The Quantum Hydrodynamic Smooth Potentialdownloads.hindawi.com/journals/vlsi/1998/089074.pdf · Anextension of the quantum hydrodynamic (QHD)model is discussed which is valid for

VLSIDESIGN1998, Vol. 6, Nos. (1-4), pp. 17-20Repnnts available directly from the publisherPhotocopying permitted by license only

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The Quantum Hydrodynamic Smooth Effective PotentialCARL L. GARDNER** and CHRISTIAN RINGHOFER

Department ofMathematics, Arizona State University, Tempe, AZ 85287-1804

An extension of the quantum hydrodynamic (QHD) model is discussed which is valid forclassical potentials with discontinuities. The effective stress tensor for the QHD equationscancels the leading singularity in the classical potential at a barrier and leaves a residualsmooth effective potential with a lower potential height in the barrier region. The smoothingmakes the barrier partially transparent to the particle flow and provides the mechanism forparticle tunneling in the QHD model.

Keywords: quantum hydrodynamic model, smooth effective potential

In his lectures on Statistical Mechanics, Feynmanderives an effective quantum potential by a Gaussian

smoothing of the classical potential. After demon-strating that the effective free energy based on theeffective potential is accurate for smooth classical

potentials like the anharmonic oscillator, he goes onto say that "it fails in its present form when the [clas-sical] potential has a very large derivative as in thecase of hard-sphere interatomic potential" or forpotential barriers in quantum semiconductor devices.In this note, we discuss an extension of Feynman’sideas to a smooth effective potential for the quantumhydrodynamic (QHD) model that is valid for the tech-

nologically important case of potentials with disconti-nuities.The QHD equations have the same form as the

classical hydrodynamic equations:

+ w--(nli) 0 (1)oxi

) )V mnuj(mnuj) + (mnuiuj- Pij) -n (2)

OW 0(uiW ujPij + qi) --nui

v

(3)

where repeated indices are summed over and where n

is the particle density, u is the velocity, rn is the parti-cle mass, Pij is the stress tensor, V is the classicalpotential energy, W is the energy density, q is the heatflux, and TO is the ambient temperature. Collisioneffects are modeled by the relaxation time approxima-tion, with momentum and energy relaxation times xpand xw. Quantum effects enter through the expressionfor the stress tensor (and for the energy densityderived from the stress tensor).

Originally the quantum correction to the stress ten-

sor in the QHD equations was given to O(h2) andinvolved second derivatives of the classical potential.In the spirit of Feynman, Ferry and Zhou derived a

* Research supported in part by the U.S. Army Research Office under grant DAAH04-95-1-0122.+ Corresponding author. E-mail [email protected]. Research supported in part by ARPA under grant F49620-93-1-0062.

18 CARL L. GARDNER and CHRISTIAN RINGHOFER

smoothed quantum potential for the QHD equationsby linearizing an equation for the equilibrium densitymatrix. The Feynman effective partition functioninvolves a smoothed potential of the form

Va2(X) ] dy { (x- y)2 ) V(y) (4)x/2aexp

2a2

where a2 oc fJh2/m and 1/T is the inverse tempera-ture. The Ferry-Zhou effective stress tensor involves

the difference between the smoothed and the localquantum potential-h272n/8mn + V. Their smoothingfunction is of the form exp{-(x-y)Z/2a2}/Ix-yl.Neither of these two approaches which involve justa spatial averaging has enough smoothing to handlediscontinuities in V, since second spatial derivativesof the smoothed potential appear in both Feynman’sfree energy and the QHD equations.

To derive the new effective stress tensor and energydensity, we construct an effective density matrix as an

O(V) solution to the Bloch equation. Then using theeffective density matrix, we take moments of the

quantum Liouville equation to derive the QHD equa-tions with the effective stress tensor and energy den-sity [2].The effective density matrix has the form

p(,X,y)

{ m (x-y)2exp -2--0.04

0.02

0

-0.02

-0.04

-0.06

Smooth------Bloch

where V is given in center-of-mass coordinates

R=. (x+y),s=x-yby

f ( 2m[5 )3/22--- dfJ’ d3X x

{ 2m[ X,2}exp(13 ,) ([3 + [3,)h2

P’s)] (6)

In constructing the effective density matrix, it is the

change 6V (max V rain V}) over a character-istic length scale that is assumed small rather than I3V.The change 6V over a characteristic length scale

may not be small at a potential step. In fact, for a 0.2eV barrier, 6V 8 at 300 K and 32 at 77 K. How-ever, numerical comparisons demonstrate excellentagreement between the first three moments of the

equilibrium full density matrix and the effective den-

sity matrix for the Bloch equation with a barrier

potential for [6V < 4 (see Fig. 1), and good qualita-tive agreement for 138V < 20 (see Fig. 2). Themoments of the O(h2) density matrix are in severe

quantitative and qualitative disagreement with themoments of the full density matrix.

Using the effective density matrix in the moment

expansion of the quantum Liouville equation, we

-150 -i00 -50 0 50 i00 150

X

FIGURE Quantum term in the energy in eV for electrons in a 0.1 eV double barrier in OaAs at T 300 K. The barriers are 25/i and thewell is 50 A wide. x is in A for all figures

THE QUANTUM HYDRODYNAMIC SMOOTH EFFECTIVE POTENTIAL 19

0.2

0 1

0

-0.1

-0.2

-0.3

-0.4

SmoothBloch

- .so - oo -so o so--x

FIGURE 2 Quantum term in the energy in eV for electrons in a 0.5 eV double barrier

obtain the QHD conservation laws as the first threemoments with

h2n 2VPij- -nTij-

4mT oa5:i3xj (7)

3 2 ----]J’2nV2vW -nT + -mnu +

8rnT(8)

where the "quantum potential" is

v(13,x)1 13 2 2m13 3/:2

fo fa3x,)h2) x

exp -(_ ,)(+,)h(X’-x) V(X’). (9)

e quantum coection to the classical stress tensorand energy density is valid to all orders of and to

first order in gV,d involves bo a smooing inte-

gration of the classical potential over space and an

averaging integration over temperature.We define the 1D smoo effective potential in the

momentum conseation equation (2) as the most sin-

gulp of V- P l"

h2 d2VU V 4

4mT dx2"(10)

e double integration (over both space and tempera-ture) provides sufficient smooing so that the Plte in the smooth effective potential actually ccels

the leading singularity in the classical potential at a

barrier (see Fig. 3), leaving a residual smooth effec-tive potential with a lower potential height in the bar-rier region. This cancellation and smoothing makesthe barriers partially transparent to the particle flowand provides the mechanism for particle tunneling inthe QHD model. Note that the effective barrier heightapproaches zero as T 0. This effect explains influid dynamical terms why particle tunneling is

enhanced at low temperatures. As T -+ o, the effec-tive potential approaches the classical double barrier

potential and quantum effects in the QHD model are

suppressed.

10.7 20.50.25 1

0

Log [T/300 K]-i00

io0

FIGURE 3 Smooth effective potential for electrons in GaAs for50/ wide unit potential double barriers and 50/, wide well as afunction ofx and logl0(T/300 K)

20 CARL L. GARDNER and CHRISTIAN RINGHOFER

ReferencesD. K. Ferry and J.-R. Zhou, "Form of the quantum potentialfor use in hydrodynamic equations for semiconductor devicemodeling," Physical Review, vol. B 48, pp. 7944-7950, 1993.

[2] C. L. Gardner and C. Ringhofer, "The smooth quantumpotential for the hydrodynamic model," Physical Review, vol.E 53, pp. 157-167, 1996.

BiographiesCarl L. Gardner is Professor of Mathematics at Ari-zona State University. His current research interestslie in classical and quantum semiconductor devicesimulation and computational fluid dynamics.

Christian Ringhofer is Professor of Mathematicsat Arizona State University. His current researchinterests include classical and quantum transportequations and moment models for semiconductordevice modeling.

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The Quantum Hydrodynamic Smooth Potentialdownloads.hindawi.com/journals/vlsi/1998/089074.pdf · Anextension of the quantum hydrodynamic (QHD)model is discussed which is valid for