Table Of ContentCotunneling current through quantum dots with phonon-assisted spin-flip processes
J¨org Lehmann∗ and Daniel Loss
Department of Physics and Astronomy, University of Basel,
Klingelbergstrasse 82, CH-4056 Basel, Switzerland
(Dated: February 2, 2008)
6
0 Weconsidercotunnelingthroughaquantumdotinthepresenceofspin-flipprocesses inducedby
0 thecoupling toacoustic phononsof thesurrounding. Anexpression for thephonon-assisted cotun-
2 neling current is derived by means of a generalized Schrieffer-Wolff transformation. The influence
ofthespin-phononcoupling on theheating of thedot is considered. Theresult is evaluated for the
n
case of a parabolic semiconductor quantum dot with Rashba and Dresselhaus spin-orbit coupling
a
and a method for thedetermination of thespin-phonon relaxation rate is proposed.
J
0
PACSnumbers: 73.21.La,05.60.Gg,72.25.Rb,03.65.Yz
3
]
l I. INTRODUCTION determination of the spin-relaxationrate.
l
a Theoutlineofthepresentworkisasfollows. InSec.II,
h we start by introducing the model for the quantum dot
Inthe pastyears,prospectiveapplicationsinspintron-
s- ics and quantum computation1,2,3,4 have lead to a grow- coupled via a tunneling Hamiltonian to two fermionic
e ing interest in the magnetic properties of nanoscale sys- leads and via a spin-phonon coupling to a bosonic envi-
m ronment. In the next section, we put forward a general-
tems. A variety of systems ranging from semiconduc-
. torquantumdotsovermolecularmagnets5,6,7,8,9 downto ized Schrieffer-Wolff transformation, which, in the limit
t
a single atomson surfaceshavebeen studiedin detail.10,11 of a weak spin-phonon coupling, allows one to eliminate
m tolowestorderthe dot-leadcoupling. We thenderive,in
Fortheinvestigationofsuchsystems,transportmeasure-
Sec.IV,expressionsfortheelasticandinelasticcotunnel-
- ments provide an excellent tool. Recently, the Zeeman
d ing current in the presence of the spin-phonon coupling.
splitting of two spin-levelshas been measuredvia inelas-
n tictunnelingspectroscopy,bothinGaAsquantumdots12 For the evaluation of the current, the knowledge of the
o
occupation probabilities of the spin-states on the dot is
c andinsingleMnatomsonasurfacewhichhavebeenad-
[ dressed by an STM tip.10 Using the same technique, the required. Their dynamics is governed by a master equa-
tion, which is discussed in Sec. V. The experimentally
singlet-tripletsplitting in few-electronquantumdots has
2
been determined.13 most relevant quantity, the differential conductance as a
v
function of the bias voltage, is evaluated in Sec. VI. In
0 For the applications mentioned at the beginning it is
Sec. VII, we apply the generalresult to a realistic model
2 crucialthateffectslikerelaxationandthelossofquantum
4 coherence resulting fromthe coupling of the nanosystem describingarecentexperimentbyKoganetal. (Ref.12).
9 to its surrounding environment are sufficiently weak. In Sec.VIII brieflydiscussesthe low-temperaturebehavior.
0 The final conclusions are presented in Sec. IX.
particular, a detailed knowledge about these effects is a
5
prerequisitefortheassessmentofthesuitabilityofaspe-
0
/ cific system for application purposes. In the case of the
t II. MODEL
a electronspin in semiconductor quantum dots, relaxation
m atlowmagneticfieldsisdominatedbythehyperfinecou-
pling to the nuclei,14,15,16 while at higher fields the spin- We considertransportacrossaquantumdotinaspin-
-
d orbit interaction induced coupling to acousticalphonons 1/2 ground-state contacted to two leads in the presence
n ismostrelevant.17,18Itturnsoutthatbotheffectsaresuf- ofanintrinsicspin-flipmechanismduetothecouplingof
o ficiently weak, and correspondingly long spin-relaxation the dot to a phonon bath (see Fig. 1). The total system
c times have been reported.19,20,21 For the single-atomex- consisting of dot, leads and phonons is described by the
:
v periments reported in Ref. 10, the strength of different Hamiltonian
i
X coupling mechanisms is not yet clear, however, and the
H =H +H +H . (1)
measurement schemes which were used in the semicon- 0 T S
r
a ductor case are not directly applicable.
Here,H istheHamiltonianoftheisolateddot,theideal
0
In the present work, we study to what extent inelas-
leads ℓ=L, R and the phonons, i.e.,
ticcotunnelingspectroscopycanyieldinformationabout
intrinsic spin-flip processes due to a coupling to bosonic H = E n +Un n + ǫ n + ~ω n . (2)
0 σ σ ↑ ↓ ℓk ℓkσ q q
degrees of the freedom in the surrounding. We put for-
σ=↑,↓ ℓkσ q
ward a general approach for the calculation of the co- X X X
tunneling current22,23 in the presence of such spin-flip The number operator of the dot spin is given by n =
σ
processes. Subsequently, we apply the generalformalism d†d , where d (d†) destroys (creates) an electron with
σ σ σ σ
to the case of cotunneling across GaAs/AlGaAs semi- spin σ = , on the dot. An externally applied mag-
↑ ↓
conductor quantum dots and propose a method for the netic field B in z-direction leads to a Zeeman splitting
z
2
E S direction)
2π
Γ (ǫ)= T 2δ(ǫ ǫ ). (4)
ℓ ~ | ℓk| − ℓk
k
X
H
T H
µ T Sofar,exceptforthepresenceoftwoinsteadofonelead,
L
the Hamiltoniancorrespondstothe one ofthe single-site
µR Andersonmodel,24 whichhasbeenextensivelystudiedin
the literature.25
∆ ↓ H
↓↑ S Finally,weassumethatthedotspininteractswiththe
↑ phonon system via a spin-phonon coupling of the form
0
H = (M σ +M σ )(a +a† ). (5)
S q,x x q,y y q −q
q
X
FIG.1: Sketchofthequantumdotcoupledviathetunneling
HamiltonianHTtotwoleadsatelectro-chemicalpotentialsµL In order to guarantee the hermiticity of the coupling
and µR and to a phonon bath via the spin-phonon coupling Hamiltonian, the coefficients have to fulfil Mq,i =M−∗q,i
Hamiltonian HS. The dot can be either empty (0), in one of for all q and i = x,y. Furthermore, we have intro-
the spin-directions ↑ and ↓, which are split by the Zeeman duced the spin operators σ = d†d + d†d and σ =
energy ∆↓↑, or in thesinglet state S. i(d†d d†d ). Again, we cxan cha↓ra↑cteriz↑e t↓he coupyling
↓ ↑− ↑ ↓
by a spectral density, which is defined as
∆ =E E = gµ B of the single-particle dot lev- π
els↓.↑ Fur↓th−erm↑ore,−dueBtoz electron-electron interaction, D(ω)= ~ |Mq,x+iMq,y|2δ(ω−ωq). (6)
q
the simultaneous occupation of the dot by two electrons X
costs the additional energy U. The leads ℓ = L,R are
Notethatthecouplingdoesnotcontaindephasingterms
modeled as Fermi liquids, i.e., by non-interacting quasi- proportionalto σ =d†d d†d or to the total number
particleswithenergyǫℓk,wherekisthewave-vectorand of electrons n =zn +↑n↑−on↓th↓e dot. For the electron
σthespin. Thenumberoperatorn =c† c forthe ↑ ↓
ℓkσ ℓkσ ℓkσ spinina quantumdotthis is justified to a verygoodap-
corresponding state is defined in terms of the fermionic
proximation,aswasshowninRef.17. Inthisregard,the
creationanddestructionoperatorsc†ℓkσ andcℓkσ,respec- present model differs, e.g., from the so-called Anderson-
tively. Eachleadisfilleduptoanelectro-chemicalpoten- Holstein model,26,27 which only contains coupling terms
tial µℓ, andthus the meanoccupation numbers obey the of this kind.
Fermi distributions f (ǫ) = exp[(ǫ µ )/k T]+1 −1.
ℓ ℓ B
{ − }
Here,anexternallyappliedbiasvoltageV mapstoadif-
ference µ µ =eV. In the present work, we consider
L− R III. SCHRIEFFER-WOLFF TRANSFORMATION
transportacrossasingly-occupieddotinthecotunneling
regime. This imposes the condition E µ E +U
σ ℓ σ
≪ ≪ Since weareinterestedintransportinthe cotunneling
for σ = , and ℓ = L,R. The last term in Eq. (2)
↑ ↓ regime where the dot occupancy is only changed virtu-
describes the phonons of the environment, which con-
ally, it is advantageous to eliminate to lowest order the
sist of a collection of modes characterized by a wave-
vector q with occupation numbers n = a†a . Here, tunneling Hamiltonian HT, which changes the number
q q q
of electrons on the dot by one. A standard method to
the bosonic operators a† (a ) create (destroy) a phonon
q q achievethisgoalistheso-calledSchrieffer-Wolfftransfor-
in the mode q. Note that for reasons of notational
mation,28 which we generalize to the case where a spin-
simplicity, a possible branch index has been suppressed
flip coupling H is present. Accordingly, we perform the
here, but can be added in the end results. At equilib- S
canonical transformation to the Hamiltonian
rium,thephononsaredescribedbytheBose-distribution
nB(ǫ)=[exp(ǫ/kBT)−1]−1. H¯ =eSHe−S =H +H + 1[S,H ]+ (H3), (7)
The coupling of the dot to the leadsis modeled by the 0 S 2 T O T
tunneling Hamiltonian
where the generatorS of the transformationhas to fulfil
the condition [S,H +H ]+H =0. In order to obtain
0 S T
HT = Tℓkc†ℓkσdσ+h.c., (3) an explicit expression for the generator S, we now use
ℓkσ that, as has been discussed in the Introduction, in all
X
relevant cases, the spin-flip coupling is weak and can be
where, for reasons of simplicity, we have assumed spin- treated perturbatively. Hence, we can expand
independent tunnel matrix elements T . For later use,
ℓk
weintroducethecorrespondingtunnelingrates(perspin- S =L−1H L−1[L−1H ,H ]+ (H2). (8)
0 T− 0 0 T S O S
3
Here,L istheLiouvillianofthedecoupleddot,leadsand Furthermore, the transformationleads to a renormaliza-
0
phonon system.36 We, thus, arrive at the transformed tion of the spin-phonon coupling
Hamiltonian
1
1 1 Hph = T 2X d†d +h.c. (17)
H¯ H +H + [L−1H ,H ]+ [L−1[L−1H ,H ],H ]. ren −2 | ℓk| ℓkσ− σ σ¯
≈ 0 S 2 0 T T 2 0 0 T S T ℓkσ
X
(9)
Finally, a two-electrontunneling term
The first commutator corresponds to the standard
Schrieffer-Wolff transformation28 and yields a contribu- 1
tion H2peh = 2 αTℓkTℓ′k′Xℓkσαdσdσ¯c†ℓ′k′σc†ℓkσ+h.c.
1[L−1H ,H ]=H +H +H . (10) ℓℓ′Xkk′σα (18)
2 0 T T dir,ex ren 2e
isgenerated,which,forthesamereasonsasabove,willbe
Here, the first term contains the direct and exchange in- disregardedtogetherwiththerenormalizationterm(17).
teraction between the dot and the leads
1 T T∗
Hdir,ex = 2 ǫ ℓk ℓE′kα′ nσα¯c†ℓkσcℓ′k′σ IV. CURRENT
ℓℓ′kk′σα ℓk− σ
X (cid:0)
−αdσ†¯dσc†ℓkσcℓ′k′σ¯ +h.c. (11) ThemeancurrentIℓ(t)acrosscontactℓ=L,Risgiven
bythe expectationvalue ofthe time-derivativeofthe to-
We denote flipped spins by ¯ = and vice(cid:1) versa. Here tal number of electrons N = n in lead ℓ multi-
↑ ↓ ℓ kσ ℓkσ
and in the following, the sums over α run over α = 1.
plied by the electron charge e, i.e.,
Furthermore,wehaveintroducedtheabbreviationsnα±= −P
σ
c(1on−trαib)/u2ti+onαtnoσEaqn.d(1E0)σαis=thEeσr+en[o(αrm+al1iz)/a2ti]oUn.oTfhtheendexott Iℓ(t)=−ehN˙ℓi=−i~eh[H¯,Nℓ]i. (19)
Hamiltonian H ,
0 Fortheevaluationofthisexpectationvalue,weswitchto
T 2 the interaction picture to obtain29,30
H = | ℓk| nαn . (12)
ren ǫ Eα σ¯ σ
ℓkσα ℓk− σ t−t0
As this contribution caXn be absorbed in the definition of Iℓ(t)= i~e dτh[N˜˙ℓ(t−t0),H˜1(t−τ −t0)]i. (20)
H0, it will be ignored in the following. The last term, Z0
which describes processes that change the electron num-
ber on the dot by two, reads Here,thetildedenotesinteractionpictureoperatorswith
respect to H and H :=H¯ H is the coupling part of
1 T T∗ 0 1 − 0
H = α ℓk ℓ′k′ d d c† c† +h.c. (13) the effective Hamiltonian and the expectation value has
2e 2ℓℓ′Xkk′σα ǫℓk−Eσα σ¯ σ ℓ′k′σ¯ ℓkσ to be taken at time t−τ −t0.
In the long-time limit t t , the mean value of the
Later, we shall focus on the sector of the Hilbert space ≫ 0
current becomes time-independent and we find that due
where the dot is singly occupied, and hence H can be
2e to charge conservation I = I . Hence, we can write
neglected, as well. L − R
Similarly,wecandecomposethephonon-mediatedcon- the total currentas I =IRL−ILR, where Iℓ′ℓ is the con-
tributionofthe electronsflowingfromleadℓintoleadℓ′.
tributions to the transformed Hamiltonian (9) into
In the cotunneling regime, where E µ E +U
σ ℓ σ
≪ ≪
12[L−01[L−01HT,HS],HT]=Hdpihr,ex+Hrpehn+H2peh. (14) fsomraσll=co↑r,r↓ecatniodnℓs,=alLw,aRy,stohcecudpotieids,buypatosienxgpleoneelnectitarlolny
and hence is characterized by the occupation probabili-
Here,the relevantcontributioncontaining directandex-
change terms reads ties pσ = nσ with p↑ +p↓ = 1. The currents Iℓ′ℓ can
then be wrhittein as23,31
1
Hdpihr,ex = 2ℓℓ′kk′σαTℓkTℓ∗′k′Xℓkσα Iℓ′ℓ =e Wℓ′σ′ℓσpσ, (21)
X σσ′
× nασc†ℓkσcℓ′k′σ¯ −αd†σdσ¯c†ℓkσcℓ′k′σ +h.c. where Wℓ′σ′ℓσ is the rateXfor an electron tunneling from
(cid:16) (cid:17) (15) leadℓintoleadℓ′whenthedotwasinitiallyinthestateσ
where we have introduced the phonon operator and is in the state σ′ afterwards. By convention, a pro-
cess with σ′ = σ is called elastic cotunneling, while the
X = Mq,iσiσσ¯ aq casewithσ′ =σ¯ is referredto asinelastic cotunneling.22
ℓkσα ǫ Eα ǫ Eα ~ω The contribution due to the cotunneling without par-
q,i=x,y ℓk− σ(cid:18) ℓk− σ¯ − q
X ticipation of phonons to the rates Wℓ′σ′ℓσ will be desig-
+ a†−q . (16) nated by Wℓ(′0σ)′ℓσ. In addition, the phonon-assisted co-
ǫℓk−Eσ¯α+~ωq(cid:19) tunneling process contributes with rates Wℓ(′1σ)′ℓσ to the
4
ttwotoalsurbasteecstiWonℓ,′σw′ℓeσdi=scuWssℓ(′0tσh)′ℓeσse+twWoℓ(c′1σo)′nℓtσr.ibuIntiotnhseinnedxet- and[mµaℓx′ℓ(kB=T, µµℓℓ′′ℓ−+∆µℓσ.′σ )/H∆eℓr′eσ,′ℓσt]e4rmshaovfe tbheeenordneer-
O | |
tail. Note that we thereby restrict ourselves to temper- glected in the curly brackets.
(cid:0) (cid:1)
atures abovethe Kondotemperature, where correlations
between the dot spin and the collective spin of the elec-
trons in the lead do not play a role. We shall come back B. Phonon-assisted elastic and inelastic
to a brief discussion of the low-temperature behavior in cotunneling
Sec. VIII.
We now come to the current contributions due to the
presenceofthespin-phononcoupling. UsingHph from
dir,ex
A. Elastic and inelastic cotunneling Eq. (15), we obtain after a straightforward but lengthy
calculation the rates for the phonon-assisted elastic co-
Evaluating the current expression (20) for the direct tunneling
and exchange terms (11), we arrive at the elastic cotun-
~
neling rate Wℓ(′1σ)ℓσ = 8π2 dǫ′Γℓ′(ǫ′) dǫΓℓ(ǫ)Λ(e1l)(ǫ,ǫ′) (27)
~ Z Z
Wℓ(′0σ)ℓσ = 2π dǫΓℓ′(ǫ)Γℓ(ǫ)Λ(σ0),el(ǫ)fℓ(ǫ)[1−fℓ′(ǫ)] ×N(ǫ−ǫ′)fℓ(ǫ)[1−fℓ′(ǫ′)] .
Z
(22) The corresponding energy denominator
wthiethinΛe(σl0a)s,etli(cǫ)co=tu(nEnσ¯el+ingUr−atǫe)−is2g+iv(eEnσb−yǫ)−2. Similarly, Λ(e1l)(ǫ,ǫ′)= (E +U ǫ)1(E +U ǫ′)
σ (cid:20) σ − σ¯ −
X
~ 1 2
Wℓ(′0σ¯)ℓσ = 2π Z dǫΓℓ′(ǫ−∆σ¯σ)Γℓ(ǫ)Λ(σ0),inel(ǫ) (23) + (Eσ −ǫ)(Eσ¯ −ǫ′)(cid:21) (28)
×fℓ(ǫ)[1−fℓ′(ǫ−∆σ¯σ)] , now depends on both the energy before and after the
where Λ(0),inel(ǫ)= (E +U ǫ)−1 (E ǫ)−1 2. In phonon emission and/or absorption. Note that as ex-
σ σ¯ − − σ¯ − plainedabove,we use the term“elastic”only to indicate
the inelastic case, the energy ∆ is deposited in or ex-
(cid:2) σ¯σ (cid:3) that the dot state before and after the cotunneling pro-
tracted from the dot. Correspondingly, the initial en-
cess is identical. From the point of view of the electron
ergyǫofthetunnelingelectroncomingfromleadℓdiffers
system, the phonon-mediated cotunneling process is, of
by the same amount from the final energy of the elec-
course, no longer elastic. Rather, the electrons either
tron in lead ℓ′. Inelastic cotunneling is, thus, strongly
emit the energy ǫ ǫ′ due to the stimulated and spon-
suppressed for temperatures and external voltages much taneous emission o−f phonons or absorb the energy ǫ′ ǫ
smaller than ∆ . −
σ¯σ fromthephononsystem. Bothprocessesarecapturedby
Toa verygoodapproximation,the energydependence
the function
of the coupling rates Γ (ǫ) can be ignored in the en-
ℓ
ergy range contributing to the integrals in the rate ex-
pressions (22) and (23). Furthermore, near one of the N(∆ǫ)=D(∆ǫ/~) [n (∆ǫ)+1]+D( ∆ǫ/~)n ( ∆ǫ).
B B
resonances, e.g., for Eσ . µℓ Eσ¯ + U, we can ex- − − (29)
≪
pand the dominant energy denominator of Λ(0),el(ǫ) and Similarly, we obtain the “inelastic” phonon-mediated
σ
Λ(0),inel(ǫ), respectively, and carry out the energy inte- cotunneling rates
σ
gration.37 This yields the approximate rates ~
Wℓ(′0σ)′ℓσ ≈ ~Γ2ℓπΓℓ′ ∆2ℓ′1σ′ℓσ(cid:26)1+ 4∆2ℓ1′σ′ℓσ Wℓ(′1σ¯)ℓσ = 8π2 Z×dNǫ(′ǫΓ−ℓ′(ǫǫ′′)−Z∆dσ¯ǫσΓ)ℓf(ℓǫ()ǫΛ)(i[n11e)−l(ǫf,ℓǫ′′()ǫ′)] (30)
with
4π2(kBT)2+(µℓ′ℓ+∆σ′σ)2 Θ( µℓ′ℓ ∆σ′σ)
× − −
h i(cid:27) (24) Λ(1)(ǫ,ǫ′)= 1
inel (E +U ǫ)(E +U ǫ′)
σ (cid:20) σ − σ −
where we have introduced the function X
2
1
ǫ . (31)
Θ(ǫ)= (25) − (E ǫ)(E ǫ′)
1 exp( ǫ/kBT) σ − σ− (cid:21)
− −
Forthefurtherevaluationoftherateformulasweagain
as well as the energies
assume energy-independent dot-lead couplings Γ . Since
ℓ
Eσ′ µℓ′ +Eσ µℓ the energy denominators in Eqs. (28) and (31) are al-
∆ℓ′σ′ℓσ = − 2 − (26) readyoffourthorderin∆σµ,wecanneglecttheirenergy
5
dependence aroundǫ=ǫ′ =µtogetaresultvalidtothe with the total rates W =Wcot+Wflip. Here, Wcot =
σ¯σ σ¯σ σ¯σ σ¯σ
same order as in Eq. (24). Carrying out one of the en- ℓℓ′Wℓ′σ¯ℓσ is the spin-flip rate due to inelastic cotun-
ergyintegrals,weobtainthe approximateresultforboth neling, which, in contrast to the current formula (21),
P
rates also contains processes involving a single lead only. Fur-
thermore,spin-flipprocessesduetothespin-phononcou-
W(1) ~ΓℓΓℓ′ 1 + 1 pling (5) have to be taken into account in the master
ℓ′σ′ℓσ ≈ 8π2 ∆2 ∆2 ∆2 ∆2 equation (39). These processes lead to the additional
(cid:18) σ¯′ℓ σℓ′ σ′ℓ σ¯ℓ′(cid:19)
rate
×Z d∆ǫN(∆ǫ)Θ(−µℓ′ℓ−∆σ′σ −∆ǫ) (32) Wσ¯flσip = ~2 N(∆σσ¯). (40)
with the energy differences ∆ = E µ . We now
σℓ σ ℓ
− In the stationary limit, the solution of the master equa-
consider a spectral density of the spin-phonon coupling
tion is given by p = [1+W /W ]−1, where we have
of the form σ σ¯σ σσ¯
used the normalizationcondition p +p =1. Note that
↑ ↓
D(ω 0)=γ~ω1−sωs (33) since elastic cotunneling leaves the dot state invariant,
≥ c besides the spin-flip rate Wflip only the rates for the in-
σ¯σ
with the dimensionless spin-phonon coupling parame- elastic cotunneling processes appear in the master equa-
ter γ, the cut-off frequency ω and an exponent s, which tion.
c
depends on the nature of the coupling. In order for our Figure 2 shows the ratio p↑/p↓ of the populations as
master-equation approach to be valid, we have to re- a function of the bias voltage V in the presence of an
strict ourselves to exponents s 1. The exponent s=1 Ohmic environment for different coupling strengths γ.
≥
corresponds to the so-called Ohmic case, and below, we Near thermal equilibrium, i.e., for voltages far below the
will find s = 3 or s = 5 for a spin-orbit mediated cou- onset of inelastic cotunneling, i.e., eV ∆↓↑ ,kBT, the
≪| |
plingtophonons. Insertingthespectraldensity(33)into total rates and thus the populations fulfil the detailed
Eq. (32), we can write the rates as balance relation pσ¯/pσ =Wσ¯σ/Wσσ¯ =exp( ∆σ¯σ/kBT).
−
For higher bias voltages, inelastic cotunneling leads to a
Wℓ(′1σ)′ℓσ ≈γ(~ωc)1−s~Γ8πℓΓ2ℓ′ ∆2 1∆2 + ∆2 1∆2 pdootp.ulTahtiiosnhoeafttihnegeexffceictetdbdecootmsteastele,sis.ep.,roanhoeuantciendgionf tthhee
(cid:18) σ¯′ℓ σℓ′ σ′ℓ σ¯ℓ′(cid:19) presence of the spin-phonon coupling, where the relax-
(kBT)1+sJs( (µℓ′ℓ+∆σ′σ)/kBT)Θ( µℓ′ℓ ∆σ′σ) ation due to the rate (40) equilibrates the system. For
× − − −
(34) a quantitativeestimate of the heating effect, we consider
the deviation h of the population ratio from its equilib-
with rium value:
1 e−a ∞ lnxs lnx+a p
J (a)= − dx sgn(lnx)| | . (35) h= ↑ exp(∆ /k T) . (41)
s a x 1 x e−a p − ↓↑ B
Z0 − − (cid:12) ↓ (cid:12)
(cid:12) (cid:12)
In the interesting cases of s=1, 3 and 5, we find for the Using the detailed(cid:12)(cid:12)balance relation for(cid:12)(cid:12)the rates Wσ¯flσip,
integral we then find
J (a)= 1 4π2+a2 (36) h= |exp(∆↓↑/kBT)−W↑c↓ot/W↓c↑ot|. (42)
1 6 1+Wflip/Wcot
↑↓ ↓↑
1(cid:0) (cid:1)
J (a)= 4π2+a2 8π2+3a2 , (37)
3 60 Thus, even for a non-zero difference in the numerator,
1 (cid:0) (cid:1)(cid:0) (cid:1) the heating effect becomes suppressedfor Wflip Wcot.
J (a)= 2π2+a2 4π2+a2 8π2+a2 . (38) ↑↓ ≫ ↓↑
5 42 In Fig. 2, we indicate the onset voltages of the heating
(cid:0) (cid:1)(cid:0) (cid:1)(cid:0) (cid:1) regime, defined by the relation Wflip = Wcot by vertical
↑↓ ↓↑
Inthegeneralcase,onecanreadilyshowthatJ (a)isan
s dashed lines.
even function, i.e., Js(a)=Js( a) for all s and a. Finally, we remark that heating effects are suppressed
−
when the dot-lead coupling is asymmetric,23 say Γ
L
≫
Γ . Inthis casethe excitationofthe dot,whichis domi-
R
V. MASTER EQUATION natedbyprocessesinvolvingthe tunneling fromonelead
to the other and which, hence, is proportional to the
For the evaluation of the current expressions given in product ΓLΓR, is negligible compared to the relaxing ef-
the previous section, we still need the occupation prob- fects resulting from the coupling to the left lead, which
abilities p . Their dynamics is governed by the master areproportionalto Γ2. Suchasituationmayoccur,e.g.,
σ L
equation32 whenmeasuringthecurrentthroughanatomormolecule
on a surface which is contacted on one side by an STM
p˙ = W p +W p (39) tip10 (cf. also discussion at the end of Sec. VII).
σ σ¯σ σ σσ¯ σ¯
−
6
105 the broadening of the dot states due to their coupling to
the leads does not play a role. Compared to a measure-
104 ment in the sequential tunneling regime, an experiment
can thus achieve a much higher precision for the deter-
103 mination of the splitting ∆↓↑ (see Refs. 10 and 12). An
↓↓ openquestion,however,is how spin-flipprocessestaking
pp
//
↑↑ place during the cotunneling process affect such a mea-
pp 102
γ=0 surement.
γ=0.001
101 γ=0.01 Let us first consider the zero-temperature case with-
out heating where the dot is always in its ground state
γ=0.1
which we assume to be the spin-up state. Evaluating
100
0 0.5 1 1.5 2 thecurrentusingtheapproximatecotunnelingrates(24)
and(34),we then find for the contributiondue to elastic
eeVV//∆∆↓↓↑↑
cotunneling
FIG. 2: Ratio p↑/p↓ of the populations as a function of the
biasvoltageV inthepresenceofanOhmicbathwithdifferent
coupling strengths γ. The parameters are ∆↓↑ =0.1E0, T = g = e2~ΓLΓR 1+ 1 4π2(k T)2+3(eV)2
0.01E0 and ~ΓL =~ΓR =0.02E0. Here, E0 =|E1−µ| is the el 2π∆2 4∆2 B
L↑R↑(cid:26) L↑R↑
energy difference between the energy E1 = E↑ = E↓ of the (cid:2) (cid:3)
degenerate dot level and the Fermi energy µ = µL = µR in + γ(~ωc)1−s ∆2L↑R↑ + ∆2L↑R↑
the leads in the absence of both the magnetic field and the 4π ∆2 ∆2 ∆2 ∆2
↓L ↑R ↑L ↓R!
bias voltage. The dashed vertical lines indicate the onset of
theheating regime defined byWflip=Wcot.
↑↓ ↓↑ (k T)s[k TJ (eV/k T)+eVJ′(eV/k T)] , (43)
× B B s B s B
(cid:27)
where J′(a) = dJ (a)/da. In order to obtain a compact
s s
VI. DIFFERENTIAL CONDUCTANCE expression for the inelastic cotunneling current, we fur-
thermore assume positive voltages V > 0 and not too
Having all the information for the evaluation of the hightemperaturesk T ∆ . Thenonlythe tunneling
B ↓↑
≪
current(21)athand,wecannowturntothe experimen- fromthelefttotherightcontactisrelevant,i.e.,thetotal
tally relevant quantity, the differential conductance g = current I is to a very good approximation given by I .
RL
dI/dV. As a function of the bias voltage, this quan- Furthermore, away from the step around V ∆ /e,
↓↑
≈
tity shows a characteristic step at the onset of inelas- we can use that up to exponentially small corrections
tic cotunneling around ∆ /e, when another transport Θ(ǫ) 0 for ǫ k T and Θ(ǫ) ǫ for ǫ k T. Thus,
↓↑ B B
| | ≈ ≪ ≈ ≫
channel becomes accessible. It is known that the width for eV ∆ , the inelastic cotunneling currentvanishes
↓↑
≪
of this step is proportional to the temperature T, while and for voltages eV ∆ , it contributes with
↓↑
≫
e2~Γ Γ 1
g = L R 1+ 4π2(k T)2+3(eV ∆ )2
inel 2π∆2 4∆2 B − ↓↑
L↓R↑(cid:26) L↓R↑ h i
γ(~ω )1−s ∆2 ∆2
+ c L↓R↑ + L↓R↑ (k T)s k TJ ((eV ∆ )/k T)+(eV ∆ )J′((eV ∆ )/k T)
4π ∆2↑L∆2↑R ∆2↓L∆2↓R! B B s − ↓↑ B − ↓↑ s − ↓↑ B (cid:27)
(cid:2) (cid:3)(44)
tothedifferentialconductance. Particularlyinterestingis candeterminethesolutionofthisnon-linearequationby
thestepinbetween,whichiscenteredaroundV =∆ /e. employing a linear approximation for the elastic cotun-
↓↑
In order to obtain an estimate for its width ∆V , we neling curve around the center of the step. Doing so, we
step
determine the onset V of inelastic cotunneling by us- obtainV =∆ /e g (∆ /e)/g′ (∆ /e). Defining
on on ↓↑ − inel ↓↑ inel ↓↑
ing a linear approximation of the differential conduc-
tance around the center of the step, g(V) g(∆ /e)+
↓↑
g′(∆ /e)(V ∆ /e) and intersecting wi≈th the purely
↓↑ ↓↑
−
elastic cotunneling curve (43) (see Fig. 3): g (V ) =
el on
g(∆ /e)+g′(∆ /e)(V ∆ /e). Approximately, we
↓↑ ↓↑ on ↓↑
−
7
gg== ddII For larger voltages, the effect of phonon-assisted inelas-
ddVV tic cotunneling becomes visible, as can be inferred from
the inset of panel (b).
1 10−3
×
γ
8 10−4
×
))
hh
∆V2step 22g/(e/g/(e/ 6×10−4 γγ==010−5
0 Von ∆↓↑/e 4 10−4 γ=10−4
VV × γ=10−3
γ=10−2
FIG.3: Sketchof theprocedureemployedforthedetermina- 2 10−4
tionoftheonsetVon andofthewidth∆Vstep oftheinelastic- × 0 0.5 1 1.5 2 2.5 3
cotunnelingstep. Thedashedlineshowstheextrapolation of (a) eeVV//∆∆
thedifferential conductancecurve below the step. The dash-
dotted line corresponds to the linear approximation of the
step shape. 1 10−3
×
γ=0
γ=0.01
the width as twice the difference Von ∆↓↑/e gives 8×10−4 γ=0.1
− γ=1 0.01
))
hh
∆Vstep = 6keBT 1−3 ∆kBT 2 1+γ 61π 22g/(e/g/(e/ 6×10−4 gg00−−gg00 0
(cid:26) (cid:18) L↓R↑(cid:19) (cid:20) gg
∆4 ∆4 4 10−4
L↓R↑ L↓R↑ ×
+ 0.01
× ∆2↑L∆2↑R ∆2↓L∆2↓R! − 0 1 2 3 4 5
2 10−4 eeVV//∆∆
3J′′(0)+2J′(0) kBT s−1 . (45) × 0 1 2 3 4 5
× s s (cid:18)~ωc(cid:19) (cid:21)(cid:27) (b) eeVV//∆∆
(cid:0) (cid:1)
As mentioned above, the width is essentially propor- FIG. 4: Cotunneling conductance g as a function of the bias
tional to the temperature. We also find that the spin- voltage V in the presence of an Ohmic bath with different
flipprocessesonlymodifythehigher-ordercorrectionsin coupling strengths γ. The parameters are ∆↓↑ =0.1E0, T =
kBT/∆L↓R↑. Here, the spin-flip coupling contribution is 0.01E0 and (a) ~ΓL = ~ΓR = 0.02E0, (b) ~ΓL = 0.2E0 and
proportionaltoγ andto apositiveprefactorofthe order ~ΓL = 0.002E0 with E0 as defined in the caption of Fig. 2.
In the inset of panel (b), the relative difference between the
ofoneandhasatemperaturedependencewhichdepends
differentialconductanceg in thepresenceof thespin-phonon
on the ratio of thermal and cutoff energy. In fact, we
find that with increasing coupling constant γ the width coupling and the differential conductance g0 without spin-
phonon coupling is depicted.
of the inelastic cotunneling step becomes reduced.
We now turn to the discussion of the heating effects
In Fig. 5, the step-width defined by the onset of in-
by taking into account the stationary probability distri-
elastic cotunneling is shownas a function ofthe spin-flip
bution according to the master equation (39). Figure 4
couplingstrengthγ fordifferentratiosΓ /Γ . Forequal
shows the differential conductance as a function of the L R
couplingto both leads,the maineffectagaincomes from
voltage in the presence of an Ohmic spin-flip coupling of
thesuppressionoftheheatingwithincreasingγ. Thein-
strength γ. In the upper panel, the case of a symmetric
setofFig.5depictstheγ-dependenceforlargerγ. Asde-
dot-leadcouplingis depicted. As discussedabove,this is
scribedbytheanalyticalexpression(45)thewidthdrops
thesituationwhereheatingeffectsaremostrelevant,and
linearly. Due to the finite temperature, there is a slight
for γ = 0 we indeed observe a noticeable deviation from
offset compared to the full solution, however.
the step-like behavior for voltages V &∆ /e. With in-
↓↑
creasing spin-flip coupling strength γ, however, the dot
is driven to equilibrium and the overshooting of the dif-
ferentialconductancecurvedisappears. As mentionedat VII. COMPARISON WITH EXPERIMENTS
the end of Sec. V, heating effects are suppressed when
the dot-leadcouplingisstronglyasymmetric. Suchasit- We now apply the general results to the specific ex-
uation is shown in panel (b) of Fig. 4. In this case, the perimental situation of Ref. 12 and analyze the rele-
influence of the spin-phonon coupling is much weaker. vance of spin-phonon coupling effects in this case. In
8
the dimensionless coupling constants17,18
0.6 ΓL/ΓR=1
0.605
0.5999 ΓL/ΓR=10 l ~ω ~ω
1 2
/e)/e) 0.5998 ΓL/ΓR=100 QR,D = λR,D (cid:20)−~ω1∓∆↓↑ + ~ω2±∆↓↑(cid:21) . (48)
∆∆
/(/( 0.5997 Thecut-offfrequenciesforthepiezo-electriccouplingare
pp 0.5 1 1.5 2
VVsteste 0.6 given by ωc,PE,L = cL/l and ωc,PE,T = cT/l. Note that
∆∆
the leading contribution to Q is linear in B , and
R,D z
consequently the rate (40) is proportional to B5 for low
z
magnetic fields B .
z
0.595 In the case of a deformation potential coupling to
0 2.5 10−4 5 10−4 7.5 10−4 1 10−3 acoustic phonons, the spectral density is proportional
× × × ×
γγ to ω5, i.e., s=s =5, and the prefactor is given by
DA
FIG. 5: Step width ∆Vstep = 2gi′nel(∆↓↑/e)/ginel(∆↓↑/e) as Ξ2 Q2 +Q2
a function of the coupling strength γ to an Ohmic bath for γDA = 024π~D̺l2c3R (49)
differentratiosofthetunnelingratesΓL,R,wheretheproduct (cid:0) L (cid:1)
~2ΓLΓR =(0.02E0)2 is kept constant. The other parameters
withΞ =6.7eV.Thecut-offisatthefrequencyω =
arelikeinFig.4. Thehorizontaldashedlineshowsthewidth 0 c,DA
c /l. Takingintoaccountthe magnetic-field-dependence
according to Eq. (45) for γ = 0. In the inset, the behavior L
for larger γ is shown together with the step width without of the prefactor γDA, the leading contribution to the
heating according to Eq.(45) (dashed). rate (40) is proportional to B7 at low magnetic fields.
z
Thus, the spin-phonon coupling due to the deformation
potentialelectron-phononinteractionis only relevantfor
rather high magnetic fields. Furthermore, we take a
order to do so, we model the 2D quantum dot by an
temperature T = 15mK and dot-lead coupling rates
isotropic parabolic lateral confinement characterized by
~Γ = ~Γ = 17.5µeV from Ref. 12. Finally, we as-
the frequency ω in the absence of a magnetic field. In a L R
0 sume a value E µ = 0.4meV, where E and µ are
magnetic field Bz perpendicular to the two-dimensional the energy of t|he1d−ege|nerate dot-level and th1e Fermi en-
electron gas (2DEG), the relevant frequencies then are
ergy, respectively, in the absence of the bias voltage and
ω = Ω ω /2 with the cyclotron frequency ω =
1,2 ∓ c c the magnetic field. An external bias is assumed to drop
eB /m∗c, where m∗ is the effective electron mass and
| | z symmetrically across both contacts.
Ω = ω2+ω2/4 corresponding to the effective lateral
0 c Figure6depictsthedifferentialconductanceasafunc-
confinpement length l = ~m∗/Ω (for more details, we tion of the bias voltage for the given set of parameters.
referthereadertoRef.18). TheparametersfortheGaAs Comparing the result of the calculation including the
p
quantum dot of Ref. 12 are g = 0.16, m∗ = 0.067me, spin-orbitinducedspin-phononcoupling(solidline)with
Bz =11T and we assume ~ω0 =1−.1meV. the one without this coupling (crosses),we can conclude
We consider the influence of both Rashba33 and Dres- thatthese effects playanegligibleroleinthe experiment
selhaus34 spin-orbitcoupling,whicharecharacterizedby byKoganetal. However,asshownintheinsetofFig.6,
spin-orbit lengths λ = λ = 8µm (see Appendix A). by reducing the dot-lead coupling by a bit more than
R D
Forapiezoelectriccouplingtolongitudinalandtransver- one order of magnitude, one reaches a regime where the
sal phonon modes in the limit ω c min(ℓ,d), where equilibration of the dot due to the spin-phonon coupling
α
≪
d 5nm is the vertical confinement length, the spectral becomes relevant. As discussed in Sec. V, the transition
de≈nsity (33) of the spin-phonon coupling is cubic, i.e., tothisregimeisdeterminedbytherelationWflip =Wcot,
↑↓ ↓↑
s = sPE = 3. The prefactors for the longitudinal and which for eV &∆↓↑ yields the tunneling rate
transverse contributions are given by
2πWflip
↑↓
γ = Q2D+Q2R eh14 2 (46) Γon,heat =|∆L↓R↑|s~Θ(eV ∆↓↑). (50)
PE,L 70π~̺c3 κ −
L (cid:18) (cid:19)
Q2 +Q2 eh 2 This relation can be employed for the determination of
γ = D R 14 (47) the spin-flip rate Wflip by means of a cotunneling trans-
PE,T 105π~̺c3 κ ↑↓
T (cid:18) (cid:19) port measurement. In order to do so, one has to de-
termine the critical tunneling rate Γ marking the
on,heat
with the speed of sound c = 4.73 105cm/s and transitionbetweentheregimeofthedotbeinginequilib-
L
c = 3.35 105cm/s of the longitudin×al and transver- rium and the one where heating effects are relevant. As
T
×
sal acoustic phonon modes, respectively. The density anindicatorforthistransitiononecanmeasuretheratio
of GaAs is ̺ = 5.319g/cm3 and the electron-phonon of the differential conductance above and below the in-
coupling parameters are eh = 1.2 107eV/cm and elastic cotunneling step. Determining this quantity as a
14
×
κ = 13.2. The spin-orbit coupling strength enters via function of the dot-lead coupling strength yields Γ
on,heat
9
0.006 temperature T . For lower temperatures, correlations
K
betweenbetweenthedotspinandacollectivespindegree
0.005
of freedom of the electrons in the leads become impor-
tantandaperturbativetreatmentofthe problembreaks
0.004 3.25
)) down. This leads to the so-called Kondo effect, which
hh 3
22/(e//(e/ 0.003 /g/g002.75 hthaes Abenednerasonnalymzeoddeiln25.greWate dneotwailbriniefltyhedilsitceursastuthree feofr-
gg gg11
0.002 2.5 fect of additional spin-flip processes due to the term HS
using the poor man’s scaling approach put forward by
2.25
0.001 10−4 10−3 10−2 10−1 Anderson35. In order to keepthe discussion as simple as
¯h¯hΓΓ//∆∆↓↓↑↑ possible, we consider a simplified version of our effective
0
Hamiltonian (9), which ignores the lead-, momentum-
0 0.5 1 1.5 2 2.5 3
andspin-dependence ofthe couplingconstantsaswellas
eeVV//∆∆
the direct interaction between the lead states:
FIG. 6: Differential conductance g as a function of the bias H¯ =H JS s+J (XSx+YSy)
voltage V for the parameters given in the main text. The 0− d· S d d (51)
J (XSx+YSy)n(0)+J (Xsx+Ysy)
crossesshowtheresultintheabsenceofthespin-phononcou- − 1 d d 2
pling. In the inset, the ratio of the differential conductance
Here, we have introduced the spin operators
g1 at V1 = 1.5∆↓↑/e (see arrow in the main panel) and the
differentialconductanceg0 atzerovoltagefordifferentvalues 1
vofertthicealdloitn-eledadesicgonuaptleisngthreatveasluΓe =givΓenL b=y ΓERq.. (5T0h)eadnadshthede s= 2 c†ℓkστσσ′cℓ′k′σ′ and (52)
ℓℓ′kk′σσ′
arrow thedot-lead coupling rate in theexperiment12. X
1
Sd = 2 d†στσσ′dσ′, (53)
σσ′
X
(cf. inset of Fig. 6) and via Eq. (50) the spin-flip rate
whereτ isthe vectorconsistingofthe threePaulimatri-
Wflip.
↑↓ ces. The density of lead electrons at the position of the
Let us finally discuss whether such a measurement
dot is given by
scheme would also apply to transport experiments
tohnreosurgehposirntegdleiantoRmefs.o1r0.moTlheceurele,sthoentausnunrefla-cceo,uep.lgin.,gthtoe n(0)= c†ℓkσcℓ′k′σ. (54)
ℓℓ′kk′σ
the two leads is strongly asymmetric, one contact being X
via the insulating surface and the other via an STM tip. ThephononoperatorsaredefinedasX = M (a +
q q,x q
As has been discussed at the end of Sec. V, heating ef-
a† ) and Y = M (a +a† ).
fects are suppressed in this case. Indeed, the form of −q q q,y q −q P
During the scalingprocedure,contributionsdue to ex-
the differential conductance-voltage characteristics seen P
citations of electrons and holes far away from the Fermi
in Ref. 10 can—above the Kondo temperature—be very
energy µ = 0, at a cutoff energy D, are successively
welldescribedby anequilibriumprobabilitydistribution ±
integrated out. If the resulting effective interaction is
ofthespinstateoftheatom. However,foranottoosmall
again of the form (51), we can absorb it in the coupling
tunnel-coupling across the insulating layer, it should be
constants. Note that in general, the coupling constants
possible to enter the heating regime by adjusting the
thereby acquire a momentum dependence. However, as
atom-tip coupling strength appropriately. If this cou-
long as the cutoff energy is still large enough, this de-
pling strength is not too far away from the critical value
pendence can be ignored for the relevant states near the
given by Eq. (50), it should be possible to measure the
Fermienergy. Doingso,weobtaintheusualscalingequa-
intrinsic spin-relaxation rate due to phonons by using a
tions for the exchange constant J:
similarschemeasthe oneproposedabove,althoughonly
theatom-STMcouplingstrengthcanbecontrolledeasily. dJ J2
On the other hand, it is in principle feasible to control = 2ν , (55)
dD − D
the molecule-surface coupling strength by modifying the
chemical binding of the molecule to the surface, as was which,inparticular,is notinfluencedby the otherterms
shown in Ref. 11, so the proposed scheme could be ap- in the Hamiltonian (51). The renormalization of the
plied to such a situation, as well. electron-phonon coupling constant is given by
dJ |ǫℓk|<D 1
S =νJJ =(4ln2) ν2JJ (56)
VIII. SCALING BEHAVIOR dD 2 D+ ǫ 2
ℓk
ℓk | |
X
So far, we have restricted the discussion to the while the coupling constants J and J remain invari-
1 2
case of temperatures well above the so-called Kondo- ant. Here, ν is the density of states in the leads. In
10
the anti-ferromagnetic case, where J > 0 scales to as with the Dresselhaus34 and Rashba33 contributions
∞
D 0, the same holds true for the spin-phonon cou-
→
pling constant J . We conclude that far away from the
S H =β ( σ P +σ P ), H =α (σ P σ P ).
active region around the Fermi energy, neither the pure D D − x x y y R R x y− y x
(A3)
spin-fliptermproportionaltoJ northephonon-assisted
S The strength of the spin-orbit coupling is conveniently
tunneling terms, i.e., the last two terms in the Hamilto-
measured in terms of the spin-orbit lengths λ =
nian(51),modifythescalingbehavioroftheKondo-type ~/m∗β and λ =~/m∗α . Finally, the orbital deDgrees
coupling of the dot spin to the leads. D R R
of freedom of the electron are coupled to the phonon
system via the electron-phonon interaction H =
e−ph
M (r)(a + a† ), where a branch index has been
IX. CONCLUSIONS q q q −q
suppressed.
P
We have theoretically studied the influence of a spin- ApplyingtotheHamiltonianH =H0+HSO+He−pha
phononcoupling onthe cotunneling throughananoscale Schrieffer-Wolff transformation which eliminates HSO to
system. By means of a Schrieffer-Wolff transformation, lowest order and projecting on the orbital ground state,
wehavederivedanexplicitHamiltonianinwhichtherel- we obtain to lowest order the spin-phonon coupling
evantcotunnelingprocessesappeartolowestorder. This
allowed us to evaluate the rates for elastic and inelastic H = 0[S,H ]0 (A4)
S e−ph
cotunneling in the presence of spin-flip processes. We h | | i
found that the width of the inelastic cotunneling step is
where the generator S of the canonical transformation
only weakly influenced by the spin-flip processes. More
has to fulfil [S,H ] = H . From Eqs. (A1)–(A3), we
important are the relaxation effects of the spin-phonon 0 SO
−
can infer that the generator can be written in the form
coupling which counteract the heating of the quantum
S = i(A σ +A σ ) with A and A being Hermitian
dot due to inelastic cotunneling. Considering a realistic x x y y x y
modelforarecentexperiment12,weproposeanewwayof operators which act on the orbital degrees of freedom
only. Hence, we obtain
determining the spin relaxation rate due to spin-phonon
coupling in a cotunneling experiment.
H =i 0[A ,H ]0 σ +i 0[A ,H ]0 σ .
S x e−ph x y e−ph y
h | | i h | | i
(A5)
Acknowledgments
Letting M = 0[Ω ,M (r)]0 for i = x,y, we thus
q,i i q
h | | i
TheauthorsthankD.Bulaev,W.A.Coish,V.N.Golo- arrive at the coupling Hamiltonian (5).
vach, F. von Oppen, D. Saraga, and P. Simon for help-
For a quantum dot with parabolic lateral confinement
ful discussion. Financial support by the EU RTN QuE-
(see Sec. VII), it is useful to choose coordinates rotated
MolNa,theNCCRNanoscience,theSwissNSF,DARPA, by 45◦ degrees around the z-axis17, i.e., we let x (x+
ARO, and ONR is acknowledged. →
y)/√2 and y (y x)/√2. In this new coordinate
→ −
frame,thegeneratorSofthe canonicaltransformationis
defined in terms of
APPENDIX A: DERIVATION OF SPIN-PHONON
COUPLING HAMILTONIAN
α m∗Ωy+p m∗Ωy p
R x x
A = ω +ω −
In this appendix, we derive the spin-phonon cou- x 2Ω(cid:20) 1 ~ω1−∆↓↑ 2 ~ω2+∆↓↑ (cid:21)
pling (5)startingfroma Hamiltoniandescribinganelec- β m∗Ωy+p m∗Ωy p
D x x
+ ω +ω − , (A6)
tron in a two-dimensional electron gas with an external 2Ω 1 ~ω +∆ 2 ~ω ∆
magnetic field B in z-direction (cid:20) 1 ↓↑ 2− ↓↑ (cid:21)
z α m∗Ωx+p m∗Ωx p
R y y
A = ω − +ω − −
H = P2 +U(r)+ 1gµ B σ (A1) y 2Ω(cid:20) 1 ~ω1−∆↓↑ 2 ~ω2+∆↓↑ (cid:21)
0 2m∗ 2 B z z βD m∗Ωx py m∗Ωx+py
+ ω − +ω . (A7)
2Ω 1 ~ω +∆ 2 ~ω ∆
where P = p + (e/c)A(r) is the kinetic momen- (cid:20) 1 ↓↑ 2− ↓↑ (cid:21)
| |
tum, U(r) is the confinement potential, and A(r) =
(B /2)( y,x,0)isthevectorpotential. Forx,y,z point- Note thatthis transformationdivergesfor magneticfield
z
ing alon−g the main crystallographic axes of the GaAs strengths with ~ω1,2 ∆↓↑ = 0. A detailed analysis of
±
crystal,and the electrongas lying in the [001]plane, the the relaxation rates in this case can be found in Ref. 18.
spin-orbit interaction assumes the form On the other hand, for ∆↓↑ ~ω1,2, the leading con-
| | ≪
tribution to the relaxation rates comes from the terms
H =H +H (A2) linear in ∆ B (see Ref. 17).
SO D R ↓↑ z
∝
