emitting LO phonons and subsequently thermalize through e-einteractions 7 . These remarkable experimental results clearly demonstrate our limitedunderstanding of the manner in which the carrier-carrier (e-e, h-h, e-h) interactionsinfluence the cooling and response to externally applied electric fields of thephotoexcited carriers at different excitation energies and levels.

In this paper, we examine the energy dependence of the role of e-e, h-h, e-h, andcarrier-phonon interactions on the cooling rates of electrons and holes. The EMCapproach is used to avoid making any assumptions on the form of the distributionfunctions for the holes and electrons which has been a standard practice in previoustheoretical attempts to investigate the cooling process 8 , 9 . With its built-in timeevolving distribution, the EMC approach makes it possible to examine the details of thecooling process on picosecond and sub-picosecond time scales. In an earlier study, weexamined the concentration dependence of an e-h plasma excited by a 200 femtosecond,1.64 ev laser pulse1 0 . It was found that the e-h interaction provided a significantchannel through which the electron energy is transferred to the lattice particularly athigh excitation levels where the electron-phonon interaction is strongly screened andthe e-h scattering events are more frequent. The cooling rates were lower than thosepredicted by a model that assumed a parameterized distribution function with equalelectron and hole temperatures and includes the hot phonon effects. More, recently wealso examined the hot phonon and e-h interaction using EMC and showed that for timesless than 2 ps the e-h interaction controlled the relaxation process while for timesgreater than 2 ps the not phonons slowed the cooling rates1 1 .

In the present study, situations where the e-h plasma is photoexcited with excessenergies of 40, 200, and 300 meV are investigated. These energies correspond tosituations where the electron energy (Ee), the hole energy (Eh), and the LO phononenergy (E0 ) satisfy the following conditions:

(1) Ee < E0 and Eh < E0 ,(2) Ee > E0 and Eh < EO,(3) Ee > E0 and Eh > E0,

respectively. Hot phonon effects are not considered. The density of the e-h plasma isassumed to be 5 x 10 1 6 cm- 3 , and the finite duration of the pulse is ignored. Fromthe results of this study, conclusions are drawn about the influence of the initialenergy distribution of the photoexcited carriers on the manner in which c-c interactionsachieve thermalization of the carrier distributions.

Theory

The EMC proceedure used in this study is the same used in investigating the role ofe-h interactionll, 1 2 . Thus only its main features will be summarized for the sake ofcompleteness. The band model consists of three nonparabolic valleys for the conductionband and a parabolic heavy hole valence band. The light hole band is ignored, partlybecause of its small density of states and to reduce the computation time. Figure 1,shows the flow chart of the EMC program used in this investigation. The first stepcorresponds to specifying the material parameters, initial energy distribution of thecarriers, pulse shape and duration, room temperature, value of the electric field, andwhich c-c interactions are active. The scattering rates for acoustic phonons,deformation potential, intervalley and intravalley scattering processes for electronsand holes are then calculated and tabulated. On the other hand, the scattering rates ofc-c and screened carrier-phonon interactions, which depend on the evolving distributionfunction of the carriers are calculated every 50 femtosecond initially and at longertime intervals as the time proceeds. For finite pulse duration, the number of thecarriers is adjusted according to the pulse shape with an initial energy determined fromthe energy of the pulse taking into account the nonparabolicity of conduction band. Thedynamics of the electrons and holes are then examined using the regular EMC approach.Coupling between the electron EMC and the hole EMC programs is accomplished through thee-h interaction which depends on the energy and momentum distribution of the electronsand holes. The average energy of the electrons and holes, energy-loss-rates through e-hand electron-phonon interactions are then calculated at the end of each iteration or atspecified time intervals.

The electrons and holes are assumed to interact among themselves and with each otherthrough an exponentially screened Coulomb potential. The screening length is calculatedself consistently in the static and long wavelength limit of the random phaseapproximation. Unlike previous models for e-h scattering, no a priori assumptions aremade on the magnitude of the energy exchange between the electrons and holes in an e-hscattering event. The expressions for the scattering rates for c-c and screenedcarrier-phonon interactions are given in reference 12. Figure 2, shows the scatteringrates for e-e, e-h, h-h interaction processes, assuming a Maxwellian distribution forthe electrons and holes at temperatures of 1000 K and 160 K respectively as a functiono- the average wavevector kz. From this figure, it is clear that the h-h scatteringrates are much stronger than the e-e rates due to the larger density of states in theheavy hole band. Also notice that the interaction of an electron with a hole plasma isstronger than the rate at which a hole interacts with a sea of electrons1 3 . This isrelated to the large density of final states for holes compared to that of electrons.The screening of the carrier-phonon interactions reduces the scattering ratessignificantly, especially at high carrier concentrations and low temperatures.Additionally, the presence of the relatively cold hole plasma, which is the case in aphotoexcitation process, leads to stronger screening even when the energy of theelectrons is sufficiently high1 2 . In the present analysis the acoustic phononscattering is assumed to be elastic which is a good approximation at lattice temperatureof 77 K. However, for excitation by total excess energy of 40 meV, the use of aninelastic acoustic scattering model is more appropriate because the relaxation throughthe optical phonon emission is severely diminished. Furthermore, the interactionbetween the L-valley electrons and holes is neglected, because the population of theelectrons the upper valleys is negligible. The elastic impurity scattering is ignoredin our calculation, since we assume that all of the carriers are generated by thephotoexcitation process. In order to examine the roles of e-e , h-h, and e-hinteractions more closely, the phonon system was assumed to be in equilibrium.

Results and discussion

The cooling rates of carriers photoexcited with total excess energies of 40, 200 and300 meV, corresponding to photon energies of 1.55, 1.71, and 1.81 eV respectively, wereexamined under four different combinations of active scattering processes. First thesimulation was carried out assuming only optical phonon interactions were active. Curve1 in Figures 3 through 5 corresponds to this situation. Notice that curve 1 in Figures3 and 4b, which represents cooling of electrons and holes with initial energies belowthe phonon emission threshold E0 (36.4 meV), does not change with time since no phononemission is possible and at 77 K and the probability for optical phonon absorption issmall. However, when the initial energy of the electrons is above E0 (Figures 2a and3a), the electrons emit integral multiple of LO phonons and end up with energies equaltc 27.9 meV and 4.25 meV at 4 picosecond after excitation by 1.71 and 1.81 eV laserpulses, respectively. similarly the holes reach a steady state energy of 5.8 meV whichcorresponds to emitting a single optical phonon (see Figure 3b). After reaching theseenergies no further cooling is possible to reach equilibrium with the lattice, since theacoustic scattering is assumed to be elastic.

The situation where the e-e and h-h interactions were active, in addition to theoptical and acoustic phonons, was examined next. Curve 2 in these figures shows thecooling process for the electrons and holes. The most interesting phenomena in Fig. 1is that, the e-e interaction enhances the cooling of the electron system from an initialenergy of 35 meV, which is just below the phonon emission threshold, to a steady stateenergy 13.5 meV that is slightly above the room temperature. This enhancement occursbecause some electrons gain enough energy from interacting with other electrons and thencool down by emitting LO phonons. On the other hand, the h-h interaction coupled withoptical phonon absorption drives the hole system to thermal equilibrium with thelattice. The average energy per hole changes slowly from an initial energy of 5 meV toa final energy 8.7 meV which is just below Eth. In this case each hole gains anaverage energy of 3.7 meV from the lattice which means that energy of a single opticalphonon is shared by about 10 holes. This is possible because the strength of the h-hinteraction spreads the distribution of the holes quickly in energy, so that when a holeabsorbs an optical phonon, it is more likely to share the gained energy with an otherhole than emitting an optical phonon. This results in two holes whose energies arebelow E0 , but are relatively hotter than the rest of the holes, so that throughfurther h-h scatterings the extra energy will be distributed among the holes. When theelectrons are excited by 1.71 and 1.81 eV laser pulses, the e-e interaction leads to aslower cooling rate. This happens because the interaction among the electrons quicklyredistributes the electrons into high and low energy regions. Consequently, manyelectrons end up with energies below the phonon emission threshold, while those whichend up at higher energy regions essentially have the same probability for emission ofoptical phonons. Those electrons which have energies below the phonon emissionthreshold can cool down only by giving up some of their energies to other electrons orgaining enough energy to emit an optical phonon. Comparing curve 2 in Figures 4a and5a, we see that the manner in which e-e interaction influences the cooling rate is alsodependent on whether the final energy after emitting the LO phonons (assuming no c-cinteractions) is below or above Eth. When the e-h plasma is excited by excess energyof 200 meV, the electrons would reach a steady state energy of 28 meV by emitting LOphonons only, which is not in equilibrium with the lattice. Consequently the electronsexchange energy among themselves and emit LO phonons in a manner similar to thesituation in Figure 3a. This explains why the cooling curve 2 in Figure 4a drops belowcurve 1 beyond 2 ps. On the other hand curve 2 always remains above curve 1 in Figure5a, and the electron ensemble reaches a steady state energy of 19.4 meV which is aboveEth. The cooling curve for the holes in Figure 4a behaves in the same way the theelectrons do in rigure 3a, because in both cases the initial energies of the carriersare less than E and greater than E . When the initial energy of the holes is aboveE0 , the energy 2xchange among the htes leads to faster equilibrium with the latticeas can be seen in Figure 5b (curve 2). The holes retain some part of the initial energyinstead of losing it to the lattice via optical phonon emission.

The case where only electron-phonon and e-h interactions were active was theninvestigated. From examining curve 3 in Figure 3b, we see that the average energy ofthe holes increases rapidly during the first picosecond from 5 meV to 12.5 meV as aresult of the energy transferred from the electrons. Also the cooling curve for curve 3in Figure 3a is below that of curve 2 reflecting the fact that e-h interaction tendsgradually shift the electron population to lower energy states by transferring theenergy to the holes. The cooling rates exhibit the same trend for all the casesexamined as can be seen from the relative positions of curves 2 arnd 3 in Figures 3through 5. Finally the more realistic situation where all the scattering mechanismse-e, e-h, h-h, and carrier-phonon are present was investigated. From curve 4 in Ficure3a it is obvious that the coupling of e-e and e-h interactions leads to a faster coolingof the electrons to room temperature. The same trend is true for the holes, so that the

C-.A-

energy of the holes is intermediate between the situations where h-h or e-h interactionis active only. However, it is clear form examining curves 3 and 4 in Figures 3 through5, that the cooling rates are primarily determined by the e-h interactions, since thecooling curve in this case closely follows the situations where only e-h andcarrier-phonon interactions were active.

Conclusion

The c-c interactions are sensitive to the initial energy distribution of the carriersand they influence the carrier-phonon interactions so as to achieve a fasterthermalization of the carrier distribution, when the initial energy of the carrier isbelow the phonon emission threshold c-c interactions stimulate the optical phononemission or absorption process depending on whether the initial energy of the carrier isabove or below the thermal energy respectively. The e-h interaction plays a dominantrole in the relaxation process even when excitation energy is below the LO phononemission threshold.

Acknowledcrment

This work was supported by the Office of Naval Research.

References

1. R. L. Fork, Physica 334B, 381-388 (1985).2. R. A. Hopfel, J. Shah, P. A. Wolf, and A. C. Gossard, Phys. Rev. Lett. 56, 2736

(1986).3. M. R. Junnamarker and R. R. Alfano, Phys. Rev. B 34, 7045-7062 (1986).4. J. Degani, R. F. Leheny, R. E. Nahory, and J. P. Heritage, Appl. Phys. Lett. 39,

569 (1981).5. D. D. Tang, F. F. Fang, M. Scheuermann, and T. C. Chen, Appl. Phys. Lett. 49,

1540 (1986).6. R. A. Hopfel, J. Shah, and A. C. Gossard, Phys. Rev. Lett. 56, 765 (1986).7. J. C. Tsang and J. A. Kash, Phys. Rev. B 34, 6003 (1986).8. W. Potz and P. Kocevar, Phys. Rev. B 28, 7040 (1980).9. M. Asche and 0. G. Sarbei, Phys. Stat. Sol. (b) 126, 607 (1984).

10. M. A. Osman, U. Ravaioli, R. Joshi, W. Potz, and D. K. Ferry, in Proc. 1BthIntern. Conf. Phys. Semiconductors, Stockholm, 1986, Ed. by 0. Engstrom, in press.

11. M. A. Osman, H. L. Grubin, P. Lugli, M. J. Kann, and D. K. Ferry, to be publishedin the proceedings of the Second Topical Meeting on Picosecond Electronics andOptoelectronics, Lake Taho, 1987.

12. M. A. Osman, Ph.D. dissertation, Arizona State University, Tempe, August 1986(unpublished).

13. M. A. Osman and D. K. Ferry to be published.

COIPTA O TOF ARIR

COPTTO rCARRIER t SCATTERING RATES

C~4UTATION OF SCREENED CARRIERPOPPHONON SCATTERING RATES

ELECTRON EflC

NOE H

Figure 1. Flow chart of the EMC program.

GaAs Th= 160 K

16 -n100c

'hh

-- eh

Eh e

-6 -4 -2 0 2 4 6

Figure 2. Carrier-Carrier scattering rates as a function of the averaqe kz.

40

(a) ELECTRONS

30-

- %c-ph

S- -c-ph+ c-c

_ -- .c-ph+e-h

" -2 c-ph+c-c+e-h

GaAs .-

10 T= 77 K

n = 5 x 1 6cm-3

AE = 40 meV

0 1 I

20

(b) HOLES

3

4.-)0 4 - " --E>- 10-n2 - - - -2-.. . .

w

0 I 2 3 4

TIME (ps)

Figure 3. Time evolution of the carrier energy for different active scatteringmechan.sms (a) electrons, (b) holes.

200

(a) ELECTRONS

GaAs

150 T= 77 Kn = 5xIO16 m-3

AE s 200 meV

<E c-ph

>- 100-- c-ph + c-c

Jz Iv\ c- ph +e- h

\\ c-ph+c-c ÷e-h

50-

C I

30

S. -

z"w (b) HOLES

0 1 I0 1 2 3 4

TIME (ps)

Figure 4. Time evolution of the carrier energy for different active scatteringmechanisms (a) electrons, (b) holes.

300

(a) ELECTRONS GOAs

T =77 K

n = 5 x 10 16cm-3

200 6E - 300 meV

>z c-ph+ c-c

00 -10-- c-ph+c-c+e-h

Sl-

40

(b) HOLES

>- 202et- 4

0 I 2 3 4TIME (ps)

Figure 5. Time evolution of the carrier energy for different active scatteringmechanisms (a) electrons, (b) holes.

Monte Carlo Investigation of Hot Photoexcited ElectronRelaxation in GaAs*

M. A. Osman and H.L. GrubinScientific Research Associates, Inc.Box 1058, Glastonbury, CT 06033

P. Lugli

Dipartimento di Fisica e Centro Interuniversitario di Struttura dellaMateria della Universita, Modena, Italy

M.J. Kann and D.K. FerryCenter for Solid State Electronics Research,Arizona State University, Tempe, AZ 85281

Recently, several authors [1-3] have investigated the dynamics ofelectron-hole plasma generated by picosecond and sub-picosecond laserpulses in GaAs. These measurements have indicated the importance ofelectron-hole interaction in the cooling process (especially in thepresence of a cold hole plasma) and the existance of non-equilibriumphonon distributions as a result of cooling of photoexcited electrons andholes. Previous theoretical studies of the cooling process have ignoredthe presence of the upper valleys even when the energy of the excitedelectrons exceeds the energy separation between the central and the uppervalley. Additionally, assumptions had to be imposed on the form of thedistribution function of the electrons and phonons. Ensemble Monte Carlo(EMC) techniques avoid these assumptions and have been used toinvestigate the effects of electron-hole (e-h) interaction [4] andnon-equilibrium phonons [5] on the relaxation rate of photoexcitedelectrons. In this study we present the first EMC calculation to accountfor both the e-h interaction and hot phonon effect.

The Monte Carlo model takes into account the e-h , carrier-phononinteractions and the non-equilibrium LO phonon-electron interactions.The band model consists of three nonparabolic valleys for the conductionband and a parabolic heavy-hole valence band presence. The parametersused in the EMC calculations are the same as those used in references[4] and [5]. Electron-electron, electron-plasmon and the screening ofelectron-phonon interactions have been negelected. Initially, to avoidthe complications arising from the transfer of electrons to the uppervalleys, we have examined the roles of hot phonons and electron-holeinteraction for excitati6-n energy of 1.8 eV. Under this situation, theenergy of the excited electrons is 250 meV while that of the holes is 40meV, so that the change in the LO phonon population is mainly due theemission of LO phonons by the electrons. The time evolution of the-LO-phonon distribution N(q) is calculated as a function of the wavevectorq from the Monte Carlo simulation using the approach outlined in [5].Instead of recalculating the scattering rates for the LO-phonons usingthe perturbed phonon distribution, we have used a Large value for N(q)and used the rejection technique to accept the final state after eachscattering event involving an LO phonon.

* Supported by the Office of Naval Research,

The relaxation of electrons excited b a pulse wasga1.8 eV laserpuswa

-3examined for an excitation level of 5x10 cm to understand themanner in which e-h interaction and hot phonons influence the coolingprocess. From Fig. 1, it is clear that when the e-h interaction isignored, the hot phono-Sn effect is significant only for times longerthan 1 ps. Curve 1 in Fig. 2 shows that between 0.6 and 2 ps theenergy-loss-rate through the emission of LO-phonons is reduced whenthe phonon heating is taken into account compared to the situationwhere hot phonons are negelcted (curve 1). When the e-h interactionis taken into account, the cooling rate slows for times less than 2 psas shown by curve 3 in Fig. 1. When both the e-h interaction and hotphonons are considered (curve 4), the cooling rate is slightly fasterfor times below 1 ps partly due the initial overestimation of theenhancement of the phonon emission process inherent in using a singlelarge value for N(q). However, it is obvious that the cooling processfor times larger than 2 ps is controlled by the hot phonons, whichlead to a slower cooling rate. Additionally, no significant change inthe energy of the photoexcited holes was noticed, when theelectron-hot-phonon interaction was considered. The situation wherehole-hot-phonon interaction is taken into account might lead to anincrease in the hole energy as a result of the phonon reabsorptionprocess. This work is underway and the results will be presentedelsewhere.

3WOGoAs

* TUTTK Fig. 1. Time evolution of then-oxK96ai"3 electron energy: 1) e-ph,tw I.S8 2) e-ph (hot), 3) e-ph + e-h,

4) e-ph (hot) + e-h.

1.0 10.0TIME (ps)

"Fig. 2. Time dependence of theelectron-energy-loss rate: 1)

I w/o hot phonons, 2) with hot•o "•phonons.

_ T-77Kn . 5 x t060,6-3 4

hP, 1.8 .V

o.O• 0 .I .0 6.0TW (ps)

Ati

The situation where the electrons were transferred to the uppervalleys, was investigated for the case of excitation by a 588 nm laserpulse using EMC and taking into account the carrier-phononinteractions only. The fraction of electrons in the central valleyis shown in Fig. 3 as a function of the time after the application ofthe pulse. From Fig. 3, it is obvious that the simple picture ofelectrons cascading down the central valley by emitting LO-phonons isnot complete. The electrons rather transfer to the upper valleys andcool down in these valleys by emitting phonons through polar-opticaland deformation potential interactions as shown in Fig. 4 (curves 2and 4). The energy loss rate through LO phonon emission is dominatedbv the central valley electrons for times greater than I ps and reacha maximum after 2 ps. Note that after 2 ps about 80% of the electronsare in the central valley. The energy of the electrons in the centralvalley (curve 1 in Fig. 5) remains above 100 meV up to 3 ps after theexcitation and an increase in the average energy occurs at 2 ps due tothe return of the electrons to the central valley. The initial rapiddrop in the average kinetic energy of the electron ensemble (curve 2in Fig. 5) is due to the transfer of a large fraction of the electronsto the upper valleys as can be seen in Fig. 3. Curve 3 in Fig. 5indicates that the cooling of the electron in the L-valley is notnegligible.

1(00

GaAs

T=77K Fig. 3. Time evolution of the

n = I'Ci-'& fraction of electrons in the

central valley.

50- - 588 nm

.0 0.1 1.0 10.TIME (ps)

Fig. 4. Time dependence of the

electron-energy-loss rate:". 1) LO (F), 2) LO (L),3) total LO

4) Intervalley scatteringTG7s \2 \

S T- TTK %In flO,,cm-, 4\ *

0.019.0150.0TIME (ps)

6Go)s

T=7T7K% %" n = lI T UT-5 Fig. 5. Mean kinetic energy of

X=588nm the electrons in: 1) centralvalley, 2) central + L-valley,

CC. 3) L-valley.

LU

-J %%

0.0`1 0.1 1.0 10.0TIME (ps)

In conclusion, have shown that the cooling process of thephotoexcited electrons is controlled by e-h interaction for times lessthan I ps and by hot phonons for times greater by 2 ps. In addition,we have demonstrated that the cooling of the electrons in the uppervalleys are important for Ligh excitation energies.

References

i. J. A. Kash, J. C. Tsang, and J. M. Hvam, Phys. Rev. Lett. 54, 2151(1985).

2. J. Shah, A. Pinczuk, A. C. Gossard, and W. Weigmann, Phys. Rev.Lett. 51_, 2045 (1985).

3. C. L. Collins and P. Y. Yu, Phys. Rev. B30, 4501 (1984).4. M. A. Osman, U. Ravaioli, R. Joshi, W. Potz, and D. K. Ferry, in

Proc. 18th Intern. Conf. Phys. Semiconductors, Stockholm, 1986,Ed. by 0. Engstrom, in press.

5. P. Lugli, C. Jacoboni, L. Reggiani, and P. Kocevar, submitted forpublication.

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,M/7-86

Ultrashort transients in semiconductors

H.L. Grubin, M.A. Osman and J.P.KreskovskyScientific Research Associates, Inc.

P.O. box 1058, Glastonbury, Connecticut 06033-6058

Abstract

Over the past decade a variety of physical phenomena have emerged which are exercisingprofound constraints on the speed of devices whose primary transient is electronic inorigin. For example, the broad high frequency extrapolations of high speed arising fromvelocity overshoot were found to be limited by the effects of space charge and theimposition of fields that take a finite time to reach their specified level. Much of thelatter problem may be minimized through optical processes, but in its place rests theonstrain~t of electron-hole interact-n eand other p;"jeu that may require quantumcoherence for il.. taticn. In the case of electron-hole interaction it has been foundthat it provides an important channel through which electron energy is transferred to thelattice, and that the electron cooling rates at high hole concentrations are higher whenthe electron-hole interaction is included, than when it is ignored. At low holeconcentrations the reverse is true. The effect of electron-hole interaction alsoinfluences velocity overshoot. In the case of quantum coherence, of interest is theeffect, e.g., of temporal scatterers (phonons) on the time dependent wave functions andmodifications of interference effects. The influence of space charge on transient velocityovershoot, the effects of electron-hole interaction on the relaxation of an electron-holeplasma, and the effects of time dependent and time independent scattering on quantuminterference will be discussed.

Introduction

The early suggestions of Ruchl and subsequent calculations of others, have suggestedthat high mobility semiconductors were capable of extraordinary high speeds (l0Scm/sec)provided they were sampled for sufficiently short time intervals. For submicron andultrasubmicron devices, this translates into time scales of 0.1 ps, and cutoff frequenciesof 10 3GHz. While these predictions are clearly impressive, of serious concern is aninability of realizing the modest goal of room temperature operation at 100GHz, and speedsof less than 5ps, within the framework of a three terminal device. The difficulty ofachieving anything close to that offered by the Ruch calculations has been discussed by anumber of workers over the past several years. Perhaps the simplest statement summarizingthese studies is that Ruch's assumptions are not realized experimentally. Theseassumptions include: 1) uniform fields within the device, 2) zero rise time of the electricfield, and 3) neglect of displacement currents. Because of the importance of theseassumptions, each will be briefly reviewed, along with the influence of nonequilibriumholes and phonons. Additionally the influence of quantum mechanical constraints ontransport will be discussed.

Nonuniforg fields

Electric fields in devices are nonuniform, as illustrated for the three terminalpermeable base transistor (PBT). The PBT as currently configured, is a micron-lengthstructure with ultra-submicron features. A section of the PDT is shown in figure 1, alongwith its an equivalent planar FET. As shown in figure 22, for a bias of one volt betweenthe emitter and the collector the potential distribution indicates the expected fieldnon-uniformity. For example, near the cathode, the field is low, increasing to a valuenear 20 kl/cm and finally approaching a low value near the collector contact. In uniformfield studies the predicted peak velocity approaches 108 cm/sec, followed by a decay tosteady state. The peak velocity is achieved with a carrier concentration that is close toits equilibrium value. Decay is dominated by electron transfer, with a weaker componentdue to momentum relaxation. In steady state the gamma valley carrier velocity is high,while the concentration of gamma valley carriers is low. In the PBT the initial high peakin current associated with velocity overshoot is not achieved. Rather the high velocity isoften achieved at the expense of carrier concentration. This in displayed in figures 3 and4 which show the distributon of carrier density and velocity for the PBT. It is worthwhilenoting that while the distribution of charge and velocity are not as encouraging as thatfrom the uniform field calculations, the current voltage levels are between a factor of 3and 4 above that obtained from the drift and diffusion equation calculations.

Finite rise tjing gontributions

A second issue is the zero rise time. In virtually all transient transportcalculations, overshoot phenomena is obtained by examining the response of carriers to anelectric field that is suddenly turned on. Under conditions of finite rise time,relaxation effects are coupled to the increasing field and the peak velocity is diminished,as shown in figure 53. We note that there is a qualitative similarity between the finiterise time anu the response of carriers to a gradual spatially-varying electric field.

Displacement current effects

To carry the discussion of non-uniform fields even further, calculations have beenperformed for the paradigm two-terminal N+N-N+ submicron structure shown infigure 64. The structure results in non-uniform fields and when transients are computedit is a current transient that is involved rather than a velocity transient. Indeed, invirtually all structures it is a current transient rather than a velocity transient that ismeasured. As shown in figure 7, there is a peak value to th. current that occurs withinthe time scale over which velocity overshoot may be expected. However, the details of thetransient show that the initial time dependent behavior is associated entirely with fieldrearrangement, and that the transient is entirely a displacement current transient 4 .

Noneauilibrium electron hole phonon interaction

In overshoot studies most inelastic collisions are treated assuming the emission of asingle phonon and that the emitted phonon is in equilibrium with the lattice. Forenergetic electrons, a sequence of as many as ten to fifteen phonons maybe emitted as theelectron is relaxing, and as a result an excess number of phonons may result in a lack ofphonon equilibrium with the lattice. This phenomena is illustrated for an unbiased slab ofgallium arsenide subject to a uniform photoexcited laser pulse of energy 1.8 ev. Underthis situation holes as well as electrons are excited. The energy of the excited electronsis 250 Mey, while that of the holes is 40 Mey. The time evolution of the resultingelectron, hole and phonon distribution is calculated using Monte Carlo techniques. Figure8 shows the relaxation of the electrons to equilibrium with and without hot phonons. Asseen, the inclusion of the hot phonons results in a delay of the relaxation to steady state.The situation in the presence of finite fields and equilbrium phonons is discussed nextand shown in figures 9 and 105. These calculations were performed assumingnonequilibrium interactions between the gamma valley electrons and holes. L-valley holeinteraction was not examined. The first point of note is that the electron holeinteraction is an inelastic collision. In addition, since both electrons and holes areresponding to the presence of the applied fields, momentum scattering of the electrons andholes can result in a reduction of the electron velocity, and in some cases turn theelectron in the "wrong" direction. The net result is a decrease in the peak electronvelocity. It is impc,.-tant to note that while the peak velocity is lower for thecalculation with the electron-hole interaction, there is a cross-over where, for a certaininterval of time, the carrier velocity in the presence of holes exceeds that obtained inthe absence of holes. This arises because the electron hole interaction is an efficientmethod of energy exchange and as a result, the electrons are not energetic enough totransfer into the L-valley. In other words, there is a greater fraction of electrons thatare retained in the gamma valley when holes are included than when they are not included.The possibility exists that the velocity of injected electrons in p-type material may, fora range of values of electric field, be higher than that of thermally generated carriers in

N-type material.

Quantum contributions

It will be noticed that all of our discussion has been confined for electric fields andscalar potentials. It has been presumed that we are dealing with a gauge invariant system,and that the choice of gauge is not relevant. As discussed by Aharanov and Bohm and 6 ,the vector potential can lead to observables such as interference patterns for electronsthat traverse ostensible field-free regions. In other words, interference patterns areaccounted for in terms of differences in the vector potentials encountered by electrons ontwo opposite sides of a flux line. To see how these effects enter into the problem it isuseful to re-examine an approach to quantum transport introduced earlier 7 .

In reference 7, which introduced the Wigner function as an approach to quantum transportIn electron devices, a set of hydrodynamic balance equations was introduced. While thehydrodynamic balance equations were developed for a system of particles, the equations maybe reduced to examine transport for carriers not subject to scattering. Thus, the singleparticle wave functions are of interest.

Determining the single particle wave functions requires sol-tions to the single particleSchrodinger's equation. Rewriting Schrodinger's equation in terms of a hydrodynamic modelrequires solutions to the following two equations:

- + divJ- 0

dx)M - -V(O+V) (2)

dt

where V(x) is a classical scaler potential and Q is the quantum potential

h 2 v2 4r (3)2m

The reference 7 study suggests that both e . V 'Vand J-ev be --egarded as the basicphysical quantities representing properties of the continuous fluid. It is noted that thevector potential has dropped out of the basic equations, which have a completely classicalform.

Philippidis, Bohm and Kaye (PBK) 8 argued that the above equations were incomplete.The most important objection arose because certain subsidiary conditions were notsatisfied. To see this, the wave function for the system is written as - Rexp (iS1h,with the carrier velocity expressed as

VS e (4)m mic

If it is now required that the wave function be single valued, then we are forced toadmit the condition

This means that the observable movements of electron transport in a device mustnecessarily be restricted in such a way that the integral of its velocity around a circuitcontaining a flux line depends upon the flux within the circuit. AB argue that there is nointuitively clear justification for the equation (8) constraint. Rather, they propose toassume that the vector potential has a physical meaning.

Figure 11 is a plot of the quantum potential for an electron passing through a dual slitin the absence of a magnetic field. The smooth parabolic hills are in the immediatevicinity of the slit. Particles subject to such a potential accelerate smoothly in theforward and sideward direction. This can be seen from the family of velocity trajectoriesdisplayed in figure 12. Note, near the slits where the wave functions are localizedpackets, the quantum potential exhibits two broad peaks. As the wave packets recede andapread, they begin to overlap significantly and their interference properties come intoplay. The shape of the quantum potential, a long way from the slits, is determined by theboundary conditions, the shape and size of the slits. It is through the force exerted bythe quantum potential on the particle that the boundary conditions are made physicallysignificant far from the slits. For the case when a flux line is place between the slits,the quantum potential is altered, as shown in figure 13. The velocity trajectories arealtered as shown in figure 14. The important point to notice is that the trajectories aredisplaced.

The above discussion reached initial device implementation by Datta, et al. 9 , whoconceptually altered the path length of a beam of particles as shown in figure 15, wherethe slit in the GaAs between the AlGaAs is exaggerated. Additionally they altered thephase through application of a magnetic field. These alterations lead to interference inthe downstream carrier wave functions. This result, as discussed in reference 9, is theappearance of oscillations in the conductance of the system.

An important point not addressed in reference 9 is the temporal duration of theconductance oscillations. it must be considered that the sporadic appearance of phononswill result in a loss of coherence and limit constructive interference to time scalesshorter than 100 featoseconds. For time scales longer than 100 femtoseconds, theconstructive interference effects will certainly be diminished. Whether they willdisappear is a matter to be determined. Here we note that cnherence is essential forquantum well devices. There are other issues that may tend to mitigate the appearance ofthe AB effect such as elastic scattering arising from surface roughness.

The driving force toward high speed devices is limited by the presence of carrier-carrier scattering, phonons and submicron feature sizes that tend to reduce the coherenceof scattered waves. While these effects do not lessen the importance of the nonequilibriumtransients, unless these effects are incorporated into the design of the device,applications may be limited.

Acknowledgements

This work was supported by the Office of Naval Research and the Air Force Office ofScientific Research.

i. J.G. Ruch, IEEE Trans. E!m:tron Devices ED-19, 652 (1982).

2. J.P. Kreskovsky and H.L. Grubin, Proceedings of NUMOS I (1987).

3. H.L. Grubin, J.P. Kreskovsky, Surface Sci. 132, 594 (1983).

4. H.L. Grubin, J.P. Kreskovsky, VLSI Electronics Microstructure 'J. 10, 237 (1985).

5. M.A. Osman, PhD Thesis, Arizona State University (1986).

6. Y. Aharonov and D. Bohn, Phys. Rev., 115, 485 (1985).

7. G.J. lafrate, H.L. Grubin, and D.K. Ferry, J. de Physique, C7, 307 (1981).

8. C. Philippidis, D. Bohn, and R.D. Kaye, 11 Nuovo Cimento, 71B, 75 (1982).

9. S. Datta, M.R. Mellcoh, S. Bandyopadhyay, and M.S. Lundstrom, Appl. Phys. Letts. 48,487 (1986).

D 0B(G)

I 1C(D)S E(S)I

LINE OF SYMMETRY

Figure 1. Schematic of FET and PBT.

1.0

0.8

VBE - 0.4

V - 1.0CE

0, 0.6

z

0.4 -

0.2

0

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

DISTANCE (microns)

Figure 2. PBT potential distribution along the central line of symmetry.

7$3 10

1.4

VBE . 0.4

v . 1.0

E Q

-o S1.0

0

cc 0.6

0.4

MOTE0.2

0 1 1

0 W~ 0.2 0.3 0.4 0.5 0,6 0.7 0.6 0.9 1.0

DISTANCE (mnicro-is)

Figure 3. PBT carrier distribution along line of symmetry.

S 0.6

0..0

0.6-C

0,4-!

-1.0

0.2

0 t I I I I 0

0 0.2 0.4 0.6 0 .6 0 . .DISTANCE (mDSronC}

Figure 4. PBT gamma valley velocity along line of symmetry.

V" • 17.6 kVcr~i

100.0

- 0.0

5.0Figure 5. Dependence of peak overshootvelocity on rise time. (From 13)).

1.0

0:5

0.02 3 4 5 6 7 8 9

(V I0*J cm/s*c

Figure 6. Schematic of structure forS *examining displacement current

CATHODE N' N- N ANODE effects. (From [4)).

'00ltoo

60

Figure 7. Initial current transient for Jo

the structure of figure 6. 40

2.0

O00 i 1 1 1

00on 02 04 06 00 tO I2 1

TIM( fto..

~U. iU

30r - -GaAs

T=77K

n x10'cm-"-5200 - h 1.8 eV

0..)

WI 0 0z

00.1 1.0 10.0

TIME (ps)

Figure 8. Energy relaxation with equilibriumphonons (1) and nonequilibriumphonons (2).

X 107

8 I I 80

T = T7K T = 77K

n z 10 '6 n = 10isCf"

3

-- h- --- -. 4..ca...

6 60

0

2 z 20

Uj

w 0

0 12 3 40 1 2 3

flmc(ps) TME (s

Figure 9. Overshoot witii and without Figure 10. Carrier distribution with and withoute-h scattering. (From (5)). e-h scattering. (From (51).

sli~t ali A

FIgure 11. Quantum potential for thetwo-slit system. (From (81).

Figure 12. Particle trajectories for thetwo-slit system. (From [8]).

Figure 13. Quantum potential for the AB effect(note the asymmetry). (From (81).

Y x

Cortacl dCootadI

IA, IS, x..O xLFigure 14. Trajectories for the A-B effect(note the shift of the overall

Fiu;ure 15. Proposed structure for quantum pattern). (From [8]).interference effect5. (From [9]))

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