Coupled dono s in quan um do s: Quan um size and dielec ic misma ch e ec s
J. L. Mo illa, A. Balles e , and J. Planelles*
Depa amen de Química Física i Analí ica, Uni e si a Jaume I, Box 224, E-12080 Cas elló, Spain
共Recei ed 17 Feb ua y 2009; e ised manusc ip ecei ed 17 Ap il 2009; published 20 May 2009兲
Spa ial and dielec ic con inemen modula ions o he spon aneous emission a es, ansi ion ene gies, and
cha ge-densi y dis ibu ions o a singly ionized double dono sys em 共D2
+兲in a sphe ical quan um do a e
calcula ed wi hin he amewo k o he e ec i e-mass en elope unc ion app oxima ion. Dipole momen s,
ene gy spli ings, ansi ion momen s, elec on-densi y dis ibu ions, and spon aneous emission a es in ol ing
bonding and an ibonding lowes -lying molecula s a es a e add essed o di e en dielec ic en i onmen s,
quan um do adii, and ela i e loca ions o he coupled impu i ies inside he do . The esul s indica e ha he
dono molecule beha es as he e opola when he spa ial con inemen b eaks he in e sion symme y, which is
pa alleled by a s ong educ ion in he exci ed-s a e adia i e li e ime. Dielec ic con inemen , ac ing on a la ge
leng h scale han spa ial con inemen , may eco e he bulklike homopola cha ac e when he do is embedded
in a low dielec ic cons an medium. In he weak spa ial con inemen egime, dielec ic e ec s can inc ease he
co esponding bulk adia i e li e imes signi ican ly and simul aneously modula e he cha ge-densi y
dis ibu ion.
DOI: 10.1103/PhysRe B.79.195319 PACS numbe 共s兲: 71.55.⫺i, 73.21.La, 77.22.Ej, 03.67.Lx
I. INTRODUCTION
The sys em buil o wo coupled semiconduc o quan um
do s 共QDs兲con aining elec ons, holes, o an exci on cons i-
u es one o he simples solid s uc u es p oposed as he
unc ional pa in a wide ange o de ice applica ions, in-
cluding spin onics,1op oelec onics,2pho o ol aics,3and
quan um in o ma ion echnologies.4,5The widesp ead in e -
es in his sys em elies on he abili y o manipula e con e-
nien ly i s molecula p ope ies, such as he ene gy spli ing
be ween he bonding 共nodeless兲and an ibonding 共noded兲
lowes -lying molecula like s a es o he spa ial dis ibu ion o
ca ie s in he sys em.6In his ega d, he asymme y o he
cons i uen do s 共ei he in insic o in en ional兲has been
iewed no as a law bu as an essen ial design choice o
p o ide addi ional con ol o e he sys em p ope ies.5,7,8
Ad ances in single-dopan ab ica ion9–11 and cha ge
de ec ion12,13 ha e led a omic scale elec onics 共pa icula ly
he ield o quan um compu ing兲 o pea inc easing a en ion
o ano he molecula sys em. Namely, he D2
+sys em, i.e., he
sys em buil o wo coupled dono impu i ies in a semicon-
duc o hos in which one o he wo excess elec ons has
been ionized. This sys em encodes he logical in o ma ion
ei he on he spin o on he cha ge deg ees o eedom o he
emaining excess elec on.14–20 Ini ial p oposals elied on
spin encoding. Howe e , di ec de ec ion o a single spin
emains oday a majo challenge.21 On he con a y, a ac-
ion o a single elec on cha ge can be easily de ec ed wi h
s a e-o - he-a single elec on- unneling de ices.12,13 This
eadou easibili y, oge he wi h he ela i e simplici y o
he manipula ion o cha ge deg ees o eedom as opposed o
spin, has aised in e es in cha ge qubi s, whe e he logical
s a es a e ep esen ed by he wo lowes -lying elec on o bi al
s a es posi ioned a he di e en dono s. All he same, he
sho cohe ence imes o o bi al s a es se e ely limi he scal-
abili y o cha ge qubi s,16 so ha mechanisms enhancing co-
he ence would be desi able.
S udies on con ined D2
+sys ems ha e been ecen ly ad-
d essed in he li e a u e.22–24 Pa icula ly, Kang e al.24 ound
ha he con inemen o he D2
+sys em in a sphe ical QD can
g ea ly enhance he ene gy di e ences be ween he wo
lowes -lying elec on s a es and mo e exci ed s a es, a o ing
he iden i ica ion o he quasi- wo-le el sys em equi ed o
quan um compu a ion pu poses.25
Qui e a la ge amoun o wo k has been done in he s udy
o he elec onic s uc u e and ela ed p ope ies o single
shallow dono s in nanoscale semiconduc o
he e os uc u es.26–28 In addi ion o he well-known quan um
size e ec , he in luence o he dielec ic con inemen 共com-
ing om he di e en dielec ic esponse o he QD and he
su ounding medium兲on he dono le els has been e ealed
as a pa icula ly no iceable e ec in colloidal quan um do s
as hey a e usually syn hesized in media wi h a dielec ic
esponse a he di e en o ha o he do ma e ial.26,29,30
P omp ed by he impo an ole o coupled impu i ies in
nano echnology, we p esen in his pape a heo e ical s udy
on he molecula p ope ies o a D2
+sys em con ined in a
sphe ical quan um do . We ocus on he in luence o bo h he
QD size and he dielec ic misma ch on he adia i e li e-
ime, he bonding-an ibonding ene gy spli ing, and he elec-
on cha ge dis ibu ions in he lowes -lying D2
+molecula
s a es, and analyze he dependence o he esul s on he ela-
i e posi ion o he dono s inside he quan um do .
The pape is o ganized as ollows. In Sec. II, we in oduce
he model employed in he desc ip ion o he con ined double
dono sys em and ou line he de ails o he nume ical p oce-
du e used o sol e he esul ing Hamil onian. The e ec s o
spa ial and dielec ic con inemen s on he double dono sys-
em a e p esen ed and discussed in Sec. III. Conclusions,
gi en in Sec. IV, end he pape .
II. MODEL AND THEORETICAL OUTLINE
The p esen s udy is ca ied ou wi hin he amewo k o
he e ec i e-mass and en elope unc ion app oxima ions.
Consequen ly, we employ a mac oscopic ea men o Cou-
lombic in e ac ions, and use a pa ame e , he dielec ic con-
PHYSICAL REVIEW B 79, 195319 共2009兲
1098-0121/2009/79共19兲/195319共6兲©2009 The Ame ican Physical Socie y195319-1
s an , o cha ac e ize he dielec ic esponse o each ma e ial
in ol ed. The alidi y o such a ea men has been well
es ablished o bo h doped and undoped semiconduc o
he e os uc u es.31 Ou model consis s o a shallow dono
pai con ined in a sphe ical quan um do wi h dielec ic con-
s an ␧QD, which is in u n imme sed in a dielec ic medium
wi h pe mi i i y ␧ou 共see Fig. 1兲. The co esponding
e ec i e-mass Hamil onian o he bound conduc ion-band
elec on eads, in a omic uni s,
H=−1
2
䉮
冉
1
mⴱ共 兲䉮
冊
+V共 兲+
␾
s共 兲+兺
i=1,2
␾
c
Ii共 , i兲.
共1兲
The i s e m on he igh -hand side o Eq. 共1兲is he
gene alized kine ic-ene gy ope a o ,32 which accoun s o
di e en elec on masses in di e en ma e ials. V共 兲is a
s eplike unc ion ep esen ing he ini e spa ial con ining po-
en ial due o he conduc ion-band o se be ween he media
in ol ed.
␾
c
Ii共 , i兲s ands o he Coulomb po en ial gene -
a ed by he impu i y loca ed a he ixed posi ion i, including
he e ec o he pola iza ion cha ges induced a he do su -
ace as a consequence o he dielec ic misma ch be ween he
QD and he su ounding medium. The elec on i sel also
induces pola iza ion cha ges a he do bounda ies, whose
e ec s a e desc ibed by he sel -pola iza ion po en ial
␾
s共 兲.
When 共as is he case he e兲a␧共 兲=␧共 兲s eplike dielec ic
unc ion is assumed,
␾
cand
␾
sadmi analy ical
exp essions.33
In o de o educe he numbe o a iables in he s udy, we
ix he QD ma e ial pa ame e s close o hose o CdS 共␧QD
=5.5 and mⴱ=0.15兲共Re . 34兲unless o he wise indica ed.
Since we a e no dealing wi h a speci ic su ounding me-
dium, we will assume a ypical dep h o V0=4 eV o he
QD con ining well and an e ec i e mass mⴱ=1 o he ex-
e nal medium. Likewise, we will concen a e on dono s
which a e aligned along an axis o he QD 共see Fig. 1兲,
al hough he conclusions ob ained make possible o en isage
he ends cha ac e izing o he con igu a ions. This acili a es
he calcula ion since he axial symme y o he sys em allows
us o in eg a e analy ically he angula coo dina e o he elec-
on in Eq. 共1兲. Then, o ob ain elec on ene gies and wa e
unc ions, we ca y ou an exac 共nume ical兲in eg a ion o
共
␳
,z兲coo dina es by means o a disc e iza ion scheme based
on he ini e di e ences me hod. The explici exp essions o
␾
cand
␾
s oge he wi h a de ailed desc ip ion o he in eg a-
ion me hod can be ound in Re . 28.
III. RESULTS
We s a by in es iga ing he e ec o he spa ial and di-
elec ic con inemen s on he elec on cha ge dis ibu ion o
he double dono lowes -lying s a es. When he cen e o
mass o he dono pai is no loca ed a he do cen e 共zCM
⫽0, see Fig. 1兲, he asymme y o he QD con inemen is
expec ed o b eak he in e sion symme y o hese s a es. To
moni o 共and quan i y兲 he symme y b eakdown, we calcu-
la e he dipola momen o he co esponding elec on cha ge
dis ibu ions wi h espec o zCM,
␮
=具i兩 −zCMuz兩i典,共2兲
whe e i=1,2 labels he g ound and i s exci ed s a es o he
sys em. The esul s a e shown in Fig. 2, whe e
␮
/dI−I共wi h
dI−I=10 nm兲is ep esen ed o hese s a es as a unc ion o
he QD adius and he dono s loca ion wi hin he do . Calcu-
la ions ha e been pe o med o wo di e en su ounding
dielec ic cons an s, namely, ␧ou =␧QD 共absence o dielec ic
con inemen 兲and ␧ou =1 共spa ial and dielec ic con ine-
men s兲. In he igu e, posi i e alues o
␮
/dI−Iindica e ha
he elec on is dis ibu ed o a g ea e ex en a ound he do-
no close o he QD bo de , whe eas nega i e alues indi-
ca e i s p e e ence o he inne mos impu i y. F om he
␧ou =␧QD panels we can obse e ha he spa ial con inemen
a o s he localiza ion o he g ound s a e in he inne dono
CM
RQD
I2
1
I
εQD
εou
dI−I
z
z
ρ
FIG. 1. Schema ic ep esen a ion o he sys em unde s udy: an
elec on bound o wo shallow dono s sepa a ed by a dis ance dI−I
wi hin a diame e o a con ining sphe ical QD wi h adius RQD and
dielec ic cons an ␧QD su ounded by a dielec ic medium o pe -
mi i i y ␧ou .zCM s ands o he cen e o mass o he coupled
dono s. FIG. 2. 共Colo online兲Dipola momen 关Eq. 共2兲兴 o he elec on
cha ge dis ibu ion o a con ined D2
+sys em wi h dI−I=10 nmasa
unc ion o RQD and he sys em loca ion in he do . Le 共 igh 兲
panels co espond o he elec on in he g ound 共 i s exci ed兲s a e.
In op 共bo om兲panels he QD is embedded in a ␧ou =␧QD 共␧ou
=1兲dielec ic medium. The o igin has been loca ed a he QD
cen e .
MOVILLA, BALLESTER, AND PLANELLES PHYSICAL REVIEW B 79, 195319 共2009兲
195319-2
and he exci ed s a e in he ou e one. This end, mo e p o-
nounced o low QD adii and la ge zCM alues, is a conse-
quence o he la ge des abiliza ion ha he po en ial ba ie
p oduces in he su oundings o he ou e mos impu i y. To
minimize i s ene gy, he elec on in he g ound s a e localizes
in he inne dono and, o p ese e o hogonali y, he exci ed
s a e concen a es in he ou e one. In o he wo ds, nonsym-
me ic spa ial con inemen b ings he homopola D2
+sys em
o beha e as he e opola , leading o he spa ial sepa a ion o
he wo lowes -lying elec on s a es.
In he case ␧ou =1 共lowe panels in Fig. 2兲, we can see
ha , excep o e y small do adii and e y la ge zCM al-
ues, he pola i y o he s a es is opposi e o ha in he ab-
sence o dielec ic con inemen . Now, he g ound s a e local-
izes he elec on densi y in he ou e dono , whe eas he inne
dono is occupied by he exci ed s a e. This e ec can be
explained as o igina ed by he con ibu ion o he pola iza-
ion cha ges induced by he impu i ies a he QD su ace.
Impu i y o -cen e ing inc eases he nonhomogenei y o he
co esponding induced cha ge dis ibu ion a he do su ace,
his lack o homogenei y being esponsible o an addi ional
s abiliza ion o o -cen e ed elec on loca ions.35 As a conse-
quence, he o al Coulomb po en ial is deepe a he posi ion
o he dono close o he su ace, so ha he g ound s a e
dis ibu es p e e ably a ound i . The e o e, spa ial and dielec-
ic con inemen s yield opposi e spa ial sepa a ion o he
elec on s a es36 and hei in e play will de e mine he inal
con igu a ion o he co esponding elec on cha ge dis ibu-
ions in each case.
Fo a mo e de ailed desc ip ion o his in e play, in Fig.
3共a兲we ha e depic ed
␮
/dI−I o he same sys em as in Fig.
2共dI−I=10 nm兲bu o he pa icula case o zCM=5 nm and
a mo e ex ended RQD ange. Fo his alue o zCM, one o he
impu i ies is loca ed a he QD cen e , a si ua ion a p io i
easie o be ob ained expe imen ally han any o he . In he
igu e, solid 共dashed兲cu es co espond o he g ound 共 i s
exci ed兲s a e. In e es ingly, he case o pu e spa ial con ine-
men 共␧ou =␧QD, hin ed lines兲shows homopola i y o la ge
RQD alues, whe eas in he p esence o dielec ic misma ch
共␧ou =1, hick blue lines兲 he sys em beha es as he e opola .
This is a signa u e o he di e en leng h scales in which
spa ial and dielec ic con inemen s ope a e and is a conse-
quence o he long ange o image cha ge Coulomb in e ac-
ions. The di e en leng h scales o hese con inemen
sou ces we e al eady epo ed by Goldoni e al.,37 who also
in oduced he concep o emo e dielec ic ailo ing o une
a will exci on-binding ene gies in semiconduc o /insula o
hyb id nanos uc u es.
Fo in e media e alues o RQD 关see e.g., RQD⬇14 nm in
Fig. 3共a兲兴, whe e bo h spa ial and dielec ic con inemen e -
ec s a e signi ican , we can obse e ha he dielec ic con-
inemen compensa es, and e en in e s, he end imposed
by he spa ial con inemen . Finally, o low RQD alues, he
spa ial con inemen domina es and he inne dono is p e-
e ed by he g ound s a e in bo h cases.
The spa ial sepa a ion o he wo lowes -lying elec on
s a es is mo e p onounced as he in e dono dis ance in-
c eases. This is appa en in Fig. 4共a兲, which co esponds o
he same sys em as in Fig. 3共wi h one o he impu i ies a
he do cen e 兲bu o dI−I=15 nm. In his case,
␮
/dI−Iis
plo ed as a unc ion o ␧ou o wo di e en do adii,
namely, RQD=20 and 23 nm. Again, solid and dashed lines
a e used o he g ound and i s exci ed s a es. As can be
obse ed om he RQD=20 nm cu es, when ␧ou ⬎␧QD he
dielec ic con inemen magni ies he s a es spa ial sepa a ion
imposed by he spa ial con inemen 共␧ou =5.5兲. On he con-
a y, his end is in e ed as ␧ou diminishes. The bulklike
homopola cha ac e is eco e ed o a gi en alue o ␧ou
共3.9 in his case兲, compensa ing he e ec o he spa ial con-
inemen . Finally, dielec ic misma ch e ec s domina e o
low enough alues o ␧ou , yielding again a spa ial sepa a ion
o he lowes -lying molecula s a es bu wi h opposi e pola -
i y.
This e olu ion can be seen in Fig. 4共d兲, whe e he modu-
lus o he wa e unc ion ⌿共 兲 es ic ed o he azimu hal
angle
␾
=0 is ep esen ed o bo h he g ound and i s ex-
ci ed s a es and h ee di e en alues o he su oundings
−0.4
−0.2
0
0.2
0.4
µ/d
I−I
0.2
0.4
0.6
0.8
1
µ12 /µ12
bulk
1
3
5
7
∆E/∆Ebulk
12 16 20 24 28
100
101
10−1
τ
/
τbulk
εou =εQD
εou =1
(a)
(b)
(c)
R
Q
D(nm)
g ound s a e
1s exci ed
(d)
FIG. 3. 共Colo online兲共a兲Dipola momen 关Eq. 共2兲兴 o he elec-
on cha ge dis ibu ion in he g ound 共solid lines兲and i s exci ed
共dashed lines兲s a es o he con ined coupled dono sys em 共dI−I
=10 nm兲wi h one o he impu i ies loca ed a he QD cen e as a
unc ion o he QD adius. Thick blue lines co espond o ␧ou =1,
whe eas hin ed lines co espond o ␧ou =␧QD 共absence o dielec ic
con inemen 兲.共b兲Co esponding ansi ion momen s, 共c兲spli ing
ene gies, and 共d兲 i s exci ed-s a e adia i e li e imes ela i e o
bulk alues.
COUPLED DONORS IN QUANTUM DOTS: QUANTUM SIZE…PHYSICAL REVIEW B 79, 195319 共2009兲
195319-3
pe mi i i y, i.e., ␧ou =5.5 共absence o dielec ic con ine-
men 兲,␧ou =3.9 共si ua ion in which he homopola cha ac e
is eco e ed兲, and ␧ou =2 共a ypical alue o o ganic capping
ma e ials30兲.
The case shown in Fig. 4 o RQD=23 nm ep esen s an
addi ional example o he long ange o dielec ic e ec s.
While in his case bulk cha ge dis ibu ions a e ha dly a -
ec ed by he QD spa ial con inemen , he p ope choice o
he QD en i onmen can s ill yield impo an changes in he
ela i e elec onega i i y o he dono s and, he e o e, in he
elec on cha ge dis ibu ions.
The ene gy spli ing 共⌬E兲be ween he g ound and i s
exci ed s a es is also in luenced by dielec ic and spa ial con-
inemen s 关see Fig. 4共b兲兴. As de i ed om he igu e, ⌬E
inc eases as he spa ial sepa a ion o he s a es is mo e p o-
nounced, p esen ing a minimum, close o he bulk alue,
when homopola condi ions a e uned. Con a y o ⌬E, he
co esponding binding ene gies38 共Eb兲a e negligibly co e-
la ed wi h he spa ial sepa a ion o he s a es 关see Fig. 4共c兲兴.
This can be de i ed om he compa ison o solid and do ed
cu es in he igu e, which co espond o zCM=7.5 nm 共one
dono a he QD cen e 兲and zCM=0 共symme ic con ine-
men 兲, espec i ely. Indeed, hough bo h cases p esen simi-
la Eb alues, o zCM=0 he sys em p esen s homopola i y
ega dless o ␧ou , whe eas his is no he case o zCM
=7.5 nm. The e o e, Ebis negligibly in luenced by he o -
cen e ing o he D2
+sys em and hen by he spa ial sepa a ion
o he s a es. Con e sely, as expec ed, Ebis s ongly a ec ed
by he dielec ic en i onmen , unde going a la ge inc ease as
␧ou dec eases.
I is well known ha he spa ial sepa a ion o logic s a es
in cha ge-based qubi ealiza ions can e icien ly educe he
decohe ence o he cha ge deg ees o eedom coding he
qubi , which makes possible o pe o m he cohe en quan-
um logic ope a ions a a ai ly high a io.8The e o e, con-
inemen is expec ed o a ec cohe ence in ou sys em. In
dono -based cha ge qubi s, decohe ence caused by in e ac-
ion wi h phonons occu s in ime scales ha can be made
much longe han all o he ime scales in he p oblem by
choosing an app op ia e dono sepa a ion.20 Then, o s udy
how con inemen a ec s decohe ence in ou sys em, we will
neglec in a i s app oxima ion acous ic and op ical phonon
sca e ing and will only ake in o accoun decohe ence com-
ing om acuum elec omagne ic luc ua ions 共 adia i e de-
cay兲.
The adia i e decay a e in a sphe ical quan um do can be
es ima ed wi hin he dipole app oxima ion by39
␶
−1 =9␧ou
5/2
共2␧ou +␧QD兲2
⌬E3
3
␲
␧0ប4c3
␮
12,共3兲
whe e cis he speed o ligh in acuum, ␧0is he acuum
dielec ic cons an ,
␶
is he adia i e li e ime, and
␮
12
=兩具1兩 兩2典兩2is he squa e o he dipole ansi ion momen be-
ween he s a es 兩1典and 兩2典, which a e in ou case he g ound
and i s exci ed s a es o he sys em.
␮
12, ela i e o i s alue in he absence o spa ial and
dielec ic con inemen s, is depic ed in Fig. 3共b兲 o he s ud-
ied sys em. Again, esul s o ␧ou =␧QD and ␧ou =1 a e
shown. As can be seen in he igu e,
␮
12 dec eases as he
sys em mo es away om homopola condi ions. By inspec-
ion o Eq. 共3兲, one could expec ha a dec ease in
␮
12 would
help in inc easing he li e ime. Howe e , he dependence o
␶
−1 on ⌬E3canno be neglec ed in his case since, as we
showed be o e, spa ial sepa a ion o he wo lowes -lying
s a es is pa alleled by an inc ease in he ansi ion ene gy
关see Fig. 3共c兲兴. On balance, he in luence o ⌬Eon he adia-
FIG. 4. 共Colo online兲共a兲G ound s a e 共solid lines兲and i s
exci ed-s a e 共dashed lines兲dipola momen s o a dI−I=15 nm con-
ined D2
+sys em wi h one dono a he QD cen e as a unc ion o
␧ou . Cu es a e shown o wo di e en QD adii. 共b兲Spli ing
ene gy dependence on ␧ou o he wo QD adii s udied. 共c兲Co e-
sponding g ound-s a e binding ene gies 共solid lines兲. Cu es co e-
sponding o zCM=0 共do ed lines兲ha e been included o compa i-
son. 共d兲Modulus o he wa e unc ion ⌿共 兲 es ic ed o he
azimu hal angle
␾
=0 o he same sys em as abo e wi h RQD
=20 nm and di e en alues o ␧ou .
MOVILLA, BALLESTER, AND PLANELLES PHYSICAL REVIEW B 79, 195319 共2009兲
195319-4
i e li e ime is la ge han ha o
␮
12. This can be seen in
Fig. 3共d兲, whe e
␶
has been ep esen ed ela i e o
␶
bulk. Re-
e ing o he esul s o ␧ou =␧QD 共whe e only he spa ial
con inemen is in luencing he sys em兲, we obse e ha
␶
d ops s ongly o small alues o RQD, whe eas i eaches
he bulk alue when he spa ial sepa a ion o he wo lowes -
lying s a es disappea s and homopola condi ions a e eco -
e ed.
The esul s o ␧ou =1 a e mo e in e es ing.
␶
shows a
p o ile simila o ha o ␧ou =␧QD calcula ions bu despi e
spa ial sepa a ion s ill pe sis s in he whole ange o RQD
unde s udy,
␶
eaches alues mo e han 1 o de o magni ude
la ge han in bulk. The enhancemen o
␶
when he QD is
su ounded by a low-dielec ic cons an medium is a conse-
quence o he in luence o local- ield e ec s,39 which ha e
been obse ed expe imen ally h ough measu emen s o ex-
ci on ecombina ion a es in QDs embedded in apola sol-
en s wi h low e ac i e indices.30
The d op o
␶
o small alues o RQD in Fig. 3共d兲is
induced, in bo h ep esen ed cases, by he spa ial con ine-
men h ough he la ge inc ease ha i yields on ⌬E. This can
be con i med in Fig. 5共le panel兲whe e he adia i e li e-
ime o ou sys em as a unc ion o ␧ou and he sys em loca-
ion wi hin a R=14 nm QD is shown. When he o cen e -
ing o he dono pai is no much p onounced 共low and
medium zCM alues兲, he in luence o he spa ial con inemen
is negligible and, in he absence o dielec ic misma ch,
␶
keeps close o he bulk alue. In hese condi ions,
␶
inc eases
no iceably wi h espec o he bulk as ␧ou dec eases. How-
e e , as he dono pai ge s close o he do su ace 共la ge
zCM alues兲, he adia i e li e ime becomes domina ed by he
e ec s o spa ial con inemen .
␶
expe iences a s ong educ-
ion, which is only pa ially compensa ed by dec easing ␧ou .
Finally, o p obe he e ec o inc easing he QD dielec ic
cons an , we ha e epea ed he calcula ions o QD pa am-
e e s simila o hose o Si 共␧QD=11.7 and mⴱ=0.26兲. The
esul s, shown in he igh panel o Fig. 5, e idence ha in
his case
␶
may inc ease up o almos 2 o de s o magni ude
wi h espec o he co esponding bulk alue.
IV. CONCLUDING REMARKS
We ha e p obed he in luence o spa ial and dielec ic
con inemen s on spon aneous emission a es, ene gy spli -
ings, and cha ge-densi y dis ibu ions in he lowes -lying
s a es o a D2
+double dono sys em con ined in a sphe ical
quan um do . We ound ha asymme ic con inemen 共p o-
duced when he dono s a e no symme ically dis ibu ed
a ound he QD cen e 兲leads he o he wise homopola dono
pai o beha e as he e opola , diminishing he in e dono un-
nel coupling and localizing he g ound and i s exci ed s a es
in di e en dono s. In his ega d, when 共as usual兲␧ou
⬍␧QD, spa ial con inemen ends o localize he g ound/ i s
exci ed s a e in he inne /ou e dono , whe eas dielec ic e -
ec s end o dis ibu e hem opposi ely. Howe e , hese con-
inemen sou ces ope a e in di e en leng h scales. While in
he la ge do size egime only dielec ic e ec s in luence he
D2
+sys em, in he small do size egime spa ial con inemen
e ec s domina e. This makes possible o une he unnel cou-
pling s eng h and cha ge-densi y dis ibu ion in he D2
+sys-
em a will, by choosing he app op ia e do size and su -
ounding medium.
We also ound ha bulk adia i e li e imes a e s ongly
educed by spa ial con inemen due o he enhanced spli ing
ene gies. On he o he hand, when spa ial con inemen e -
ec s a e no longe in ol ed 共la ge do sizes兲, adia i e li e-
imes may inc ease o de s o magni ude when he QDs a e
embedded in low-dielec ic cons an media.
Summing up, he con inemen o he D2
+sys em in a quan-
um do p o ides addi ional mechanisms o mold i s p ope -
ies; i is wo h o no ice he spa ial dis ibu ion o he
lowes -lying s a es, he co esponding ene gy spli ings, and
he adia i e li e imes. The 共wide ange兲 uning capabili y o
his las p ope y could be o in e es o inc ease he ela-
i ely low bulk cohe ence imes o o bi al s a es in cha ge
qubi ealiza ions.
We close by s essing ha con inemen migh also be use-
ul o imp o e he pe o mance o op ically d i en dono -
based cha ge qubi schemes.17,18 In such schemes, he deco-
he ence caused by he ioniza ion and spon aneous emission
o he exci ed s a es ac ing as connec ing channels be ween
he logical s a es could be minimized h ough he p ope
choice o QD size and dielec ic en i onmen . On one hand,
quan um con inemen would li he quasidegene acy among
nea -con inuum s a es, inc easing he co esponding ioniza-
ion h eshold and exci a ion selec i i y. On he o he hand,
he employmen o low dielec ic cons an su ounding ma-
e ials would help in educing hei 共o he wise limi ing兲
spon aneous emission a es.
ACKNOWLEDGMENTS
Suppo om MICINN unde P ojec No. CTQ2008-
03344 and Bancaixa unde P ojec No. P1-1B2006-03 is ac-
knowledged.
FIG. 5. 共Colo online兲Fi s exci ed-s a e adia i e li e ime 共 ela-
i e o bulk兲 o a dI−I=10 nm D2
+sys em con ined in a R
=14 nm QD as a unc ion o ␧ou and he sys em loca ion wi hin he
QD. Hos ma e ial pa ame e s a e simila o hose o CdS 共le 兲and
Si 共 igh 兲. Dashed lines indica e he absence o dielec ic con ine-
men in each case.
COUPLED DONORS IN QUANTUM DOTS: QUANTUM SIZE…PHYSICAL REVIEW B 79, 195319 共2009兲
195319-5

*[email p o ec ed]
1S. A. Wol , D. D. Awschalom, R. A. Buh man, J. M. Daugh on,
S. on Molná , M. L. Roukes, A. Y. Ch chelkano a, and D. M.
T ege , Science 294, 1488 共2001兲.
2Nano-op oelec onics: Concep s, Physics and De ices, edi ed by
M. G undmann 共Sp inge -Ve lag, Be lin, 2002兲.
3A. J. Nozik, Physica E 共Ams e dam兲14,115共2002兲.
4B. Baye , P. Haw ylak, K. Hinze , S. Fa a d, M. Ko kusinski, Z.
R. Wasilewski, O. S e n, and A. Fo chel, Science 291, 451
共2001兲.
5E. A. S ina , M. Scheibne , A. S. B acke , I. V. Ponoma e , V.
L. Ko ene , M. E. Wa e, M. F. Do y, T. L. Reinecke, and D.
Gammon, Science 311, 636 共2006兲.
6J. R. Pe a, A. C. Johnson, C. M. Ma cus, M. P. Hanson, and A.
C. Gossa d, Phys. Re . Le . 93, 186802 共2004兲.
7A. S. B acke , M. Scheibne , M. F. Do y, E. A. S ina , I. V.
Ponoma e , J. C. Kim, L. J. Whi man, T. L. Reinecke, and D.
Gammon, Appl. Phys. Le . 89, 233110 共2006兲.
8X. Q. Li and Y. A akawa, Phys. Re . A 63, 012302 共2000兲.
9J. L. O’B ien, S. R. Scho ield, M. Y. Simmons, R. G. Cla k, A. S.
Dzu ak, N. J. Cu son, B. E. Kane, N. S. McAlpine, M. E. Haw-
ley, and G. W. B own, Phys. Re . B 64, 161401共R兲共2001兲.
10 S. R. Scho ield, N. J. Cu son, M. Y. Simmons, F. J. Ruess, T.
Hallam, L. Obe beck, and R. G. Cla k, Phys. Re . Le . 91,
136104 共2003兲.
11 T. Shinada, S. Okamo o, T. Kobayashi, and L. Ohdoma i, Na u e
共London兲437, 1128 共2005兲.
12 R. J. Schoelkop , P. Wahlg en, A. A. Hozhe niko , P. Delsing,
and D. E. P obe , Science 280, 1238 共1998兲.
13 A. Aassime, G. Johansson, G. Wendin, R. J. Schoelkop , and P.
Delsing, Phys. Re . Le . 86, 3376 共2001兲.
14 M. J. Calde ón, B. Koille , and S. Das Sa ma, Phys. Re . B 75,
125311 共2007兲.
15 M. J. Calde ón, B. Koille , X. Hu, and S. Das Sa ma, Phys. Re .
Le . 96, 096802 共2006兲.
16 L. C. L. Hollenbe g, A. S. Dzu ak, C. Wella d, A. R. Hamil on,
D. J. Reilly, G. J. Milbu n, and R. G. Cla k, Phys. Re . B 69,
113301 共2004兲.
17 A. V. Tsukano , Phys. Re . B 76, 035328 共2007兲.
18 L. A. Openo , Phys. Re . B 70, 233313 共2004兲.
19 B. Koille , X. Hu, and S. Das Sa ma, Phys. Re . B 73, 045319
共2006兲.
20 S. D. Ba e and G. J. Milbu n, Phys. Re . B 68, 155307 共2003兲.
21 F. H. L. Koppens, C. Buize , K. J. Tiel ooij, I. T. Vink, K. C.
Nowack, T. Meunie , L. P. Kouwenho en, and L. M. K. Vande -
sypen, Na u e 共London兲442, 766 共2006兲.
22 B. Li, B. Pa oens, F. M. Pee e s, and W. Magnus, Phys. Re . B
79, 085306 共2009兲; Mic oelec on. J. 40, 446 共2009兲.
23 S. Kang, Y. M. Liu, and T. Y. Shi, Eu . Phys. J. B 63,37共2008兲.
24 S. Kang, Y. M. Liu, and T. Y. Shi, Commun. Theo . Phys. 50,
767 共2008兲.
25 X. Hu and S. Das Sa ma, Phys. Re . A 61, 062301 共2000兲.
26 R. Tsu and D. Babić, Appl. Phys. Le . 64, 1806 共1994兲.
27 M. Lannoo, C. Dele ue, and G. Allan, Phys. Re . Le . 74, 3415
共1995兲; G. Allan, C. Dele ue, M. Lannoo, and E. Ma in, Phys.
Re . B 52, 11982 共1995兲; J. M. Fe ey a, P. Bossha d, and C. R.
P oe o, ibid. 55, 13682 共1997兲; B. Li, A. F. Slachmuylde s, B.
Pa oens, W. Magnus, and F. M. Pee e s, ibid. 77, 115335
共2008兲.
28 J. L. Mo illa and J. Planelles, Phys. Re . B 71, 075319 共2005兲;
Compu . Phys. Commun. 170, 144 共2005兲.
29 M. B uchez, J ., M. Mo onne, P. Gin, S. Weiss, and A. P. Ali i-
sa os, Science 281, 2013 共1998兲; R. Tsu, Mic oelec on. J. 34,
329 共2003兲.
30 S. F. Wuis e , C. M. Donegá, and A. Meije ink, J. Chem. Phys.
121, 4310 共2004兲.
31 C. Dele ue, M. Lannoo, and G. Allan, Phys. Re . B 68, 115411
共2003兲; X. Ca oixà and L. W. Wang, Phys. Re . Le . 94,
236804 共2005兲; D. Ninno, F. T ani, G. Can ele, K. J. Hameeuw,
G. Iadonisi, E. Degoli, and S. Ossicini, Eu ophys. Le . 74, 519
共2006兲.
32 D. BenDaniel and C. B. Duke, Phys. Re . 152, 683 共1966兲;P.
Ha ison, Quan um Wells, Wi es and Do s 共Wiley, Chiches e ,
2001兲.
33 P. G. Bolca o and C. R. P oe o, J. Phys.: Condens. Ma e 13,
319 共2001兲.
34 A. D. Yo e, Ad . Phys. 50,1共2001兲.
35 A ho ough analysis o he dependence o he pola iza ion po en-
ials on he impu i ies o -cen e ing can be ound in Re . 28.
36 This is ue as long as ␧ou ⬍␧QD 共 he usual si ua ion兲. When
␧ou ⬎␧QD, bo h sou ces o con inemen a o he localiza ion o
he g ound s a e in he inne dono .
37 G. Goldoni, F. Rossi, and E. Molina i, Phys. Re . Le . 80, 4995
共1998兲.
38 The binding ene gy has been calcula ed as he di e ence be-
ween he elec on g ound-s a e ene gies in he absence and in
he p esence o he impu i ies.
39 A. Th änha d , C. Ell, G. Khi o a, and H. M. Gibbs, Phys. Re .
B65, 035327 共2002兲; R. J. Glaube and M. Lewens ein, Phys.
Re . A 43, 467 共1991兲.
MOVILLA, BALLESTER, AND PLANELLES PHYSICAL REVIEW B 79, 195319 共2009兲
195319-6