Quantum interference in resonant tunneling and single spin measurements

0.17 MB·English

by  S. A. Gurvitz

IEEETRANSACTIONSONNANOTECHNOLOGY,VOL.4,NO.1,JANUARY2005 1
Quantum interference in resonant tunneling and
single spin measurements
5 Shmuel A. Gurvitz
0
0
2

n Abstract—Weconsidertheresonanttunnelingthroughamulti- whereΓ=(ΓL+ΓR)/2isahalfofthetotalwidth.Weassume
a levelsystem.Itisdemonstratedthattheresonantcurrentdisplays thatΓ =Γ (asymmetricdot),andthatthetunnelingwidths
J quantum interference effects due to a possibility of tunneling L R
arethesameforbothlevels,yettheBreight-Wigneramplitudes
1 through different levels. We show that the interference effects
candifferinphase.Wethereforeintroducedthefactorη in(2)
3 arestronglymodulatedbyarelativephaseof statescarryingthe
current.Thismakesitpossibletousetheseeffectsformeasuring which denotes the relative phase of these amplitudes. It can

] thephasedifferencebetweenresonantstatesinquantumdots.We beshown[1]thatη cantakeonlytwo values±1,theso-called
ll extendourmodelforadescriptionofmagnetotransport through “in” or “out-of-phase” resonances, respectively.
a theZeemandoublets.Itisshownthat,duetospin-fliptransitions,
h the quantum interference effects generate a distinct peak in the
- shot-noisepowerspectrumatthefrequencyof Zeeman splitting. µ
s
e Thismechanismexplainsmodulationinthetunnelingcurrentat L E2 I
m theLarmorfrequencyobservedinscanningtunnelingmicroscope Γ Γ
experiments and can be utilized for a single spin measurement. L R µ
. E
t 1
a Index Terms—Magnetotransport, quantuminterference, reso- R
m nant phase, resonant tunneling, shot-noise spectrum, single-spin
measurement,Zeeman splitting.
- Fig.1. Interferenceeffectintheresonanttunnelingthroughtwolevels.Here
d ΓL,R denotes thepartial widthofeach ofthelevels duetotunneling tothe
n leftorrightreservoir.
o I. INTRODUCTION
c
[ THE RESONANT tunneling through quantum dots (or It follows from (2) that, if the resonances do not overlap,
impurities) has been investigated both theoretically and Γ ≪ E2 − E1, the total resonant current is a sum of the
3
v experimentally in large amount of works, yet most investiga- resonant currents flowing through the levels E1 and E2.
0 tions concentrated on the resonant tunneling through a single However, if Γ ∼ E2 − E1, the interference plays a very
1 quantum level. In the case of the resonant tunneling through importantrole in the totalresonantcurrent.Indeed,oneeasily
0 many levels one usually consideredthe total current as a sum finds that, in the case of constructive interference, η = 1,
6 ofcurrentsthroughindividuallevels.Ingeneral,however,this the total current increases with Γ as I ∼ Γ and, in the
0
4 procedure cannot be correct due to the quantum interference case of destructive interference, η = −1, the total current
0 effects. We illustrate this point with a simple example. decreases with Γ as 1/Γ. Since in the case of quantum dots
/ Let us consider the resonant tunneling through a quantum thetunnelingwidthsΓL,R canbe variedbythe corresponding
t
a dot coupled with two reservoirs with different chemical po- gate voltage, one can use this interference effect in order
m to measure the relative phase of different levels. This can
tentials, µ . We assumethattwo levelsofthe dot,E , are
L,R 1,2
- insidethepotentialbiasµ −µ (seeFig.1).Thentheelectric provide an alternative method for a measurement of this
d L R
quantity,inadditiontothatwhichutilizedtheAharonov-Bohm
n current flows from the left (emitter) to the right (collector)
oscillations[2].
o reservoirs through the two levels. If we neglect the Coulomb
c repulsion between the electrons, the total current is given by The interference effects described above are related to the
v: the Landauer formula stationary current. We can also anticipate the interference
i effects in temporal characteristics of the current. Indeed, it
X e
I = T(E)dE, (1) is known that the average resonant current trough a double-
r 2π dot system would display damped oscillations generated by
a Z
quantum interference[3]. Since any double-well potential can
whereT(E)isthetotaltransmission.Sinceanyelectronfrom
be mappedto a single well with two levels, one can expectto
the left reservoir can tunnel to the right reservoirs via these
observe similar damped oscillations in the resonant current
two levels (Fig. 1), the total transmission is given by a sum
flowing through two levels of a single dot. The presence
of two Breight-Wigner amplitudes:
of oscillations in the average current is usually reflected in
Γ Γ 2 the current shot-noise power spectrum density, S(ω). For
T(E)= +η , (2)
E−E +iΓ E−E +iΓ instance, the resonant current through a double-dot structure
(cid:12) 1 2 (cid:12)
(cid:12) (cid:12) would develop a dip in the S(ω) at the Rabi frequency[4].
Based on work pre(cid:12)(cid:12)sented at the 2004 IEEENTCQuantum(cid:12)(cid:12)Device Tech- Similarly, one can think that the current flowing through two
nologyWorkshop.
levels (Fig.1) would develop the same structure in S(ω) at
S.A.GurvitziswiththeDepartmentofParticlePhysics,WeizmannInstitute
ofScience, Rehovot76100,Israel. ω = E2 −E1. However, the result should strongly depend
IEEETRANSACTIONSONNANOTECHNOLOGY,VOL.4,NO.1,JANUARY2005 2
on the relative phase of two levels. This phase-dependenceof STM measurements. This provides us with a possibility of a
the spectral density has not been discussed in the literature, single nuclearspin detection.SectionIV providesa summary.
although this effect can have important applications.
The interference effects in resonant tunneling can also be II. RESONANTTUNNELING THROUGH DIFFERENTLEVELS
anticipatedinthemagnetotransport[5].Indeed,inthepresence
Let us consider resonant transport in a multilevel system.
ofmagneticfield,alllevelsinquantumdotorimpurityaresplit
We shall treat this problem in the framework of a tunnel
(Zeeman splitting). Therefore, electrons in the left reservoir
Hamiltonian approach. Therefore, we introduce the following
withdifferentorientationofspin(paralleloranti-paralleltothe
tunneling Hamiltonian describing the electron transport from
magneticfield)wouldtunneltotherightreservoirthroughthe
the emitter to the collector via different levels of a quantum
differentZeemansublevelsofthedot.Thisalonecannotresult
dot (impurity) E (Fig. 1), H =H +H +H +H , with
j L R D T
in quantum interference, since the corresponding spin states
are orthogonal.However,if the g-factor in the dot is different H = E a† a , H = E d†d +Uˆ ,
L(R) l(r) l(r) l(r) D j j j C
from that in the reservoirs, then the spin-orbit interaction Xl(r) Xj
generates the spin-flip of an electron traveling through the
H = Ω(j)d†a +l ↔r +H.c.. (3)
quantum dot[5]. As a result, the same electron from the left T l j l
reservoir flows to the right reservoir through two Zeeman (cid:16)Xl,j (cid:17)
sublevels(Fig.1). Similar to thepreviouscase, onecan expect Here a† (a ) is the creation (annihilation) operator of an
l,r l,r
that the related interference effects would be reflected in the electron in the reservoirs and d†(d ) is the same operator
j j
behavior of the current spectral density S(ω). In particular, it for an electron in the dot (we omitted the spin indices). The
wasarguedin[5]that,inthisway,onecanexplainthepuzzled operator UˆC = jj′(UC/2)d†jdjd†j′dj′ denotes the Coulomb
oscillations at Larmour frequency observed in scanning tun-
interaction of between electrons in the dot and Ω(j)(Ω(j)) is
neling microscope (STM) experiments[6], [7] and considered P l r
a coupling between the states E (E ) and E of the reservoir
as a promising tool for a single spin measurement[8], [9]. l r j
and the dot, respectively. This coupling is related to the
In this paper, we investigate the interference effects in
corresponding tunneling width by Γ(j) = 2πρ |Ω(j)|2,
resonanttunnelingthroughmultilevelsystemsasquantumdots L,R L,R l,r
where ρ is the density of states in the corresponding
orimpurities.Atfirstsight,thetreatmentoftheseeffectslooks L,R
reservoir. (In the absence of magnetic field, one can always
rather straightforward in terms of single electron description
choose the gauge such that all couplings Ω are real).
[(1) and (2)]. However,this is notthe case when the electron-
All parametersof the tunneling Hamiltonian (3) are related
electron repulsion inside the dot is taken into account. In
to the initial microscopic description of the system in the
fact, this effect can never be disregarded, and it always
configuration space (x). For instance, the coupling Ω(j) is
plays a very important role in the electron transport. For l(r)
given by the Bardeen formula[16]
this reason, one uses the Keldysh nonequilibrium Green’s
efuffneccttioinnttehcehneilqeucter[o1n0]t,ra[1n1sp]ofrotr. Tanheascecocuanlctuolaftitohnesi,nhteorwacetvioern, Ω(l(jr)) =−21m x φj(x)∇↔n χl(r)(x)dσ , (4)
Z ∈Σl(r)
are rather complicated and are usually performed only in a
where φ (x) and χ (x) are the electron wave functions
weakcouplinglimit.Inthispaper,weuseadifferent,simpler, j l(r)
inside the dot and the reservoir, respectively, and Σ is a
and more transparenttechnique developedby us in Ref. [12]-
surface inside the potentialbarrier that separates the dot from
[15] that consists of reduction of the Schro¨dinger equation
the corresponding reservoir. In one-dimensional (1-D) case
to Bloch-type rate equations for the density matrix obtained
φ (x)≡ φ (x) and χ (x)≡χ (x), (4) can be rewritten
by integrating over the reservoir states. Such a procedure can j j l(r) l(r)
as[17]
be carried out in the strong nonequilibrium limit without any
stochastic assumptions and valid beyond the weak coupling Ω(j) =−(κ /m)φ (x¯ )χ (x¯ ), (5)
l(r) j j l(r) l(r) l(r)
limit. The resulting equations can be used straightforwardly
forevaluatingthecurrentin amultilevelsystem anditspower where κ = 2m(V(x¯)−E ). The point x¯ should be
j j l(r)
spectrum, with the Coulomb repulsion inside the dots taken taken inside the left (right) barrier and far away from the
into account. classical turninpg points where Ω(j) becomes practically in-
l(r)
The remainder of this paper is as following. In Section dependent of x¯[18].
II we study the resonant tunneling through two levels of Itwasdemonstratedin [12]-[14] thattheSchro¨dingerequa-
the quantum dot. We obtain the generalized quantum rate tion i∂ |Ψ(t)i = H|Ψ(t)i, describing the quantum transport
t
equations describing the entire system, including the electric through a multidot system, can be transformed to the Bloch-
current. Special attention is paid to effect of the relative type rate equation for the reduced density-matrix σn (t) ≡
αβ
phase of resonances on the average current and on the shot- σnn(t),where|αi, |βi,...arethediscretestatesofthesystem
αβ
noise power spectrum. In Section III, we concentrate on the in the occupation number representation and n is the number
magnetotransport through quantum dots or impurities. We of electrons arriving at the corresponding reservoir by time
derive the rate equations for this case and evaluate the shot- t. This reduction takes place after partial tracing over the
noise power spectrum. The obtained results suggest a natural reservoir states, and it becomes the exact one in the limit
explanationof a peakat the Larmordensity and the hyperfine of large bias µ − µ ≫ Γ without explicit use of
L R L,R
splitting due to interaction with nuclear spin found in new any Markov-type or weak coupling approximations. As a
IEEETRANSACTIONSONNANOTECHNOLOGY,VOL.4,NO.1,JANUARY2005 3
result, the off-diagonalin n density matrix elements, σnn′(t), of the Ω(1) may be the opposite one with respect to the sign
αβ L,R
becomes decoupled from the diagonal in n terms, σαnβ(t), of Ω(L2,)R. Note that, in the 1-D case, a sign of the product
in the equations of motion[15]. Finally, one arrives at the Ω(j)Ω(j) isdeterminedbyasignoftheproductφ (x¯ )φ (x¯ ),
following Bloch-type equations describing the entire system L R j l j r
[see (5)]. The latter can be positive or negative,dependingon
[14]:
thenumberofnodesofφ (x) insidethedot[1].Thus,fora 1-
j
D dot,the productΩ(j)Ω(j) changesits sign when j →j+1.
σ˙n =iǫ σn +i σn Ω˜ − Ω˜ σn L R
αβ βα αβ αγ γ→β α→γ γβ Thereasonisthatthecorrespondingwavefunctionsφ (x)and
! j
Xγ Xγ φj+1(x) differ by an additional node. Hence, the ratio
− πρ(σn Ω Ω +σn Ω Ω )
αγ γ→δ δ→β γβ γ→δ δ→α
γ,δ Ω(j+1)Ω(j+1)
+X πρ(Ω Ω +Ω Ω )σn−1, (6) η = L R (9)
γ→α δ→β γ→β δ→α γδ Ω(j)Ω(j)
γ,δ L R
X
is −1 for a 1-D dot. However, in the case of a three-
where ǫ = E − E and Ω denotes one-electron
βα β α α→β
dimensional (3-D) quantum dot, where the corresponding
hoppingamplitudethatgeneratesα→β transition.Wedistin-
guishbetweentheamplitudesΩ˜ andΩofone-electronhopping coupling Ω is given by (4), this condition does not hold.
Taking into account (9) one obtains from (6) the follow-
among isolated states and among isolated and continuum
ing quantum rate equations describing the electron transport
states, respectively. The latter transitions are of the second
order in the hopping amplitude ∼ Ω2. These transition are through two levels
produced by two consecutive hoppings of an electron across
σ˙n =−2Γ σn +Γ (σn−1+σn−1)
continuum states with the density of states ρ. 00 L 00 R 11 22
+ηΓ (σn−1+σn−1)(10a)
Solving (6), we can determine the probability of finding n R 12 21
eallelocwtrsonussitnotdheetecromlliencetothr,ePanve(tr)ag=e currjeσnjntj(t). This quantity σ˙1n1 =−ΓRσ1n1+ΓLσ0n0−ηΓ2R(σ1n2+σ2n1) (10b)
P Γ
I(t)=e nP˙(t), (7) σ˙2n2 =−ΓRσ2n2+ΓLσ0n0−η 2R(σ1n2+σ2n1) (10c)
Xn σ˙n =iǫσn −Γ σn +Γ σn −ηΓR(σn +σn ),(10d)
and the current power spectrum. The latter is given by the 12 12 R 12 L 00 2 11 22
McDonald formula[19], [5] where σn = (σn )∗ and ǫ = E −E . In these equations,
21 12 2 1
∞ d we assumed that Ω(1) = Ω(2), so that η = Ω(1)/Ω(2). In the
S(ω)=2e2ω dtsin(ωt)dtNR2(t), (8) caseofadifferentgLauge,ΩL(1) =Ω(2) andη =RΩ(1)R/Ω(2),the
Z0 R R L L
factor η would appear only in front of the width Γ in (10d).
where N2(t)= n2P (t). L
R n n This of course does not affect the final result.
Consider again the resonant tunneling through the two
P Equations (10a)-(10d can be interpreted in terms of “loss”
levels, (Fig.1). Let us assume that the Coulomb repulsion of
and “gain” terms, and, therefore, they represent the quantum
electrons inside the dot U is large such that two electrons
C rate equations. For instance, the first (loss) term in (10a)
cannot occupy the dot. Then, there are only three available
describes decay of state (0) in Fig.2 due to tunneling of one
states of the system, shown in Fig. 2.
electron from the left reservoir to the dot. The second (gain)
termofthesameequationdescribesdecayofstates(1)and(2)
µL E2 n to state (0). The last (gain) term describes decay of the linear
Ω Ω
L E R µ superposition of states (1) and (2). It is given by the product
1 R
of the correspondinghoppingamplitudesfromthe levelsE
1,2
(0) tothe collectorreservoir.Since these amplitudescandifferby
a sign, this term is proportional the relative phase η between
µ n µ n
L L the states E1 and E2.
µ µ It is important to note that all transitions in (10) take
R R
placethroughavailablecontinuumstates.Therefore,theterms
(1) (2) σn and σn in (10b) and (10c) can couple with the off-
11 22
diagonalmatrixelementsσn throughthe rightreservoironly.
Fig.2. Threeavailablestatesofthesystem.Herendenotesthenumberof 12
electrons arriving atthecollector bytimet. The coupling via the left reservoir would be possible for
noninteractingelectronsthroughanewstate(3)corresponding
Let us apply (6) by assigning α,β ={0,1,2},in an accor- to two electrons occupying the levels E and E . The rate
1 2
dance with the states shown in Fig.2. Since the states 1 and 2 equationsinthiscasewouldbetotallysymmetricwithrespect
arenotdirectlycoupled,thecorrespondinghoppingamplitude to an interchange of Γ and Γ , and the result will coincide
L R
Ω˜ =0 in (6). However,these states can be connectedthrough with that of the single electron description, [see (1) and (2)].
thereservoirs[thethirdandtheforthtermsof(6)].Weassume Note also that, in the case of η = −1, the two-level system,
thatthecorrespondingcouplingsareweaklydependenton the shown in Figs. 1,2, can be mapped to a coupled-dot system.
energy, so that |Ω(1)| = |Ω(2)| = |Ω |. However, the sign Then (10) turn into the system of quantum rate equations,
l,r l,r L,R
IEEETRANSACTIONSONNANOTECHNOLOGY,VOL.4,NO.1,JANUARY2005 4
η
1 =1
found earlier for a description of electron transport through
I/
∋
the coupled-dotsystem[12], [20].
0.8
On can find that the factor η = ±1, in (10) has the (a)
same meaning as the relative phase η of two Breit-Wigner 0.6
η
amplitudes in Eq. (2). Indeed, it is always −1 for two =-1
0.4
subsequent resonances in one dimensional case. However, in
a 3-D quantum dot, the two subsequent resonances can be 0.2
found in the same phase, depending on particular properties Γ /
R ∋
of the quantum dot. One even predicts a whole sequence 1 2 3 4 5
1
of the resonances with the same phases[1], [21]. Thus, a I/
∋
measurement of the resonance phase η could supply us with η (b)
0.8 =1
additional information on a quantum dot (impurity) structure,
complementary to spectroscopic measurements. 0.6
Consider first the total current, I(t) = aI (t) + bI (t),
L R 0.4
where I (t), [see (7)] are the currents in the left or in the
L,R
rightreservoirs.Thecoefficientsaandbwitha+b=1depend 0.2 η=-1
on each junction capacitance[22]. For simplicity we consider Γ /
L ∋
onlya casewherethecurrentintherightreservoirdominates, 1 2 3 4 5
b≫a. One easily obtains from (10) that
Fig. 3. Total current through two resonance levels (a) as a function of
I(t)=eΓ [σ (t)+σ (t)+2ηReσ (t)], (11) the width ΓR for ΓL = 0.5ǫ and (b) as a function of the width ΓL for
R 11 22 12 ΓR=0.5ǫ.
where σ (t)= σn (t).
αβ n αβ
Performing summation over n in (10) and solving these I(t) / ε
P
equations in the stationary limit, t→∞, one easily finds for
the stationary current I =I(t→∞) 0.2 η=1 η=-1
0.175
2ǫ2Γ Γ
I/e= L R . (12)
ǫ2Γ +2Γ [ǫ2+(1−η)Γ2] 0.15
R L R
0.125
(Note that the stationary current is independenton the capac-
itance of junctions, a and b). 0.1
Asexpected,whentheresonancesbeginoverlap,thecurrent
0.075
becomesvery sensitive on a sign of the relative phase η. This
0.05
is illustrated Fig. 3(a), where we plot the stationary current
I as a function of the widths Γ . One finds that the current 0.025
R
I decreases with Γ if η = −1 and increases with Γ if
R R
10 20 30 40 50
η =1. However, the dependence of the total current I on the t [in units of 1 /ε ]
width Γ [Fig.3(b)] is rather unexpectable.One finds that the
L
current increases for both values of η. This is very different Fig.4. Timedependenceoftheresonantcurrentflowingthroughtworesonant
from the case of non-interactionelectrons[(1) and (2)] where levels E1 and E2, forΓL =ǫ and ΓR =0.1ǫ.The solid line corresponds
the current is symmetric under an interchange of Γ and Γ . to the resonances in phase, η = 1, and the dashed line to the off-phase
L R resonances, η=−1.Thedotisemptyfort=0.
Such an asymmetry in the case of interacting electrons is a
result of the Coulomb blockade effect[12], [20]. Indeed, an
electronentersthedotfromtheleftreservoirwiththerate2Γ .
L
phaseηonthequantumoscillationsisverysubstantial.Indeed,
However, it leaves it with the rate Γ , since the state where
R
the oscillations related to different values of η are shifted by
the two levels E are occupied is forbiddendue to electron-
1,2
half of the periodand, moreover,the correspondingdampings
electronrepulsion.Theseresultscanbeverifiedexperimentally
are quite different.
in the case of a quantum dot, where the width Γ can be
L,R
varied by changing the corresponding gate voltage. Then the The oscillations in the average current are reflected in
relative phase η can be obtained from observing the behavior the shot-noise power spectrum given by S(ω) = aSL(ω)+
of the resonant current with ΓR [Fig. 3(a)]. bSR(ω)−abSQ(ω)[5]. Here SL,R is the current power spec-
Thequantuminterferenceeffectsappearaswellinthetime- trumintheleft(right)reservoir,[(8)]andSQ(ω)isthecharge
dependent current. Let us calculate I(t) [(11)] by solving correlationfunctionofthequantumdot.Thelattercanalsobe
[(10)]withtheinitialconditionsσjj′(0)=δj0δj′0 correspond- obtained from (10). Again, we take for simplicity the case of
ing to the empty dot. The time-dependent average current is b≫a, so thatS(ω)=SR(ω). Thenoneeasily findsfrom(8)
shown in Fig. 4 for Γ =ǫ and Γ =0.1ǫ for two values of and (10) that
L R
the relativephase,η =±1.Onefindsfromthisfigurethatthe
current displays strong oscillations in contrast with resonant S(ω)=2e2ωΓ Im[Z (ω)+Z (ω)+Z (ω)+Z (ω)],
R 11 22 12 21
tunneling through a single level. The influence of the relative (13)
IEEETRANSACTIONSONNANOTECHNOLOGY,VOL.4,NO.1,JANUARY2005 5
where Zeeman splitting (Fig. 6). Then an electron with spin-up can
∞ tunnel only through the upper level (Fig. 6). Respectively, an
Z (ω)= (2n+1)σn (t)exp(iωt)dt. (14)
αβ αβ electron with spin-downtunnels only throughthe lower level.
Z0 Xn No interferencetakes place in this case. However,if g-factors
These quantities are obtained directly from (10) by reducing in the quantum dot and in the reservoirs are different, the
them to the system of linear algebraic equations. tunnelingtransitionsareaccompaniedbythespinflip[5].Then
Using (13), we calculate the ratio of the shot-noise power thesameelectroncantunnelfromthelefttotherightreservoir
spectrumtotheSchottkynoiseS(ω)/2eI (Fanofactor),where via two level (cf. with Fig. 1). This process would generate
I is given by (12). This quantity is shown in Fig. 5 for oscillations in the resonant current in the same way as was
ΓL = ǫ and ΓR = 0.1ǫ, which are the same parameters as discussed in the previous section.
in Fig. 4, and η =±1. As expected,the quantuminterference
Let us evaluate the corresponding tunneling amplitudes,
isreflectedin theshot-noisepowerspectrum.We findthatthe
which we denote as Ω and δΩ , respectively, for no
L,R L,R
correspondingFanofactorshowsapeakatω =ǫinthecaseof
spin-flipandspin-fliptransitions(Fig.6).Thiscanbedoneby
“in-phase” resonances and a dip for out-of-phase resonances.
using(4).Considerforthedefinitenesstheelectrontransitions
Although Fig. 5 displays the Fano factor for an asymmetric
between the dot and the right reservoir. The corresponding
quantum dot, ΓL > ΓR, such a strong influence of the phase reservoir wave function χ (r) of (4) is represented by a
R
on the shot-noise power spectrum pertains in a general case. Kramers doublet χ (x) = u (r)| ↑i+v (r)| ↓i, where u
R R R R
Theeffectismerelymorepronouncedfortheasymmetricdot. and v are functions of spatial coordinate r only. Therefore,
R
ThereasonistheCoulombrepulsionthatpreventstwoelectron
the tunnelingmatrixelementscorrespondingto the transitions
from occupying the dot (c.f. with Fig. 3). For noninteracting
from the resonantlevel to the rightreservoir without spin flip
electrons (U =0), however, the effect is mostly pronounced
C and accompanied by spin flip are[5]
for a symmetric case, Γ =Γ .
L R
S(ω )/2eI Ω ↔ u (r)
2.5 (cid:18)δΩRR(cid:19)=−1/(2m)Zr∈ΣRφ(r)∇n (cid:18)vRR(r)(cid:19)dσ (15)
2
η
=1 η Forrelativelysmalldeviationsofg factorintherightreservoir
=-1
1.5 from 2, |v| ∼ O(|∆gu|), ∆g = g −2,[24], and so the two
transition amplitudes are related as |δΩ |∼O(|∆gΩ |). For
R R
1 ∆g >1, the two componentsur and vr are ofthe same order
ofmagnitudeandsoδΩ ∼Ω .Thecorrespondingtunneling
R R
0.5 amplitudes from the resonant level and the left reservoir are
evaluated in the same way.
ω/ε
0.5 1 1.5 2 2.5
µ
Fig. 5. Fano factor versus ω for the resonant current through two levels.
L Ω Ω n
Theparameters arethesameasinFig.4.
L ε R
δΩ δΩ µ
Obviously, the shot-noise spectrum of resonant current
L R R
through a double-dot system should be similar to that shown
in Fig. 5 for η = −1. Indeed, such a system is mapped to
a single dot with two levels, corresponding to the symmetric Fig. 6. Electron current through an impurity in the presence of magnetic
(nodeless) and antisymmetric(one-node)states. Thereforethe field.Hereǫdenotes Zeemansplitting andnisthenumberofelectrons that
havearrived attherightreservoir(collector) bytimet.
correspondingshot-noise powerspectrum would always show
a dip at Rabi frequency (cf.[4]), in contrast with earlier
evaluations, which predicted a peak[23].
Now we can obtain the quantum rate equations for magne-
Our results suggest that the measurement of shot-noise
totransport through the Zeeman doublet (Fig. 6). We denote
spectrum can be used for a measurement of the relative
δΩ = α Ω , where the coefficients α are of the
L,R L,R L,R L,R
phase η. Technically, it would be more complicated than the
orderof ∆g/g.One findsthat, althoughsign[δΩ ]=±1,the
L,R
measurementof the total currentas a functionof ΓR (Fig. 3), product δΩ δΩ >0. Thus, the resonances belonging to the
L R
which also determines η, yet the measurement of S(ω) does Zeeman doublet are always in phase (η =1). It is convenient
not distort the dot, and the phase η can be determined even to write rate equations separately for electrons polarized up
for non-overlappingresonances, ΓL,R ≪ǫ. and down in the emitter and collector. Let us consider the
polarized up current in the emitter and the collector (Fig. 6).
III. INTERFERENCEEFFECTSIN MAGNETO-TRANSPORT Using (6), we obtain the following rate equations for the
Consider now the electron transportthrougha quantumdot reduced density matrix σn (t) described the spin-polarized
αβ
or impurity in the presence of magnetic field. In this case, transportthrogh the Zeeman doublet (the index n denotesthe
all of the levels of the quantum dot are doubled due to the numberofelectronswithspinupthathavearrivedattheright
IEEETRANSACTIONSONNANOTECHNOLOGY,VOL.4,NO.1,JANUARY2005 6
reservoir by time t): We argued in[5] that the interference effect in the resonant
tunneling through impurities, considered in the present study,
σ˙n =−Γ (1+α2)σn
00 L L 00 can explain coherent oscillations with Larmor frequency in
+Γ (σn−1+σn )+α2Γ (σn +σn−1) the STM current. These oscillations were observed in a set
R 11 22 R R 11 22
−α Γ (σn−1+σn−1−σn −σn ) (16a) of STM experimentsas a peak in the tunnelingcurrentpower
R R 12 21 12 21
spectrum[6], [7], probablyin the spin-polarizedcomponentof
σ˙n =−Γ (1+α2)σn +Γ σn (16b)
11 R R 11 L 00 the current[9]. In fact, there have been several attempts to
σ˙2n2 =−ΓR(1+α2R)σ2n2+α2LΓLσ0n0 (16c) explain the experiments[6], [25]-[28]. All these explanations
σ˙n =iǫσn −Γ (1+α2)σn −α Γ σn (16d) were based on an assumption that the oscillations of the
12 12 R R 12 L L 00
tunneling current are generated by precession of a localized
Here we took into accountthatthe spin-fliptransitionsampli-
spin 1/2, interactingwith tunnelingelectrons. In contrastwith
tudes,δΩ,fromtheupperandlowerlevelsofthequantumdot
these models, we suggest that it is not the impurity spin but
(Fig.6)areoftheoppositesign.Similarto(10)oftheprevious
the current itself that develops coherent oscillations due to
section, the quantum interference is generated by transitions
tunnelingofelectronswithviatheresonantlevelsofimpurity,
between the states of the Zeeman doublet via the reservoirs.
spilt by the magnetic field. Indeed, these oscillation would
Using (7) and (16), one obtains for the spin-up polarized
look like those generated by a single spin precession, since
current in the right reservoir
the Zeeman splitting coincides with the Larmor frequency.
However, there is no precessing spin in our explanation, but
I(t)=Γ [σ (t)+σ (t)−α σ (t)−α σ (t)], (17)
R 11 22 R 12 R 21 only the interference effect of electrons moving through two
where σ (t) = σn (t). The corresponding shot-noise different states[5].
αβ n αβ
power spectrum S(ω) is given by the McDonald formula (8). An essential requirement for our explanation should be a
Using (16), we obtPain sizable spin-orbit coupling effect. This would imply that the
g-factor near impurity is different from those inside the bulk
S(ω)=2e2ωΓRIm Z11(ω)+α2RZ22(ω) and in the tip. This might be due to low space symmetry of
−α [Z (ω)+Z (ω)]},(18) an impurityon the surface[29]. Also, the nature of the tip can
(cid:8) R 12 21
playa majorrole,sothatthe g-factorofthe tipwoulddepend
whereZαβ(ω)is givenby(14).The corresondingFanofactor strongly on the tip radius[30].
S(ω)/2eI, where I = I(t → ∞), is therefore determined by It follows from our arguments that the peak in the STM
(17) and (18). currentspectrumisnotanevidenceof a singlespin detection,
We display in Fig. 7 the Fano factor as a function of ω butratheran effectofcoherentresonantscattering(tunneling)
for an asymmetric quantum dot, with the parameters ΓL =ǫ, onimpurity.Nevertheless,theabovedescribedspin-coherence
ΓR = 0.1ǫ and αL = αR = 0.2. This quantity shows a clear mechanismcan be used for a single nuclearspin detection,as
peak at frequencyclose to the Zeeman splitting[5]. Similar to was suggested in[5]. Indeed, due to the hyperfine coupling,
the previouscase, discussed in Section II, the effectis mostly each electronic level will be split into a number of sub-
pronounced for an asymmetric dot due to the influence of levels.Then,accordingtoourmodel,thepeakinSTMcurrent
Coulomb repulsion. Also, we would like to emphasize that spectrumwouldbesplitinanumberofpeakscorrespondingto
the two resonances of the Zeeman doublet are “in phase”, so transitions between various hyperfine levels. Such a splitting,
that η = 1. Therefore, the shot-noise power spectrum cannot in fact, has already been observedin recentmeasurements[9].
be compared with that of the current through a couple-dot The data clearly displays differentpeaks in the current power
structure. The latter corresponds to η = −1 and, therefore, spectrum–evidenceofhyperfinesplitting.Theseexperimental
the corresponding current spectrum would always display a resultsstronglysupportsourexplanationandopensanewway
dip[4], as shown in Fig. 5. for a measurement of single nuclear spin[5], [9], [30].
ω
S( )/2eI
IV. CONCLUSION
1
In this paper, we studied the interference effects in quan-
0.95 tum transport through quantum dots or impurities, where the
transportis carried via several levels. In our investigation,we
0.9
usedanewmethodofquantumrateequationswhichismostly
suitablefortreatmentofthistypeofproblemsandaccountsthe
0.85
Coulombrepulsioninasimpleandpreciseway.Wefoundthat
0.8 theinterferenceeffectsstronglyaffectthetotalcurrentaswell
asthecurrentpowerspectrumanddependontherelativephase
0.75
of the levels, carryingthe current.For instance, in the case of
ω/ε
0.7 out-of-phase resonances, the total current drops down when
0 1 2 3
the coupling with the collector increases. This contraintuitive
Fig.7. Fanofactorversusωforthespin-polarizedmagnetotransportcurrent result represents an effect of the destructive interference. On
throughtheZeemandoublet, showninFig.6. theotherhand,nodestructiveinterferenceeffectwouldappear
IEEETRANSACTIONSONNANOTECHNOLOGY,VOL.4,NO.1,JANUARY2005 7
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