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UTILIZATION OF QUANTUM DISTRIBUTION FUNCTIONS FOR ULTRA-SUBMICRON DEVICE
TRANSPORT
G. Iafrate, H. Grubin, D. Ferry
To cite this version:
G. Iafrate, H. Grubin, D. Ferry. UTILIZATION OF QUANTUM DISTRIBUTION FUNCTIONS
FOR ULTRA-SUBMICRON DEVICE TRANSPORT. Journal de Physique Colloques, 1981, 42 (C7),
pp.C7-307-C7-312. �10.1051/jphyscol:1981737�. �jpa-00221674�
JOURNAL DE PHYSIQUE
Colloque C7, supplément au n°10, Tome 42, octobre 1981 page C7-307
Résumé - La représentation de Wigner est considérée comme une méthode utilisable pour décrire les phénomènes de transport quantique dans la physique des disposi- tifs 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.
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.
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 equa- tion 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 equa- tions; 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 compu- tationally 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 region are coming to fruition due to the advent of MBE processing so that the need 2 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 JOURNAL DE PHYSIQUE
p h y s i c s 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 d e f i n e d over t h e phase space of t h e system, i n t h e quantum f o r m u l a t i o n of t r a n s p o r t p h y s i c s , t h e concept of a phase s p a c e 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 uncer- t a i n t y p r i n c i p l e ) p r e c l u d e s 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 p o i n t i n phase space.
However, w i t h i n t h e m a t r i x f o r m u l a t i o n of quantum mechanics, i t i s p o s s i b l e t o con- s t r u c t a " p r o b a b i l i t y " 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 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 .
001 0 I
EFFECTIVE MASS(rnYrnd
Fig. 1: Thermal d e B r o g l i e wavelength v e r s u s e f f e c t i v e mass.
There i s y e t a n o t h e r approach t o t h e f o r m u l a t i o n 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 u s e i n ultra-submicron d e v i c e 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 i n f o r m a t i o n about t h e s t a t e of t h e system y e t h a s 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 s t u d y i s t o e x p l o r e 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) d e v i c e 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 equa- t i o n s u s i n g t h e WDF. It i s shown t h a t t h e moment e q u a t i o n s c o n t a i n quantum c o r r e c - t i o n s t o t h e c l a s s i c a l moment e q u a t i o n s ; 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 d e B r o g l i e 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 f u n c t i o n 3 i s g e n e r a l l y d e f i n e d i n terms of a l l t h e g e n e r a l i z e d c o o r d i n a t e s and momenta of t h e system a sHowever, 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 b e 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 adap- t 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 gener- a l i z a t i o nwhere P i s t h e p r o b a b i l i t y t o b e i n s t a t e "n". (For example, f o r a system i n n
c o n t a c t w i t h a h e a t bath a t c o n s t a n t temperature, Pn = e-EnikT.)
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; w h i l e 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 f o l l o w s from Eqs. (4a,b) t h a t , f o r an o b s e r v a b l e 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 a l o n e , 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 addi- t i v e combination t h e r e i n , t h e e x p e c t a t i o n v a l u e of t h e o b s e r v a b l e i s given by 3
< W >
= jj
W 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 e x p r e s s i o n f o r t h e average v a l u 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 t h e o r y i n t o quantum t r a n s p o r t t h e o r y 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 f u n c t i o n . 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 a n a l o g 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 non- p o s i t i v e d e f i n i t e and non-unique.F u r t h e r 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 e q u a t i o n 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 e q u a t i o n f o r a system 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 e q u a t i o n 2 3
JOURNAL DE PHYSIQUE
where
2
a
Alternately, 0 . P W can be expressed as
8 P w ' - - li [
sin- {-& $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 = - - 3
so that Eq. (6) reduces to thew
ax
a pclassical collisionless Boltzmann equation.
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). One other distribution function which we have examined in detail
where $(p) is given by Eq. (4b). In actuality the non-uniqueness and non-positive definiteness of the Wigner distribution function can be traced to the non-commutation of the position and momentum operators.
Moment Equations
-
In this section, we derive the first three moments of a "Wigner- Bo1tzmann"-like transport equation,This equation was constructed to include an ad-hoc collision term which may not necessarily express the same phenomenology as that of the classical Boltzmann trans- port equation, since P is not a true probability distribution function. These
W
problems are conceptually reduced when dealing with moments in a relaxation approxi- mation. The moment equations are obtained by multiplying Eq. (8) by an appro- priate function of momentum, #(p), and then integrating over all momenta to obtain: #-
a+ l a z
+ - -
< + p >at m
a x
(9).-,
a 2 n + l
n =O col I.
where < > refers to an integration over momentum. In making the assumptions that +(p) be an analytic function of momentum and that Pw(x,p) vanish at the momentum limits, it follows that
and that Eq. (9) becomes
2 n t l r /
+ f
n - o(&,'"I
( 2 n t 1 ) !(cv(x$<$&>
a X 2 n + l= < ~ ( a )
C O ~ 1 . >For specific values of qp)=pO, p, and P
-
Eq. (11) becomes 2m'where upon reduction
thereby showing the dependence of <pn>. FOT n=0,1,2,3 we show <pn> explicitly:
Note that use of Eqs. (144in Eq. @2a) results in the correct quantum mechanical continuity equation.
In order to see the transition between the quantum and classical regimes, we invoke a wavefunction of the form
q (
x , t )=
A ( x , t ) e is(x,t)/ti,and so,
C7-312 JOURNAL DE PHYSIQUE
where p(x,t) is the probability density and v(x,t) if the ensemble velocity. Insert- ing the wavefunction of Eq. (15) into Eq. (14) results in
It is evident from Eq. (16) that the terms possessing an explicit dependence on
I ,
K
1 1 are the quantum corrections to the classical momentum-density moments. Further-more, insertion of these momentum-moments into Eqs. (12) results in a set of moment equations which contain explicit quantum corrections as well. Of course, as h + O , these moment equations reduce to the classical, zero temperature, moment equations.
The logarithmic derivative term appearing in Eqs. (16) is not uncommon; such a term appears in the real part of the Schrodinger equation when the wavefunction associated with Eq. (15) is used 6
.
In the limit where this term is negligible, theSehrodinger equation goes over to the Hamilton-Jacobi equation.
We have made order of magnitude estimates of the strengths of the quantum correc- tion term appearing in Eq. (16). Using a Gaussian spatial variation for P(x), and
1 2
a thermal ensemble valuefor-mv, typical of central valley GaAs electrons, we find 2
that the quantum correction is substantial for widths of the order of 80
i.
Acknowledgment
This work was supported by the U.S. Army Research Office.
References
1. Barker, J.R. and D. K. Ferry, Solid St. Electron.
23,
519 (1980).2. Malik, R.J., K. Board, L.F. Eastman, G.E.C. Wood, T.R. AuCoin and R.L. Ross, The Institute of Physics, Inst. Phys. Conf. Ser. No. 59, Ch. 9.
3. Wigner, E., Phys. Rev.
5,
749 (1932).4. Grubin, H.L., D.K. Ferry, G.J. Iafrate, and J.R. Barker, in "VLSI Electronics:
Microstructure Science", Ed. by N.G. Einspruch (Academic, New York) To be published.
5. Iafrate, G.J., H.L. Grubin and D.K. Ferry, Bull. Am. Phys. Soc.
26,
458 (1981).(Complete text under preparation.)
6. Messiah, A., "Quantum Mechanics", Vol. 1, p. 222, Interscience Publishers, Inc., New York, 1961.
