UTILIZATION OF QUANTUM DISTRIBUTION FUNCTIONS FOR ...

JOURNAL DE PHYSIQUE

Colloque C7, supplément au n°10, Tome 42, octobre

1981

page C7-307

UTILIZATION OF QUANTUM DISTRIBUTION FUNCTIONS FOR ULTRA-SUBMICRON DEVICE TRANSPORT G.J. Iafrate, H.L. Grubin and D.K. Ferry

U.S. Army Electronics Technology and Devices Laboratory, Fort Monmouth, New-Jersey 07703, U.S.A. *Scientific Research Associates, Inc. P.O. Box 498, Glastonbury, Connecticut 06033, U.S.A. Colorado State University, Ft. Collins, Colorado 80523, U.S.A. Résumé - La représentation de Wigner est considérée comme une méthode utilisable pour décrire les phénomènes de transport quantique dans la physique des dispositifs submicroniques. La fonction de distribution de Wigner est rappelée ici et sa non unicité est discutée, ainsi que sa généralisation aux autres fonctions de distribution quantique. L'équation du mouvement pour la fonction de distribution de Wigner est décrite et un nouveau résultat, les moments de cette équation du mouvement, est obtenu. On montre que les équations du moment contiennent des corrections quantiques par rapport aux équations classiques et que ces termes quantiques ne sont pas négligeables lorsque les longueurs de transit des porteurs sont de l'ordre de la longueur d'onde de de Broglie. Abstract - The Wigner representation is considered as a method for describing quantum transport phenomena in connection with submicron device physics. The Wigner distribution function is reviewed, its non-uniqueness discussed; its generalization to other quantum distribution functions is examined. The equation of motion for the Wigner distribution function is described and a new result, the moments of this equation of motion, is derived. It is shown that the moment equations contain quantum corrections to the classical moment equations; these quantum terms are seen to be non-negligible when the carrier transit lengths are of the order of the particle deBroglie wavelength. Introduction - As semiconductor technology continues to pursue the scaling-down of IC device dimensions into the submicron (less than ten thousand Angstroms) regimej many novel and interesting questions will emerge concerning the physics of charged particles in semiconductors.

One of the more important topics to be considered is

that of the appropriate transport

picture to be used for a given spatial and tem-

poral regime. Moreover, from the point of view of device physics, it is most desirable to have a microscopic description of semiconductor transport which is computationally manageable or at least amenable to phenomenological treatment so that its properties can be meaningfully incorporated into device simulations.

In this paper

attention is focused on a useful transport methodology for the ultra-submicron regime.

At present, electronic devices with transit lengths in the ultra-submicron 2 region are coming to fruition due to the advent of MBE processing so that the need for an appropriate description of transport in this regime is indeed imperative. Semiconductor transport in the ultra-submicron regime approaches the category of quantum transport.

This is suggested by the fact that within the effective mass

approximation the thermal deBroglie wavelength for electrons in semiconductors (See Fig. 1) is of the order of ultra-submicron dimensions. Whereas classical transport

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981737

C7-308

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physics i s based on t h e concept of a p r o b a b i l i t y d i s t r i b u t i o n f u n c t i o n which i s defined over t h e phase space of t h e system, i n t h e quantum formulation of t r a n s p o r t physics, t h e concept of a phase space d i s t r i b u t i o n f u n c t i o n i s not p o s s i b l e inasmuch a s t h e non-cormnutation of t h e p o s i t i o n and momentum o p e r a t o r s ( t h e Heisenberg uncert a i n t y p r i n c i p l e ) precludes t h e p r e c i s e s p e c i f i c a t i o n of a point i n phase space. However, w i t h i n t h e matrix formulation of quantum mechanics, i t is p o s s i b l e t o cons t r u c t a "probability" d e n s i t y m a t r i x which i s o f t e n i n t e r p r e t e d a s t h e analog of the c l a s s i c a l distribution function.

Fig. 1:

001

Thermal deBroglie wavelength v e r s u s e f f e c t i v e mass.

0I EFFECTIVE MASS(rnYrnd

There i s y e t another approach t o t h e formulation of quantum t r a n s p o r t based on t h e concept of t h e Wigner d i s t r i b u t i o n f u n c t i o n (WDF)

3

.

This formalism i s p a r t i c u -

l a r l y a t t r a c t i v e f o r use i n ultra-submicron device t r a n s p o r t i n t h a t i t c o n t a i n s a l l of t h e quantum mechanical information about t h e s t a t e of t h e system y e t has elements of t h e c l a s s i c a l p i c t u r e i m p l i c i t l y b u i l t i n .

Thus, t h e i n t e n t of t h i s study i s t o

explore t h e p o t e n t i a l u s e f u l n e s s of t h e WDF a s w e l l a s o t h e r p o s s i b l e quantum d i s t r i b u t i o n f u n c t i o n s f o r d e s c r i b i n g quantum (ultra-submicron) device t r a n s p o r t i n semiconductors.

To t h i s end we f i r s t review t h e s a l i e n t f e a t u r e s of t h e WDF and

then d i s c u s s a new r e s u l t , t h e d e r i v a t i o n of t h e f i r s t t h r e e quantum moment equat i o n s using t h e WDF.

It i s shown t h a t t h e moment equations c o n t a i n quantum correc-

t i o n s t o t h e c l a s s i c a l moment equations; t h e s e quantum terms a r e non-negligible when t h e t r a n s i t l e n g t h s a r e of t h e o r d e r of t h e c a r r i e r deBroglie wavelength. Wigner D i s t r i b u t i o n Function

-

The Wigner d i s t r i b u t i o n function3 i s g e n e r a l l y defined

i n terms of a l l t h e generalized c o o r d i n a t e s and momenta of t h e system a s

However, f o r s i m p l i c i t y h e r e i n t h i s paper we d i s c u s s t h e p r o p e r t i e s of a s i n g l e c o o r d i n a t e and momentum WDF:

where

$(XI

r e p r e s e n t s t h e s t a t e of t h e system i n t h e c o o r d i n a t e r e p r e s e n t a t i o n .

Although we w i l l be t r e a t i n g t h e WDF f o r t h e s p e c i a l c a s e of pure s t a t e s , t h e adapt a t i o n of t h i s formalism t o i n c l u d e mixed s t a t e s i s accomplished through t h e generalization

(For example, f o r a system i n i s t h e p r o b a b i l i t y t o be i n s t a t e "n". n c o n t a c t with a h e a t bath a t constant temperature, Pn = e-EnikT.)

where P

The d i s t r i b u t i o n f u n c t i o n of Eq. (2) has i n t e r e s t i n g p r o p e r t i e s i n t h a t i n t e g r a t i o n over a l l momenta l e a d s t o t h e p r o b a b i l i t y d e n s i t y i n r e a l space; while i n t e g r a t i o n over a l l c o o r d i n a t e s l e a d s t o t h e p r o b a b i l i t y d e n s i t y i n momentum space3:

where

It follows from Eqs.

(4a,b) t h a t , f o r an observable w(x,p) which is e i t h e r a

f u n c t i o n of t h e momentum o p e r a t o r alone, t h e p o s i t i o n o p e r a t o r a l o n e , o r any addi3 t i v e combination t h e r e i n , t h e expectation v a l u e of t h e observable i s given by

=

jjW

average value.

PW(X, p ) d x y p

which i s analogous t o t h e c l a s s i c a l expression f o r t h e

Herein l i e s t h e i n t e r e s t i n g a s p e c t of t h e WDF; i t i s p o s s i b l e t o

t r a n s f e r many of t h e r e s u l t s of c l a s s i c a l t r a n s p o r t theory i n t o quantum t r a n s p o r t theory by simply r e p l a c i n g t h e c l a s s i c a l d i s t r i b u t i o n by t h e Wigner d i s t r i b u t i o n function.

However, u n l i k e t h e d e n s i t y m a t r i x , t h e WDF i t s e l f cannot be viewed a s

t h e quantum analog of t h e c l a s s i c a l d i s t r i b u t i o n f u n c t i o n s i n c e i t i s g e n e r a l l y nonp o s i t i v e d e f i n i t e and non-unique. Further resemblance of t h e Wigner d i s t r i b u t i o n f u n c t i o n t o t h e c l a s s i c a l d i s t r i b u t i o n f u n c t i o n i s apparent by examining t h e equation of time e v o l u t i o n f o r Pw(x,p). Upon assuming t h a t $(x) i n Eq. (2) s a t i s f i e s t h e Schrodinger equation f o r a system 3 2 w i t h Hamiltonian H=p /2m+V(x), i t h a s been shown t h a t Pw s a t i s f i e s t h e equation

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where

Alternately, 0 . P

W

can be expressed as

2

8Pw' - li [ sin

a {-&

$11

V'X'P~'~.P'

where it is understood that the position gradient operates only on the potential energy, V(x).

In the limit%+O,

8.p = w

classical collisionless Boltzmann equation.

3 ax a p

so that Eq. (6) reduces to the

The Wigner distribution function is not the quantum analog of the classical distribution function4 as evidenced by the fact that it is generally non-positive defnite and non-unique.

The non-uniqueness is clearly evident by noting that many

quantum distribution functions can be constructed which obey the sum rules of Eqs. (4a,b).

where $(p)

One other distribution function which we have examined in detail

is given by Eq. (4b).

In actuality the non-uniqueness and non-positive

definiteness of the Wigner distribution function can be traced to the non-commutation of the position and momentum operators. Moment Equations

-

In this section, we derive the first three moments of a "Wigner-

Bo1tzmann"-like transport equation,

This equation was constructed to include an ad-hoc collision term which may not necessarily express the same phenomenology as that of the classical Boltzmann transport equation, since PW is not a true probability distribution function. These problems are conceptually reduced when dealing with moments in a relaxation approximation. The moment equations are obtained by multiplying Eq. (8) by an appropriate

a+ at

#-

function of momentum, #(p),

+

and then integrating over all momenta to obtain:

l a z - - m ax

.-,

(9) a 2 n +l

n =O

where
refers to an integration over momentum.

col I. In making the assumptions that

+(p) be an analytic function of momentum and that Pw(x,p) limits, it follows that

vanish at the momentum

and that Eq. (9) becomes

2ntl r/

=