Excess conductance of a spin-filtering quantum dot

0.13 MB·English

by  C. W. J. Beenakker

Excess conductance of a spin-filtering quantum dot
C. W. J. Beenakker
Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands
(Dated: January 2006)
6 TheconductanceGofapairofsingle-channelpointcontactsinseries,oneofwhichisaspinfilter,
2
0 increases from 1/2 to2/3×e /h with more andmore spin-flipscattering. This excess conductance
0 wasobservedinaquantumdotbyZumbu¨hletal.,andproposedasameasureforthespinrelaxation
2 time T1. Here we present a quantum mechanical theory for the effect in a chaotic quantum dot
n (mean level spacing ∆, dephasing time τφ, charging energy e2/C), in order to answer the question
a whether T1 can be determined independently of τφ and C. We find that this is possible in a
time-reversal-symmetry-breaking magnetic field, when the average conductance follows closely the
J
2 −1
formula hGi=(2e /h)(T1+h/∆)(4T1+3h/∆) .
7
2
PACSnumbers: 72.25.Dc,72.25.Rb,73.23.-b,73.63.Kv

]
l
al The study ofspinrelaxationinthe presenceofchaotic spin-flipping
h scattering is a challenge for theorists and experimental- voltage probe
- ists. Thecommongoalistoidentifytransportproperties
s
e thatcanbereadilymeasuredandthatdependasdirectly
m as possible on the spin relaxation time (T1). One line
of research is to study how quantum interference effects chaotic
.
at such as weak localization or universal conductance fluc- I quantum dot I
m tuations are modified by spin relaxation [1]. A direct
spin-polarized unpolarized
- relation with T1 in that context is hindered by the fact
d that dephasing (both of the orbital and of the spin de- current source current drain
n greesof freedom)alsomodifies the quantuminterference
o
effects. Another line ofresearchis tostudy spin-resolved spin-conserving
c
[ currentnoise[2]. ThereadirectrelationwithT1 ispossi- voltage probe
ble,butthecomplicationsinvolvedinthemeasurementof

1 both time- andspin-dependent currentfluctuations have
v FIG. 1: Illustration of the model. A current I is passed
sofarpreventedanexperimentalrealization. Ideally,one
8 throughaquantumdotviatwosingle-channelleads,atavolt-
3 would like to relate T1 to the time averagedcurrent in a age difference V. Spin-flip scattering and decoherence (with
way which is insensitive to dephasing. It is the purpose
6 relaxation times T1 and τφ) are introducedby meansof ficti-
1 of this work to present such a relationship. tiousvoltageprobes,separatedfromthequantumdotbytun-
0 Ourresearchwasinspiredby the proposalofZumbu¨hl nelbarriers(dashedlines). Thelower(ferromagnetic)voltage
6 etal.ofanewtechniquetomeasurespinrelaxationtimes probe reinjects an electron into the quantum dot with the
0 inconfinedsystems[3]. Theseauthorsreportedmeasure- samespinbutarandomphase(contributingonlytoτφ). The
t/ ments of the conductance of an open two-dimensional upper(normalmetal)voltageproberandomizesbothspinand
a GaAs quantum dot in a parallel magnetic field. One of phase (contributingto both T1 and τφ).
m
the two point contacts was set to the spin-selective e2/h
-
d conductanceplateau. The otherpointcontactwassetto
n transmit both spins. In this configuration, the classical down electrons. The filtering property of a QPC can
o series conductance of the two point contacts is 1 e2/h be turned on and off by adjusting its local electrostatic
2
c if there is no spin relaxation and 2 e2/h if t×here is potential (via a gate voltage). The polarity of the spin
: 3 ×
v strong spin relaxation. What we will show here is that filter is fixed by the direction of the magnetic field. The
i the ensemble averaged conductance in a time-reversal- conductance becomes sensitive to spin-flip scattering if
X
symmetry-breaking magnetic field varies between these onepointcontactisaspinfilterwhiletheothertransmits
r
a twolimitsasarationalfunctionoftheproductofT1 and both spin directions. (To be definite, we will take the
the mean level spacing ∆ — largely independent of the current source as the spin filter, but it does not matter
presence or absence of dephasing. which is which in the linear response regime.)
ThegeometryoftheproblemissketchedinFig.1. We We assume that the magnetic field is sufficiently weak
discuss its various ingredients. that we may neglect the spin dependence of the Fermi
Electrons in a two-dimensional electron gas (2DEG) wave length inside the quantum dot (where the Fermi
enter and leave the quantum dot via two single-channel energy is much greater than in the point contact). The
quantum point contacts (QPC). A QPC can operate as effect of the magnetic field on the orbital motion will
a spin filter in a magnetic field [4, 5], as a result of the typicallybreaktimereversalsymmetry(symmetryindex
slightlydifferentFermiwavelengthsofspin-upandspin- β =2),ifthefieldisorientedperpendiculartothe2DEG.
2
We contrast this with the case β = 1 of preserved time The electron reservoirs connected to the quantum dot
reversalsymmetry,appropriateformoderatelyweakpar- haveelectrochemicalpotentialsµ ,withX =s(source),
X
allel fields (until the finite thickness of the 2DEG drives X = d (drain), X = n (normal metal voltage probe),
β = 1 2 even for a parallel field [6]). The mean dwell and X = f (ferromagnetic voltage probe). In the latter
7→
timeinthequantumdotisassumedtobesmallcompared casewedistinguishthetwospinpolarizationsbyasuper-
to the spin-orbit scattering time, so that spin-orbit cou- script: µ↑,µ↓. We choose the zero of energy such that
f f
pling can be neglected. Landau level quantization inside µ = 0, hence µ = eV. Both the temperature and the
d s
the quantum dot is assumed to be insignificant. The ef- applied voltage V are assumed to be small compared to
fects of a finite charging energy will be assessed at the ∆, so that we may neglect the energy dependence of the
end of the paper. scattering processes.
Two independent time scales characterizethe spin de- Thepotentialsofthevoltageprobesaredeterminedby
cay, the time scale T1 onwhich the spin directionis ran- demanding that no current is drawn from the quantum
domizedandthe time scaleT2 2T1 onwhichthe phase dot [13],
≤
of the spin-dependent part of the wave function is ran-
domized [7, 8, 9]. In closed GaAs quantum dots, hyper- 0 = (2N T↑ T↓ )µ T↑ eV T↑ µ↑
fineinteractionwithnuclearspinsisthedominantsource n− n→n− n→n n− s→n − f→n f
of spin decay for weak magnetic fields, with T2 ≃µs and −Tf↓→nµ↓f, (2)
T1increasingfromµstomswithincreasingmagneticfield 0 = (N T↑ )µ↑ T↑ eV T↑ µ , (3)
[10, 11]. For the transport problem in an open quantum f − f→f f − s→f − n→f n
dotconsideredhere,thedecoherencetimeτφ ofthewhole 0 = (Nf −Tf↓→f)µ↓f −Tn↓→fµn. (4)
wave function, rather than just its spin-dependent part,
is the relevant quantity. Typically, τ is dominated by ThecurrentI throughthequantumdotthenfollowsfrom
φ
dephasing of the orbital degrees of freedom by electron-
electron interactions.
h
indEexppeenrdimenetnctson[1d2u]ctoinonthheaveeffebcetenofaanafilynziteedτiφn tohnesppainst- eI =(1−Ts↑→s)eV −Tn↑→sµn−Tf↑→sµ↑f. (5)
usingBu¨ttiker’svoltageprobemodel[13,14,15]. Exten-
Here T↑ and T↓ denote the transmission proba-
sions to spin-dependent conduction have been proposed X→Y X→Y
bilities, summed over all channels, from reservoir X to
more recently [16, 17]. As described in Ref. [16], one
reservoir Y with spin up or down. They satisfy the sum
needs two types of voltageprobes to describe spin relax-
rules [13]
ationanddecoherence. Onetypeofvoltageprobeiscon-
nected to a normal metal reservoir,while the other type
of voltage probe is connected to a pair of ferromagnetic TXσ→Y = TYσ→X =NX, (6)
reservoirs(ofoppositepolarization,paralleltothe polar- Y=Xs,d,n,f Y=Xs,d,n,f
izationofthespinfiltersinthequantumpointcontacts).
with N =N 1. For later use we define
Foreachreservoir,anelectronthatentersitisreinjected s d
≡
into the quantum dot with a random phase. The ferro-
↑ =2 T↑ T↑ T↑ T↑ , ↓ =1 T↓ .
magnetic reservoirs conserve the spin (contributing only R − s→s− d→d− s→d− d→s R − d→d
(7)
to τ ), while the normal metal reservoir randomizes the
φ
Because of the spatial homogeneity of the coupling of
spin (contributing both to T1 and τφ).
Each voltage probe is connected to the quantum dot the quantum dotto the voltageprobes,the transmission
byatunnelbarrier. Thenormalmetalvoltageprobehas probabilitiesfornormalandferromagneticprobesarere-
N↑ = N↓ N channels for each spin direction and lated by ratios of tunnel conductances,
n n ≡ n
tchheanfnererlso.mEaagcnhetbiacrvrioelrtahgaestpurnobneelsphraovbeabNilf↑ity=ΓNpfe↓r≡chaNnf- TXσ→n = Tnσ→X = γn, if X s,d , (8)
Tσ Tσ γ ∈{ }
nel and per spin direction. By taking the limit Γ 0, X→f f→X f
N ,N at fixed (dimensionless) tunnel con→duc- Tσ δ N (1 Γσ ) γ γ
n f → ∞ X→Y − XY X − eff = X Y ,
tcaanycpersoγcenss=es aNrneΓs,paγtfial=ly hNofmΓogweeneeonussur[1e5t].haTthethdeecdaey- TXσ′→Y′ −δX′Y′NX′(1−Γσeff) γX′γY′
if X,Y n,f . (9)
times are ∈{ }
h h h The effective tunnel probability Γσ = (Γ∆)ρσ differs
T1 = γ ∆, τφ = (γ +γ )∆ ≡ γ ∆, (1) from the bare tunnel probability Γeffbecause the density
n n f φ
of states ρσ in the quantum dot has spin and energy
with ∆ the mean spacing of spin-degenerate levels and
dependent fluctuations around the average 1/∆.
γ γ +γ . These time scales should be compared
wφith≡thne spinf-dependent mean dwell time τσ in the With the help of these relations the solution of Eqs.
dwell (2–5) for the conductance G = I/V can be written in
quantum dot without voltage probes, given by τd↑well = terms of transmission probabilities between source and
h/2∆, τ↓ =h/∆. drain,
dwell
3
e2
G = 1 T↑ (1 T↑ T↑ )(1 T↑ T↑ ) , (10)
h − s→s−Q − s→s− d→s − s→s− s→d
(cid:2) (cid:3)
Γ↑ Γ↓ γ (γ +γ )+Γ(Γ↑ +Γ↓ )γ ↓
eff eff n n f eff eff f
= R . (11)
Q Γ↑ Γ↓ γ (γ +γ )( ↑+ ↓)+Γ(Γ↑ +Γ↓ )γ ↑ ↓
eff eff n n f eff eff f
R R R R
The transmission probabilities between source and In the opposite limit γ 1 the fluctuations in the den-
φ
drain are constructed from two scattering matrices S↑ sity of states can be negl≫ected, so that
and S↓, one for spin-up and one for spin-down. The
spin-up scattering matrix is a 2 2 matrix, Γσeff/Γ ρσ∆=1, if γφ 1. (17)
× ≡ ≫
r t′ Thesetwolimitsaresufficientforthepurposeofcompar-
S↑ = , (12)
(cid:18)t r′(cid:19) ing coherent and incoherent regimes.
We calculate the average conductance G separately
such that T↑ = r2, T↑ = r′ 2, T↑ = t2, and in the regime γ 1 of strong orbital dhepihasing and
s→s | | d→d | | s→d | | f ≫
T↑ = t′ 2. The matrix S is symmetric for β = 1, the regimeγf 1ofweakorbitaldephasing. Forstrong
mde→ansing |th|at Ts→d = Td→s in that case. For β = 2 the dephasing we h≪ave Γσeff → Γ, R↑ → 2, R↓ → 1, τ1,τ2 →
two transmission probabilities are not related. Because 1, hence
of the voltage probes, S is sub-unitary. The eigenvalues
τ1,τ2 [0,1]ofthematrix11 S↑S↑† givetheprobability G = 2e2 1+γn , if γ 1. (18)
∈ − f
to enter one of the voltage probes. h i h 4+3γn ≫
The statistics of the matrix S↑ in an ensemble of
This incoherent regime is insensitive to the presence or
chaotic quantum dots was calculated in Ref. [15] using
absence of time-reversal symmetry. By writing Eq. (18)
the methods of random-matrix theory. It is given in
as G =(e2/h)[2(1 p)+3p]−1 withp=γ /(1+γ ),we
terms of the polar decomposition h i − 2 n n
canunderstanditasaclassicalseriesresistance,weighted
S↑ =u √1−τ1 0 u′, (13) by the probability p of a spin-flip scattering event.
(cid:18) 0 √1 τ2(cid:19) Turning now to the phase coherent regime, we find to
− linear order in γ and γ the expansions
f n
with unitary matrices u′ = uT if β = 1 and u′ inde-
pendent of u if β =2. These matrices are uniformly dis- h 1 +0.14γ + 1 γ , if β =1,
tributedintheunitarygroup. ThedistributionPβ(τ1,τ2) e2hGi=(cid:26) 3 12 +0.1n0γn2,4 f if β =2. (19)
is the Laguerre ensemble for γ 1 and a more compli-
φ
≪
cated (but known) function for larger γφ. Note the absence of a term linear in γf for β = 2. The
InadditiontoS↑wealsoneedS↓. Thisisasinglecom- difference between the zeroth order terms 1/3 and 1/2
plex number, such that T↓ = S↓ 2. It is constructed inthepresenceandabsenceoftime-reversalsymmetryis
from the coefficients r,t,t′d→indEq.|(13|) by reflecting spin- known as weak localization or coherent backscattering.
down from the source contact, In the absence of any orbital dephasing, γf = 0, we
obtain the results plotted in Fig. 2 (dashed curves).
eiαtt′ Comparisonwith the incoherentresult(18) (solidcurve)
S↓ =r′+ . (14)
1 eiαr shows that the presence or absence of orbital dephasing
−
does not change G by more than a few % if β = 2
(Thephaseshiftαneednotbespecifiedbecauseitdrops h i
(no time-reversalsymmetry). For β =1,in contrast,the
out upon averaging over u and u′.) Using Eq. (14) the
dependence on T1 is entirely different with and without
statistics of S↓ follows from the statistics of S↑.
orbital dephasing.
To complete the random-matrix theory, we need to
Sofarwehavenotincludedtheeffectsofafinitecharg-
knowthestatisticsofthedensityofstatesρσ oftheopen ing energy e2/C (with C the capacitance of the quan-
quantumdot,whichdeterminestheeffectivetunnelprob-
tum dot). These results therefore apply to the regime
abilities. For weak decoherence we have the relation[18] e2/C ∆. In the opposite, more realistic, regime
2 ≪
e /C ∆ the charging energy introduces a weight fac-
≫
Tr(11−SσSσ†)=ρσγφ∆+O(γφ2). (15) t[1o9r]e.qTuhailstowethigehdtefnascittoyrocfosntvaetretssinthtehgereannsde-mcabnleonavicearlaagves-
Since the left-hand-side of Eq. (15) equals σ by defini- erage (considered so far) into a canonical average:
R h···i
tion (7), we have
Γσeff/Γ≡ρσ∆=Rσ/γφ, if γφ ≪1. (16) h···icanonical = 12∆h···(ρ↑+ρ↓)i. (20)
4
The incoherent result (18) is the same in the canoni-
cal and grand-canonical ensembles, because the fluctu-
ations in the density of states are suppressed by deco-
herence. In order to assess the importance of density-of-
states fluctuations in the coherent regime, we approxi-
mate (∆/2)(ρ↑+ρ↓) ( ↑+ ↓)/ ↑+ ↓ . This for-
≃ R R hR R i
mulainterpolatessmoothly betweenthe twoexactlimits
(16)and(17)ofweakandstrongdecoherence. Asshown
in Fig. 2 (dotted curves), the effect on the average con-
ductance remains relatively small.
In conclusion, we have calculated the dependence on
the spin relaxation time T1 of the average conductance
G ofaquantumdotwithaspin-filteringquantumpoint
h i
contact. In the incoherent regime there is a simple one-
to-one relationship (18) between G and T1. The pres-
h i
ence or absence of orbital dephasing was found to be
insignificant for β = 2, so that the value of T1 can be
FIG. 2: (color online) Dependence of the average conduc- extracted from G with good accuracy — without re-
tance hGi on the spin relaxation time T1, normalized by the quiring knowledhgeiof coherence time or charging energy.
meanlevelspacing∆. Thesolidcurve(red)istheincoherent
Forβ =1,incontrast,the interplaywiththe weaklocal-
result (18), valid for strong orbital dephasing. The dashed
ization effect obscures the effect of spin relaxation.
and dotted curves are the results of random-matrix theory
forweakorbitaldephasing,inthetwocasesofbroken(β =2,
black) and unbroken (β = 1, blue) time reversal symmetry.
2
Thedashedcurvesaregrand-canonical averages(e /C ≪∆)
and the dotted curves are canonical averages (e2/C ≫ ∆).
The black and red curves lie close together, demonstrating I am indebted to C. M. Marcus for drawing my at-
that T1 can be determined accurately from hGi for β = 2. tention to Ref. [3] and for valuable comments on the
Thelargedifferencebetweentheblueandredcurvesprevents manuscript. This research was supported by the Dutch
this for β =1. Science Foundation NWO/FOM.
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