Controlling inelastic cotunneling through an interacting quantum dot by a circularly-polarized field

0.44 MB·English

by  Bing Dong

February 6, 2008
Controlling inelastic cotunneling through an interacting quantum dot by a
circularly-polarized field
Bing Dong and X. L. Lei
Department of Physics, Shanghai Jiaotong University, 1954 Huashan Road, Shanghai 200030, China
7
0
We study inelastic cotunneling through a strong Coulomb-blockaded quantum dot subject to
0
a static magnetic field and a perpendicular circularly-polarized magnetic field using a quantum
2
Langevin equation approach. Our calculation predicts an interesting controllable cotunneling cur-
n rent characteristic, splitting–zero-anomaly–splitting transition of the differential conductance with
a increasing the driving frequency, ascribing to the role of photon-assisted spin-flip cotunneling pro-
J
cesses.

5
PACSnumbers: 72.40.+w,73.63.Kv,72.10.Fk

]
l
l
a Coherent control of spin dynamics in a quantum dot +c† c S−+c† c S+ +H , (1c)
h ηk↑ η′k′↓ ηk↓ η′k′↑ dir
(QD) using circularly-polarized magnetic field (CPF)1,2
- H = J c† +c† c +c(cid:3) , (1d)
s in a solid-state environment has been a subject of in- dir 0 Lkσ Rkσ Lkσ Rkσ
me creasing interest in recent years.3 This interest has led Xσ (cid:0) (cid:1)(cid:0) (cid:1)
t. tsopina rlaesrogneaanmceou(EntSRof)winorakQoDn,4d,5e,t6e,7ctaionndoinfitthiaelizeilnecgtrtohne wopheerraetoθr=foωrcetl,ecct†ηrkoσns(cwηkitσh)misotmheenctruematikon, s(painnn-σihailnadtioenn)-
a
m sptoaltaeriozfedelleicgthrto8n,9,s1p0i.nRveicaenotpltyi,cathleabsisnogrlpet-ieolnecotrfocnirEcuSlRarilny esprginyoǫηpkeriantolresadofηel(e=ctrLo,nRs)i,nSth≡e Q(SDx,[SSy±,Sz)Saxre PiSauy]l,i
- aQD-leadsystemwithsizableZeemansplittinghasbeen ≡ ±
d andJηη′ istheexchangecouplingconstant. ∆0 =gµBB0
theoretically reported to carry pure spin flow, which can
n isthestaticmagnetic-fieldB -inducedZeemanterm,and
0
o be used as a fundamental element and/or a spin source ∆ = gµ B describes the spin-flip scattering caused by
c device in the spintronic circuit.11,12 These papers have B 1
the CPF. H is the potential scattering term, which is
[ studiedthetransportthroughaQDinresonanttunneling dir
decoupled from the electron spin due to the number of
regime by the nonequilibrium Green’s function (NGF)11
2 electrons in the dot level being one. As a result, this
v andthe quantum rateequationapproach,12 respectively. term has no influence on the dynamical evolution of the
3 Theeffectofthes-dexchangeinteractionbetweenthelo-
electronspin and behaves only as a direct bridge to con-
7 calized spin and conduction electrons on electron trans-
nect the left and right leads. For an Anderson model
5 mission probability in linear transport regime has been
1 withsymmetricalcouplingto the leadst,wehaveJηη′ =
exploited for a mesoscopic ring embedded with an mag-
1 netic impurity by the NGF.13 In this paper, we focus 2J0 =t2/ǫd. As in our previous paper16, we can rewrite
6 the tunneling term, Eq. (1c), as a sum of three products
0 our studies, for the first time, on the influence of a driv- oftwovariables: H =QzSz+Q+S−+Q−S++Qˆ1 with
/ ing CPF upon cotunneling (non-resonant coherent tun- I
t thesamedefinitionsofQz(±)asinRef.16andQˆ1 =H .
a neling) through a strongly interacting QD in the weak- dir
m tunneling limit,14,15 predicting that the cotunneling cur- It is noted that our model describes the non-resonant
tunneling through a localized magnetic impurity involv-
- rent can be easily controlled by tuning the driving fre-
d quency. ing s-d exchange interaction with the conduction elec-
n trons in the presence of both a static magnetic field B
0
co subCjoetcutntnoelainngamthbrioeungthcoanssitnagnlte-mleavgenleǫtdicinfiteelrdacBtingaloQnDg achnadngaerionttaetriancgtimonagtnoetbiec wfieeladkBso1.thWatenaossKumonedotheeffeexct-
: 0
v the z-axis and a driving external (magnetic) CPF with emerges, and assume that charge fluctuation completely
Xi frequencyωc whosedirectionrotatesintheplaneperpen- vanishes.
dicular to the z-axis, B (t) = B (cosω t,sinω t,0), can It is well-known that the isolated spin-1/2 electron
ar be described by the Ham1iltonian1H =HcB+Hc0+HI:14 uBnd(te)r (tthhee iRnflabuienpcreobolfemb1o)t,hEBq.0(a1nbd), ahnaseaxnterannaallyCtiPcaFl
1
solution.2,17 Intherotatingframe,x=cosθSx sinθSy,
y =sinθSx+cosθSy, and z =Sz, the QD Ha−miltonian
HB = εηkc†ηkσcηkσ, (1a) Eq. (1b) takes the form
ηkσ
X H = ∆x δz, (2)
H = ∆ Sz 1∆(eiθS++e−iθS−), (1b) 0 − −
0 0
− − 2 withδ =∆ ω . Correspondingly,theHeisenbergequa-
HI = Jηη′ c†ηk↑cη′k′↑−c†ηk↓cη′k′↓ Sz tions of mo0t−ionc(EOeM’s) of these free rotating coordi-
ηη′,kk′ nates become: x˙ = δy, y˙ = δx+∆z, and z˙ = ∆y.
X (cid:2)(cid:0) (cid:1) − −
2
Solvingtheseresultingordinarydifferentialequations,we δ δ δ
+ 1+ C(ω +Ω) 1 C(ω Ω) ,
c c
obtain the free evolutions of these rotating coordinates: Ω Ω − − Ω −
(cid:20)(cid:18) (cid:19) (cid:18) (cid:19) (cid:21)
(7d)
r(t′) = M(τ)r(t), [r(t) (x(t),y(t),z(t))T], (3)
δ∆ ∆ δ
≡
δ2a++ ∆2 iδa− δ∆(a+ 1) Γzx = 2Ω2C(ωc)+ Ω 1− Ω C(ωc−Ω)
M(τ) = Ω2iδa−Ω2 −aΩ+ −Ω2i∆a−− , δ (cid:20)(cid:18) (cid:19)
 δ∆(Ωa+ 1) i∆a− ∆2−a+Ω+ δ2  − 1+ Ω C(ωc+Ω) , (7e)
−Ω2 − Ω Ω2 Ω2 (cid:18) (cid:19) (cid:21)
  2 2
∆ δ
with τ = t − t′, a± = 12(e−iΩτ ± eiΩτ), and the Rabi Γzz = 2 Ω C(ωc)+ 1− Ω C(ωc−Ω)
frequency Ω=√∆2+δ2. (cid:18) (cid:19) (cid:18) (cid:19)
δ 2
Furthermore, the transformed interacting Hamilto- + 1+ C(ω +Ω), (7f)
c
Ω
nian, Eq. (1c), reads in terms of these rotating frame: (cid:18) (cid:19)
2 2
1 ∆ δ
H =Qxx+Qyy+Qzz+Qˆ1, (4) γz = 2 R(ωc)+ 1 R(ωc Ω)
I 2" (cid:18)Ω(cid:19) (cid:18) − Ω(cid:19) −
withQx =e−ieθQ−+eiθQ+andQy =i(e−iθQ− eiθQ+). δ 2
The Heisenberg EOM’s for the spin operators a−re: + 1+ Ω R(ωc+Ω) , (7g)
(cid:18) (cid:19) #
x˙ = δy Qzy+Qyz, (5a) with the reservoir correlation function, C(ω), and the
−
y˙ = δx+∆z+Qzx Qxz, (5b) response function, R(ω), defined as
− −
z˙ = ∆y Qyx+Qxy. (5c) π ω
C(ω) = (g +g )Tϕ
− − 2 LL RR T
It is clear that the spin dynamics, apart from free evo- π ω+V (cid:16)ω (cid:17)V
+ g T ϕ +ϕ − , (8a)
LR
lutions, are perturbatively modified by the weak tunnel 2 T T
(cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
coupling. Toobtainthemodifieddynamics,weemploya π
R(ω) = (g +g +2g )ω, (8b)
generic quantum Langevin equation approach.16,18,19 In 2 LL RR LR
ovuoirrsdearrievafitrisotn,exoppreersasteodrsfoorfmtahlelyQbDy isnptiengraantdiotnhoefrtehseerir- with gηη′ ≡ Jη2η′ρ20 (ρ0 is the constant density of states
of both electrodes) and ϕ(x) xcoth(x/2). V is the
Heisenberg EOM’s, Eq. (5), exactly to all orders of Jηη′. bias-voltageappliedsymmetrica≡llytothetwoelectrodes,
Next, under the assumption that the time scale of decay µ = µ =V/2. We use units with ~=k =e=1.
processes is much slower than that of free evolutions, we L − R B
Notice that these quantities of Γ describe the dissi-
replace the time-dependent operators involved in the in- αβ
pation of cotunneling processes to the dynamics of the
tegrals of these EOM’s approximately in terms of their
QD spin, in conjunction with the effect of CPF: transi-
free evolutions, Eq. (3). Thirdly, these EOM’s are ex-
tionbetweentwospin-splittingstates Ω/2(spinanalog
pandedinpowersofJηη′ uptosecondorder. Tothisend, of charge optical Stark effect) due to±photo-absorption
we can establish the Bloch-type dynamicalequations for
and/or emission. Also, it is easy to see that the Bloch
the averagedspin variables r as:
equations, Eq. (6), is exactly reduced to the results of
Ref. 16 in the case of vanishing CPF, ∆=0 and ω =0.
x˙ Γ δ Γ x γ c
− xx xz x The nonequilibrium steady-state spin projections of
y˙ = δ Γ ∆ y + 0 ,
   − − yy     the QD in the rotating frame, r∞ = (x∞,y∞,z∞),
z˙ Γ ∆ Γ z γ
zx − − zz z can be readily obtained from Eq. (6). As is evident,
      (6) x∞ = y∞ = 0, while z∞ = 1R(∆ )/C(∆ ) is identi-
in which 2 0 0
cal to the previous theoretical result of nonequilibrium
∆ 2 δ 2 magnetization16,20 in absence of driving CPF. On the
Γ = 2 C(Ω)+2 C(0) otherhand,inthecaseofvanishingstaticmagneticfield,
xx
Ω Ω
(cid:18)δ (cid:19) (cid:18) (cid:19) δ ∆0 = 0, the nonzero driving CPF can induce an ad-
+ 1 C(ω Ω)+ 1+ C(ω +Ω), (7a) ditional spin orientation z∞ = 0 along the rotating di-
− Ω c− Ω c rection of the CPF, together6with the nonzero x∞ (not
(cid:18) (cid:19) (cid:18) (cid:19)
δ∆ shown here). At equilibrium, this optically-induced spin
Γ = 2 [C(0) C(Ω)], (7b)
xz Ω2 − orientationphenomenon has historicallybeen termed, in
∆ δ δ literature, as either the inverse Faraday effect21 or as
γx = 2R(Ω) 2 R(ωc) 1 R(ωc Ω) the Zeeman light shift,22 which is ascribed to the CPF-
2Ω − Ω − − Ω −
(cid:20) (cid:18) (cid:19) induced spin-splitting (ac spin Stark effect). In addi-
δ
+ 1+ R(ωc+Ω) , (7c) tion, one can interestingly observe a nearly vanishing y-
Ω
(cid:18) (cid:19) (cid:21) componentofthe spinpolarizationevenatCPFdriving.
∆ 2 δ 2 For nonzero static magnetic field, previous theoretical
Γ = 2 [C(Ω)+C(ω )]+2 C(0)
yy Ω c Ω studies show thatcoherentsuppressionoftunneling may
(cid:18) (cid:19) (cid:18) (cid:19)
3
0.6
z
x ∆/∆ =1.41
0.4
0
0.4
0.3
z
0.2
0.2
z
,
y
x, V/∆ =0 0.1
0 0
y
1.0
2.0 0
y
−0.2 ∆/∆ =0.2 4.0
0 −0.1
(a) (b)
x
−0.4 −0.2
0 0.5 1 1.5 2 0 1 2 3 4 5
ω ∆ ω ∆
/ /
c 0 c 0
FIG. 1: The nonequilibrium spin projection r∞ vs. ωc/∆0 (∆0 =1) with increasing bias-voltage under a weak driving CPF,
∆=0.2∆0 (a),andastrongdrivingfield,∆=√2∆0 (b). Thearrowindicatesthedirection ofincreaseofbias-voltageV. The
other parameters are: gLL =gRR =gLR =0.05 and T =0.02.
takeplaceonlyiftheRabifrequencyΩmatcheswiththe where the definitions of Qzσσ and Q± can be found in
ηη′ ηη′
driving frequencyωc, which correspondsto the following our previous paper16 and those terms in the last line
relationship of the driving field24: are stemming from the direct tunneling term H =
dir
J Qzσσ. The linear-response theory gives
∆2+∆2 0 ηη′,σ ηη′
Ω∗ =ω = 0 . (9)
c 2∆ P
0 t
I = J (t) = i dt′ [J (t),H (t′)] , (11)
In Fig. 1, we plot the nonequilibrium spin projection h L i − h L I −i0
r∞ under the influence of both nonzero static magnetic Z−∞
field and driving CPFs. Different from the results of wherethe statisticalaverage isperformedwithre-
0
∆0 =0,wefindanobviousnonzeropolarizationofthey- spect to decoupled two subsyh·s·te·mi s, QD and reservoirs.
component of QD spin near the resonant field, ωc =∆0, InsertingEqs.(1c)and(10)intoEq.(11),onecanderive
for the weak driving field ∆ = 0.2∆0 [Fig. 1(a)]. Inter- the explicit expression for steady-state current in terms
estingly, we observe z∞ =0 at ωc =∆0 for any strength ofthesteady-statespinprojectionsr∞ throughalengthy
of CPF, meaning that the resonant frequency of a CPF but straightforwardcalculation. In particular, we note
may overcome the Zeeman-splitting of a static magnetic
field. At the same time, x∞ becomes unpolarized for [(Qz↑↑ Qz↓↓)Sz,H ] = 0, (12)
weak driving field [Fig. 1(a)], while reaches the maxi- h LR − LR dir −i0
mum value for strong driving field, which decreases with h[(QzR↑L↑−QzR↓L↓)Sz,Hdir]−i0 = 0, (13)
increasing of the bias-voltage [Fig. 1(b)]. Moreover, we [Q± S∓,H ] = 0, (14)
notice that (1) x∞ =z∞ =0 atω =2Ω∗; and(2) z∞ is h ηη′ dir −i0
c
negative if ∆0 <ωc <2Ω∗ for strong driving CPF. and
We now proceed with the calculation of tunneling
current. The current operator through the QD is de- [(Qz↑↑ Qz↑↑+Qz↓↓ Qz↓↓),H ]
fined as the time rate of change of charge density N = h LR − RL LR − RL I i0
η = [(Qz↑↑ Qz↑↑+Qz↓↓ Qz↓↓),H ] =0,(15)
c† c in lead η (we choose the left lead as an h LR − RL LR − RL dir i0 6
k,σ ηkσ ηkσ
example):
P because the QD is connected to two normal leads in the
systemunderconsideration.16Theseresultsindicatethat
J (t) = N˙ =i[N ,H]
L − L L − the contribution of the direct tunneling term to the cur-
= i(Qz↑↑ Qz↑↑)Sz i(Qz↓↓ Qz↓↓)Sz
LR − RL − LR − RL rent is independent of the dynamics of the QD and only
+i(Q−LR−Q−RL)S++i(Q+LR−Q+RL)S− depends on the bias-voltage and temperature of the two
+i(Qz↑↑ Qz↑↑+Qz↓↓ Qz↓↓), (10) electrodes. To this end, the current I is23
LR − RL LR − RL
4
I 1 δ 2 δ ∆ 2 ω +V ω V
= 4V T +1 ++ −+ ϕ c ϕ c− z∞
πgLR − (2"(cid:18)Ω(cid:19) #I ΩI (cid:18)Ω(cid:19) (cid:20) (cid:18) T (cid:19)− (cid:18) T (cid:19)(cid:21))
δ∆ ∆ δ∆ ω +V ω V ∆ Ω+V Ω V
2T ++ − ϕ c ϕ c− + ϕ ϕ − x∞, (16)
− 4Ω2Ic 4ΩIc − 2Ω2 T − T 2Ω T − T
(cid:26) (cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21) (cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)(cid:27)
with (a,b) and a strong driving field ∆/∆ = √2 (c,d) in
0
Fig. 3. If ω = 0, the I-V curve reduces to the ordinary
ω +Ω+V ω Ω+V c
± = ϕ c ϕ c− cotunneling characteristics under an ambient constant
I T ± T
(cid:18) (cid:19) (cid:18) (cid:19) magnetic field:16 a jump at V = Ω = ∆2+∆2
ϕ ωc+Ω−V ϕ ωc−Ω−V . (17) [ ∆ in Fig. 3(a) due to ∆ ∆± and±= √30∆ in
− T ± T ≈ 0 ≪ 0 p 0
(cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21) Fig. 3(c), respectively], ascribing to the energetic inacti-
Different from the previous results of the nondriving vation (activation) of spin-flip processes at bias-voltage,
QDsystem,16 thenonzerox-componentofthestationary V < (>)Ω. While for nonzero driving frequency, such
| |
spin polarization has additional contribution to the cur- as ω /∆ = 0.5, the CPF effectively suppresses spin-
c 0
rentbesides the z-component. The linearly bias-voltage- splitting Ω/∆ 0.54 at the case of weak driving field,
0
≈
dependent term in Eq. (16) stems from the cotunnel- thus leading to two jumps, one of which is located at
ing processes, in which the spin projection remain un- V = Ω due to the pure activated spin-flip process, and
±
changed. The CPF has no effect on these processes. another of which appears at V = (Ω + ω ) ∆ ,
c 0
± ≈ ±
Moreimportantly,one observesthat the currentformula whichcanbeascribedtoopeningofanadditionalchannel
Eq. (16) includes some new factors involving ω Ω V for electron transfer cotunneling due to the one-photon-
c
± ±
[Eq.(17)],whichcanbeintuitivelyascribedtothephoto- emission-assisted spin-flip resonant event. In contrast,
assistedspin-flipcotunnelingprocesses. Forinstance,the the strong driving field ∆ = √2∆ makes Ω = 1.5∆
0 0
terminvolvingΩ ω resultsfromthecontributionofco- and generates three jumps: the first one corresponds to
c
−
tunneling accompanied by an absorption of one-photon theΩ ω process;thesecondoneisduetothepureelec-
c
−
[Fig. 2(a)], while the term involving Ω+ω corresponds tronic spin-flip event (Ω); and the third one results from
c
totheprocessassistedbyaspontaneousemissionofone- theΩ+ω process. Whenthefrequencyincreasestoabit
c
photon [Fig. 2(b)]. Therefore, we simply name these co- higher than the resonantfrequency [1.0∆ (Ω∗ =1.5∆ )
0 0
tunnelings as Ω ω processes respectively below. Be- for the weak (strong) driving field], a zero-bias peak
c
±
sides, the factors Ω V in Eq. (16) stem from the spin- emerges for both cases, which can be understood by the
±
flip cotunneling events without spontaneous emission or fact that the proper high frequency CPFhas enoughen-
absorption of photon, while the role of driving field is ergy to spur the spin-flip cotunneling even at equilib-
reflected through the effective spin-splitting Ω. In the rium [see Fig. 2(a)], while the rising voltage, V Ω
≥ ±
following, we will see that the joint effects of these new [Figs. 3(a, b)] or (Ω ω ) [Figs. 3(c, d)], contrarily
c
± −
terms are responsible for the controllable patterns of co- suppressesitsactivationuntilapurebias-drivenspin-flip
tunneling through a QD by the presence of CPF. processisexcitedatV = ∆ or Ω. Recently,thezero-
0
± ±
anomaly (ZA) behavior of cotunneling current has been
alsoreportedforaQDconnectedtotwoanti-parallelfer-
romagneticelectrodes.25,26 Here,itshouldbepointedout
that the appearance of ZA in the present system is due
tophoton-assistedinelasticspin-flipscatteringbecoming
resonantinthepresenceofexternalmagneticfield,which
isdifferentfromthepreviousmechanism,elasticspin-flip
event in the case of polarization leads without any mag-
netic field.25,26 As a result, the present system has more
richtransportfeatures by tuning strengthandfrequency
of driving CPF.
For the weak CPF field with ω 2∆ , the photon-
c 0
FIG. 2: Schematic diagrams of photon-absorption- (Ω−ωc) assistedexcitationofspin-flipcotunn≈elingmergescoinci-
(a) and photon-emission- (Ω+ωc) (b) assisted spin-flip co-
dentlyintopurebiasexcitation,leadingtothedisappear-
tunnelingprocesses.
ance of ZA behavior [Fig. 3(b)]. While for strong CPF
field [Fig. 3(d)], the ZA also vanishes if ω 2Ω∗, which
c
We plot the differential conductance G = ∂I (in unit isduetothe peculiarfeaturesofthe nonequ≥ilibriumspin
∂V
of 2πg ) vs bias-voltage as functions of different driv- projectionsx∞ andz∞[seeFig.1(b)]. Forhigherdriving
LR
ing frequencies ω for a weak driving CPF ∆/∆ = 0.2 frequency, the cotunneling exhibits splitting differential
c 0
5
ω /∆ =1.2
1.2 1.2 c 0
1.3
1.5
1 1 1.8
V
2.0
∂
/
I
∂ 0.8 ω /∆ =0 0.8
c 0
0.5
0.8
0.6 (a) 0.6 (b)
0.9
1.0
−2 −1 0 1 12.1 −2 −1 0 1 2
V/∆ V/∆
0 0
ω /∆ =0,0.5,1.0,1.4,1.5,1.6,1.8,2.0
c 0
ω /∆ =2.5,2.8,3.0,3.2,3.5,4.0,4.5,5.0,6.0
1.2 1.2 c 0
1 1
V
∂
/
I
∂ 0.8 0.8
0.6 (c) 0.6 (d)
−3 −2 −1 0 1 2 3 −2 −1 0 1 2
V/∆ V/∆
0 0
FIG.3: ThecalculateddifferentialconductancedI/dV vs. bias-voltageV/∆0 (∆0 =1.0)underweakdrivingCPF∆/∆0 =0.2
(a,b)andstrongCPF∆/∆0 =√2(c,d)withseveraldrivenfrequenciesatanonzerostaticmagneticfield. Thearrowindicates
thedirection of increase of driven frequency ωc. The other parameters are thesame as in Fig. 1.
conductance and eventually tends to the pattern in the gleQDunderastaticmagneticfieldandaperpendicular
presenceofastaticmagneticfieldalone,becausetheQD CPF,revealingacontrollableI-V pattern,thetransition
spin cannot, from physical point of view, catch the de- betweenZAandsplittingofthedifferentialconductance,
tailsofthedrivingfieldwithconsiderablyhighfrequency. due to photon-assisted spin-flip inelastic cotunneling.
Finally, we point out that the ZA feature is robust over
a wide region of temperature (not shown here). This work was supported by Projects of the National
Conclusion.—In summary, we have presented an ana- Science Foundation of China and the Shanghai Munici-
lytical study of the inelastic cotunneling, including the palCommissionofScienceandTechnology,theShanghai
nonequilibrium spin projections and currents, in a sin- Pujiang Program,and NCET.
1 I.I. Rabi, Phys.Rev. 51, 652 (1937). H.W. Jiang, Nature430, 435 (2004).
2 M. Grifoni and P. H¨anggi, Phys. Rep. 304, 229 (1998). 5 L.N. Bulaevskii, M. Hruˇska, and G. Ortiz, Phys. Rev. B
3 Semiconductor Spintronics and Quantum Computation, 68, 125415 (2003).
eds. D.D. Awschalom, D. Loss, and N. Samarth. Series 6 H.-A.EngelandD.Loss,Phys.Rev.Lett.86,4648(2001);
on Nanoscience and Technology, Springer(2002). Phys. Rev.B 65, 195321 (2002).
4 Y.Manassen,R.J.Hamers,J.E.Demuth,andA.J.Castel- 7 O.Gywat, H.-A.Engel, D.Loss, R.J. Epstein, F.M. Men-
lano,Jr.,Phys.Rev.Lett.62,2531(1989);C.Durkanand doza, D.D. Awschalom, Phys. Rev. B 69, 205303 (2004);
M.E. Welland, Appl. Phys. Lett. 80, 458 (2002); I. Mar- M.V.Gurudev Dutt, J. Cheng, B. Li, X. Xu, X. Li, P.R.
tin, D. Mozyrsky, and H.W. Jiang, Phys. Rev. Lett. 90, Berman, D.G. Steel, A.S. Bracker, D. Gammon, S.E.
18301 (2003); M. Xiao, I. Martin, E. Yablonovitch, and Economou,R.B.Liu,andL.J.Sham,Phys.Rev.Lett.94,
6
227403 (2005). 16 Bing Dong, N.J. M. Horing, and H.L. Cui, Phys. Rev. B
8 Optical Orientation, edited by F. Meier and B.P. Za- 72, 165326 (2005).
kharchenya, Modern Problems in Condensed Matter Sci- 17 L.AllenandJ.H.Eberly,opticallResonanceandTwo-level
ences Vol.8 (North-Holland, Amsterdam, 1984). Atoms (Wiley, New York,1975).
9 A.Shabaev,Al.L. Efros, D.Gammon, and I.A.Merkulov, 18 J.R. Ackerhalt and J.H. Eberly, Phys. Rev. D 10, 3350
Phys. Rev.B 68, 201305(R)(2003). (1974).
10 C.E. Pryor and M.E. Flatt´e, Phys. Rev. Lett. 91, 257901 19 G.F. Efremov and A.Yu. Smirnov, Zh. E´ksp. Teor. Fiz.
(2003). 80, 1071 (1981) [Sov. Phys. JETP 53, 547 (1981)]; A.Yu.
11 B. Wang, J. Wang, and H. Guo, Phys. Rev. B 67, 92408 Smirnov,Phys. Rev.B 60, 3040 (1999).
(2003); H.K. Zhao and J. Wang, Eur. Phys. J. B 44, 93 20 O.ParcolletandC.Hooley,Phys.Rev.B66,85315(2002).
(2005). 21 J.P.vanderZiel,P.S.Pershan,andL.D.Malmstrom,Phys.
12 Bing Dong, H.L. Cui, and X.L. Lei, Phys. Rev. Lett. 94, Rev.Lett.15, 190 (1965); P.S.Pershan, J.P. vanderZiel,
66601 (2005). and L.D. Malmstrom, Phys.Rev. 143, 574 (1966).
13 A. Aldea, M. Tolea, and J. Zittartz, Physica E 28, 191 22 C. Cohen-Tannoudji and J. Dupont-Roc,Phys. Rev. A 5,
(2005). 968 (1972).
14 K.A. Matveev, Zh. Eksp. Teor. Fiz. 99, 1598 (1991) 23 We took no account of the contribution of the direct tun-
[Sov. Phys. JETP 72, 892 (1991)]; A. Furusaki and K.A. neling term in our previous paper, Ref. 16. If this term
Matveev, Phys. Rev.B 52, 16676 (1995). is included, the linearly bias-voltage relevant term in the
15 S.De Franceschi, S. Sasaki, J.M. Elzerman, W.G. van der charge current formula, Eq. (36), becomes 4V in stead of
Wiel,S.Tarucha,andL.P.Kouwenhoven,Phys.Rev.Lett. 3V.
86, 878 (2001); A. Kogan, S. Amasha, D. Goldhaber- 24 J. Shao and P. H¨anggi, Phys.Rev. A 56, R4397 (1997).
Gordon, G. Granger, M.A. Kastner, and H. Shtrikman, 25 I. Weymann, J. Barna´s, J. K¨onig, J. Martinek, and G.
Phys. Rev.Lett. 93, 166602 (2004); D.M. Zumbu¨hl, C.M. Sch¨on,Phys. Rev.B 72, 113301 (2005).
Marcus, M.P. HansonandA.C.Gossard, Phys.Rev.Lett. 26 Bing Dong, X.L. Lei, and N.J.M. Horing, unpublished.
93, 256801 (2004).