transition by the Wigner-function approach
Lucio Demeio
Dipartimento di ScienzeMatematiche
UniversitaPolitecnica delleMarche
Via BrecceBianche1,I-60131 Ancona,Italy
Paolo Bordoneand Carlo Jacoboni
INFM- National Research Centeron
nanoStructures and bioSystems at Surfaces (S 3
)and
Dipartimento diFisica
Universitadi Mo dena eReggio Emilia
Via Campi 213/A,I-41100 Mo dena, Italy
Septemb er22, 2003
Abstract
Inthisworkwepresentarecentlydevelop edtransp ortmo del,basedonthe
Wigner-functionapproach and allowingfor non-parab olicband pro les. Two
scatteringmechanismsareincludedbymeansofaBoltzmann-likecollisionop-
erator,describing thecollisionsb etween electronsand p olaropticaland inter-
valleyphonons. Thetransp ort equationfor the Wigner functionis solvedby
theSplitting-Scheme algorithm. We have chosen a1-dimensionalmo delband
pro le, which exhibits satellitevalleys,b esides theminimumat thecenter of
theBrillouinzone,similartothebandpro leofGaAs.
Key-words: Wigner function, Semiconductors, Quantum Transp ort,Non-parab olic
Electron Transp ort.
TheWigner-function formalismhasb ecome a very common to olin the description
of the transp ort prop erties of semiconductors and electronic devices [1, 2, 3]. All
the relevant and realistic applications of this approach, however, have considered
until nowonly thosepro cesses whichcanb eadequatelydescrib ed within thesingle-
parab olic-band approximation. Recently, thegeneralization oftheWigner-function
formalismto a multi-band, non-parab olic transp ort mo del was intro duced [4,5, 6]
fora collisionless system.
In this work, we add a Boltzmann-like collision op eratorto thetransp ort equa-
tions, representing thecollisions of electronswith two scattering mechanisms. One
isthe standardp olar optical phononinteraction. A second mo del scattering mech-
anism isintro duced with phonons with constant energy and deformationp otential
interaction. Since this scattering allows large momentum transfers, it will b e re-
sp onsible forelectron transitionstothe bandregions ofthesecondaryminima. For
analogy with the many valleys mo del we shell refer to this mechanism as interval-
ley. Theab ove mo del will b eused tostudy numerically thetransp ort prop erties of
electronsin a semiconductor.
2 The Transport Model
Let f(x;p;t) b e the Wigner function of an ensemble of conduction electrons mov-
ing in a semiconductor. The single-band evolution equation, in the presence of an
arbitrarybandpro leandundertheactionofanexternalp otential,isgivenby[4,7]
@f
@t
(x;p;t)+(Af)(x;p;t)+(f)(x;p;t)=(Qf)(x;p;t); (1)
wherethetransp ortop eratorA,describingtheactionofthep erio dicp otentialofthe
lattice and the pseudo di erential op erator , describing the actionof theexternal
p otential,are de ned by
(Af)(x;p;t)= i
h
X
2L b
"()
f(x+
2
;p;t) f(x
2
;p;t)
e ip=h
(f)(x;p;t)= i
h
Z
ÆV(x;) b
f(x;;t)e ip =h
d:
and we have added a collision op erator on the right hand side. Here, f(x;;t)
is the Fourier transform of the Wigner function with resp ect to the momentum
variable, b
"(), 2 L (b eing L a vector of the crystal lattice), are the Fourier
co eÆcients of the energy band and ÆV(x;) = V(x+=2) V(x =2) is the
symb ol of the pseudo di erential op erator, with V(x) the external p otential. We
imp ose the b oundary conditions f(x;p
max
) =f(x; p
max
)=0, withp
max
=h=a,
andf( L;p)=f(L;p)=0,thatistheweassumethattheWignerfunctionvanishes
atthe b oundaryofthesimulation region. Thechoice oftheb oundaryconditions in
momentumspace,requiresthatwefollowthetimeevolutionoftheWignerfunction
only until all comp onents of the electron p opulation remain well inside the rst
Brillouinzone. Two scatteringmechanismsareincluded in thecollisionop eratoras
describ edintheprevioussection. Forthesakeofsimplicity,asa rstapproximation,
a1-dimensional(1-D)Boltzmann-likecollisionop eratorisconsidered. Thereduction
of the 3-D pronlem to1-D is obtained by integrating the transp ort equation along
theothertwodirections, wherea thermaldistribution andaconstant e ectivemass
m
haveb een used[1,8]. Thecollision op eratorthenresults tob e:
(Qf)(x;p;t)= Z
h =a
h =a [(p
0
;p)f(x;p 0
;t) (p;p 0
)f(x;p;t)]dp 0
(2)
The scattering cross section (p;p 0
) is given by the sum of theintervalley and the
p olaroptical crosssections, (p;p 0
)=
int (p;p
0
)+
opt (p;p
0
),where
int (p;p
0
)=S
int h
(N
0 +1)e
0
(p;p 0
)=(k
B T)
+N
0 e
0
+ (p;p
0
)=(k
B T)
i
(3)
opt (p;p
0
)=S
opt
(N
1
+1)I (p;p 0
)+N
1 I
+ (p;p
0
)
(4)
withS
int
=m
D 2
=(4hah!
0 ),S
opt
=(Q 2
=
0 )(
1
1
1
r
)(2m
=h 2
)(h!
1
=k
B
T)=(ha)
and
I
(p;p
0
)= Z
1
0 e
=(k
B T)
H(+
)
q
(k k 0
) 4
+(2m
=h 2
) 2
+(4m
=h 2
)(k k 0
) 2
(2+
)
d:
Here, H is the Heavyside function, k = p=h, 0;1
(p;p
0
) = max[ 0;1
(k ;k
0
);0],
0;1
(k ;k
0
) = "(k ) "(k 0
)h!
0;1
and N
0;1
= [exp( h!
0;1
=(k
B
T) 1]
1
are the
phonono ccupationnumb ers. Also,T =77Kisthetemp erature,D=10 9
eV cm 1
the deformation p otential, = 5:33 g cm 3
is the volumic mass, m
= 0:067m
e
is the electron e ective mass, h!
0
=0:0278 eV and h!
1
= 0:035eV are the ener-
giesof theintervalley andof thep olaroptical phonons,resp ectively,
0
thevacuum
1 r
p ermittivities, resp ectively.
3 Numerical results
Inthisexample, westudythetimeevolutionoftheWigner functionofan ensemble
of electronsunder theaction of a constant external eld and in presence of a non-
parab olicbandpro le. Weusethemo delofSection2andsolvenumerically equation
(1)with amo di ed versionof theSplitting-Scheme algorithm [9, 10]. Thecollision
op eratorisincludedintheverticalshifttogetherwiththepseudo di erentialop erator
thatdescrib estheexternal eld. Wehavechosenabandpro lesomewhatsimilarto
thebandpro leofGaAs,withaminimumatk=0andasecondminimumatab out
halfwayofthe Brillouinzoneedge. The bandshap e,showninFigure1,isgiven by
the nite Fourierexpansion "(k )= P
2L b
"
m ()e
ik
,with =l aand b
"()=b
l e
i
l
.
Here, b
0
= 2 eV, b
1
= 0:05 eV, b
2
= 0:05eV and b
3
= 0:25 eV, b
l
= 0, l > 3
and
l
=0, l =0;:::. Also, we take a =5:6510 8
cm (GaAs lattice p erio d), and
L=5000ade nesthespatialsimulationregion. An external eldE 21kVcm 1
actson a spatial regionslightly smallerthan thesimulation region.
The initial Wigner function corresp onds to a statistical mixture and is given
by f(x;p;0)=1=()e (
2
=
2
)(x x
0 )
2
(p hk
0 )
2
=( h) 2
; where x
0
is the initial average
p osition, h k
0
the initial average momentum, the initial momentum spread and
= the initial p osition spread. Also, thenormalization jjfjj= R R
f(x;p)dxdp=1
has b een used. In this example we have x
0
= 0:1 L, = L=100, k
0
= 0 and
=0:04=a.
We havep erformedtwosimulations andcomparedtheresults. Inone simulation
(referred to as R1), only the external eld is present (the scattering mechanisms
are removed). In a second simulation (R2),we use thefull mo del describ ed in the
previoussection,including thetwoscatteringmechanisms andtheexternal eld. A
casesimilar to(R1) hasalready b eenpresented in [10].
The results ofoursimulationsare shown in Figures2and 3. Figure 2showsthe
averagemomentum(Figure2A)andthemomentumspread (Figure2B)forthetwo
cases;thesolid lines referto(R2)and thedashed lines to(R1). Finally,Figures3A
and3B showtheWigner function f(x;p)inphase space att=2ps forsimulations
graphof theband pro le (inthe pvariable) is alsoshownforreference.
These results show that, when the collisional mechanisms are notincluded, the
Wignerfunctiontendstoremainorganizedandits shap edo esn'tchangemuchfrom
the initial Gaussian (see Figure 3A and Figure 2B, which shows that the momen-
tum spread remains constant overthis timescale). The second valley atp ositive p
b ecomes p opulated, in this case, only by e ect of the electric eld; the increase in
theaveragemomentumislinear atall times,while theaveragep osition (notshown
here),from ab out1psonward,su ersadecelerationdue tothereversaloftheslop e
of the energy band. In the presence of the collisional mechanisms, b oth valleys at
p > 0 and at p < 0 b ecome p opulated (see Figure 3B), due to the e ect of the
phonon-induced transitions, and themomentum spread growsmuch largerin time,
whilethe averagemomentum,afterthe initial rise, remainsalmostconstant.
Acknowledgement. Work partially supp orted by the U.S. OÆce of Naval
Research (contract No. N00014-98-1-0777) and by the Italian National Research
ProjectCOFIN2002\Mathematical problems ofKinetic Theories".
References
[1] W.R.Frensley, Rev. Modern Phys.,62:745-791(1990).
[2] P. Bordone, M.Pascoli, R. Brunetti, A. Bertoni and C. Jacob oni, Phys.Rev.
B59(4)(1999) 3060-3069.
[3] C.Jacob oni,R.Brunetti,P.Bordone,andA.Bertoni,Quantumtransp ortansits
simulationwiththe Wigner-function approach,in Topicsin High FieldTrans-
port in Semiconductors, K. Brennan and P.Paul Ruden, ed. (World Scienti c,
Singap ore 2001),pp.25-61.
[4] L. Demeio, L. Barletti, P. Bordone and C. Jacob oni, Transp ort Theory and
Statistical Physics, 32(3-4),321-339(2003).
[5] L. Demeio, P. Bordoneand C.Jacob oni, Pro c. XXI I I Symp osium on Rare ed
GasDynamics, Whistler,BC, Canada,July 20-25,2002.
on Transp ort Theory, London, July 2001, in TransportTheory Stat. Phys., in
press.
[7] P.A.Markowich,N.J.Mauserand F.Poupaud,J.Math. Phys.,35:1066-1094
(1994).
[8] B.Niclot,P.DegondandF. Poupaud,J.Computat.Phys.78: 313-349(1988).
[9] A.Arnold and C.Ringhofer, SIAMJ. of Num. Anal.33: 1622-1643(1996).
[10] L. Demeio, Pro c. 9th Int. Workshop Computat. Electronics, to app ear in the
Journal of Computational Electronics.
Figure 1.Band pro le"(k) ineV, ka.
Figure 2. (A) Average momentum < p > (t) and (B) momentum
spread p(t)in units of h =a, 0 t2 ps.Dashed lines,simulation
(R1); solid lines,simulation(R2).
Figure 3. f(x;p;t) at t = 2 ps, L x L, h=a p h =a,
for (A) simulation (R1) and (B) simulation(R2). The band pro le
is shown on the o or for reference.
-3 -2 -1 0 1 2 3
k
1.4 1.8 2.2
E
Figure 1: Bandpro le"(k )ineV, k a.
0 0.5 1 1.5 2
t
0.5 1 1.5 2
<p>
A
0 0.5 1 1.5 2
t
0.5 1 1.5
<dp >
B
Figure2: (A)Average momentum<p>(t) and(B)momentumspread p(t)inunits of
h=a,0t2ps.Dashedlines, simulation(R1);solidlines,simulation(R2).
A
-4000 0
4000
x
-2 0
2
p
0 0.008
f
0
B
-4000 0
4000
x
-2 0
2
p
0 0.001
f
0
Figure3: f(x;p;t)at t=2ps, LxL, h =aph =a, for(A)simulation(R1)
and(B)simulation(R2).Thebandpro leisshownonthe o orforreference.
