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Using the wigner function for quantum transport in device simulation
Mathl. Comput.
Pergamon
Vol. 25, No. 12, pp. 33-53, 1997 Copyright@1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved Modelling
08957177/97
$17.00 + 0.09
PII: SO8957177(97)00093-9
Using the Wigner F’unction for Quantum Transport in Device Simulation M. NEDJALKOV AND I. DIMOV Center for Inform&& and Computer Technology Acad. G. Bountchev Str. B125A, Sofia, Bulgaria P.
BORDONE, R. BRUNETTI AND C. JACOBONI Istituto Nazionale per la Fisica della Materia Dipartimento di Fisica, Universitit di Modena Via Campi 213/A, 41100 Modena, Italy (Received March 1997; accepted April 1997)
Abstract-The Wigner function was introduced as a generalization of the concept of distribution function for quantum statistics. The aim of this work is pushing further the formal analogy between quantum and classical approaches. The Wigner function is defined as an ensemble average, i.e., in terms of a mixture of pure states. From the point of view of basic physics, it would be very appealing to be able to define a Wigner function also for pure states and the associated expectation values for quantum observables, in strict analogy with the definition of mean value of a physical quantity in classical mechanics; then correct results for any quantum system should be recovered as appropriate superpositions of such “pure-state” quantities. We will show that this is actually possible, at the co& of dealing with generalized functions in place of proper functions,
Keywords-Wigner function, Quantum transport, Convergence procedure, Pure-state Wigner function, Generalized functions.
1. INTRODUCTION Transport in mesoscopic systems has been widely investigated in recent years [l-4]. This increasing interest is mainly related to the fundamental problems of the basic physics involved, as well as to possible utilization of such systems in the electronic device production. Since the dimensions of mesoscopic systems are comparable with typical electron lengths, a correct analysis of transport phenomena in such systems requires a detailed
coherence quantum-
mechanical treatment. The general case of interest is that of a mesoscopic semiconductor structure coupled with two leads considered as reservoirs at equilibrium. The potential can have an arbitrary profile inside the structure, while it takes constant and different values in the two leads. Thus, the system is open and, in the general case, far from equilibrium. The transport properties of the electron system are provided by quantum equations originating from the Liouville-von Neumann equation for the density matrix operator. In order to study such structures, the Wigner function [5-81 is a valid tool. In fact, it provides a rigorous quantum-mechanical approach and it constitutes a direct link between quantum and The authors would like to thank T. Gramchev (University of Cagliari) for helpful discussions and comments. work has been partially funded by A.R.O. and O.N.R. through E.R.O. and 1501, MM449 of MSETB.
33
This
M. NEDJALKOV
34
et al.
classical descriptions of the evolution of the system in the phase space through a distribution (z), the Wigner function is defined as the Weyl-Wigner transform f(r, K, t). In one dimension of the density matrix operator
where
]q) is the state
of the system
and the bar means
ensemble
average.
f”
has several
interesting properties. It is a real function that, when integrated over K, provides the correct probability distribution for the space coordinate and also, when integrated over Z, the correct quantum mechanical probabilities for the momentum K. Furthermore, one may get the correct expectation values of any observable A as
where A(z, K) is the Weyl-Wigner
transform
s
of the operator
+oO
A(x, K)
=
-co
emiKq
t+;qlA+
A:
> .
The Wigner function was introduced as a generalization of the concept of distribution function to quantum statistics [5]. Here, we are interested in pushing the formal analogy between quantum (l), the and classical approach further on. Since an ensemble average is included in equation Wigner function is defined in terms of a mixture of pure states. From the point of view of basic physics, it would be very interesting to be able to extend the concept of Wigner function also for pure states; the associated expectation values of quantum observables are then in strict analogy with the mean values of the classical mechanics. The correct quantum average values for the physical system of interest should be recovered as suitable superpositions of such “pure-state” quantities. Wigner functions for pure states have been considered in the literature [9,10] mainly in connection with localized states of a discrete spectrum. In particular, Berry [lo] used them to define a “spectral Wigner function”. Since, however, we are interested in open systems, in order to carry on the project described above we must face a fundamental analytical problem. The Wigner function and, as a consequence, the observable expectation values, are defined in terms of correlation functions, which do not vanish at infinity for pure states leading to nonconverging integrals. In this work, we will show that it is actually possible to define the Wigner function and the expectation value of an observable for pure states, at the cost of dealing with generalized functions in place of proper functions. The generalized-function formalism has been used recently for the calculation of some quantities related to the Wigner picture of quantum transport [11,12]. In [ll], the definition of the potential-scattering kernel-function in the Wigner equation for a step potential has been obtained. In [12], the explicit form of the Wigner function for a square barrier eigenstate (with equal left and right potential values) is presented. Usually, the perturbation theory is used in dealing with quantum transport. The form of the quantum equation describing the dynamics of the electron system depends on the choice of the unperturbed Hamiltonian Ho. For example, Ho (in single-electron approximation [13]) can describe the free electron dynamics, or it can also include the potential profile, or at least part eigenstate set of He is used to obtain the appropriate form of the of it [14]. The corresponding evolution equation and to define the initial and boundary conditions. Since the system is open, the eigenstates above a certain energy are degenerate, and proper linear combinations of such eigenstates has to be chosen. It has been recognized that a natural basis of quantum states for such systems is provided by the so-called “scattering states” [15]. It
Quantum Transport
has also been demonstrated solution
of the Wigner
[16] that
transport
boundary
equation
35
values that
assure the stability
are easily expressed
of the numerical
in terms of this basis.
The orthonormality property of the degenerate scattering states, very important for their use as an eigenstate set, has been subject of discussion in the literature [15,17]. The difference between the potential values in the two leads places such problem outside the traditional scattering theories. In [17], a correction to the Esaki-Tsu’s expression [18] for the resonant-tunneling current due to an assumed nonorthogonality of the degenerate scattering states is proposed. In [15], however, an orthonormality proof of the scattering states is presented, and the correction proposed in [17] is rejected. This proof has been given following an involved and quite indirect procedure. We present, in Section 2, such proof in a straightforward way, performing the direct calculation with the scattering states. In Section 3, the concept of pure-state Wigner function in terms of generalized discussed, and the explicit form is given for the case of a potential step. In Section
4, we show that it is possible
to use the obtained
pure-state
expression for the expectation value of an observable in the Wigner Conclusions and final remarks are drawn in Section 5. 2.
PHYSICAL
SYSTEM
AND
Wigner
functions
functions
is
in the
picture.
SCATTERING
STATES
We consider a one-dimensional problem for a mesoscopic potential with arbitrary profile and two leads with constant but different potential values. The scattering states are formed by a plane wave in each lead incident towards the mesoscopic region, plus a reflected component and a transmitted part outgoing in the other lead. For a fixed value of the incoming wave vector, the reflected
and transmitted
parts depend
on the mesoscopic
potential.
We consider here a step-like potential profile having a positive value V for z < 0 and 0 for z > 0. The scattering states will be labelled by a value k indicating the momentum of the incoming wave. Thus, positive k will indicate particles coming from the left with kinetic energy ti2k2/2m above V, negative k from 0 to -ko = -dm indicate particles coming from the right with k below the step V, and negative k from -ko to -oc correspond to particles coming from the right with kinetic energy higher then the potential step. Being solutions of the Schrodinger equation, scattering states corresponding to different energy values are orthogonal. Here, we propose a direct and simple proof that also the degenerate scattering states are orthogonal. The states with energies below V are nondegenerate,
so we consider
the states
having
energy
above the step value z > 0, z < 0, z > 0, t < 0,
for k < -KC,, (4) for k > 0,
(m being th e e 1ec t ron effective mass), k’ = - dm for k < -ko, k’ = where ko = dm ,/m for k > 0. Due to the absence of translational invariance of the system, the scattering states are not eigenvectors of the momentum operator, and k is here a label for the incoming wave with kinetic energy h2k2/2m. The coefficients r and t axe given by:
t(k) =
&,,
k - k’ r(k) = k+lc”
Let us consider the scalar product of the scattering and k2 > 0. To develop the calculations, we introduce B(ft)eikzdt
= d(k)
t(k) - r(k) = 1. states @k,(z) and ‘$kl(z) the formal expression f iP
with
kl < -kc,
M. NEDJALKOV et al.
36
where P means “principal value” and 0 is the step function. Then the scalar product above can be written as
J
+m 6,
-lXl
WkJZ)
dz =
&
{ t(k1)nb
(k:
-
+ t(w+2)qk;
k2) +
it(kl)P
(&)
+ k2) + it(w(~2)P
& (
+
@2)7r6
+
r(k1)t(kz)nb
(h
-
Icg -
(h
it(k2)P
+ q
(
&
1
2>
2>
+ +(h)t(lcz)P
(G&J}*
Since kl < 0, ki < 0 and k2 > 0, kh > 0, the terms with the S-functions S(ki - k2) and 6(klgive no contribution, Then, by substituting equation (5) into equation (7)) we obtain S+m~;,(L)~~r(.)dZ=~{2n~~~(k:+k2)+2*~~~(kl+k~) 12 -m
2
21
k2 - k;
kl + m 1 kl - k; --- kz k2 + k; kl - k; +iiyqmk:
m k2
(7)
k!J
1
1 k’, + k2
(8)
1 11 *
With a straightforward calculation, the argument of the principal value can be shown to be zero, that means no contribution of the principal value to the scalar product. From the a-function properties it follows: 6(ki + k2) = lk2/k116(kl + ki) , and recalling that ICI< 0, we get that, substituted into equation (8) yields
Being ICI< 0 and k2 > 0, we have kl/jkl( = -1 and k2 = Ik2l, that leads to /+mti;,(Z)lh(Z)~Z= --M
(-~&+&~)6(kltk;) =-
kz k2 + ki
kl + kh ”
s (kl 1 k4)
(k: + kd kl + k’,
(10)
The S-function on the right-hand side (RHS) of equation (10) is different from zero only for ICI+ ki = 0, and this implies: ki + k2 = 0. Thus, when the argument of the B-function is zero also the coefficient in front of it vanishes. We can thus conclude that the RHS of equation (10) is identically zero, that proves the orthogonality of the scattering states. To check the normalization of the scattering states, let us consider the csse of two waves incident from the left (k 1,2 > 0) (the same proof is valid for ICI,2< 0). Using again equation (6)) we obtain
J_+_m G;, (zhz(z) dz = & {nWz - h) - iP (&-) ++W(kl - r(kl)d(kl + Gl)G2Mh
+t(kl)t(ka)d
-
k2) -
+ k2)
+ kz) - ir(kl)P
Wl)r(k2)P
(k; - k’l) + it(kl)t(k2)P
Quantum Since ki and k2 are positive, the &function properties be rewritten as
/+m -cm
1(1& (‘+h(~) dz =
6(kh - ki) = (ki/kl)d(kz
& {fla (kl + ip
(
in equation
kz)
j$q
-dkMkz) Using the relations
37
S(kl + kz) gives no contribution.
the &function
it follows:
Transport
(1+
- kl). Then the scalar product
r(k&oz)
+ T(kd& -&-
(5) and remembering
2
Furthermore,
from can
+t(k$($$) - r(kd& .
+ t(kMkz)&12
that k! - ki = kf - kc, the argument
of the
principal value can be shown to vanish. Therefore, considering that the first term on the RHS of equation (12) is different from zero only for ICI= kp, and using again equation (5) we are left with +O” $2, (z)& I --oo
(z) do = b(kr - kz),
(13)
that proves the normalization of the scattering states to the S-function. The right-incident states with energies below V are defined as z > 0, z < 0, where
-ko < k < 0, k’ = dp,
and
t(k) = In this case, the states side, the normalization
(14
&,
k + ik’ r(k) = k-
so the orthogonality are nondegenerate, can easily be obtained, to a &function
is already assured. On the other since the z < 0 contribution is
negligible.
Figure 1. Wigner function for a step potential with V > 0 for I < 0 and V = 0 for z > 0. The given cases are reported.
(15)
M. NEDJALKOV et al.
38
(b) V = 0.1 eV.
(c) V = 0.01 eV. Figure 1. (cont.)
The left- and right-incident scattering-state normalization corresponds to the normalization of left- and right-incident plane waves, in agreement with the Lipmann-Schwinger equation [15,19].
3. WIGNER
FUNCTION
FOR PURE
SCATTERING
STATES
In Figure 1, the Wigner function for a step potential is presented for three different values of the Step V. The associated density matrix is obtained under the hypotheses of local equilibrium in both coupled reservoirs (we consider Boltzmann statistics), and neglecting the electron-phonon
QuantumTransport
39
lb) Figure 2. Density matrix for a step potential with V = 0.3eV (V > 0 for z < 0 and V = 0 for t > 0). Parts (a) and (b) refer to real and imaginary parts, respectively.
interaction. Under these conditions, ive may assume that the density matrix is diagonal in the scattering states P(.zl17) = 2-l
~~dk~*(2+~P)Y’;(z-~7))S(k).
where g(k) = e-hak2/2mkBT and 2 is the partition function.
(16) To proceed in the calculation,
M. NEDJALKOV et al.
Flea1part, k = 8 x lo6 m-l
(k > 0).
Imaginary part, k = 8 x 10’ m-l (k > 0).
Figure 3. Real and imaginary parts of the correlation function in equation (16) for a step potential with V = 0.3eV for three different values of the wav&vector k. Positive values of k correspond to particles moving from the negative to the positive side of the t-axis. Negative k between 0 and ko (Part (c) of the figure) correspond to particles moving from the positive to the negative side of the c-axis, reflected back from the step since their energy is below V.
the k integration has to be divided into three parts, corresponding to the regions --OO < k 5 -ko, -k. c k _< 0, and.0 < k < +co, respectively, where the scattering states have different
Quantum Transport
Real part, k = -1 x log m-l
Imaginary
41
(k < +).
part, k = -1 x log m-l
(Ic < -/CO).
(b) Figure 3. (cont.)
expressions for different shown in Figure 2 as a with different k (see the oscillations generated by
ranges of the k value. The density matrix for the case V = 0.3eV is function of z and 77. The weighted superposition of scattering states integral in equation (16)) leads to a progressive smoothing out of the interference phenomena with increasing correlation length 17.
M. NEDJALKOV et al.
42
Real part, /c = -5 x lOa m-l
Imaginary part, k = -5 x lo8 m-l
(0 > k > -ko, ko = dm
(0 > k > -ko, ko = dm
= 7 x lO’m_‘).
= 7 x 107mq1).
(cl Figure 3 (cont.)
The Fourier transform in equation (1) imposes some requirements on the behavior of p as a function of 17. The basic question we face now is whether such requirements are fulfilled, i.e., whether equa, tions (1) and (2) still hold, when a pure state is used and there is no weight function g to assure the convergence of the integrals. In [ll], it has been shown that the convergence can be achieved
Quantum Transport
43
Realpart,o:=lJ.
Imaginary part, (Y = 0.
(4 Figure 4. Comparison between the correlation function in equation (16) without (Case (a), (Y = 0) and with (Case (b), (2 = 10’) the presence of a convergence factor for a step potential with V = 0.3eV and assuming k = -1 x log m-l.
for pure states if the eigenstates are normalized to unity. Here, however, the eigenstates axe normalized to a b-function. As a consequence, the required behavior of p can be provided only by the presence of a weight function g. 1101 25:12-O
M. NEDJALKOV
et al.
Real part, 0 = 10’ m-l. -
c(4)
>bV $0
c!f
Imaginary part, c2 = 10’ m-l
@I Figure 4. (cont.)
the meaning of a pure-state Wigner function As for the case of a pure-state density matrix po, can be recovered by using an appropriate asymptotic procedure in the basic definitions and relations. be written in terms of the pure-state density matrix The pure-state Wigner function can p&z, k, r]) as follows
Quantum Transport
45
(17) where the weight function equation (2) is well defined.
g is chosen in such a way that p(z,q) E L1 and the integral in The integration with respect to k has to be performed first. Here the
Wigner function is written as a mixture of pure-state density a mixture of “pure-state Wigner functions”, we must exchange
(a) k=SX
lO*m-‘.
(b) k = -1
x log m-‘.
Figure 5. Pure-state Wigner function (@(.z, and V = 0.3eV at the given values of k.
matrices. In order to define it as the order of the two integrations.
K, ka)) for the case a = 1.
X
lo7 m-l
46
M. NEDJALKOV et al.
(c) k = -1 X lOEm-‘. Figure 5. (cont.)
According to F!rbini theorem this is possible if the integral Js )~e(z, k, v)g(k)j dq dk exists. This is, unfortunately, not true in our case, since the correlation function Ipe(z, k, q)l = I~.J~z+~/~)$$(Eq/2)1 for large 9 behaves as a constant and convergence cannot be achieved. On the other hand, if the integration order of equation (17) is maintained, the inner integral, due to our hypotheses, gives a function in Lr . Since for a given f(z) E Lr, the following relation holds:
(a) a = 1. x lo8 m-l. Figure 6. Purestate Wigner function for a step potential with V = 0.3eV and k = 8 x lo* m-l at the given values of CL
Quantum Transport
(b) a = 8.
x
47
106m-‘.
Figure 6. (cont.)
Jli
[h(k,a) =J.%“~‘f(*)dz] =h(k) = /dk”f(*)&,
(18)
we can write
PO(Z, k, q)g(k)dk >
dq.
(19)
48
M. NEDJALKOV et al.
Once the factor e-al’rl is included, the Fubini theorem can be applied, and the integration order can be inverted to give
where
is a generalized function that can be defined as the pure-state Wigner function. The pure-state density matrix pe(z, Ic,q) is presented in Figure 3 for three different values of the wave-vector k. For states with energy above the potential step, a strong oscillating behavior of such function is observed, that does not allow convergence of the integral with respect to the correlation length 77. In Figure 4, the effect of the converging factor e-Ql’Jl on pe(t., k, 77)is shown for the case (Y = lo7 m-l and k = -1 x 10gm-l (corresponding to a particle moving from the positive to the negative part of the z-axis with kinetic energy higher than the step energy). Oscillations are damped by the exponential function, assuring the convergence of the integral used in equation (21) to define the pure-state Wigner function. It should be noticed that Q = 107m-r corresponds to a damping over a correlation length of about 800 A. A comparison between Figure 4 and Figure 2 allows us to understand the physical meaning of the approximation introduced by using the converging factor. In Figure 4, the converging factor smooths out the oscillations of the pure-state density matrix in correspondence of values of the correlation length 77for which, even for the case of the density matrix obtained from equation (16), the superposition of the states of the basis set has already significantly smoothed out the oscillations. For this reason, the physical properties of the system are not influenced, in practical calculations, by the introduction of a suitable converging factor that guarantees the convergence of the involved integrals.
(a) 0 = 1. X lo7 m-l. Figure 7. Wigner function for a step potential different values of the converging factor.
with V = 0.3eV
ss obtained
with
Quantum Transport
49
(b) a = 5. x 107m-‘.
(c)
a = 1.
x lo8 m-l
Figure 7. (cont.)
Pure-state Wigner functions are presented in Figure 5 for (Y = 10’ m-l at three values of k. The variation of the pure-state Wigner function with decreasing cx is displayed in Figure 6 for k = 8 x 10’ m-‘. When LYis large (Case (a)), the structures of the Wigner function are smeared out and broadened. Decreasing cx (Cases (b) and (c)) means increasing and sharpening f$‘(~, K, k, 0). In the limit of vanishing (Y f$‘(z, K, k, a), tends to the infinitely steep and narrow generalized function. Still, according to the properties of the generalized functions, the integral on the RHS of equation (20) is convergent. This can be seen in Figure 7, where the total Wigner function, as obtained for three specific values of cx, is presented. For a = 1. x lO’m_‘, the Wigner
se auI8saq? aq 0% VI0 sum!J ‘daw Isgua$od aq$ ahoqs &?laua Y!J!M !@$.I ay$ IIIOLJr)uappu!ap?s %!~aa??leDs aq? ~03 uo!ssaJdxa %u!puodsa~~oa aq;L '(9)UOil8nbau! UaA@ al8 (y)~puv (y).i aJay!
‘ 9
[zPi -
9 12(v +
,Y)lso3 (YMYN+
,a so3(Y)W+
(zz)
saop suog81n3p3 sMoqs uos!zedwo:,
p+aurnu syj&
‘(~1)
u! ampaDoad Su$haAuos uogvnba
aq? JO asn ay$ ‘S,D lp3tus LI3ua!3ylns Jo3 ‘38y7
uy uaA@ uoggap
ap? %I!sn pazgt3a.T ‘81 a.In%iJ u! pa$uasaJd au0 ay% I#M
ay? II! uoy8~a~u~ $uappu~oD ‘Qwg~rd
30 raplo ~8uo~~uaAuo:, %p8aJp
s! uoy~~ry
Quantum
Transport
4. MEAN
51
VALUES
In equation (20), fc(z, K, k) is used to obtain f”(z,K). Our goal in this section is to prove that it is possible to obtain the mean value of an observable in terms of a weighted average of mean values calculated with pure-state Wigner functions. Let us replace f”( z, K) in equation (2) using equation (20). For the sake of simplicity, A is chosen to depend only on K and, to avoid the z integration, we consider a position-dependent mean value (A(z)) =/A(K)
(~~~+Sg(“)~~(~,K,k,a)dk)
dK.
(24)
In order to use f$‘(z, K, k) in the evaluation of an observable mean value in the same way as we do with fW(z, K) in equation (2), we need to assume the validity of the following identity:
that is, once more we have to exchange the order of the two integrations. Since the inner integral on the RHS of equation (25) represents the expectation value of the observable A on the pure state k, in this way an observable mean value would be defined as an overlap with proper weights of “pure-state mean values”. Unfortunately equation (25) does not hold in the general case without making some assumptions on the function A(K) [20]. To give an example, equation (25) is verified when l (A( dK < +CCL But A(K) can also be a polynomial such that J A(K)fg(z, K, k) dK does not have meaning. This analytical difficulty can be again overcome by applying the same asymptotic procedure used in deriving equation (20). Since equation (24) is still well defined if A(K) is a polynomial, we can introduce again a converging factor e-fllrcl and write (A(z))
= hl+
/eTPIKjA(K)
(,lim+ /g(k)f,r(z,
K, k,a)dk)
Due to the presence of the e- Ml factor, we can invert the lim _,s+ with respect to K and use the Fubini theorem to obtain (A(z))
=
lim
lim
/Lo+ a-+0+
J g(k) (I
e-PIK’A(K)fg(z,
dK.
(26)
operation with the integral
K, k, a) dK
JA(K)(J~~+ J dk)fo”(z, K, k a>dk> dK K, k) dK dk, = Js(k) (ii?+ Je -P’K’A(K)fo”(z, >
>
dk
(27)
=
which is a generalization of equation (25) and gives exactly equation (25) when A(K) is a “wellbehaved” function. In order to give a practical example of the validity of the above equation, the results obtained for the mean value of particle density 72kand current jl, of a left-incident scattering state using equation (4) in the conventional Wigner function theory are compared with the same quantities evaluated with the use of fg(z, K, k). The first calculation leads to the following results:
nk =
& ((1 +2rcos(2kz)
+ r’(k))
0(-z)
+ t’(k)e(z))
, (28)
jk
= &
(k (1 - r2(k)) 0(-z)
+ k’t2(k)Q(z))
.
M. NEDJALKOV
52
et&.
The current is constant with respect to the position, as it can be easily seen using equation (5). The same quantities, as obtained from the superposition of pure-state Wigner functions are 12= lim M f+‘lc’f;(Z, P-+0s -_oo
K, Ic) dK,
(29)
j = lim bo e-DIKI ~fo ‘K w(z, K, k) dK. P-0 s -_oo The results in equation (28) are recovered by substituting by using the following asymptotic expressions: 00
e-fllKI sin[(K - a)21 dK=
lim
K-a
B--r0/ -_oo 00
equation (22) into equation (29) and
COsinK KdK=r,
s --oo
,-+lrcl cos [(K - a)zl lim P (K-a) dK=O’ D-o+ J -_m lim
J
co
e-pllcl sin [(K - a)~] dK = 0,
p-+0+ -_03
where a is a parameter.
5. CONCLUSIONS In the present paper, the use of the Wigner function for open systems has been critically reviewed. In particular, for the case of a density matrix diagonal in the basis set we have shown that it is possible to define a “pure-state” Wigner function and a “pure-state” expectation value of a quantum observable which can be suitably used to describe the Wigner function and the expectation values for the quantum system of interest. We have adopted the scattering states as basis set of the system and specialized our analysis to the case study of a step potential. The above results have been obtained by introducing suitable convergence factors. The use of these factors, both in the definition of the necessary generalized functions and as a practical tool to perform numerical integrations, has been justified by a detailed analysis of the properties of the Wigner function of real quantum systems. It may be noticed that the complete Wigner function discussed in the present paper (see Figure 1) corresponds to two electron ensembles in equilibrium with their native contacts. The presence of phonon scattering inside the device would tend to perturb the situation mainly by relaxing the extra kinetic energy gained by the electrons at the potential step. The rigorous inclusion of phonon scattering in the Wigner formalism is at present under investigation [14].
REFERENCES 1. D.K. 2. 3. 4. 5. 6. 7. 8. 9. 10.
Ferry,
J.R.
Barker
and C. Jacoboni,
Editors,
Granular
Nanoelectronics,
Plenum Press,
New York,
(1990). W.P. Kirk and M.A. Reed, Editors, Nanostructures and Mesoscopic Systems, Academic Press, San Diego, CA, (1992). D.K. Ferry, H.L. Grubin, C. Jacoboni and A.P. Jauho, Editors, Quantum lknaport in Ultnasmall Devices, Plenum Press, New York, (1994). S. Datta, Electronic 2%ansport in Mesoscopic Systems, Cambridge Press, Cambridge, (1995). E. Wigner, Phys. Rev. 40, 749, (1932). V.I. Tatarskii, Sov. Phys. Usp. 26, 311, (1983). M. Hillery, R. O’Connell, M. Scully and E. Wigner, Phys. Report 106, 121, (1984). G.J. I&ate, The Physics of Submicron Semiconductor Devices, (Edited by H.L. Grubin, D.K. Ferry and C. Jacoboni), p. 521, Plenum Press, New York, (1988). P. Carruthers and F. Zachariasen, Rev. Mod. Phys. 55, 245, (1983). M.V. Berry, Proc. R. Sot. Lond. A 423, 219, (1989).
Quantum Transport
53
11. M. Nedjalkov, I. Dimov, F. Rossi and C. Jacoboni, Convergency of the Monte Carlo algorighm for the solution of Wigner quantum-transport equation, Mathl. Comput. Modelling 23 (8/g), 159, (1996). 12. J.G. Muga, R. Sala and S. Brouard, Solid State Comm. 94, 877, (1995). 13. J. Rammer, Rev. Mod. Phys. 63, 781, (1991). (Edited by K. Hess), Plenum 14. R. Brunetti, C. Jacoboni and M. Nedjalkov, Hot Carriers in Semiconductors, Press, New York, (1996). 15. A.M. Kriman, N.C. Kluksdahl and D.K. Ferry, Phys. Rev. B 36, 5953, (1987). 16. W.R. Frensley, Rev. Mod. Phys. 62, 745, (1990). 17. D.D. Coon and H.C. Liu, Appl. Phys. Lett. 47, 172, (1985). 18. Ft. Tsu and L. Esaki, Appl. Phys. Lett. 22, 562, (1973). 19. E. Merzbacher, Quantum Mechanics, Wiley and Sons, New York, (1961). 20. Vladimirov, Equations of Mathematical Physics, Nauka, Moskow, (1971).
Pergamon
Vol. 25, No. 12, pp. 33-53, 1997 Copyright@1997 Elsevier Science Ltd Printed in Great Britain. All rights reserved Modelling
08957177/97
$17.00 + 0.09
PII: SO8957177(97)00093-9
Using the Wigner F’unction for Quantum Transport in Device Simulation M. NEDJALKOV AND I. DIMOV Center for Inform&& and Computer Technology Acad. G. Bountchev Str. B125A, Sofia, Bulgaria P.
BORDONE, R. BRUNETTI AND C. JACOBONI Istituto Nazionale per la Fisica della Materia Dipartimento di Fisica, Universitit di Modena Via Campi 213/A, 41100 Modena, Italy (Received March 1997; accepted April 1997)
Abstract-The Wigner function was introduced as a generalization of the concept of distribution function for quantum statistics. The aim of this work is pushing further the formal analogy between quantum and classical approaches. The Wigner function is defined as an ensemble average, i.e., in terms of a mixture of pure states. From the point of view of basic physics, it would be very appealing to be able to define a Wigner function also for pure states and the associated expectation values for quantum observables, in strict analogy with the definition of mean value of a physical quantity in classical mechanics; then correct results for any quantum system should be recovered as appropriate superpositions of such “pure-state” quantities. We will show that this is actually possible, at the co& of dealing with generalized functions in place of proper functions,
Keywords-Wigner function, Quantum transport, Convergence procedure, Pure-state Wigner function, Generalized functions.
1. INTRODUCTION Transport in mesoscopic systems has been widely investigated in recent years [l-4]. This increasing interest is mainly related to the fundamental problems of the basic physics involved, as well as to possible utilization of such systems in the electronic device production. Since the dimensions of mesoscopic systems are comparable with typical electron lengths, a correct analysis of transport phenomena in such systems requires a detailed
coherence quantum-
mechanical treatment. The general case of interest is that of a mesoscopic semiconductor structure coupled with two leads considered as reservoirs at equilibrium. The potential can have an arbitrary profile inside the structure, while it takes constant and different values in the two leads. Thus, the system is open and, in the general case, far from equilibrium. The transport properties of the electron system are provided by quantum equations originating from the Liouville-von Neumann equation for the density matrix operator. In order to study such structures, the Wigner function [5-81 is a valid tool. In fact, it provides a rigorous quantum-mechanical approach and it constitutes a direct link between quantum and The authors would like to thank T. Gramchev (University of Cagliari) for helpful discussions and comments. work has been partially funded by A.R.O. and O.N.R. through E.R.O. and 1501, MM449 of MSETB.
33
This
M. NEDJALKOV
34
et al.
classical descriptions of the evolution of the system in the phase space through a distribution (z), the Wigner function is defined as the Weyl-Wigner transform f(r, K, t). In one dimension of the density matrix operator
where
]q) is the state
of the system
and the bar means
ensemble
average.
f”
has several
interesting properties. It is a real function that, when integrated over K, provides the correct probability distribution for the space coordinate and also, when integrated over Z, the correct quantum mechanical probabilities for the momentum K. Furthermore, one may get the correct expectation values of any observable A as
where A(z, K) is the Weyl-Wigner
transform
s
of the operator
+oO
A(x, K)
=
-co
emiKq
t+;qlA+
A:
> .
The Wigner function was introduced as a generalization of the concept of distribution function to quantum statistics [5]. Here, we are interested in pushing the formal analogy between quantum (l), the and classical approach further on. Since an ensemble average is included in equation Wigner function is defined in terms of a mixture of pure states. From the point of view of basic physics, it would be very interesting to be able to extend the concept of Wigner function also for pure states; the associated expectation values of quantum observables are then in strict analogy with the mean values of the classical mechanics. The correct quantum average values for the physical system of interest should be recovered as suitable superpositions of such “pure-state” quantities. Wigner functions for pure states have been considered in the literature [9,10] mainly in connection with localized states of a discrete spectrum. In particular, Berry [lo] used them to define a “spectral Wigner function”. Since, however, we are interested in open systems, in order to carry on the project described above we must face a fundamental analytical problem. The Wigner function and, as a consequence, the observable expectation values, are defined in terms of correlation functions, which do not vanish at infinity for pure states leading to nonconverging integrals. In this work, we will show that it is actually possible to define the Wigner function and the expectation value of an observable for pure states, at the cost of dealing with generalized functions in place of proper functions. The generalized-function formalism has been used recently for the calculation of some quantities related to the Wigner picture of quantum transport [11,12]. In [ll], the definition of the potential-scattering kernel-function in the Wigner equation for a step potential has been obtained. In [12], the explicit form of the Wigner function for a square barrier eigenstate (with equal left and right potential values) is presented. Usually, the perturbation theory is used in dealing with quantum transport. The form of the quantum equation describing the dynamics of the electron system depends on the choice of the unperturbed Hamiltonian Ho. For example, Ho (in single-electron approximation [13]) can describe the free electron dynamics, or it can also include the potential profile, or at least part eigenstate set of He is used to obtain the appropriate form of the of it [14]. The corresponding evolution equation and to define the initial and boundary conditions. Since the system is open, the eigenstates above a certain energy are degenerate, and proper linear combinations of such eigenstates has to be chosen. It has been recognized that a natural basis of quantum states for such systems is provided by the so-called “scattering states” [15]. It
Quantum Transport
has also been demonstrated solution
of the Wigner
[16] that
transport
boundary
equation
35
values that
assure the stability
are easily expressed
of the numerical
in terms of this basis.
The orthonormality property of the degenerate scattering states, very important for their use as an eigenstate set, has been subject of discussion in the literature [15,17]. The difference between the potential values in the two leads places such problem outside the traditional scattering theories. In [17], a correction to the Esaki-Tsu’s expression [18] for the resonant-tunneling current due to an assumed nonorthogonality of the degenerate scattering states is proposed. In [15], however, an orthonormality proof of the scattering states is presented, and the correction proposed in [17] is rejected. This proof has been given following an involved and quite indirect procedure. We present, in Section 2, such proof in a straightforward way, performing the direct calculation with the scattering states. In Section 3, the concept of pure-state Wigner function in terms of generalized discussed, and the explicit form is given for the case of a potential step. In Section
4, we show that it is possible
to use the obtained
pure-state
expression for the expectation value of an observable in the Wigner Conclusions and final remarks are drawn in Section 5. 2.
PHYSICAL
SYSTEM
AND
Wigner
functions
functions
is
in the
picture.
SCATTERING
STATES
We consider a one-dimensional problem for a mesoscopic potential with arbitrary profile and two leads with constant but different potential values. The scattering states are formed by a plane wave in each lead incident towards the mesoscopic region, plus a reflected component and a transmitted part outgoing in the other lead. For a fixed value of the incoming wave vector, the reflected
and transmitted
parts depend
on the mesoscopic
potential.
We consider here a step-like potential profile having a positive value V for z < 0 and 0 for z > 0. The scattering states will be labelled by a value k indicating the momentum of the incoming wave. Thus, positive k will indicate particles coming from the left with kinetic energy ti2k2/2m above V, negative k from 0 to -ko = -dm indicate particles coming from the right with k below the step V, and negative k from -ko to -oc correspond to particles coming from the right with kinetic energy higher then the potential step. Being solutions of the Schrodinger equation, scattering states corresponding to different energy values are orthogonal. Here, we propose a direct and simple proof that also the degenerate scattering states are orthogonal. The states with energies below V are nondegenerate,
so we consider
the states
having
energy
above the step value z > 0, z < 0, z > 0, t < 0,
for k < -KC,, (4) for k > 0,
(m being th e e 1ec t ron effective mass), k’ = - dm for k < -ko, k’ = where ko = dm ,/m for k > 0. Due to the absence of translational invariance of the system, the scattering states are not eigenvectors of the momentum operator, and k is here a label for the incoming wave with kinetic energy h2k2/2m. The coefficients r and t axe given by:
t(k) =
&,,
k - k’ r(k) = k+lc”
Let us consider the scalar product of the scattering and k2 > 0. To develop the calculations, we introduce B(ft)eikzdt
= d(k)
t(k) - r(k) = 1. states @k,(z) and ‘$kl(z) the formal expression f iP
with
kl < -kc,
M. NEDJALKOV et al.
36
where P means “principal value” and 0 is the step function. Then the scalar product above can be written as
J
+m 6,
-lXl
WkJZ)
dz =
&
{ t(k1)nb
(k:
-
+ t(w+2)qk;
k2) +
it(kl)P
(&)
+ k2) + it(w(~2)P
& (
+
@2)7r6
+
r(k1)t(kz)nb
(h
-
Icg -
(h
it(k2)P
+ q
(
&
1
2>
2>
+ +(h)t(lcz)P
(G&J}*
Since kl < 0, ki < 0 and k2 > 0, kh > 0, the terms with the S-functions S(ki - k2) and 6(klgive no contribution, Then, by substituting equation (5) into equation (7)) we obtain S+m~;,(L)~~r(.)dZ=~{2n~~~(k:+k2)+2*~~~(kl+k~) 12 -m
2
21
k2 - k;
kl + m 1 kl - k; --- kz k2 + k; kl - k; +iiyqmk:
m k2
(7)
k!J
1
1 k’, + k2
(8)
1 11 *
With a straightforward calculation, the argument of the principal value can be shown to be zero, that means no contribution of the principal value to the scalar product. From the a-function properties it follows: 6(ki + k2) = lk2/k116(kl + ki) , and recalling that ICI< 0, we get that, substituted into equation (8) yields
Being ICI< 0 and k2 > 0, we have kl/jkl( = -1 and k2 = Ik2l, that leads to /+mti;,(Z)lh(Z)~Z= --M
(-~&+&~)6(kltk;) =-
kz k2 + ki
kl + kh ”
s (kl 1 k4)
(k: + kd kl + k’,
(10)
The S-function on the right-hand side (RHS) of equation (10) is different from zero only for ICI+ ki = 0, and this implies: ki + k2 = 0. Thus, when the argument of the B-function is zero also the coefficient in front of it vanishes. We can thus conclude that the RHS of equation (10) is identically zero, that proves the orthogonality of the scattering states. To check the normalization of the scattering states, let us consider the csse of two waves incident from the left (k 1,2 > 0) (the same proof is valid for ICI,2< 0). Using again equation (6)) we obtain
J_+_m G;, (zhz(z) dz = & {nWz - h) - iP (&-) ++W(kl - r(kl)d(kl + Gl)G2Mh
+t(kl)t(ka)d
-
k2) -
+ k2)
+ kz) - ir(kl)P
Wl)r(k2)P
(k; - k’l) + it(kl)t(k2)P
Quantum Since ki and k2 are positive, the &function properties be rewritten as
/+m -cm
1(1& (‘+h(~) dz =
6(kh - ki) = (ki/kl)d(kz
& {fla (kl + ip
(
in equation
kz)
j$q
-dkMkz) Using the relations
37
S(kl + kz) gives no contribution.
the &function
it follows:
Transport
(1+
- kl). Then the scalar product
r(k&oz)
+ T(kd& -&-
(5) and remembering
2
Furthermore,
from can
+t(k$($$) - r(kd& .
+ t(kMkz)&12
that k! - ki = kf - kc, the argument
of the
principal value can be shown to vanish. Therefore, considering that the first term on the RHS of equation (12) is different from zero only for ICI= kp, and using again equation (5) we are left with +O” $2, (z)& I --oo
(z) do = b(kr - kz),
(13)
that proves the normalization of the scattering states to the S-function. The right-incident states with energies below V are defined as z > 0, z < 0, where
-ko < k < 0, k’ = dp,
and
t(k) = In this case, the states side, the normalization
(14
&,
k + ik’ r(k) = k-
so the orthogonality are nondegenerate, can easily be obtained, to a &function
is already assured. On the other since the z < 0 contribution is
negligible.
Figure 1. Wigner function for a step potential with V > 0 for I < 0 and V = 0 for z > 0. The given cases are reported.
(15)
M. NEDJALKOV et al.
38
(b) V = 0.1 eV.
(c) V = 0.01 eV. Figure 1. (cont.)
The left- and right-incident scattering-state normalization corresponds to the normalization of left- and right-incident plane waves, in agreement with the Lipmann-Schwinger equation [15,19].
3. WIGNER
FUNCTION
FOR PURE
SCATTERING
STATES
In Figure 1, the Wigner function for a step potential is presented for three different values of the Step V. The associated density matrix is obtained under the hypotheses of local equilibrium in both coupled reservoirs (we consider Boltzmann statistics), and neglecting the electron-phonon
QuantumTransport
39
lb) Figure 2. Density matrix for a step potential with V = 0.3eV (V > 0 for z < 0 and V = 0 for t > 0). Parts (a) and (b) refer to real and imaginary parts, respectively.
interaction. Under these conditions, ive may assume that the density matrix is diagonal in the scattering states P(.zl17) = 2-l
~~dk~*(2+~P)Y’;(z-~7))S(k).
where g(k) = e-hak2/2mkBT and 2 is the partition function.
(16) To proceed in the calculation,
M. NEDJALKOV et al.
Flea1part, k = 8 x lo6 m-l
(k > 0).
Imaginary part, k = 8 x 10’ m-l (k > 0).
Figure 3. Real and imaginary parts of the correlation function in equation (16) for a step potential with V = 0.3eV for three different values of the wav&vector k. Positive values of k correspond to particles moving from the negative to the positive side of the t-axis. Negative k between 0 and ko (Part (c) of the figure) correspond to particles moving from the positive to the negative side of the c-axis, reflected back from the step since their energy is below V.
the k integration has to be divided into three parts, corresponding to the regions --OO < k 5 -ko, -k. c k _< 0, and.0 < k < +co, respectively, where the scattering states have different
Quantum Transport
Real part, k = -1 x log m-l
Imaginary
41
(k < +).
part, k = -1 x log m-l
(Ic < -/CO).
(b) Figure 3. (cont.)
expressions for different shown in Figure 2 as a with different k (see the oscillations generated by
ranges of the k value. The density matrix for the case V = 0.3eV is function of z and 77. The weighted superposition of scattering states integral in equation (16)) leads to a progressive smoothing out of the interference phenomena with increasing correlation length 17.
M. NEDJALKOV et al.
42
Real part, /c = -5 x lOa m-l
Imaginary part, k = -5 x lo8 m-l
(0 > k > -ko, ko = dm
(0 > k > -ko, ko = dm
= 7 x lO’m_‘).
= 7 x 107mq1).
(cl Figure 3 (cont.)
The Fourier transform in equation (1) imposes some requirements on the behavior of p as a function of 17. The basic question we face now is whether such requirements are fulfilled, i.e., whether equa, tions (1) and (2) still hold, when a pure state is used and there is no weight function g to assure the convergence of the integrals. In [ll], it has been shown that the convergence can be achieved
Quantum Transport
43
Realpart,o:=lJ.
Imaginary part, (Y = 0.
(4 Figure 4. Comparison between the correlation function in equation (16) without (Case (a), (Y = 0) and with (Case (b), (2 = 10’) the presence of a convergence factor for a step potential with V = 0.3eV and assuming k = -1 x log m-l.
for pure states if the eigenstates are normalized to unity. Here, however, the eigenstates axe normalized to a b-function. As a consequence, the required behavior of p can be provided only by the presence of a weight function g. 1101 25:12-O
M. NEDJALKOV
et al.
Real part, 0 = 10’ m-l. -
c(4)
>bV $0
c!f
Imaginary part, c2 = 10’ m-l
@I Figure 4. (cont.)
the meaning of a pure-state Wigner function As for the case of a pure-state density matrix po, can be recovered by using an appropriate asymptotic procedure in the basic definitions and relations. be written in terms of the pure-state density matrix The pure-state Wigner function can p&z, k, r]) as follows
Quantum Transport
45
(17) where the weight function equation (2) is well defined.
g is chosen in such a way that p(z,q) E L1 and the integral in The integration with respect to k has to be performed first. Here the
Wigner function is written as a mixture of pure-state density a mixture of “pure-state Wigner functions”, we must exchange
(a) k=SX
lO*m-‘.
(b) k = -1
x log m-‘.
Figure 5. Pure-state Wigner function (@(.z, and V = 0.3eV at the given values of k.
matrices. In order to define it as the order of the two integrations.
K, ka)) for the case a = 1.
X
lo7 m-l
46
M. NEDJALKOV et al.
(c) k = -1 X lOEm-‘. Figure 5. (cont.)
According to F!rbini theorem this is possible if the integral Js )~e(z, k, v)g(k)j dq dk exists. This is, unfortunately, not true in our case, since the correlation function Ipe(z, k, q)l = I~.J~z+~/~)$$(Eq/2)1 for large 9 behaves as a constant and convergence cannot be achieved. On the other hand, if the integration order of equation (17) is maintained, the inner integral, due to our hypotheses, gives a function in Lr . Since for a given f(z) E Lr, the following relation holds:
(a) a = 1. x lo8 m-l. Figure 6. Purestate Wigner function for a step potential with V = 0.3eV and k = 8 x lo* m-l at the given values of CL
Quantum Transport
(b) a = 8.
x
47
106m-‘.
Figure 6. (cont.)
Jli
[h(k,a) =J.%“~‘f(*)dz] =h(k) = /dk”f(*)&,
(18)
we can write
PO(Z, k, q)g(k)dk >
dq.
(19)
48
M. NEDJALKOV et al.
Once the factor e-al’rl is included, the Fubini theorem can be applied, and the integration order can be inverted to give
where
is a generalized function that can be defined as the pure-state Wigner function. The pure-state density matrix pe(z, Ic,q) is presented in Figure 3 for three different values of the wave-vector k. For states with energy above the potential step, a strong oscillating behavior of such function is observed, that does not allow convergence of the integral with respect to the correlation length 77. In Figure 4, the effect of the converging factor e-Ql’Jl on pe(t., k, 77)is shown for the case (Y = lo7 m-l and k = -1 x 10gm-l (corresponding to a particle moving from the positive to the negative part of the z-axis with kinetic energy higher than the step energy). Oscillations are damped by the exponential function, assuring the convergence of the integral used in equation (21) to define the pure-state Wigner function. It should be noticed that Q = 107m-r corresponds to a damping over a correlation length of about 800 A. A comparison between Figure 4 and Figure 2 allows us to understand the physical meaning of the approximation introduced by using the converging factor. In Figure 4, the converging factor smooths out the oscillations of the pure-state density matrix in correspondence of values of the correlation length 77for which, even for the case of the density matrix obtained from equation (16), the superposition of the states of the basis set has already significantly smoothed out the oscillations. For this reason, the physical properties of the system are not influenced, in practical calculations, by the introduction of a suitable converging factor that guarantees the convergence of the involved integrals.
(a) 0 = 1. X lo7 m-l. Figure 7. Wigner function for a step potential different values of the converging factor.
with V = 0.3eV
ss obtained
with
Quantum Transport
49
(b) a = 5. x 107m-‘.
(c)
a = 1.
x lo8 m-l
Figure 7. (cont.)
Pure-state Wigner functions are presented in Figure 5 for (Y = 10’ m-l at three values of k. The variation of the pure-state Wigner function with decreasing cx is displayed in Figure 6 for k = 8 x 10’ m-‘. When LYis large (Case (a)), the structures of the Wigner function are smeared out and broadened. Decreasing cx (Cases (b) and (c)) means increasing and sharpening f$‘(~, K, k, 0). In the limit of vanishing (Y f$‘(z, K, k, a), tends to the infinitely steep and narrow generalized function. Still, according to the properties of the generalized functions, the integral on the RHS of equation (20) is convergent. This can be seen in Figure 7, where the total Wigner function, as obtained for three specific values of cx, is presented. For a = 1. x lO’m_‘, the Wigner
se auI8saq? aq 0% VI0 sum!J ‘daw Isgua$od aq$ ahoqs &?laua Y!J!M !@$.I ay$ IIIOLJr)uappu!ap?s %!~aa??leDs aq? ~03 uo!ssaJdxa %u!puodsa~~oa aq;L '(9)UOil8nbau! UaA@ al8 (y)~puv (y).i aJay!
‘ 9
[zPi -
9 12(v +
,Y)lso3 (YMYN+
,a so3(Y)W+
(zz)
saop suog81n3p3 sMoqs uos!zedwo:,
p+aurnu syj&
‘(~1)
u! ampaDoad Su$haAuos uogvnba
aq? JO asn ay$ ‘S,D lp3tus LI3ua!3ylns Jo3 ‘38y7
uy uaA@ uoggap
ap? %I!sn pazgt3a.T ‘81 a.In%iJ u! pa$uasaJd au0 ay% I#M
ay? II! uoy8~a~u~ $uappu~oD ‘Qwg~rd
30 raplo ~8uo~~uaAuo:, %p8aJp
s! uoy~~ry
Quantum
Transport
4. MEAN
51
VALUES
In equation (20), fc(z, K, k) is used to obtain f”(z,K). Our goal in this section is to prove that it is possible to obtain the mean value of an observable in terms of a weighted average of mean values calculated with pure-state Wigner functions. Let us replace f”( z, K) in equation (2) using equation (20). For the sake of simplicity, A is chosen to depend only on K and, to avoid the z integration, we consider a position-dependent mean value (A(z)) =/A(K)
(~~~+Sg(“)~~(~,K,k,a)dk)
dK.
(24)
In order to use f$‘(z, K, k) in the evaluation of an observable mean value in the same way as we do with fW(z, K) in equation (2), we need to assume the validity of the following identity:
that is, once more we have to exchange the order of the two integrations. Since the inner integral on the RHS of equation (25) represents the expectation value of the observable A on the pure state k, in this way an observable mean value would be defined as an overlap with proper weights of “pure-state mean values”. Unfortunately equation (25) does not hold in the general case without making some assumptions on the function A(K) [20]. To give an example, equation (25) is verified when l (A( dK < +CCL But A(K) can also be a polynomial such that J A(K)fg(z, K, k) dK does not have meaning. This analytical difficulty can be again overcome by applying the same asymptotic procedure used in deriving equation (20). Since equation (24) is still well defined if A(K) is a polynomial, we can introduce again a converging factor e-fllrcl and write (A(z))
= hl+
/eTPIKjA(K)
(,lim+ /g(k)f,r(z,
K, k,a)dk)
Due to the presence of the e- Ml factor, we can invert the lim _,s+ with respect to K and use the Fubini theorem to obtain (A(z))
=
lim
lim
/Lo+ a-+0+
J g(k) (I
e-PIK’A(K)fg(z,
dK.
(26)
operation with the integral
K, k, a) dK
JA(K)(J~~+ J dk)fo”(z, K, k a>dk> dK K, k) dK dk, = Js(k) (ii?+ Je -P’K’A(K)fo”(z, >
>
dk
(27)
=
which is a generalization of equation (25) and gives exactly equation (25) when A(K) is a “wellbehaved” function. In order to give a practical example of the validity of the above equation, the results obtained for the mean value of particle density 72kand current jl, of a left-incident scattering state using equation (4) in the conventional Wigner function theory are compared with the same quantities evaluated with the use of fg(z, K, k). The first calculation leads to the following results:
nk =
& ((1 +2rcos(2kz)
+ r’(k))
0(-z)
+ t’(k)e(z))
, (28)
jk
= &
(k (1 - r2(k)) 0(-z)
+ k’t2(k)Q(z))
.
M. NEDJALKOV
52
et&.
The current is constant with respect to the position, as it can be easily seen using equation (5). The same quantities, as obtained from the superposition of pure-state Wigner functions are 12= lim M f+‘lc’f;(Z, P-+0s -_oo
K, Ic) dK,
(29)
j = lim bo e-DIKI ~fo ‘K w(z, K, k) dK. P-0 s -_oo The results in equation (28) are recovered by substituting by using the following asymptotic expressions: 00
e-fllKI sin[(K - a)21 dK=
lim
K-a
B--r0/ -_oo 00
equation (22) into equation (29) and
COsinK KdK=r,
s --oo
,-+lrcl cos [(K - a)zl lim P (K-a) dK=O’ D-o+ J -_m lim
J
co
e-pllcl sin [(K - a)~] dK = 0,
p-+0+ -_03
where a is a parameter.
5. CONCLUSIONS In the present paper, the use of the Wigner function for open systems has been critically reviewed. In particular, for the case of a density matrix diagonal in the basis set we have shown that it is possible to define a “pure-state” Wigner function and a “pure-state” expectation value of a quantum observable which can be suitably used to describe the Wigner function and the expectation values for the quantum system of interest. We have adopted the scattering states as basis set of the system and specialized our analysis to the case study of a step potential. The above results have been obtained by introducing suitable convergence factors. The use of these factors, both in the definition of the necessary generalized functions and as a practical tool to perform numerical integrations, has been justified by a detailed analysis of the properties of the Wigner function of real quantum systems. It may be noticed that the complete Wigner function discussed in the present paper (see Figure 1) corresponds to two electron ensembles in equilibrium with their native contacts. The presence of phonon scattering inside the device would tend to perturb the situation mainly by relaxing the extra kinetic energy gained by the electrons at the potential step. The rigorous inclusion of phonon scattering in the Wigner formalism is at present under investigation [14].
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