DIPARTIMENTO DI SCIENZE MATEMATICHE
QUADERNO N. 4- MAGGIO2006
Luigi Barletti,Lucio Demeio,
Giovanni Frosali
Multiband Quantum Transport Models
for Semiconductor Devices.
Dipartimentodi ScienzeMatematiche
UniversitaPolitecnicadelle Marche
ViaBrecceBianche1
Devices
LuigiBarletti 1
,LucioDemeio 2
andGiovanniFrosali 3
1
Dipartimento diMatematica\U.Dini"UniversitadegliStudidiFirenze-VialeMorgagni67/A,
2
Dipartimento diScienzeMatematiche UniversitaPolitecnica delleMarche-ViaBrecceBianche1,
I-60131Ancona,Italy [email protected]
3
Dipartimento diMatematica\G.Sansone" UniversitadiFirenze-ViaS.Marta3,I-50139Firenze,
1 Introduction
The mo delingof semiconductor devices, which isa very active and intense eld of research, has
to keep up with the sp eed at which the fabricationtechnology pro ceeds; the devices of the last
generationsb ecomesmallerandsmallerandtheyhavereachedasizesosmallthatquantume ects
dominate their b ehaviour.Quantum e ects such as resonant tunnelingand other size-quantized
e ectscannotb edescrib edbyclassicalorsemiclassicaltheoriesandneedafullquantumdescription
[Fre90, JAC92, KKFR89, MRS90,RBJ91,RBJ92].A very imp ortantfeature, thathas app eared
inthedevices ofthelast generationandwhichrequiresafullquantumtreatment,is thepresence
of theinterband current, that isacontributionto thetotalcurrent which arisesfromtransitions
b etweentheconductionandthevalencebandstates.Resonantinterbandtunnelingdio des(RITD)
areexamplesofsemiconductordeviceswhichexploitthisphenomenon;theyareofbigimp ortance
innanotechnologyfortheirapplicationstohigh-sp eedandminiaturizedsystems[YSDX91,SX89].
Inthebanddiagramstructure ofthesedio desthereisasmallregionwherethevalencebandedge
liesab ovetheconductionbandedge(valencequantumwell),makinginterbandresonancep ossible.
So far, most part of the existing literature has b een devoted to quantum transp ort mo dels
whereonlyconductionbandelectronscontributetothecurrent owandundertheparab olicband
approximation,withonlyasmallregionoftheBrillouinzoneneartheminimumofthebandb eing
p opulated. Inbip olarmo dels,thecontribution ofthevalenceband(the currentdueto theholes)
isalsoincludedatthemacroscopiclevel.Quantummo delswhichinclude theinterband resonance
pro cess are called \multibandmo dels", and have largely b een formulated and analyzed only in
divided intwoclasses: Schrodinger-based mo delsand Wigner-function-based (or density-matrix-
based)mo dels.Theformeronesaimatthecalculationofthewavefunctionforthesystemordevice
understudy,andcontainnostatistics.Thelatteronesinvolveelectronstatisticsortransp orttheory
concepts.
Hydro dynamicmo delshavealsob eenformulatedanddiscussed[Gar94];again,forthemostpart
only single-band hydro dynamics has receivedattention. Only recently, multibandhydro dynamic
mo dels,basedonthemultibandkineticmo delsmentionedab ove,haveapp eared.
Inthisreview pap er,we describ e themultibandmo delsthathave recently b eenformulatedin
b othclasses.Attentionisgiventothede nitionsoftherelevantquantitieswhichcharacterizeeach
mo delandtotheadvantagesanddisadvantagesofeachmo delcomparedtoothers. Thetechnical
details of thederivations ofthevariousmo dels,as well as therigorouspro ofsof consistency and
existenceofthesolutions,arediverteddirectlytothepap erswherethemo delshaveb eendescrib ed.
Thispap er isorganized asfollows:inSection 2we brie y recalltheBlo ch theoryof electrons
moving ina p erio dic p otential; Section 3 is devoted to the envelop e-function theory; in Section
4 we dealwith the multibandmo delsbased on theSchrodinger equation; Section 5containsthe
statistical kinetic mo dels based on the Wigner-function approach and in Section 6 we give an
outlineofthehydro dynamicmo dels.
2 The Schrodinger equation and the wave function in a periodic
potential
Thestartingp ointofanytheoreticaldescriptionofaquantumsystemistheSchrodingerequation,
whichwenowdiscussforap erio dicHamiltonian[RS72, MRS90].
We consider an ensemble of electrons moving in a semiconductor crystal. The electrostatic
p otentialgeneratedbythecrystalionsisrepresentedbyap erio dicp otentialV
L
(x),thep erio dicity
b eing describ ed asfollows:
V
L
(x+a)=V
L
(x); 8a2L;
where L is the p erio dic lattice of the crystal. The quantum dynamics of a single electron is,
therefore,generatedbytheHamiltonian
H=H
0
+V(x); (1)
where H
0
=p 2
=2m+V
L
(x) is the p erio dic part of the Hamiltonian,which contains the kinetic
energy andthep erio dic p otential.Also,p= i~ris themomentumop erator, m isthe electron
mass,~isPlanck'sconstantover2andV(x)isthep otentialduetoexternal elds,suchasbarriers
orbias.Thep erio dicHamiltonianH
0
hasacompletesystemofgeneralizedeigenfunctionsb
n (x;k ),
theso-calledBrillouinzone. Thisisde nedasthecenteredfundamentaldomainoftherecipro cal
latticeL
,i.e.
B=
k2R 3
kiscloser to0thantoanyotherp ointofL
:
TheBlo chwavessatisfythegeneralizedeigenvalueequation
H
0 b
n
(x;k )=
n (k )b
n
(x;k ); (2)
(or H
0
jnk i=
n
(k )jnk i inDirac's notation),where thegeneralized eigenfunctions
n
(k )arethe
energybandsofthecrystal.Accordingly,theintegerniscalled\band-index".
UsingDirac'snotation,wecho ose thefollowingnormalizationoftheBlo ch functions:
hnkjn 0
k 0
i=jBjÆ
nn 0
Æ(k k 0
); (3)
so thatanywave function canb e decomp osedas follows
(x)= X
n Z
B dk
jBj
n (k )b
n
(x;k ); (4)
where
n (k )=
Z
R 3
dxb
n
(x;k ) (x): (5)
Itis wellknownthattheBlo chwavescanb ewritten intheform
b
n
(x;k )=e ik x
u
n
(x;k ); (6)
whereu
n
(x;k ),calledBlochfunctions,areL-p erio dicinxandhavetheprop ertythatfu
n
(;k )jn2Ng
isanorthonormalbasisofL 2
(C)forany xedk2B,whereCdenotesthefundamentalcellofthe
direct latticeL.Inparticular,
Z
C u
n (x;k )u
n 0(x;
k )dx=Æ
nn 0
; k2B: (7)
The electron p opulation of the semiconductor material is partitioned into the energy bands
of the Hamiltonian. The highest o ccupied energy band usually contains only a small electron
p opulationandtherefore ithas manyuno ccupied states; this istheconduction band.Thestates
of all other (lower energy) bands are instead fullyo ccupied and formthe valence bands. Inthe
older devices, based on resonanttunneling, onlytheelectrons of theconduction band contribute
to the owofthecurrentacrossthedevice.Insomeofthedevicesofthelastgeneration,instead,
theresonanttunnelingo ccursb etween statesb elongingtodi erentbands,sothatalsothecarrier
p opulationofthevalenceband contributestothe owofcurrent.Forthedescription oftheselast
devices, multibandmo delsmustb eused.
3 Envelope function theory
The wave function of an electron moving under the action of a p erio dic p otential, which we
have describ ed inthe previous Section, isa fastoscillating object (b oth intimeand space) and
is therefore not well suited for numerical computations. A widely used metho dology is that of
smo othingoutthesefastoscillations,thusleadingtothe\envelop efunction"approach.Envelop e
functions can b eintro duced inbasicallytwodi erentways, one due toWannier[Wan62] (called
the Wannier-Slater envelop e functions) and one due to Luttinger and Kohn [LK55] (called the
Luttinger-Kohn envelop e functions). The Luttinger-Kohn envelop e functions are the building
blo cks of the Kane mo del, which will b e describ ed in the next Section. Here, we intro duce the
de nitions andoutlinethemostimp ortantprop erties ofb othkindsofenvelop efunctions.
3.1 Wannier-Slaterenvelop efunctions
TheWannier-Slater(W-S)envelop e functions[Wan62]arede ned asfollows:
f
n (x)=
1
(2 ) 3=2
Z
B
n (k )e
ixk
dk ; (8)
where
n
(k )isgivenby(5). NotethattheW-Senvelop e functionsareinverse Fouriertransforms
to which fastoscillations due to thep erio dic p otentialhave b een removed.In other words,each
envelop efunctionf
n
hastheprop ertythatitsFouriertransformissupp ortedintheBrillouinzone
B. TheW-Senvelop e functions areeasily expressed intermsofthewave functionbyintro ducing
\continuous-indexWannierfunctions"
a
n (x;x
0
)= 1
(2 ) 3=2
Z
B b
n (x;k )e
ix 0
k
dk : (9)
Using(5), (8)and(9)we get
f
n (x)=
Z
R 3
a
n (x
0
;x) (x 0
)dx 0
; (10)
and,conversely,
(x)= X
n 1
jBj Z
R 3
a
n (x;x
0
)f
n (x
0
)dx 0
: (11)
To b etter understand the meaning of the W-S envelop e functions, consider the (discrete-index)
Wannierfunctions[Wan62],whicharetheFouriercomp onentsoftheBlo chwaveswithresp ect to
k :
a
n
(x )= Z
B dk
jBj b
n (x;k )e
ik
= Z
B dk
jBj u
n (x;k )e
ik (x )
; (12)
where is a p oint of the p erio dic lattice L. The most imp ortant prop erty of the Wannier
functions is that they are lo calized at the sites of the lattice, with an exp onential decay away
oscillatoryb ehaviourthroughoutR.TheWannierfunctions,liketheBlo chwaves,formacomplete,
generalized, orthonormalbasisandanywave functioncan b eexpandedas
(x)= X
n X
2L f
n ()a
n
(x );
where
f
n ()=
Z
B dk
jBj
n (k )e
ik
:
If the lengthscale of the crystallattice is smallwith resp ect to themacroscopic scale describ ed
bythevariablex,theBrillouinzoneb ecomesverylargeandtheFourierco eÆcientsf
n
()can b e
replacedbythecontinuousFouriertransform,yieldingde nition(8).
ThedynamicsoftheW-Senvelop e functionscanb ededucedfrom(10)and(11),andfromthe
Schrodingerequation
i~
@
@t
(x;t)=H (x;t);
whereHistheHamiltonianop erator(1).Thisyields(see[Bar03b]forthedetailsofthederivation):
i~
@
@t f
n
(x;t)=e
n
( ir)f
n (x;t)+
X
n 0
Z
R 3
V WS
nn 0
(x;x 0
)f
n 0
(x 0
;t)dx 0
: (13)
Here,
V WS
nn 0
(x;x 0
)= 1
jBj Z
R 3
a
n
(y ;x)V(y )a
n (y ;x
0
)dy (14)
are matrix-elements of the external p otential with resp ect to the continuous-index Wannier
functions and e
n
( ir) are pseudo-di erential op erators asso ciated to the energy bands with a
cut-o outside theBrillouinzone,namely
e
n
( ir)f
n (x;t)=
1
(2 ) 3
Z
R 6
I
B (k )
n (k )f
n (x
0
)e ik (x x
0
)
dx 0
dk :
whereI
B
isthecharacteristicfunctionoftheBrillouinzoneB.
3.2 Luttinger-Kohnenvelop efunctions
Ageneral de nitionofenvelop efunctions inthesense ofLuttingerandKohn[Bur92,LK55] may
b egivenasfollows.Letfv
n
(x)jn2Ngb e L-p erio dicfunctionsthatformanorthonormalbasisof
L 2
(C).Then,theLuttinger-Kohn(L-K) envelop e functionsofawave function ,withresp ect to
thebasisv
n
(x)arefunctions F
n
(x)suchthat
(i) (x)= P
n F
n (x)v
n (x);
(ii) theF
n
areslowlyvaryingwithresp ect tothelatticep erio dicity,namely
supp
^
F
n
B; n2N; (15)
where
^
F denotestheFouriertransformofF .
Usually thebasis functions v
n
arechosen to b e theBlo ch functionsu
n
(x;k )evaluatedat k =0,
so that
(x)= X
n F
n (x)u
n (x;0);
but, of course, other choices are p ossible. It can b e provedthat the L-K envelop e functions are
uniquelydeterminedbythetwoconditions(i)and(ii)andthattheParseval-likeequality
k k 2
= 1
jCj X
n kF
n k
2
(16)
holds.ItisnotdiÆculttoseethattheL-Kenvelop e functionsareeasilyexpressed intermsofthe
wave functionas follows:
f
n (x)=
Z
B dk
jBj 1=2
Z
R 3
dyX
n (y ;k )e
ik x
(y ); (17)
where
X
n
(y ;k )= 1
jBj 1=2
v
n (y )e
ik y
; y2R 3
; k2B; n2N: (18)
isa(generalized)Luttinger-Kohnbasis[LK55].Byusingtheab overelationsitisp ossibletodeduce
thedynamicsofL-Kenvelop e functions.Inthecasev
n (x)=u
n
(x;0)we have[Wen99]
i~
@
@t F
n
(x;t)=
n (0)F
n (x;t)
~ 2
2m
F
n (x;t)
~ 2
m X
n 0
K
nn 0
rF
n 0
(x;t)
+ X
n 0
Z
R 3
V LK
mn 0(x;x
0
)F
n 0
(x 0
;t)dx 0
: (19)
Here,
n
(0)isthem-thenergy bandevaluatedatk=0and
K
nn 0
= Z
C u
n
(x;0)ru
n 0(x;
0)dx= K
n 0
n
(20)
are the matrixelementsof the gradientop erator b etween Blo ch functions (which,we recall, are
real-valued).Thematrix-elementsoftheexternalp otentialaregivenby
V LK
nn 0
(x;x 0
)= 1
(2 ) 3
Z
B dk
Z
R 3
dy Z
B dk
0
n
e ik x
X
n
(y ;k )V(y )X
n 0
(y ;k 0
)e ik
0
x 0
o
; (21)
where, ofcourse,u
n
hastob e usedinde nition(18)inplaceofv
n .
4 Pure-state multiband models
The equations of envelop e-function dynamics, eqs. (13) and (19), are still to o complicated for
mo delingpurp osesand,therefore,theyshouldb econsideredasstartingp ointsforbuildingsimpler
First ofallwenote that,iftheexternal p otentialisslowlyvaryingwith resp ect tothe lattice
p erio d,thentheL-p erio dicfunctionu
n (y ;0)u
n
0(y ;0)in(21)(seede nition(18))canb esubstituted
byitsaverageonap erio diccell.Hence,wecan write
V LK
nn 0(x;x
0
) 1
jBjjCj(2 ) 3
Z
C u
n (z)u
n 0(z)
dz
Z
B dk
Z
R 3
dy Z
B dk
0 n
e ik
0
x
e
iy (k k 0
)
e ik
0
x 0
V(y ) o
andso,using(7) ,jCjjBj=(2 ) 3
,andBR 3
,
V LK
nn 0
(x;x 0
)Æ
nn 0
Æ(x x 0
)V(x x 0
): (22)
In other words, ifthe p otential V is smo oth enough,the complicated p otential term ineq. (19)
can b e approximatedby thesimplemultiplicationbyV(x)of each F
n
.Thesameprop erty holds
forV WS
nn 0
(x;x 0
)(see de nition(14)) andthepro ofissimilar.
Anothertypicalapproximationisthee ective-massdynamics.Thiscanb eeasilydeducedfrom
theWannier-Slaterequations(13)bysimplysubstituting theenergy-band function
n
(k )withits
parab olic approximationnearastationary p oint(thatwe assumetob e alwaysk=0forthesake
ofsimplicity).This,togetherwiththeapproximation(22)yieldsacompletelydecoupleddynamics
oftheform
i~
@
@t f
n (x;t)=
~ 2
2 rM
1
n rf
n
(x;t)+V(x)f
n (x;t)
whereM isthee ective-masstensor:
M 1
n
=rr
n (k )j
k =0 :
The e ective-mass mo del is widely used in semiconductor mo deling and it has b een rigorously
studied, asanasymptotic dynamics,inRefs. [AP05], [BLP78]and [PR96].However,ifinterband
e ects have to b e included, then we have to go b eyond thee ective-massapproximationand to
include atleasttwocoupledbands.
4.1 Thetwo-bandKane mo del
Asimplemultibandmo delwas intro duced byKane [Kan56] intheearly60'sinordertodescrib e
theelectron transp ortwithtwoallowedenergy bandsseparated byaforbiddenregion.TheKane
mo delisasimpletwo-bandmo delcapableofincludingoneconductionbandandonevalenceband
anditisformulatedastwocoupledSchrodinger-likeequationsfortheconduction-bandandvalence-
band envelop e functions[BFZ03].ThecouplingtermistreatedbythekP p erturbation metho d
[Wen99],whichgivesthesolutionsofthesingleelectronSchrodingerequationintheneighb orho o d
of electrons and holes,resp ectively, are concentrated. TheKane mo delis very imp ortantforthe
mo delingoftheRITDdevices,andiswidelyusedinliterature [SX89,YSDX91].
Fromourp ointof view,theKane mo delcan b e viewedasanapproximateevolutionequation
forL-Kenvelop e functionsarisingfromeq.(19)whenusingthefollowingapproximations:
1. theexternalp otentialkernel(21)issubstitutedbythelo calanddiagonalapproximation(22);
2. onlytwobands(conductionandvalence) areincluded;
3. theb ottomofconduction bandE
c
=
c
(0) andtopofvalenceband E
v
=
v
(0)are viewedas
functionsofthep ositionx(thisallowstomo delbandheterostructures) .
Thus, using the indices c for conduction and v for valence, we have a two-term L-K envelop e
functionexpansion
(x)=
c (x)u
c (x)+
v (x)u
v (x);
ofthewavefunction andthefollowingevolutionequationsfor
c and
v :
i~
@
@t
c
(x;t)=(E
c
+V)(x)
c (x;t)
~ 2
2m
c (x;t)
~ 2
m Kr
v (x;t);
i~
@
@t
v
(x;t)=(E
v
+V) (x)
v (x;t)
~ 2
2m
v (x;t)+
~ 2
m Kr
c (x;t);
(23)
whichisthetwo-bandKanemo del.NotethatthequantityK,calledKanemomentum,isgivenby
K=K
cv
= K
v c
= Z
C u
c (x)ru
v (x)dx
(see (20) and recall that the Blo ch functions u
c and u
v
are real-valued). A wordof caution has
to b e sp ent on thenotation:
c and
v
are not really band-projections (sp ectral projections) of
the wave function, not only b ecause of the envelop e function approximation but also b ecause
the Hamiltonianop erator de ned by the right-hand side of eq. (23)is not diagonal,even inthe
absence of external p otentials. The identi cation of
c and
v
with sp ectral projections is only
approximatelytruefork0.
TheKanemo delintheSchrodinger-likeform(23)hasb eenrecentlystudiedbyJ.Ke ,[Kef03],
andintheWigner-equationformbyBorgioli,FrosaliandZweifel[BFZ03].
4.2 TheMorandi-Mo dugnomultibandmo del
In this section we brie y intro duce themultibandenvelop e function mo del, intro duced recently
byMo dugnoandMorandi(M-M);forthecompletederivationofthemo delwerefer thereader to
[MM05].
Thestartingp ointistheW-Senvelop efunctiondynamics(13).Whenthep otentialV issmo oth
enough, we can approximatethematrix-elementsV WS
nn 0
inthesamewayas we deduced eq.(22),
obtaining
V WS
nn 0
(x;x 0
) 1
jBj(2 ) 3=2
Z
R 3
dk Z
R 3
dk 0
n
e ik x
B
nn 0(k ;k
0
)
^
V(k k 0
)e ik
0
x 0
o
(24)
where
B
nn 0(k ;
k 0
)= 1
jCj Z
C u
n (z;k )u
n 0(z;
k 0
)dz: (25)
By usingtheeigenvalueequation(2)oneobtains
B
nn 0
(k ;k 0
)= 1
jCj
~
m (k k
0
) P
nn 0
(k ;k 0
)
E
nn 0(k ;
k 0
)
; forn6=n 0
,
where
P
nn 0(k ;k
0
)= Z
C u
n
(x;k )( i~r)u
n 0(x;k
0
)dx (26)
and
E
nn 0(k ;
k 0
)=
n (k )
n 0(k
0
)
~ 2
2m (k
2
k 02
):
Moreover,ascan b ededucedfromeq.(3),thediagonaltermsaresimplygivenby
B
nn (k ;k
0
)= jBj
(2 ) 3
= 1
jCj
: (27)
Using(24),(25)and(27)ineq.(13)(andrecallingthatBR 3
)weget
i~
@
@t f
n
(x;t)=
n
( ir)f
n
(x;t)+V(x)f
n (x;t)
+
~
m X
n 0
6=n Z
R 3
dk Z
R 3
dk 0
e ik x
(2 ) 3
P
nn 0(k ;k
0
)
E
nn 0(k ;
k 0
)
^
V(k k 0
)
^
f
n 0(k
0
;t)
where a diagonal part and a non-diagonal part of the dynamics can b e clearly distinguished.
Assuming, for the sake of simplicity, that the stationary p oint of each band is k = 0 and that
the crystal momentumk remains small during the whole evolution, we can expand the term
P
nn 0=E
nn
0, which characterizes the interband coupling, to rst order in k and k 0
. After some
manipulationsbymeansofstandardp erturbation techniques, wegetthemultibandequation
i~
@
@t f
n
(x;t)=
n
( ir)f
n
(x;t)+V(x)f
n (x;t)
i~
m
rV(x) X
n 0
6=n P
nn 0
E
nn 0
f
n 0(x;
t)
~
m
rV(x) X
n 0
6=n M
nn 0
E
nn 0
rf
n 0
(x;t)
~
m X
0 M
nn 0
E
nn 0
r 2
V(x)f
n 0
(x;t)+rV(x)rf
n 0(x;
t)
: (28)
whereweput P
nn 0
P
nn 0(0;
0),E
nn 0
E
nn 0
(0;0)and
M
nn 0 =
~
m X
n 00
6=n 0
P
nn 00P
n 00
n 0
E
n E
n 00
; M
n 0
n
=
~
m X
n 00
6=n 0
P
nn 00P
n 00
n 0
E
n 0 E
n 00
aree ective-massterms.Asimpletwo-bandmo delcan b ebuiltusing thefollowingassumptions:
1. onlytwobands(candv ) areincluded;
2. theenergybandop erator
n
( ir)issubstitutedbyitsparab olicapproximation(e ective-mass
energyband);
3. the interband terms of order greater than 2 in k are neglected (this amounts to neglecting
termsprop ortionalto thematricesM
nn 0);
4. theb ottomoftheconductionbandandthetopofthevalencebandarefunctionsofthep osition
x(asinthetwo-bandKanemo del).
Thisyields
i~
@
@t
c
(x;t)=(E
c
+V)(x)
c (x;t)
~ 2
2 rM
1
c r
c (x;t)
i~
mE
g (x)
rV(x)P
v (x;t);
i~
@
@t
v
(x;t)=(E
v
+V)(x)
v (x;t)
~ 2
2 rM
1
v r
v (x;t)
i~
mE
g (x)
rV(x)P
c (x;t);
(29)
whereE
g
(x)=E
c (x) E
v
(x)istheband-gap.ContrarilytotheKanemo del(23),inthetheM-M
mo del (29)the envelop e functions
c and
v
are true band-functions, to the extent that inthe
absence ofexternalp otentials(V =0)thedynamicsisdiagonal.
5 Statistical multiband models:density matrix and Wigner function
Wenowturnourattentiontothemultibandmo delsthatmakeuseofstatisticalconcepts, mainly
of the Wigner-functionapproach [Wig32, MRS90, BJ99, JBBB01 , Bar03a]. A multiband mo del
involvingthedensitymatrixwasalreadyintro ducedbyKriegerandIafrate[KI87,IK86]bytaking
matrixelementsofthedensityop eratorb etweenBlo chstates.Subsequently,anumb erofmultiband
mo delsbased onWigner-functionapproach weredevelop ed. In[Bar03b,Bar04a, BD02]envelop e
functions wereusedto construct themultibandWignerfunction;in[BFZ03]aWignerversion of
theKanemo delwasintro duced;in[DBBBJ02,DBBJ02,DBJ03a,DBJ03b]themultibandWigner
functionwasobtainedbyusingtheBlo ch-staterepresentation ofthedensitymatrix.
We recall that statistical states in quantum mechanics are describ ed either in terms of the
densityop erator ortheWignerfunctionf(x;p),[Fey72].Thedensityop eratorisusuallyde ned
by a statistical mixture of states, say f
j
jj 2 Ng, where
j
(x) are the wave functions that
characterize each state of the mixture. If
j
0 is the probability distribution of the states,
then P
j
j
=1andthedensityop eratorisgivenby
= X
j
j j
j ih
j
j (30)
inDirac'snotation,andthedensitymatrixinthespacerepresentation isgivenby
(x;x 0
)= X
j
j
j (x)
j (x
0
)= X
j
j hxj
j ih
j jx
0
i: (31)
TheWignerfunctionf(x;p)isde nedbytheWigner-Weyltransformofthedensityop erator,that
is
f(x;p)= Z
d
(2 ~) 3
x+
2
;x
2
e ip =~
: (32)
In the theoretical mo dels based on the solution of the Schrodinger equation (pure states),
the calculation of the current across the device, j(x), followsthestandard quantum-mechanical
de nition
J(x)=
~
m
Im (x)r (x)
: (33)
In the statistical mo dels, instead, the current is expressed interms of the density matrix or in
termsoftheWignerfunction.Inthe rstcase thecurrentis
J(x)= i~
2m (r
x r
x 0
)(x;x 0
)j
x=x 0
; (34)
and,intheWignerpicture,by
J(x)= 1
m Z
pf(x;p)dp; (35)
anexpressionwhichis,remarkably,identicaltotheclassicalexpressionforthecurrentinstatistical
systems.Itcanb eeasilyshownthat,inthecaseofpurestates,thesetwoexpressions coincidewith
(33).
5.1 Wigner-functionbased statisticalmo dels
A suitable partition of the Wigner function among the energy bands can b e obtained by using
the completeness of the Blo ch states inequation (32). We adopt hereafter Dirac's notation and
consider,forthesakeofsimplicity,theone-dimensionalcaseonly.Byde ningtheco eÆcients
mn
andtheintegralkernel W
mn ,
mn (k ;k
0
;x;p)= Z Z
d
2 ~ D
x+
2
nk
ED
nk 0
x
2 E
e ip =~
(36)
W
mn (x;p;x
0
;p 0
)= Z
B 2
dk dk 0
mn (k ;k
0
;x;p)
mn (k ;k
0
;x 0
;p 0
); (37)
the Wigner function can b e written as a sum of projections over the Flo quet subspaces of the
energy bands(see [DBBJ02]fordetails):
f(x;p)= X
mn f
mn
(x;p); (38)
where
f
mn
(x;p)= Z
B 2
dk dk 0
mn (k ;k
0
)
mn (k ;k
0
;x;p): (39)
By expressing as afunctionof f, we can write f
mn
=P
mn
f, where P
mn
is thelinear integral
op erator
(P
mn
f)(x;p) 1
2 ~ Z Z
dx 0
dp 0
W
mn (x;p;x
0
;p 0
)f(x 0
;p 0
):
Here,
mn (k ;k
0
)=hmkjjnk 0
iarethematrixelementsofthedensityop eratorintheBlo ch-state
representationandthelinearintegralop eratorP
mn
isaprojectionop eratorandyieldstheWigner
projectionsf
mn
fromthetotalWignerfunctionf.
Thetimeevolution of theWigner function is givenby the sumof the timeevolutionsof the
band projections,
i~
@f
@t
(x;p;t)= X
mn i~
@f
mn
@t
(x;p;t);
givenby[DBBJ02]
i~
@f
mn
@t
= X
2L h
b
m ()f
mn (x+
2
;p;t) b
n ()f
mn (x
2
;p;t) i
e ip=~
+ Z Z
dx 0
d
c
W
mn (x;p;x
0
; )ÆV(x 0
; ) b
f(x 0
; ;t); (40)
where c
W
mn
istheFouriertransformofW
mn
withresp ect tothemomentumvariable:
c
W
mn (x;p;x
0
; )= 1
2 ~ Z
dp 0
W
mn (x;p;x
0
;p 0
)e ip
0
=~
: (41)
Equation(40)istheequationthatgovernsthetimeevolutionoftheFlo quetprojectionsf
mn ofthe
Wignerfunctionforanensembleofelectronsmovinginasemiconductorcrystalinthepresenceof
external eldsandallowingforenergybandsofarbitraryshap e.The rstterm,containingthesum
over the latticevectors, refers to theaction of thep erio dic p otential of thecrystal lattice,while
the last term,written intheformof an integral op erator, refers to the actionof theexternal or
self-consistent eldsactingontheelectrons. The rstterm,asshownin[DBBJ02],reducestothe
thistermastothestreamingterm,whilethesecondtermwillb ecalledtheforceterm,inanalogy
withthe corresp ondingforce termoftheBoltzmannequation. Theseequations showthat,inthe
absence ofexternal elds,di erentbandsremaindynamicallyuncoupled andeachcontributionto
the Wigner function evolves indep endently. Inthe case V(x) 0, these equations were already
writtenbyMarkowich,MauserandPoupaud[MRS90,MMP94]forasingleband.Itcanb eshown
that,inthecase ofasingleparab olicband,eq.(40)reducesto theusualWignerequation inthe
e ective massapproximation
@f
@t +
p
m
@f
@x +
i
~
[ÆV]f=0;
wherem
isthe(one-dimensional)electrone ective massintheselectedband and
( [ÆV]f)(x;p)= 1
2 ~ Z
R 2
e i(p p
0
) =~
ÆV(x;)f(x;p 0
)ddp 0
(42)
isapseudo-di erential op eratorwithsymb ol
ÆV(x;)=V
x+
2
V
x
2
: (43)
A multiband mo del for electron transp ort in semiconductors, based on the density-matrix
approach, was intro duced by Krieger and Iafrate in [KI87, IK86].They considered a statistical
ensembleof electrons movingunder theactionof anexternaltime-dep endentelectric eld.Here,
we brie y summarizethis mo delinasimpli edform.Their mo delisobtainedby expandingthe
densitymatrixelementsinBlo chfunctions:
(y ;z)= X
mn Z
B 2
dk dk 0
mn (k ;k
0
)b
m (k ;y )b
n (k
0
;z); (44)
where
mn (k ;k
0
) = hmkjjnk 0
i are the already-intro duced matrix elements b etween Blo ch
functions,whoseevolutionisgivenby
i~
@
mn (k ;k
0
)
@t
=[
m (k )
n (k
0
)]
mn (k ;k
0
)
+ X
l Z
B dk
00
[V
ml (k ;k
00
)
ln (k
00
;k 0
) V
ln (k
00
;k 0
)
ml (k ;k
00
)]: (45)
Here, V
mn (k ;k
0
) = hmkjVjnk 0
i are the matrix elementsof the external p otentialin the Blo ch
representation. The main source of diÆculty with this approach lies exactly in these matrix
elements,whichareill-de nedformostp otentialsofpracticalinterest.
TheWigner-functionformalismhasalsob eenusedbyBuotandJensen[Buo74,Buo76,Buo86,
BJ90]toformulatemultibandmo delswithintheframeworkoftheLattice-Weyltransform,inwhich
a non-canonical de nition of the Wigner function, based on a discrete Fourier Transform,was
intro duced.Thisde nitionoftheWignerfunctionmakesuseoftheWannierfunctionsintro duced
eq.(12));here, Listhedirect latticeandthevectors are elementsofthedirect lattice.We can
consider matrixelementsofthedensity op eratorintheWannierrepresentation,
mn
(;)=hmjjni
with;2L.A Wignerfunctionisthenintro duced by
f
mn
(k ;)=N X
v 2L
mn
(+v ; v )e 2ik v
; (46)
whereN isanormalizationfactor,2Lisalatticevectorandk2B.Thisde nitionoftheWigner
function is sometimescalled discrete Wigner-Weyltransform,and has asimilar structure of the
de nitiongivenin(32);therearehoweversomeimp ortantdi erences: theWignerfunctionisonly
de ned onthelatticep oints;itisde nedbyaFourierseries,ratherthattheFouriertransform;it
isafunctionofthecrystalmomentum,whichhasnotb eenintegratedover.Accordingto(35),the
currentdensityisthengivenby
J()= X
mn Z
B dk
jBj p
m f
mn
(k ;); (p=~k )
andis alsode ned onthelatticep oints.
5.2 ReducedWigner-Blo ch-Floquetmo dels
Equations(40)arethemostgeneraltimeevolutionequationsthatcanb e writtenfortheFlo quet
projections of the multiband Wigner function in presence of external elds and in absence of
collisions.The actionof the p erio dic p otential is describ ed by the rst term,which containsthe
Fourierco eÆcientsoftheenergybands,andwhichreducestotheusualfree-streamingop eratorin
theparab olic-bandapproximation.Thesecondtermdescrib estheactionoftheexternalp otential.
We note that, while the rst term requires only the knowledge of the energy band functions,
the second term requires the knowledge of the Blo ch eigenfunctions of the material of interest.
Therefore,themo delequations(40)areveryhardtosolveinfullgeneralityinpracticalapplications,
andthederivationofasetofsimpli edmo delsis needed.Inthefollowingsubsections,weoutline
someofthereducedmo delswhichhave b eenderivedwithintheBlo ch-Flo quetapproach.
Two-bandmo delin theparab olicband approximationwithoutexternal elds
It is interesting to consider a simple two-bandmo del in the parab olic band approximationand
without external elds, in order to study the o -diagonal Flo quet projections of the Wigner
function, which arise in this case. In atwo-band mo del, the Wigner function and its evolution
n = 0;1.The Wigner function is given by the sumof four contributions, f
00 , f
01 , f
10 and f
11 .
It can b e seen easily from equations (36), (39) and (37) that f
01
= f
10
, while f
00 and f
11 are
real. Each of the four contributions evolves according to equations (40). In the parab olic band
approximation,thedi erentialequationsforf
00 andf
11 are:
@f
00
@t +
p ~k
0
m
0
@f
00
@x
=0
@f
11
@t +
p ~k
1
m
1
@f
11
@x
=0;
(47)
wherem
0 andm
1
arethee ective massesforband0andband1resp ectivelyandk
0 andk
1 arethe
valuesofthecrystalmomentumatwhichband0andband1attaintheirminimum.Theevolution
equationsforf
01 andf
10
=f
01
haveinsteadadi erentstructure.Asimplecalculationshowsthat:
i~
@f
01
@t
=
0 (k
0 )+
(p ~k
0 )
2
2m
0
1 (k
1 )+
(p ~k
1 )
2
2m
1
f
01 (x;p)+
i~
2
p ~k
0
m
0 +
p ~k
1
m
1
@f
01
@x 1
8
~ 2
m
0
~ 2
m
1
@ 2
f
01
@x 2
: (48)
whichfollowsfromequation (40)afterexpanding f
mn
(x =2;p;t) inTaylorseriesab out =0
andusingparab olic pro lesforthetwobands. Byintro ducingthefrequencies
!
01
=(
0 (k
0 )
1 (k
1 ))=~
01 (p)=!
01
+(p ~k
0 )
2
=(2m
0
~) (p ~k
1 )
2
=(2m
1
~)
andthenewfunction
g
01
(x;p;t)=f
01
(x;p;t)e i01(p)t
;
equation(48)canb e castinthemoreelegantform
@g
01
@t +
1
2
p ~k
0
m
0 +
p ~k
1
m
1
@g
01
@x i~
8
1
m
0 1
m
1
@ 2
g
01
@x 2
=0: (49)
Notethatinthede nitionoftheWignerfunction(38)f
01 andf
10
app earonlyinthecombination
f
01 +f
10
, consistently with the Wigner function b eing real. Equation (48)shows that the time
evolutionof f
01
isgivenby threecontributions: anoscillatory term,afree streaming termand a
di usive term with imaginarydi usion co eÆcient(Schrodinger-like term). The frequency of the
oscillatory term,
01
,is prop ortionalto thedi erence ofthetotalenergies of theparticlesofthe
twobands; the velo city of the free streaming termis an average of the relative velo cities of the
particlewithresp ect to thetwominimaandtheimaginarydi usionco eÆcientvanisheswhenthe
twoe ective massesareequal.
Equations(47) and(49) completelydescrib e the timeevolution of all thecomp onentsof the
Wignerfunctioninatwo-bandmo delwiththeparab olic-bandapproximationand intheabsence
Multibandmo delin theLuttinger-Kohnapproximation
As we have already seen in Section 3.2, the Luttinger-Kohn mo del [LK55]considers the carrier
p opulations near minima (or maxima) of the energy bands and it is therefore to b e used in
conjunction with theparab olic-band approximation.For theBlo ch states near theminimum(or
maximum)oftheband,theBlo chfunctionsu
n
(x;k )arereplacedwiththesetoffunctionsu
n (x;k
n ),
i.e.theBlo chfunctionsattheb ottom(ortop)oftheband,hereassumedatk=k
n
.Thefunctions
e ik x
u
n (x;k
n
), after a suitable normalization,also form a complete set (see Ref. [LK55] and see
alsoSection3.2)andanywavefunctioncanb eexpandedintheirbasis.InthisSection, weusethe
Luttinger-Kohn basis forexpressing theFlo quet projections f
mn
ofthe Wignerfunctionand for
writingtheevolutionequations.TheactionofthefreeHamiltonianistreatedintheparab olic-band
approximation.
Ifthen thbandhasanextremumat k=k
n
,we canapproximatetheBlo chwavesas
hxjnk i=b
n
(x;k )u
n (x;k
n )e
ik x
: (50)
Since thefunctions u
n (x;k
n
)are p erio dicfunctionswithp erio d a,wecan intro ducetheirFourier
expansion,
u
n (x;k
n )=
1
X
n 0
= 1 b
U n
n 0e
iK
n 0x
;
where, K
n
= 2 n=aare vectors oftherecipro cal latticewith K
n
= K
n
. After evaluatingthe
co eÆcients
mn
andthe integralkernel W
mn
inthisbasis, andaftercarryingouttheintegration
overthemomentumvariablesk andk 0
,oneobtainsfortheFlo quet projectionf
mn
of theWigner
function:
f
mn
(x;p)=4
X
m 0
n 0
m 00
n 00
b
U m
m 0
b
U n
n 0
b
U m
m 00
b
U n
n 00
e i(K
m 0 K
n 0)x
H
a
p
~ K
m 0
+K
n 0
2
Z
dx 0
sin2[ =a jp=~ (K
m 0+K
n
0)=2j](x x 0
)
x x
0
f
x 0
;p ~ K
m 0
+K
n 0
K
m 00
K
n 00
2
e i(K
m 00 K
n 00)x
0
;
where the integrals are p erformedover the whole real line andH is theHeaviside function.The
evolutionequationshave b een formulatedforthecase oftwoenergy bands intheparab olic band
approximation.If m
0 and m
1
arethee ective massesfor band 0and band 1resp ectively andk
0
andk
1
arethevaluesofthecrystalmomentumatwhichband0andband1attaintheirminimum,
wehave that
@t +
m
0
@x +
~ (
00
f)(x;p)=0 (51)
@f
11
@t +
p ~k
1
m
1
@f
11
@x +
i
~ (
11
f)(x;p)=0; (52)
i~
@f
01
@t
=
0 (k
0 )+
(p ~k
0 )
2
2m
0
1 (k
1 )+
(p ~k
1 )
2
2m
1
f
01 (x;p)
i~
2
p ~k
0
m
0 +
p ~k
1
m
1
@f
01
@x 1
8
~ 2
m
0
~ 2
m
1
@ 2
f
01
@x 2
+(
01
f)(x;p); (53)
where
mn
isanop erator actingonthewholeWignerfunctionf and,recallingde nition(42),is
givenby
(
mn
f) (x;p;t)= Z Z
dx 0
d
c
W
mn (x;p;x
0
; )ÆV(x 0
; ) b
f(x 0
; ;t)
=4
X
m 0
n 0
m 00
n 00
b
U m
m 0
b
U n
n 0
b
U m
m 00
b
U n
n 00e
i(K
m 0
K
n 0
)x
H
a
p
~ K
m 0
+K
n 0
2
Z
dx 0
e i(K
m 00
K
n 00
)x
0sin2[ =a jp=~ (K
m 0+K
n
0)=2j](x x 0
)
x x
0
Z
d ÆV(x 0
; )e i(p (K
m 0+K
n 0 K
m 00 K
n 00)=2) =~
b
f(x 0
; ;t): (54)
A two-bandmo delwith empty-latticeeigenfunctions
Adi erentsimpli cationofthetransp ortequationscanb eobtainedbyusingtheBlo ch functions
ofthe\emptylattice",that isp erio dic planewaves.Here, we consideronlythetwolowestenergy
bands, givenby
0 (k )=
~ 2
k 2
2m
(55)
1 (k )=
~ 2
2m
[H(k )(k K) 2
+H( k )(k+K) 2
]; (56)
withK=2 =aandmthebareelectronmass,and whoseeigenfunctionsare
0k
(x)=hxj0k i= 1
p
2
e ik x
(57)
1k
(x)=hxj1k i= 1
p
2
(H(k )e iKx
+H( k )e iKx
)e ik x
: (58)
By using this basisin thede nition (39)of the multibandWigner function, one obtains forthe
band projectionsf
mn
(see [DBJ03b]forthedetails):
f
00
(x;p)= 1
H
K
2
p
~
Z
sin2(K =2 jp=~j)(x x 0
)
x x
0
f(x 0
;p)dx 0
(59)
f
01
(x;p)= 1
Z
e
H
3~K
4
;p;0
e i(
1 +
2
+K)(x x 0
) sin(
2
1 )(x x
0
)
x x
0
+ e
H
0;p;
3~K
4
e
i( 3+ 4 K)(x x 0
) sin(
4
3 )(x x
0
)
x x
0
f(x 0
;p)dx 0
(60)
f
11
(x;p)= 1
Z
e
H
~K ;p;
~K
2
sin2(K =4 jp=~+3K =4j)(x x 0
)
x x
0
+ e
H
~K
2
;p;~K
sin2(K =4 jp=~ 3K =4j)(x x 0
)
x x
0
+2H
K
4
p
~
sin2(K =4 jp=~j)(x x 0
)
x x
0
cos 3
2
K(x x
0
)
f(x 0
;p)dx 0
: (61)
wherethefunction e
H (a;x;b)H(x a)H(b x)hasb eenintro duced,and
1 (p)=
K
2 +
p
~ +
K
2
2 (p)=
K
4
p
~ +
K
4
3 (p)=
K
4 +
p
~ K
4
4 (p)=
K
2
p
~ K
2
:
ThetimeevolutionoftheFlo quetprojections oftheWignerfunctionisgivenby:
@f
00
@t +
p
m
@f
00
@x +
i
~ (
00
f)(x;p)=0
@f
11
@t +
p
m
@f
11
@x +
i
~ (
11
f)(x;p)=0;
@f
01
@t +
p
m
@f
01
@x +
i
~ (
01
f)(x;p)=0;
whereisanop erator actingonthetotalWignerfunctionf andisgivenby
(
00
f)(x;p)=
H
a
~
x x
0
Z
ÆV(x 0
; ) b
f(x 0
; ;t)e ip =~
ddx 0
(
01
f)(x;p)= 1
Z
e
H
3~K
4
;p;0
e
i( 1+ 2+K)(x x 0
) sin(
2
1 )(x x
0
)
x x
0
+ e
H
0;p;
3~K
4
e i(
3 +
4
K)(x x 0
) sin(
4
3 )(x x
0
)
x x
0
Z
ÆV(x 0
; ) b
f(x 0
; ;t)e ip =~
ddx 0
(
11
f)(x;p)= 1
Z
e
H
~K ;p;
~K
2
sin2(K =4 jp=~+3K =4j)(x x 0
)
x x
0
+ e
H
~K
2
;p;~K
sin2(K =4 jp=~ 3K =4j)(x x 0
)
x x
0
+2H
K
4
p
~
sin2(K =4 jp=~j)(x x 0
)
x x
0
cos 3
2
K(x x 0
)
Z
ÆV(x 0
; ) b
f(x 0
; ;t)e ip =~
ddx 0
Equations(59)-(61)showthattheFlo quetprojectionsoftheWignerfunctiongivenbythismo del
arefunctions withcompact supp ortand cover di erentp ortionsof thephase space. Thesupp ort
of the projection f
00
on the lower band, for example, corresp onds to the rst Brillouin zone;
the supp orts of theother projections are larger and extend b eyond the rstBrillouinzone. The
equations ofthistwo-bandmo delarevery hardtoapproach numerically,b ecause ofthepresence
ofconvolutionintegralsofhighlyoscillatoryfunctions.
5.3 Envelop e-functionbased statisticalmo dels
Analternativeapproachtostatisticalmo delsbasedontheWignerpicturestartsfromanenvelop e-
functionmo del,suchastheKanemo del(23)ortheM-Mmo del(29)andthenapplyingtheWigner
transformation(32)directlytotheenvelop efunctions (
c and
v
intheformercase,
c and
v in
thelatter).
Foratwo-bandmo delweneeda22matrixofWignerfunctions(Wignermatrix),de ned as
thecomp onent-wiseWignertransform:
w
ij
(x;p)=(W
ij
)(x;p); i;j2fc;v g;
where W denotes theWignertransformation(32)and
ij
is anenvelop e-functiondensity matrix
(i.e., in the pure-state case, it is given by
ij (x;x
0
) =
i (x)
j (x
0
) for the Kane mo del and by
ij (x;x
0
)=
i (x)
j (x
0
)fortheM-Mmo del).Theself-adjointnessofthedensityop eratorimplies
ij (x;x
0
)=
ji (x
0
;x) =) w
ij
(x;p)=w
ji (x;p):
TheevolutionequationfortheWignermatrixinthecase oftheKanemo del(23)is
@
@t +
p
m
r
x +
i
~
[V
cc ]
w
cc
= i~K
2m
r
x (w
cv w
v c )
Kp
m (w
cv +w
v c )
@
@t +
p
m
r
x +
i
~
[V
cv ]
w
cv
= i~K
2m
r
x (w
cc +w
v v )+
Kp
m (w
cc w
v v )
@
@t +
p
m
r
x +
i
~
[V
v c ]
w
v c
= i~K
2m
r
x (w
cc +w
v v )+
Kp
m (w
cc w
v v )
@
@t +
p
m
r
x +
i
~
[V
v v ]
w
v v
= i~K
2m
r
x (w
cv w
v c )+
Kp
m (w
cv +w
v c )
(62)
whereweput
V
ij
(x;)=(E
i +V)
x+
2
(E
j +V)
x
2
; i;j2fc;v g; (63)
andthepseudo-di erentialop erator,inthepresentthree-dimensionalcase,isgivenby
( [ ]f)(x;p)= 1
(2 ~) 3
Z
R 6
e i(p p
0
) =~
(x;)f(x;p 0
)ddp 0
: (64)
The system (62) has b een studied from a mathematicalp ointof view in [BFZ03]. The Wigner
matrixdescribingthermalequilibriumof theKane mo delhasb een analyzedinRef.[Bar04a].
TheevolutionequationfortheWignermatrixinthecase oftheM-Mmo del(4.2)is
@
@t +pM
1
c r
x +
i
~
[V
cc ]
w
cc
= [F ]w
cv
[F
+ ]w
v c
@
@t +p
M 1
c +M
1
v
2 r
x i~
4 r
x
M
1
c M
1
v
2 r
x +
ip
~
M
1
c M
1
v
2 p
w
cv
= i
~
[V
cv ]w
cv
+ [F ]w
cc
[F
+ ]w
v v
@
@t +p
M 1
c +M
1
v
2 r
x +
i~
4 r
x
M
1
c M
1
v
2 r
x ip
~
M
1
c M
1
v
2 p
w
v c
= i
~
[V
v c ]w
v c
[F
+ ]w
cc
+ [F ]w
v v
@
@t +pM
1
v r
x +
i
~
[V
v v ]
w
v v
= [F
+ ]w
cv
+ [F ]w
v c
(65)
whereweput
F
(x;)=
rV P
mE
x
2
;
and thesymb ols V
ij
arestill given by (63). Fromeq.(26) we see that that P and, consequently,
F
arepurelyimaginary,sothat thefollowingrelationshold:
[F
]w
ij
= [F
]w
ji
; i;j2fc;v g:
Inthesp ecialcaseofconstantandopp osite e ective-masses,
M
c
=m
I; M
v
= m
I;
theab ovesystemreducesto
@
@t +
p
m
r
x +
i
~
[V
cc ]
w
cc
= [F ]w
cv
[F
+ ]w
v c
@
@t i~
4m
r
2
x +
ip 2
~m
+
i
~
[V
cv ]
w
cv
= [F ]w
cc
[F
+ ]w
v v
@
@t +
i~
4m
r
2
x ip
2
~m
+
i
~
[V
v c ]
w
v c
= [F
+ ]w
cc
+ [F ]w
v v
@
@t p
m
r
x +
i
~
[V
v v ]
w
v v
= [F
+ ]w
cv
+ [F ]w
v c
;
(66)
(see also Ref. [FM05]). The negative e ective-mass intro duced in this mo del has the e ect of
makingtheHamiltonianunb oundedfromb elow.Asitiswellknown,suchaHamiltonianisnotvery
go o d,esp ecially forstatistical purp oses(the thermal equilibriumstates are ill-de ned).However,
the correct interpretationis that (66)should b e considered just as anapproximationofthe true
dynamicsforsmallvaluesofthemomentump.
6 Hydrodynamicmodels
It is universally recognized that the hydro dynamic approach presents imp ortantprop erties b oth
from the theoretical and the numerical p oint of view b ecause it gives an interpretation of the
transp ort phenomenon by macroscopic quantities and it pro duces many advantages from the
computationalp ointofview.
Theliteratureonhydro dynamicmo delsisverybroad,b othinclassicalaswellasinsemiclassical
andquantumframework.
Someveryinterestingresultshaveb eenachieved,prop osingquantumhydro dynamicequations,
able to describ e the b ehaviour of nanometric devices like resonant tunneling dio des. Here, we
restrict ourselves to describing the hydro dynamic versions of the Kane mo del and of the M-M
Mostoftheresultspublishedintheliteraturerefertosingle-bandproblems.Thegeneralization
to multibandmo delspresents several diÆculties.Amongothers,thede nitionofthemacroscopic
quantitieswitharealisticphysicalmeaningandthediÆcultyinimp osingb oundaryconditions.
Inthis reviewwe give aninsightintotheclassicalderivation ofatwo-bandquantum uid.As
wehavesaid,theab ove-mentionedmultibandmo delsarebasedonthesingleelectronSchrodinger
equation, and the resultingequations areessentially linear.By applyingthe WKB metho d,it is
p ossibleto derive azero-temp erature hydro dynamicversion oftheSchrodinger two-bandmo dels.
Whenitisdesirable tomo delthedynamicsofafamilyof electrons,thestatistical description
requires theintro duction ofasequence of mixedstates,with anattachedo ccupation probability.
In this case, the WKB metho d leads to a sequence of hydro dynamic equations. Starting from
it, it is p ossible to derive a set of equations for certain macroscopic averaged quantities. These
hydro dynamicequations share asimilarstructure with the corresp onding equations for asingle
electron, the only di erence b eing the app earance of terms that can b e interpreted as thermal
tensors,andofadditionalsourceterms.Thesenewtermsdep endonallstates,sothesystemisnot
closed unless appropriateclosureconditionsareprovided.It isclearthat the nal hydro dynamic
mo delwithtemp erature isbynomeansequivalenttotheoriginalquantummo del.We couldsay
thatthenonlinearityoftheresultinghydro dynamicmo delis\genuine"andisthepricetopayfor
keepingonlya nitenumb erofequations.
6.1 Thehydro dynamicquantities
Inordertoobtainhydro dynamicversionsofthekineticmo delsdescrib ed intheprevioussections,
one p ossibility is to follow the general hydro dynamic approach to quantum mechanics due to
Madelung [LL77]. This approach consists in writing the wave function in the quasi-classical
form aexp iS
, where a is called the amplitude and S= the phase. With this approach, the
hydro dynamic limitis valid only for pure states, that is to say, it is valid only for a quantum
systematzero temp erature.Inthecaseofatwo-bandmo del,we have
a (x;t)=
p
n
a
(x;t)exp
iSa(x;t)
; a=c;v : (67)
where the squared amplitudehas the physical meaning of the probability density of ndingthe
\particle"atsomep ointinspace,andthegradientofthephasecorresp ondstotheclassicalvelo city
ofthe\particle".
Intheframeworkoftwo-bandmo dels,thedensities
n
ab
=
a b
;
are intro duced, where
a
, with a = c;v , is the envelop e function for the conduction and the
valence band, resp ectively. When a= b, thequantities n = n = j j 2
are real andrepresent
thep ositionprobabilitydensitiesoftheconductionbandandofthevalencebandelectrons,alb eit
onlyinanapproximatesense,since
c and
v
areenvelop e functionswhich mixtheBlo chstates.
Nevertheless, n=
c c
+
v v
isexactly thetotalelectron density intheconduction and inthe
valence band,and, as exp ected, itsatis es acontinuityequation. Whena6=b,the density
a b
is acomplexquantity,which do es not haveaprecise physical meaning.Despiteof this, asit will
b ecome clearinthenext section, the complexquantities
a b
app ear explicitly intheevolution
equationforthetotaldensityn.
Itiscustomary,after(67),towritethecouplingtermsinamoreconvenientway,byintro ducing
thecomplexquantity
n
cv
=
c v
= p
n
c p
n
v e
i
; (68)
where isthephase di erence de nedby
= S
v S
c
: (69)
Inthis way,inorder tostudy azero-temp erature quantumhydro dynamicmo del,we need touse
onlythethreequantitiesn
c
;n
v
andtocharacterize thezeroordermoments.
Thesituationis moreinvolved forthe current densities. In analogyto the one-band case,we
intro ducethequantummechanicalelectron currentdensities
J
ab
=Im
a r
b
: (70)
Itis naturaltorecovertheclassicalcurrentdensities,
J
c
= Im
c r
c
=n
c rS
c
; J
v
= Im
v r
v
=n
v rS
v
; (71)
whose physicalmeaningisclear.
Theintro duction ofthecomplexquantity(68)allowstowrite
a r
b
in(70)as
c r
v
=n
cv u
v
;
v r
c
=n
cv u
c
; (72)
wherethecomplexvelo citiesu
c andu
v
aregivenby
u
c
=u
os;c +iu
el;c
; u
v
=u
os;v +iu
el ;v
: (73)
withu
os;c andu
os;v
theso-calledosmotic velo cityandcurrent velo citygivenby
u
os;a
=
r p
n
a
p
n
a
; u
el;a
=rS
a
= J
c
n
a
;a=c;v : (74)
Inanalogywiththesingle-bandcasewehavede nedtheosmoticandcurrentvelo citiesascomplex
quantities which can b e expressed solely bymeansof n
c
;n
v
;J
c and J
v
.Inaddition,thecoupling
termn hasb eende ned byintro ducingthephase di erence .We notethat
rn
cv
=n
cv (u
c +u
v
): (75)
Coming back to the choice of the hydro dynamic quantities, we maintain that, for a zero-
temp eraturequantumhydro dynamicsystem,itissuÆcienttotake theusualquantitiesn
c ,n
v ,J
c
andJ
v
,plusthephase di erence .Thiswillb e con rmedinthenextsection.
6.2 Hydro dynamicversion of theKane system
TheKanemo delwasintro ducedinSection4.1byusingenvelop efunctions.Beforeintro ducingthe
hydro dynamicform,werewrite itbyusingdimensionlessvariables.Tothis aimwe intro ducethe
rescaled Planckconstant =~= ,where thedimensionalparameter isgivenby =mx 2
R
=t
R ,
by usingx
R andt
R
as characteristic (scalar) lengthandtimevariables. Theband energy can b e
rescaled by taking new p otential units V
0
= mx 2
R
=t 2
R
. A dimensional argument shows that the
originalcouplingco eÆcient isarecipro cal ofacharacteristic lenght,thusthe co eÆcientis scaled
byKx
R
,comp onentwise.
Hence,droppingtheprimesandwithoutchangingthenameofthevariables,wegetthefollowing
scaled Kanesystem,whichwillb e theobjectofourstudy:
i
@
c
@t
=
2
2
c +V
c c
2
Kr
v
i
@
v
@t
=
2
2
v +V
v v +
2
Kr
c
;
(76)
whereKistherescaledcouplinginterbandco eÆcient,istherescaledPlanckconstant,V
c
=E
c + V
and V
v
=E
v
+V. IntheKane mo delthe coupling parameterhasto b e considered constant. In
realistic heterostructure semiconductor devices, the parameter K, approximativelyexpressed in
termsofthee ectiveelectronmassandtheenergygap,dep ends onthelayercomp ositionthrough
thespatialco ordinates.
Takingintoaccountthewaveform(67)andusingtheequationsofsystem(76),timederivation
ofn
a
;a=c;v gives immediately
@n
c
@t
+rJ
c
= 2K Im( n
cv u
v )
@n
v
@t
+rJ
v
= 2K Im (n
cv u
c );
(77)
where (71) hasb een used forJ
c and J
v
. We remark that theright-hand sideof (77), containing
the termsn
cv u
v andn
cv u
c
, can b e expressed interms ofosmotic andcurrent velo cities, andthe
Addingtheequationsin(77)andusingtheidentity Im(
c r
v
) Im(
v r
c
)=rImn
cv
;
weobtainthebalancelawforthetotaldensity
@
@t (n
c +n
v
)+r(J
c +J
v
+2K Imn
cv )=0;
whichisjust thequantumcounterpart oftheclassicalcontinuityequation.
Thederivationoftheequationsforthephases S
c ,S
v
,andconsequentlyforJ
c andJ
v
,ismore
involved.
Referringthereadertotheoriginalpap er[AF05]formoredetails,theequationsforthecurrents
take theform
@J
c
@t
+ div
J
c J
c
n
c +
2
r p
n
c
r
p
n
c
2
4
rrn
c
+ n
c rV
c
= 2
Re
r(
c
(Kr
v
)) 2r
c
(Kr
v )
: (78)
@J
v
@t
+ div
J
v J
v
n
v +
2
r p
n
v
r
p
n
v
2
4
rrn
v
+ n
v rV
v
=
2
Re
r(
v
(Kr
c
)) 2r
v
(Kr
c )
: (79)
The left-handsides of the equations for the currents can b e put in a morefamiliar formby
using theidentity
div
r p
n
a
r
p
n
a 1
4
rrn
a
= n
a
2 r
p
n
a
p
n
a
; a=c;v :
Thecorrection terms
2
2
p
n
a
p
n
a
; a=c;v;
canb eidenti edwiththequantumBohmp otentialsforeachband,b ecausetheycanb einterpreted
as internal self-consistent p otentials, in analogywith the single-band case. The right-hand sides
can b efurtherexpressed intermsofthehydro dynamicquantities,obtainingthe nalsystem
@J
c
@t
+ div
J
c J
c
n
c
n
c r
2
p
n
c
2 p
n
c
+n
c rV
c
= rRe(n
cv Ku
v
) 2Re(n
cv Ku
v u
c );
@J
v
@t
+ div
J
v J
v
n
v
n
v r
2
p
n
v
2 p
n
v
+n
v rV
v
= rRe(n
cv Ku
c
)+2Re(n
cv Ku
c u
v ):
(80)
It is evident that the equations for the conduction and the valence band are coupled. Also,
oftheclassicalMadelung uidequationstoatwo-bandquantum uid. Inthis context, we cho ose
thefollowingconstraint
r= J
v
n
v J
c
n
c
: (81)
Nowwe areinp ositiontorewrite thehydro dynamicsystem(80)asfollows
@n
c
@t
+divJ
c
= 2K Im(n
cv u
v );
@n
v
@t
+divJ
v
= 2K Im(n
cv u
c );
@J
c
@t
+ div
J
c J
c
n
c
n
c r
2
p
n
c
2 p
n
c
+n
c rV
c
= rRe(n
cv Ku
v
) 2Re( n
cv Ku
v u
c );
@J
v
@t
+ div
J
v J
v
n
v
n
v r
2
p
n
v
2 p
n
v
+n
v rV
v
= rRe(n
cv Ku
c
)+2Re(n
cv Ku
c u
v );
r= J
v
n
v J
c
n
c
;
(82)
wheren
cv
;u
v ,andu
v
areexpressedinthetermsofthehydro dynamicquantitiesn
c
;n
v
;J
c
;J
v ,and
by(68)and(73).
6.3 Thenonzero-temp eraturecase
The extension of the previous analysis to an electron ensemble requires a quantum statistical
mechanicstreatment.Accordingtothegeneraldiscussionattheb eginningofSection5,itisp ossible
to represent an electron ensemble as amixed quantum mechanicalstate given by asequence of
pure states
k
, with o ccupation probabilities k
0,so that P
k
k
=1. Inthe two-band case,
each pure state is represented by a couple of envelop e-functions, k
c and
k
v
and, therefore, we
shallextend thede nitionofthehydro dynamicquantitiesas asup erp osition, withweights k
,of
the corresp onding pure-state quantities. For example,the density will b e n
a
= P
k
k
k
a k
a , for
a=c;v .
Inthesequelwe shallworkattheformallevel,andwe referto theequationsfoundin[AF05].
Thek -thstate fortheKanesystemisdescrib ed bythesolutionsofthesystem
i
@ k
c
@t
=
2
2
k
c +V
c k
c
2
Kr k
v
;
i
@ k
v
=
2
k
v +V
v k
v +
2
Kr k
c :
(83)
