TRANSPORT IN SEMICONDUCTORS
Lucio Demeio
Dipartimentodi Matematica\V.Volterra"
Universita'degli Studi diAncona
ViaBrecce Bianche1, 60131 Ancona- Italy
e-mail: demeio@p op csi.unian.it
Luigi Barletti
Dipartimentodi Matematica\U.Dini"
Universita'degli Studi diFirenze
VialeMorgagni 67/A, 50134 Firenze- Italy
Paolo Bordone and Carlo Jacob oni
INFMeDipartimento diFisica
UniversitadiMo dena e Reggio Emilia
Via Campi 213/A, 41100 Mo dena -Italy
ABSTRACT
Inthisworkwepresentaone-dimensionalmo delofquantumelectrontransp ort
in semiconductorsthat makesuse of the Wigner function formalismand that
takes into account the full band structure of the medium for energy bands
of any shap e. We intro duce a multi-band Wigner function and derive the
evolution equations for each comp onent, with and without external elds, by
using a Blo chstates representationof the density matrix.
1 Introduction
The Wigner-function approachto quantumelectrontransp ort insemiconduc-
tors iswidelyused to describ e the prop erties of electronicdevicessuch as the
spaceconcepts,itpresentsacloseanalogytothe classicalBoltzmann-equation
approach; this has the advantage that many of the analytical and numerical
techniquescommonlyused forthe Boltzmannequationcan b e adapted to the
Wigner function. In particular, the numerical complexity of other quantum
statistical approaches to electron transp ort, such as the density-matrix ap-
proach [4, 5, 6 ] and the Green's-function approach [7, 8 , 9 ], is signi cantly
reduced in the kinetic mo dels that make use of the Wigner function. Also,
when dealing with space dep endent problems in nite domains, it is always
diÆcult to devise the correct b oundary conditions to b e imp osed; b ecause of
theanalogywiththeclassicalBoltzmannequation,thisdiÆcultyismoreeasily
overcomewith the Wigner-function approach, since one can rely on imp osing
classical b oundaryconditions inaregionsuÆcientlyfarfrom thequantumre-
gion, where classicale ectsdominate[3]. Also,adding collisionsto the mo del
equationsthat governthe evolutionof theWigner functionislesscomplicated
thanincludingcollisionsintotheotherstatisticalmo delsofquantumtransp ort
[2].
Allexistingtransp ort mo delsforsemiconductorsbasedon theWignerfunc-
tion, however, rely on the following two approximations: 1) that only single-
conduction-band electrons contribute to the current ow, and 2) that only a
smallregionofthe Brillouinzone nearthe minimumofthe bandisp opulated,
leadingtothe parab olicbandapproximation. Undertheseconditions,conduc-
tion electrons can b e considered as semiclassical particles having an e ective
mass relatedto the curvatureof the energy bandfunction near the minimum.
The evolution equation for the Wigner function of the conduction electrons
then b ecomesthe evolutionequation for free particles with an e ective mass.
This allows the inclusionof any elds (barriersor bias)by means ofthe stan-
dard pseudo di erentialop erator [1].
In the case of devices in which interband transitions or non-parab olicity
isfactory. A correctly de ned Wigner function for these phenomena should
include the p opulations of all bands involved in the transp ort pro cesses and
the evolution equationthat governs the timedep endence of the Wigner func-
tionshouldtakeintoaccountp ossiblenon-parab olicitye ects. Inthiswork,we
removethesingle-bandapproximation andtheparab olicbandapproximation,
byintro ducing aWigner functionwhichincludesthep opulations of allenergy
bands and derive an evolution equation whichallows for energy bands of any
shap e. The resulting equations provide an exactmo del for the description of
collisionlesselectrontransp ort insemiconductorswithout the single-bandand
the e ective mass approximations. Collisions can b e added to the mo del in
standard ways. We also discuss the derivation of the e ective mass approx-
imation starting from the exact equations. Some of our results have already
app eared in the literature for particular cases. In absence of external elds,
the evolution equations are a generalization of an earlier result for a single
bandbyMarkowich,Mauserand Poupaud [10,11]. Astatisticaldescriptionof
multibandtransp ort was formulatedbyKriegerandIafrate[12,13 ]by making
use of acceleratedBlo chstatesinamo delbased onthe densitymatrix. Atwo
band kineticmo del without external elds was develop edby Kuhn and Rossi
[14] and by Hess and Kuhn [15] for the description of semiconductorlasers.
2 The density matrix and the Wigner function
In this work, we consider an ensemble of electrons moving in a semiconduc-
tor crystal, which is considered as an in nite homogeneous medium. The
main quantity of a quantum statistical description of the electron ensemble
in phase space is the Wigner function, which is de ned by a suitable Fourier
transformation of the density matrix. If isthe singleparticle density op era-
tor, the corresp onding density matrixin the space representation is given by
function [1] isderivedfromthe density matrixby rst considering atransfor-
mationfromthevariablesrandsto thevariablesx=(r+s)=2 (centerofmass
variable)and =r s (relativevariable),and thentaking Fouriertransforms
with resp ect to :
f(x;p)= Z
d<x+
2 jjx
2
>e ip =h
: (1)
This relation can b e inverted and the elements of the density matrix can b e
written intermsof the Wigner function:
<r jjs>=
1
2h Z
dpf
r+s
2
;p
e
ip(r s)=h
(2)
In general, an electron in the crystal will b e found in a sup erp osition of
Blo chstatesb elonging to di erentenergybands. Ifwewish to usethe Wigner
function approach in situations in which band to band transitions o ccur, the
Wignerfunctionhastocontaintheinformationab outtheelectronp opulations
of all energy bands and their dynamics. A suitable partition of the Wigner
function among the energy bands is obtained by using the Blo ch states and
the matrixelementsofthedensityop erator intheBlo chstatesrepresentation.
Let H
0
b e the Hamiltonian containing only the kineticenergy and the p e-
rio dic p otential V
p
, and givenby H
0
=p 2
=2m+V
p
, and let
m
(x;k),m 2N,
k 2 B, with B the Brillouinzone [10], b e the eigenfunctions and
1 (k),
2 (k),
...,
m 1 (k),
m
(k),...,the eigenvalues of H
0 :
H
0
m
(x;k)=
m (k)
m (x;k);
or, using Dirac'snotation,
H
0
jm;k>=
m
(k)jm;k>: (3)
The eigenfunctions are givenbythe Blo chfunctions
<xjmk>
m
(x;k)=e ik x
u
mk
(x) (4)
mk
of the Blo ch function for the m-th band and the states fjmk >g, m 2 N,
k 2 B are the Blo ch states. If a is the p erio d of the lattice, we have that
u
mk
(x+a)=u
mk
(x) and
e ik
<xjmk>=<x+jmk > (5)
8 2 L, with L the direct lattice (therefore = (l ) = l a, l 2 Z). The
functions
m
(k) are the energy bands of the material and are p erio dic in the
crystal momentumk with p erio d 2=a. They can b e expanded in a Fourier
series,
m (k)=
X
2L b
m ()e
ik
; (6)
where b
m () =
b
m
( ), since the
m
(k) are real functions of k. By using the
completenessofthe Blo chstatesfjmk >ginequation(1),theWignerfunction
can b e written as a double sumof contributions fromall energy bands:
f(x;p)= X
m 0
m 00
f
m 0
m 00
(x;p) (7)
where
f
m 0
m
00(x;p) = Z
B 2
dk 0
dk 00
m 0
m 00(k
0
;k 00
)
Z
d<x+
2 jm
0
k 0
><m 00
k 00
jx
2
>e ip =h
(8)
and
m 0
m 00
(k 0
;k 00
)=<m 0
k 0
jjm 00
k 00
> (9)
are the elements of the density op erator in the Blo ch states representation.
Equation (8) can b e written in a more compact form, by intro ducing a pro-
jection op erator P
m 0
m
00 such that, (P
m 0
m
00f)(x;p) = f
m 0
m
00(x;p). Let us rst
de ne the co eÆcients
m 0
m 00
(k 0
;k 00
;x;p)= Z
d <x+
2 jm
0
k 0
><m 00
k 00
jx
2
>e ip =h
: (10)
1
2h Z Z
dxdp
m 0
m 00
(k 0
;k 00
;x;p)=Æ
m 0
m 00
Æ(k 0
k 00
); (11)
which can b e easily derivedfrom the orthogonality of the Blo chstates. After
substituting into(8) we have:
f
m 0
m 00
(x;p) = Z
B 2
dk 0
dk 00
m 0
m 00
(k 0
;k 00
)
m 0
m 00
(k 0
;k 00
;x;p): (12)
The co eÆcients
m 0
m 00
(k 0
;k 00
)of the density matrixcan b e writtenin termsof
the Wignerfunction by using (2). Aftersome algebra, we have:
m 0
m 00(k
0
;k 00
)=< m 0
k 0
jjm 00
k 00
>=
= Z Z
dr ds<m 0
k 0
jr><r jjs><sjm 00
k 00
>=
= 1
2h Z Z
dxdpf (x;p) Z
d<m 0
k 0
jx+
2
><x
2 jm
00
k 00
>e ip =h
=
= 1
2h Z
dx Z
dp
m 0
m 00
(k 0
;k 00
;x;p)f(x;p):
Finally, by intro ducing the transfer function
W
m 0
m 00
(x;p;x 0
;p 0
)= Z
B 2
dk 0
dk 00
m 0
m 00
(k 0
;k 00
;x;p)
m 0
m 00
(k 0
;k 00
;x 0
;p 0
); (13)
the bandprojection f
m 0
m 00
can b e writtenas
f
m 0
m 00
(x;p) = 1
2h Z Z
dx 0
dp 0
W
m 0
m 00
(x;p;x 0
;p 0
)f(x 0
;p 0
)(P
m 0
m 00
f)(x;p):
(14)
The last equality de nes the linear integral op erator P
m 0
m 00
, which yields the
projectionsf
m 0
m 00
fromthetotalWignerfunctionf. ItiseasytoseethatP
m 0
m 00
is a projection op erator. The functions f
m 0
m 00
, which are the projections of f
onto the band subspaces, and the co eÆcients
m 0
m 00(k
0
;k 00
) are similarto the
ones intro ducedin [17 ,18, 19 ].
Themacroscopicquantitiessuchasparticledensity,currentandenergy,are
likewise expressed as a sum of band terms. It can b e shown that only the
diagonal termscontribute to the totalnumb erof particles,that is
Z Z
dxdpf(x;p) = X
m Z Z
dxdpf
mm (x;p):
This follows fromthe factthat dxdpf
m 0
m 00
(x;p) =0form 0
6=m 00
,as results
from equation(11).
3 General evolution equations
The time evolution of the Wigner fuction is given by the sum of the time
evolutions of the band projections,
ih
@f
@t
(x;p;t)= X
m 0
m 00
ih
@f
m 0
m 00
@t
(x;p;t);
and it must follow from the time evolution of the density matrix, which is
governed by the Liouville-von Neumannequation
ih
@
@t
=[H ;]: (15)
We shall analyze separatelythe contribution to the timeevolution due to the
p erio dic p otential, and determinedby the HamiltonianH
0
, and the contribu-
tion due to the external p otentialV. Weshall write explicitely
@
@t
=
@
@t
!
0 +
@
@t
!
V
; (16)
where ih(@=@t)
0
=[H
0
;] and ih(@=@t)
V
=[V;]. Similarly,we shall write
@f
@t
=
@f
@t
!
0 +
@f
@t
!
V
(17)
for the Wignerfunction.
We b egin by considering the time evolution of the Wigner function due to
the p erio dicp otential. Fromequation (12) we have:
ih
@f
m 0
m 00
@t
!
0
(x;p;t)= Z
B 2
dk 0
dk 00
ih
@
m 0
m 00
@t
!
0 (k
0
;k 00
;t)
m 0
m 00
(k 0
;k 00
;x;p)
(18)
and, by using equations(3) and (15),
ih
@
m 0
m 00
@t
!
0 (k
0
;k 00
;t)=<m 0
k 0
j[H
0
;]jm 00
k 00
>=
=[
m 0(k
0
)
m 00(k
00
)]
m 0
m 00(k
0
;k 00
;t):
ih
@f
m 0
m 00
@t
!
0
(x;p;t)=
Z
B 2
dk 0
dk 00
[
m 0
(k 0
)
m 00
(k 00
)]
m 0
m 00
(k 0
;k 00
;t)
m 0
m 00
(k 0
;k 00
;x;p)=
= Z
B 2
dk 0
dk 00
X
2L [
b
m 0
()e ik
0
b
m 00
()e ik
00
]
m 0
m 00
(k 0
;k 00
;t)
m 0
m 00
(k 0
;k 00
;x;p): (19)
In orderto obtaina closedsystemof equationsfor thef
m 0
m 00
's,theright hand
side of equation (19) must b e rewritten in terms of the f
m 0
m 00
's themselves.
By using (5) and following [11 ], we have that e ik
< x+=2jmk >=< x+
+=2jmk > and e ik
< mkjx =2 >=< mkjx =2 > and, after
substituting in (19) and using the explicitexpressionfor the co eÆcients(10),
wehave:
ih
@f
m 0
m 00
@t
!
0
(x;p;t)= X
2L Z
B 2
dk 0
dk 00
m 0
m 00(k
0
;k 00
;t)
b
m 0
()
m 0
m 00
(k 0
;k 00
;x+
2
;p) b
m 00
()
m 0
m 00
(k 0
;k 00
;x
2
;p)
e ip=h
:
After rearranging termswe nally obtain:
ih
@f
m 0
m 00
@t
!
0
(x;p;t)=
= X
2L
b
m 0
()f
m 0
m 00
(x+
2
;p;t) b
m 00
()f
m 0
m 00
(x
2
;p;t)
e ip=h
(20)
Equations (20) for the band projections f
m 0
m 00
are the equations that govern
the timeevolution of the Wigner function of an ensembleof electronsmoving
in a semiconductor crystal in the absence of external elds and of collisions,
and that allows for energy bands of any shap e. Note that, as was to b e
exp ected, the p opulations of the bands do not interactand eachcontribution
to the Wignerfunctionevolvesindep endentlyintheabsenceofexternal elds.
Equation (20) is a generalization to the multi-band case of an earlier result
obtained by Markowich, Mauser and Poupaud [11 ] for a single band. In the
the e ectivemass approximation.
Next, we consider the time evolution of the Wigner function due to the
external p otentials:
ih
@f
@t
!
V
(x;p;t)= Z
d<x+
2
j[V;]jx
2
>e ip =h
: (21)
Werecallthat the righthand sideof equation(21), canb e castinthe form[1]
Z
d <x+
2
j[V;]jx
2
>e ip =h
=((ÆV)f)(x;p;t) (22)
where(ÆV)isthe pseudo di erentialop eratorwith symb olÆV(x;)=V(x+
=2) V(x =2),
((ÆV)f)(x;p;t)= Z
dÆV(x;)
^
f(x;;t)e ip =h
;
and
^
f(x;;t)= 1
2h Z
dpf(x;p;t)e ip =h
istheFouriertransformoftheWignerfunctionwithresp ecttothemomentum
variable. Expressions (21)-(22) remain valid also inthe multiband case, with
f the total Wigner function. We are howeverinterestedin the timeevolution
ofthe singlecomp onentsf
m 0
m
00,whichcanb eobtainedbyactingon b othsides
of equation (21) with the op erator P
m 0
m 00
. Fromequation(22) wehave:
P
m 0
m 00
((ÆV)f)(x;p;t)=
= 1
2h Z Z
dx 0
dp 0
W
m 0
m 00
(x;p;x 0
;p 0
)((ÆV)f)(x 0
;p 0
;t)=
= 1
2h Z Z
dx 0
dp 0
W
m 0
m
00(x;p;x 0
;p 0
) Z
dÆV(x 0
;)
^
f(x 0
;;t)e ip
0
=h
=
= 1
2h Z
dx 0
Z
dp 0
W
m 0
m 00
(x;p;x 0
;p 0
)e ip
0
=h Z
dÆV(x 0
;)
^
f(x 0
;;t)=
= Z Z
dx 0
d
c
W
m 0
m 00
(x;p;x 0
; )ÆV(x 0
;)
^
f(x 0
;;t);
where
c
W
m 0
m 00
(x;p;x 0
;)= 1
2h Z
dp 0
W
m 0
m 00
(x;p;x 0
;p 0
)e ip
0
=h
isthe FouriertransformofW
mm
withresp ectto the last variable,and nally
ih
@f
m 0
m 00
@t
!
V
(x;p;t)= Z Z
dx 0
d
c
W
m 0
m 00
(x;p;x 0
; )ÆV(x 0
;)
^
f(x 0
;;t):
(23)
The full time evolution of the band projection f
m 0
m 00
of the Wigner func-
tion, due to b oth the p erio dic p otentialof the crystal latticeand the external
p otential, is obtained by adding the two contributions of equation (20) and
equation (23):
ih
@f
m 0
m 00
@t
=
= X
2L
b
m 0()f
m 0
m 00(x+
2
;p;t) b
m 00()f
m 0
m 00(x
2
;p;t)
e ip=h
+
+ Z Z
dx 0
d
c
W
m 0
m
00(x;p;x 0
; )ÆV(x 0
;)
^
f(x 0
;;t): (24)
Equation (24) gives the full time evolution of each comp onent of the Wigner
function, in presenceof an external eld and in the absenceof collisions.
3.1 Quantum e ects for linear and quadratic potentials
Itiswellknownthatinthecaseoflinearandquadraticp otentialsthepseudo d-
i erentialop erator ofthetransp ort equationforthestandard Wignerfunction
reduces to the classical di erential op erator of the Boltzmann equation. We
now show that this prop erty holds inthe multibandcase with aslightmo di -
cation.
In the case of linear and quadratic p otentials, the symb ol ÆV of the pseu-
do di erentialop erator factors in the form
ÆV(x;)=V(x+=2) V(x =2) = F(x)
where F(x) is the force. For a linear p otential, V(x) = Ex and we have
F(x)=E;foraquadraticp otential,V(x)= x 2
=2 andwe haveF(x)= x.
