Shot noise in magnetic tunneling structures with two-level quantum dots

0.31 MB·

by  T. Szczepanski

Shot noise in magnetic tunneling structures with two-level quantum dots
T. Szczepan´ski,1 V. K. Dugaev,1 J. Barna´s,2,3 I. Martinez,4
J. P. Cascales,4,5 J.-Y. Hong,6 M.-T. Lin,6,7 and F. G. Aliev4
1Department of Physics and Medical Engineering, Rzesz´ow University of Technology,
al. Powstan´c´ow Warszawy 6, 35-959 Rzesz´ow, Poland
2Department of Physics, Adam Mickiewicz University, Umultowska 85, 61-614 Poznan´, Poland
3 Institute of Molecular Physics, Polish Academy of Sciences,
ul. M. Smoluchowskiego 17, 60-179 Poznan´, Poland
7 4Dpto. de Fisica de la Materia Condensada, C-III, IFIMAC and INC,
1
Universidad Autonoma de Madrid, 28049, Madrid, Spain
0 5Francis Bitter Magnet Lab, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
2 6Department of Physics, National Taiwan University, 10617 Taipei, Taiwan

n 7Institute of Atomic and Molecular Sciences, Academia Sinica, Taipei 10617, Taiwan
a (Dated: January 10, 2017)
J
We analyze shot noise in a magnetic tunnel junction with a two-level quantum dot attached to
9
the magnetic electrodes. The considerations are limited to the case when some transport channels

are suppressed at low temperatures. Coupling of thetwo dot’s levels totheelectrodes are assumed
]
l to be generally different and also spin-dependent. To calculate the shot noise we apply the ap-
l
a proachbasedonthefullcountingstatistics. Theapproachisusedtoaccountforexperimentaldata
h obtained in magnetic tunnel junctions with organic barriers. The experimentally observed Fano
- factors correspond tothesuper-Poissonian statistics, andalso dependon relativeorientation of the
s
e electrodes’ magnetic moments. Wehavealso calculated thecorresponding spin shot noise, which is
m associated with fluctuations of spin current.
.
t PACSnumbers: 72.25.-b;73.40.Rw;85.75.-d
a
m
- I. INTRODUCTION low-temperatures, where the corresponding experimen-
d
tal data show that it does not depend on temperature
n
and is also constant in the low-frequency range.2 The
o The problem of current fluctuations has been attract-
c ing recently more and more attention due to increasing mechanismofshotnoise is relatedto the quantizationof
[ chargeandspinofparticles,thataretransferredthrough
role of fluctuations of various physical quantities in the
the system.
1 nanoworld.1–3 Inprinciple,thisisratherobviousbecause
v the fluctuations strongly increase with decreasing num- The methods used for theoretical treatment of the
3 ber of particles in the system.4 Starting from the pio- noisearealsodifferent,dependingontheroleofCoulomb
5 neering article by Schottky5 and several famous papers interaction,phonons, disorder,etc. It turned out that in
2
of Khlus,6 Lesovik7 and Bu¨ttiker et al.,8,9 the theoreti- some cases one can formulate a general approach which
2
0 calstudyofcurrentfluctuationsbecameanexcitingfield is based on the master equation describing dynamics of
. of research in statistical physics. One of the most im- quantum states of the system, so that the correlation
1
pressive achievements of the theory is the full counting functions (so-called cumulants) describing current corre-
0
statistics10–17, which allows to calculate the correlation lations in all orders (not only pair correlations) can be
7
1 functions of any order and to identify the type of statis- derived from a single generating function. The method
: tics of current correlations. of such calculations of cumulants is known as the full
v
i In addition, recent progress in experimental methods countingstatistics11,13 (FCS),anditprovidesacomplete
X hasresultedinmodernmeasurementtechniqueswhichal- description of fluctuations in the system. In particular,
r lowtostudyexperimentallythecurrentnoiseandextract one can find the mean value of current, zero-frequency
a
fromthenoiseevenmoreinformationthanfromtheusual pair correlation function (shot noise), and also establish
measurementofthe averagecurrent.18–20 Obviously,this the statistics of fluctuations – whether it is Poissonian
concerns not only the fluctuations of current, but also oranyother(super-Poissonianorsub-Poissonian). Some
fluctuations of any transport-related quantity like, for examples of using this method are presented in Ref. 14.
example, spin or pseudospin current, spin torque, heat The approach based on FCS was used to explain the
fluxes, and others. super-Poissonian shot noise, for which the Fano factor
It is well known that there are various sources of the F in a tunnel junction with quantum dot16 is higher,
noise. Correspondingly, the dominant mechanism al- F > 1, than the corresponding Fano factor for Pois-
ways depends on a specific problem under consideration sonian statistics (F = 1). Here, we consider a similar
and on various additional internal and external factors. problem of current and spin current noise in a magnetic
Here we consider the shot noise which has purely quan- tunnel junction with a nonmagnetic quantum dot, but
tum character. The shot noise is mostly observed at the dot is attachedto two ferromagneticelectrodes. The
2
experiments on organic tunnel junctions with ferromag-
netic contacts demonstrated super-Poissonianshot noise
which additionally depends on magnetic polarization of
the electrodes.21 It was assumed that the model based
on transfer of electrons through two discrete levels of
molecules is sufficient to describe statistics of the fluc-
tuations in such a system.21
It should be also noted that the problem of spin shot
noise has been already considered in many papers22–36
andforvarioussystems. Themaininterestoftheseworks
was focused on how the discreteness of spin affects the
current fluctuations.
In this paper we present a theoretical description of
the model used for explanation of the experimental data
on magnetic tunnel junctions with organic molecules.21
FIG.1: Schematicofthetunneljunctionwithtwo-levelquan-
Apart from charge fluctuations, we also consider spin tum dot considered in this paper.
fluctuations which influence the electric current. More-
over, we also consider how the spin fluctuations affect
the spincurrentinthe system. InSec.IIwedescribethe cesses to the left, which should be taken into account.
model and the theoretical method used to calculate the This is accountedfor by a temperature-dependentfactor
noise. Current shot noise is calculated in Sec. III, while x which describes tunneling to the left at the energy, at
the spin noise is calculated in Sec. IV. The relation with whichallelectronstatesintheleftelectrodesarefilledat
theexperimentisdiscussedinSec. V,andthediscussion T =0(Fig.1),butmaybeemptyathighertemperatures.
of results and final conclusions are in Sec. VI. We consider the case of T =0 and assume that the den-
6
sity of temperature-activated holes in the left electrode
is relatively small, so the parameter x can be evaluated
II. MODEL AND THEORETICAL METHOD as x exp[(ε E )/k T] 1.
− F B
∼ − ≪
Tocalculatetheshotnoiseinjunctionsunderconsider-
Themodelconsideredinthispaperisbasedonaquan- ation,wefollowthemethodofFCScalculationsproposed
tum dot with two discrete electron levels37 coupled via by Bagrets and Nazarov.14 First, we need to find the
tunneling processes to the left and right magnetic elec- probability of quantum dot to be in one of the possible
trodes. Weassumethatthedirecttunnelingbetweenthe quantum states, which can be found from the following
electrodes (so-called cotunneling) is very small as com- master equation describing dynamics of the dot’s states:
pared to the sequential tunneling through the levels of
dP
the quantum dot, and therefore will be ignored. Apart =MˆP, (1)
from this, Coulomb interaction of electrons localized at dt
the dot is assumed to be strong enough to completely
where
suppress the states with two electrons in the dot. This
model is a direct generalization of the model studied in
PT = P−, P−, P+, P+, P (2)
Ref.16tothecaseofamagneticjunction–twomagnetic ↑ ↓ ↑ ↓ 0
leadsanda non-magneticquantumdot. Accordingly,we (cid:16) (cid:17)
assume(i)differentprobabilitiesfortunneling ofspin-up is a vector whose components describe probabilities of
and spin-down electrons from the dot to the leads (and the dotto be inthe state with one spin-σ electronin the
vice versa), and (ii) different probabilities of tunneling low-energy level (P−), one spin-σ electron in the high-
σ
from/tothelow-energyandhigh-energylevelsofthedot. energy level (P+), and the probability of the state with
σ
The system under consideration is shown schematically no electrons in the dot (P0).
inFig.1. Thecentralpartpresentsthe two-levelsystem, Asalreadymentionedabove,thestatewithtwoexcess
and both energy levels are coupled to the leads via the electrons in the dot is assumed to have rather high en-
hopping terms. We consider the situation when the sys- ergyduetostrongelectroncorrelations,soitisruledout
temisbiasedasshowninFig.1,soelectronstunnelfrom fromtheconsiderations. Thisassumptioniswelljustified
right to left. when QDs are sufficiently small. In our case we consider
Thekeypropertyofthemodel16 isanassumptionthat tunneling through short molecules which play a role of
thelow-energylevelε ofthedotisbelowtheFermilevel QDs. Coulomb energy of doubly charged molecules is
−
of left electrode (and thus also of the right electrode), as then sufficiently large, so the above assumption is rea-
shown schematically in Fig. 1. Hence, at T = 0 there is sonable and well justified.
notunnelingofelectrontotheleft(andalsototheright) ThematrixMˆ ontherightsideofthemasterequation
from the dot, and the junction is completely blocked. (1) includes the rates Γ± and Γ± of electron tunneling
Lσ Rσ
Atnonzerotemperaturestherearepossiblehopping pro- from the dot to the left electrode and from the right
3
electrode to the dot, respectively, generating function S(χ) of the current correlators,
xΓ− 0 0 0 Γ− S(χ)= t λ (χ), (5)
− 0L↑ xΓ− 0 0 Γ−R↑ − 0 0
Mˆ = 0 − 0L↓ Γ+ 0 Γ+R↓ , (3) where t0 is the period of transfer of a charge, and λ0(χ)
− L↑ R↑ is the lowest eigenvalue of the matrix Zˆ(χ),
 0 0 0 Γ+ Γ+ 
 − L↓ R↓ 
 xΓ−L↑ xΓ−L↓ Γ+L↑ Γ+L↓ −ΓΣ  det[Zˆ(χ)−λ]=0. (6)
wherewealsointroducedthenotationΓΣ =Γ+R↑+Γ−R↑+ In the case of x = 0 (which corresponds to T = 0)
Γ+ +Γ− . Since the electrodes are ferromagnetic, the oneobtainsfromEq.(6)thattheminimumeigenvalueof
R↓ R↓
tunneling probabilities are assumed to be dependent on Zˆ(χ) is λ =0. Thus, for small x, x 0, one may look
0
→
the electron spin orientation. The signs ascribed to the for a solution which is linear in x, λ = xλ˜. Using then
0
elements of the matrix Mˆ correspond to increasing or Eqs.(4) and (6) we find the following algebraicequation
decreasingprobabilityofthecorrespondingdotstatedue for λ˜:
totherespectivetunnelingprocesses. Thefactorxinthis
matrix was already defined above and is assumed to be (Γ− +λ˜)(Γ− +λ˜)[(Γ+ +Γ+ )( 1+eiχ) Γ− Γ− ]
L↑ L↓ R↑ R↓ − − R↑− R↓
small, x≪1. +eiχ(Γ− +λ˜)Γ− Γ− +eiχ(Γ− +λ˜)Γ− Γ− =0. (7)
To distinguish between the probabilities of electron L↑ L↓ R↓ L↓ L↑ R↑
tunneling from the right electrode to the upper or to Thisisaquadraticequationforλ˜,whichcanbepresented
the lower energy level of the dot, we introduced differ- as λ˜2+2bλ˜+c=0, where
ent parameters Γ+ and Γ− . This difference can be at-
Rσ Rσ
tributedtodifferentshapesofthe electronicorbitalscor- (Γ− Γ− +Γ− Γ− )eiχ
responding to the dot’s states. Transmissionofelectrons b = L↑ R↑ L↓ R↓
2[(Γ+ +Γ+ )(eiχ 1) Γ− Γ− ]
in the tunneling structure shownin Fig. 1 is a stochastic R↑ R↓ − − R↑− R↓
process, which consists of random hoppings of electrons Γ− +Γ−
L↑ L↓
+ , (8)
between electrodes and QD at random times τ . There-
i 2
fore, the calculation of mean current, say through the (eiχ 1)(Γ+ +Γ+ +Γ− +Γ− )
left junction, as well as of current correlation functions c = Γ− Γ− − R↑ R↓ R↑ R↓ .(9)
implyaveragingoverprocessesζs withanarbitrarynum- L↑ L↓ (Γ+R↑+Γ+R↓)(eiχ−1)−Γ−R↑−Γ−R↓
bersofsequentialtransitionswithelectrontransferinall
Thus, the FCS generating function in the limit of low T
possible channels. The probability Q of the process ζ
s s (small x) can be written as
is determined by the probabilities of system to stay in
certain states during the time between transitions and
S(χ)= t x b b2 c , (10)
by the probability of single transitions at τi (i = 1,...s) − 0 − ± −
specified by the process ζs. The probabilities of partic- with the parametersb an(cid:0)d c definped by E(cid:1)qs.(8) and (9).
ular transitions are the matrix elements in Eq. (3). To
find the generatingfunction S(χ) ofcumulantexpansion
one has to average the exponent exp i dτχ(τ)Iˆ(τ) , III. ELECTRIC CURRENT SHOT NOISE
{ }
where Iˆ(τ) is an instanteneous current at τ and χ(τ) is
R
the source field introduced to find current cumulants by The mean value of electric current, I, and the corre-
using the generating function S(χ). The key point of lator of current fluctuations (shot noise), S , are deter-
2
the theory in Ref. 14 is that averaging of expression for mined by the first two cumulants C (n = 1,2) of the
n
the generating function with source field χ induces χ- generating function, C = ( i)n[dnS(χ)/dχn] , i.e.
n χ=0
− − |
dependent probabilities Qχ which differ from Q by an explicitly
s s
exponential factor eiχ(τ) in the probability of tunneling
I =ieS′(χ) (11)
through the considered (left) junction. All the details of χ=0
|
this derivation can be found in the cited work. S =(I I)2 =2e2S′′(χ) , (12)
2 χ=0
Thus, following the method of Ref. 14, we consider − |
eigenvalues of the matrix Zˆ(χ) defined as respectively, where S′(χ) and S′′(χ) stand for the first
and second derivative of S with respect to χ. Obvi-
xΓ− 0 0 0 Γ−
− L↑ R↑ ously, the FCS method gives the possibility to calculate
0 xΓ− 0 0 Γ− all higher current correlationfunctions, S , S , etc.
Zˆ(χ)= 0 − 0L↓ Γ+ 0 ΓR+↓ . (4) Using Eqs. (8)-(12) one finds the follow3ing4expression
− L↑ R↑
 0 0 0 Γ+ Γ+  for the mean value of electric current (we take the units
xΓ−L↑eiχ xΓ−L↓eiχ Γ+L↑eiχ Γ−+L↓eLi↓χ −ΓR↓Σ  with t0 =1):
 
As compared to Mˆ, the matrix Zˆ(χ) includes an addi- exΓ−L↑Γ−L↓(Γ+R↑+Γ+R↓+Γ−R↑+Γ−R↓)
I = . (13)
tional phase factor eiχ, which allows to determine the Γ− Γ− +Γ− Γ−
L↑ R↓ L↓ R↑
4
We recall that the above expression is valid in the low
temperature limit, where x ≪ 1. Similarly, once can 3.5 P R=1
also determine the relevant shot noise S . Since the cor- 2
2
responding formula is relatively long, we present it in 3
3.0 4
Appendix, see Eq.(A12),wherewe alsogivemoredetails
5
on its derivation. Having found the shot noise, one can
F 6
determine the corresponding Fano factor, 2.5
C2 S2 2(Γ+R↑+Γ+R↓)+Γ−R↑+Γ−R↓ 2.0
F = = =
C1 2eI (Γ−R↑+Γ−R↓)
2(Γ− Γ− +Γ− Γ− )(Γ+ +Γ+ +Γ− +Γ− ) 1.5
L↑ R↑ L↓ R↓ R↑ R↓ R↑ R↓ -1.0 -0.5 0.0 0.5 1.0
+
(Γ− +Γ− )(Γ− Γ− +Γ− Γ− ) +
R↑ R↓ L↑ R↓ L↓ R↑ log
2Γ− Γ− (Γ+ +Γ+ +Γ− +Γ− )(Γ− +Γ− )
L↑ L↓ R↑ R↓ R↑ R↓ R↑ R↓
.(14)
− (Γ− Γ− +Γ− Γ− )2 3.5
L↑ R↓ L↓ R↑ =1
AP R
2
In the nonmagnetic case, Γ±L↑ =Γ±L↓ =ΓL and Γ±R↑ = 3.0 3
Γ± = Γ , we obtain the results of Ref. 16, with the 4
loRw↓est twoRcumulants and the Fano factor equal 2.5 5
F 6
C =2xΓ , C =6xΓ , and F =C /C =3. (15) 2.0
1 L 2 L 2 1
Thus, the corresponding shot noise is then super- 1.5
Poissonian, with F = 3. If we take into account the
spindependenceofelectrontunneling,butassumeΓ− = 1.0
Γ+ , then we find from Eq. (15) that the Fano factRoσr is -1.0 -0.5 0.0 0.5 1.0
Rσ +
evenlargerthan3,F >3,foranychoiceofotherparam- log
eters.
One can describe the shot noise and the Fano factor
(15) by a certain number of parameters which quantify FIG.2: Fanofactorintheparallel(top)andantiparallel(bot-
tom) magnetic configurations as afunction of α+. Theother
therelevantasymmetryineachofthetransportchannels.
−
To do this let us define the junction resistance R± parameters are xR =0.3, xL =1, βL =4, α =0.2, and βR
L,Rσ as indicated.
for each level- and spin-channel. The resistance R±
L,Rσ
is inversely proportional to the corresponding tunneling
rate Γ± . Accordingly, we introduce the parameters
L,Rσ Variation of the Fano factor F with the parameter
α+ = R+ /R+ and α− = R− /R− to describe the α+ =R+ /R+ inthe P andAP configurationsis shown
R↑ L↑ R↑ L↑ R↑ L↑
right-left asymmetry, in the spin-up channel associated in Fig. 2 for different values of the parameter β . Two
R
with the high-energy and low-energy dot’s levels. Apart features immediately follow from this figure. First, the
from this, we also define the parameters βR =RR−↓/RR−↑ shot noise and thus also the Fano factor are strongly en-
and β = R− /R− for the spin asymmetry in the cou- hancedwhenα+ <<1,i.e. forR+ <<R+ . Thisisbe-
L L↓ L↑ R↑ L↑
pling of the low-energy dot’s level to the leads. To de- causeaspin-upelectrontunnelingfromthesource(right)
scribeadifference inthe couplingofthe twolevelsofthe electrode to the high-energy level spends relatively long
dot to the right electrode, we introduce the parameter time before tunneling further to the sink(left) electrode,
x defined as x =R− /R+ . Similar parameter is also blocking this way electronic transport via other chan-
R R R↑ R↑
introduced to describe asymmetry of the coupling of the nels. Second,theFanofactorintheparallelconfiguration
two levels to the left electrode, x =R− /R+ . is generally larger than in the antiparallel state. Note,
L L↑ L↑
In case of magnetic electrodes, we also distinguish be- that for β = 1 the parallel and antiparallel configu-
R
tween the parallel (P) and antiparallel (AP) arrange- rations are equivalent (right electrode is then nonmag-
ments of the magnetic moments of both electrodes. For netic). Then, whenβ >1,the Fanofactorinthe paral-
R
definiteness, we define the spin-uporientationas the ori- lelconfigurationis lowerwhile inthe antiparallelstateis
entation of majority spins in the left electrode (i.e., op- higher,whichisinagreementwithearlierobservations.21
posite to magnetizationvectorinthe left electrode), and In turn, dependence of the Fano factor on the parame-
assume that magnetic moment of the right electrode is ters x = R− /R+ is shown in Fig. 3 for both mag-
R R↑ R↑
reversed in the AP configuration. Thus, in the AP con- netic configurations. The noise is super-Poissonian and
figuration the spin-up and spin-down electrons in the the Fano factor is relatively large for x >> 1, i.e. for
R
rightelectrodecorrespondtothespin-minorityandspin- R− >>R+ . Again, the noise is smaller in the antipar-
R↑ R↑
majority electrons, respectively. allel configuration.
5
12 2.0
=1
R P
10 2 1.8
3
8 4 1.6
5
F 6 6 F1.4
+=0.1
4 1.2 0.3
1.0
2 1.0 3.0
10
0 0.8
-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.0 0.05 0.1 0.15 0.2
log x x
R
12
=1 FIG. 4: Dependence of the Fano factor on the temperature
R AP −
10 2 factor x in the P configuration for α = 0.3 and different
3 values of α+. The other parameters are xR = 0.3, xL = 1,
8 4 βR =2, and βL =4.
5
F 6 6
values of the matrix Zˆ (χ), which we define as
s
4
xΓ− 0 0 0 Γ−
2 − L↑ R↑
0 xΓ− 0 0 Γ−
0 Zˆ (χ)= 0 − 0L↓ Γ+ 0 Γ+R↓ .(16)
-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 s − L↑ R↑
 0 0 0 Γ+ Γ+ 
log xR  xΓ− eiχ xΓ− e−iχ Γ+ eiχ Γ−+ eL−↓iχ ΓR↓ 
 L↑ L↓ L↑ L↓ − Σ 
 
IncontrasttoEq.(4),wecountherethehoppingthrough
FIG.3: Fanofactorintheparallel(top)andantiparallel(bot-
tom) configurations as a function of xR for α+ = α− = 1, theleftjunctionofspin-upandspin-downelectrons,cor-
responding to the plus and minus sings in the exponents
xL=1, βL=4, and βR as indicated.
inthe bottomrow,respectively. Thismeansthatwecal-
culate the spin current as a difference of the fluxes of
electrons in the spin-up and spin-down channels.
When the temperature increases, the parameter x in
All the calculations are similar to those in the case of
Eq. (3) also increases, which leads to the temperature
electric current, so we will not repeat the details, but
dependence of the Fano factor. The simple algebraic
present only some results. In the low-temperature limit
method presented above can not be used now. Hence,
(x 0) the first two cumulants can be written in the
we calculated numerically the eigenvalues of the matrix →
form
Zˆ(χ), Eq. (4), and used the lowest eigenvalue λ of the
0
matrix Zˆ(χ) to determine the first two cumulants and xΓ− Γ− (Γ+ +Γ− Γ+ Γ− )
Cs L↑ L↓ R↑ R↑− R↓− R↓ , (17)
thus the Fano factor, F =C2/C1. The dependence of F 1 ≃ Γ− Γ− +Γ− Γ−
L↑ R↓ L↓ R↑
on the temperature-dependent parameter x is presented
inFig.4. Thelow-temperaturelimitoftheFanofactorF Cs 2xΓ−L↑Γ−L↓ 1(Γ+ +Γ− +Γ+ +Γ− )
corresponds to x 0. The magnitude of Fano factor es- 2 ≃ Γ− Γ− +Γ− Γ− 2 R↓ R↓ R↑ R↑
→ L↑ R↓ L↓ R↑ (cid:26)
sentially decreases with increasing temperature. This is
Γ+ Γ+
relatedtode-blockingoftheconductionchannelthrough +(Γ+ +Γ− Γ+ Γ− ) R↓− R↑
the low-energylevelE . Note,the systemmaygotothe R↓ R↓− R↑− R↑ Γ− +Γ−
− R↑ R↓
sub-Poissonianregime with increasing temperature.
Γ+ Γ+
+ Γ− Γ− Γ− Γ− +(Γ− Γ− +Γ− Γ− ) R↓− R↑
" L↓ R↓− L↑ R↑ L↓ R↓ L↑ R↑ Γ−R↑+Γ−R↓#
Γ+ +Γ− Γ+ Γ−
IV. SPIN CURRENT NOISE R↓ R↓− R↑− R↑ Γ− Γ− (Γ− +Γ− )
× Γ− Γ− +Γ− Γ− − L↑ L↓ R↑ R↓
L↑ R↓ L↓ R↑
TheFCSmethodforcalculationofcurrentandcurrent Γ+ +Γ− Γ+ Γ− 2
R↓ R↓− R↑− R↑ . (18)
nanodisespcianncbuerreeanstilnyogiseen.eTraolidzeodthtoisswtuedcyonthsiedesrpitnhecuerigreennt- × Γ−L↑Γ−R↓+Γ−L↓Γ−R↑ ! 

6
V. EXPERIMENTAL DATA ON SHOT NOISE
0.65
IN MAGNETIC TUNNEL JUNCTIONS
P
0.6
Experimental measurements of shot noise have been
performed in magnetic tunnel junctions with molecular
/I 0.55 perylene-teracarboxylic dianhydride (PTCDA) organic
s
I
barriers. The molecularlayerwasup to 5 nmthick. The
shot noise measurements have been done at 0.3 K and
0.5
for the bias up to 10 mV. Detailed description of the
preparation method of the tunnel junctions and of the
0.45 experimental technique used to measure shot noise have
1 2 3 4 5 6 been published elsewhere.38
R Representative experimental results are shown in
Fig. 7. More experimental data can be found in Ref. 21.
We have measured not only the shot noise and the cor-
0.65
responding Fano factor, but also the tunneling magne-
AP
0.6 toresistance (TMR). The latter is defined as the rela-
tive difference in the junction resistances in antiparallel
0.55
and parallel magnetic configurations. As one can see in
/I 0.5 Fig. 7a, the organic magnetic tunnel junctions (OMTJs)
s
I show TMR ratio ranging between 10% and 40%, with
0.45 the lowest value of TMR observed in the PTCDA-free
samples, i.e. in the sample with no PTCDA layer, but
0.4
0.35
1 2 3 4 5 6
R =2
10 R P
3
4
FIG. 5: Spin polarization of current Is/I as a function for 8 5
βR in P (top) and AP (bottom) configurations. The other 6
parameters are xR = 0.3, xL = 1, α+ = 0.3, α− = 0.2, and Fs
βL =4. 6
4
-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6
log x
R
Accordingly, the mean spin current can be calculated
as Is =C1s, while the spin current noise as S2s =2C2s. In -1
Fig.5 wepresentthe spinpolarizationofelectriccurrent AP
Is/I = C1s/C1 in the P and AP configurations. As we -2
see, the polarization strongly depends on the parameter
βR describing asymmetry between the spin-up and spin- -3
down channels. s
F
-4 =2
R
3
4
In Fig. 6 we show the calculated spin Fano factor, de- -5
5
fined as F = Cs/Cs, for both parallel and antiparal-
s 2 1 6
lel magnetic configurations. These Fano factors are pre- -6
sented as a function of x . In the parallel configuration -1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6
R
the FanofactorincreaseswithincreasingxR while inthe log xR
antiparallel state it decreases with increasing x . Note,
R
the spin Fano factor is positive in the P configuration
and negative in the AP state. This difference is associ- FIG.6: SpinFanofactorintheparallel(top)andantiparallel
ated with different signs of the spin current in the two (bottom)configurationsasafunctionofxR forα+ =α− =1,
configurations. xL=1, βL =4, and βR as indicated.
7
of the junction resistance. As already reported earlier,
(a) the Fano factors in the AP state are smaller than in the
100
P one.
In order to account for the experimental observation

%) of the shot noise in OMTJs, we have proposed21 a theo-
( 10 reticalmodelbasedontunnelingthroughatwo-levelsys-
R tems, likethat presentedabove. Takinginto accountthe
M factthatthe super-Poissonianshotnoiseappearsmainly
T TMR at larger voltages, such a two-level system may be at-
tributed to interfacial states of the PTCDA molecules
1
in a biased junction. Indeed, the experimental results
1000 10000 100000 can be quite well explained qualitatively and also quan-
R( ) titativelyinterms ofthe modelbasedonspin-dependent
(b) electron tunneling through an interacting two-level sys-
tem, described in detail in the preceding sections. In
2 orderto qualitatively account for the experimentally ob-
servedsituationwiththeFanofactorintheAPstate(on
the average,we observe F =1.5) being smaller than the
O Fano factor in the P state (1.7), we did numerical calcu-
N
A1 lations based on the model presented above, see Fig. 2,
F and from fitting to the experimental data we evaluated
P state the parameters that reproduce the Fano factors in both
AP state
configurations.
0
100 1000 10000 100000
R ( ) VI. DISCUSSIONS
FIG. 7: (a) Tunneling magnetoresistance of OMTJs with Theresultsofourcalculationsareinqualitativeagree-
PTCDA barriers with different PTCDA thickness ranging
ment with the physical interpretation given in Ref. 16.
from 0 nm (1.2 nm of AlOx buffer layers only) to 5 nm and Indeed,consideringthe simplesttwo-levelmodel(Fig.1)
plotted as a function of device resistance (low bias junction
itwasconcludedthatthegeneratingfunctionS(χ)canbe
resistance in the P state). Measurements have been done at
presented as a sum of independent Poissonian processes
10 K and with applied biasof 1 mV (b)Fano factor in theP
of transferring ne charges with probability of (1/2)n
and AP states as a function of OMTJs resistance measured
at T=0.3 K and averaged for the bias range about 3-10 mV. with n = 1 to . In turn, the process of transfer-
∞
Dashed lines are guides for theeye. ring ne charges with large n during one cycle is possi-
ble because the tunneling to the left lead from the lower
level is strongly suppressed by the temperature factor
with1.2nmAlOx tunnelbarrieronly. The experimental (1 f(ε−)). In other words, several electrons can be
−
values of TMR are in agreement with the model calcu- quickly transferred through the upper level till the cycle
lations for the parameter β 1.6.21 Note, the TMR is stopped by an electron at the lower level. This is a
R
≃
ratio in Fig. 7 is shown as a function of low bias junc- super-Poissonianprocess,andtheFanofactorisequalto
tion resistance in the P state. Previous measurements 3.
indicated approximately exponential dependence of the In our calculations we used the model of QDs with
junction resistance on the PTCDA thickness.21 twoenergylevels,when one ofthem is locatedbelow the
The measurements of shot noise reveal super- Fermi level of left electrode ε , and the other one is
FL
Poissonian tunneling statistics, with the Fano factor between the Fermi levels of left and right electrodes. In
ranging between 1.5 and 2 when the barrier includes reality the QD or molecule can have many energy levels,
the PTDCA layer (see also the preliminary report21). with part of them situated below ε and another part
FL
The control sample (i.e., the sample without PTCDA between ε and ε . It is rather obvious that this is
FL FR
but with 1.2 nm AlO tunnel barrier only) shows the not so important for the mechanism of super-Poissonian
x
lowest resistance and also the lowest Fano factor of the noise related to blocking of electron transport through
orderofF=0.3(whichcorrespondstothesub-Poissonian thelowenergylevel. Generalizationtothemultilevelsys-
statistics), as expected for disordered metals. Hence, we temwithN upperandN lowerlevels(bunchedintwo
+ −
concludethatthesuper-Poissonianshotnoiseobservedin blockswiththesametunnelingprobabilityineachblock)
OMTJs is most likely associatedwith tunneling through can change the statistics, so that F = (1+p)/(1 p),
−
discrete states. The measuredFano factorsinbothmag- where p = N /(N +N ). In particular, for p = 1/2
+ + −
netic configurationsareshowninFig.7bfor3-10mVbi- we obtain again F = 3. In this multilevel model, one
asedjunctions. Thedataarealsopresentedasafunction can also get F = 2 with p = 1/3, which corresponds to
8
N = 2N (e.g., lower level is twice degenerate and the (S2013/MIT-2850). J.P.C. acknowledges support from
− +
upper oneisnondegenerate). Inthe caseofnonmagnetic the Fundacion Seneca (Region de Murcia) posdoctoral
system, each of the levels is spin degenerate. Thus, as- fellowship (19791/PD/15)
suming equal tunneling probabilities for the spin-up and
spin-down electrons, one would get p=1/2 and F =3.
We also assumed that the tunneling probabilities are Appendix A: Calculation of the shot noise
differentfor the lowerandupper levels. This changeses-
sentiallytheresultfortheFanofactorbecausetheproba- Using the expression for λ˜(χ)
bilityoftransferringneelectronsincludesnowtheweight
factoroftheratio(Γ−/Γ )n sincetheprobabilityoftun-
R R λ˜(χ)= b b2 c (A1)
nelingofasingleelectronfromtherightleadtotheupper − ± −
level is not equal to 1/2 anymore. In other words, the p
we find
transferofelectronsthroughtheupperlevelcanbenotso
quickduetoalowerprobabilityofthecorrespondingtun- 2bb′ c′
neling,andthispartiallysuppressesthesuper-Poissonian λ˜′ = b′ − , (A2)
− ± 2√b2 c
processasasumofPoissonianprocesseswiththetransfer −
2(b′)2+2bb′′ c′′ (2bb′ c′)2
of multiple charges. λ˜′′ = b′′ − − .(A3)
Within this approach one can also consider electron − ± 2√b2 c ∓ 4(b2 c)3/2
− −
tunnelingthroughachainofmoleculesinrelativelythick
In the limit of χ 0 we get
junctions. Now the energy levels of different molecules →
are not exactly at the same energy. First, because there
Γ Γ− +Γ Γ−
is a potentialslope within the junction, which shifts cor- b L↑ R↓ L↓ R↑, (A4)
respondinglyalltheenergylevelsinthejunction. Second, ≃ 2(Γ− +Γ− )
R↑ R↓
due to inevitable disorder, there exist some fluctuations
c 0, (A5)
of potential. This means that the intermolecular tunnel- ≃
i(Γ Γ− +Γ Γ− )
ing can be possible only due to emission or absorption b′ L↑ R↑ L↓ R↓
of appropriatephonons. In this situation one canexpect ≃− 2(Γ− +Γ− )2
R↑ R↓
thatthereisonlyone’optimal’pathoftheelectrontrans-
(Γ +Γ +Γ− +Γ− ), (A6)
fer through the chain of molecules, which uses a chosen × R↑ R↓ R↑ R↓
number of the energy levels. The probability of charge (Γ− Γ− +Γ− Γ− )(Γ− +Γ− +Γ+ +Γ+ )
b′′ = L↑ R↑ L↓ R↓ R↑ R↓ R↑ R↓
transferthroughotherpathesisexponentiallysmallsince (Γ− +Γ− )2
it requires substantial energy change at each intermolec- R↑ R↓
ular tunneling. Hence, we come back to a Poissonian 1 Γ+ +Γ+
R↑ R↓
+ , (A7)
process of the transfer of a single charge through the × 2 Γ−R↑+Γ−R↓!
molecular chain. In this case we naturally obtain F =1.
iΓ Γ (Γ +Γ +Γ− +Γ− )
It is also worth noting, that the super-Poissonian c′ L↑ L↓ R↑ R↓ R↑ R↓ , (A8)
noise can appear due to other physical mechanisms as ≃− Γ− +Γ−
R↑ R↓
well, for example due to electron-phonon or electron-
Γ Γ (Γ +Γ +Γ− +Γ− )
electroninteractions.2However,themechanismproposed c′′ L↑ L↓ R↑ R↓ R↑ R↓
by Belzig16 andbasedontunneling throughtwo ormore ≃ (Γ− +Γ− )2
R↑ R↓
discretelevelsisthemostappropriateoneinourcase. In- [2(Γ +Γ )+Γ− +Γ− ]. (A9)
deed,theassumptionoftunnelingthroughdiscretelevels × R↑ R↓ R↑ R↓
(with one low-energy level) is physically reasonable and
Then we obtain the cumulants
justified. Moreover,thismodelexplainsthepossibilityof
a rather strong enhancement of the Fano factor, and is xΓ Γ (Γ +Γ +Γ− +Γ− )
alsoable to accountfor the experimentalobservationsin C1 =iS′(χ)χ=0 = L↑ L↓Γ RΓ↑− +RΓ↓ Γ−R↑ R↓ , (A10)
the studied system. L↑ R↓ L↓ R↑
C =S′′(χ) =xΓ Γ (Γ +Γ +Γ− +Γ− )
Acknowledgments 2 χ=0 L↑ L↓ R↑ R↑ R↑ R↓
2(Γ +Γ )+Γ− +Γ−
R↑ R↓ R↑ R↓
This work was supported by the National Science ×"(Γ−R↑+Γ−R↓)(ΓL↑Γ−R↓+ΓL↓Γ−R↑)
Center in Poland as a research project No. DEC-
2(Γ Γ− +Γ Γ− )(Γ +Γ +Γ− +Γ− )
2012/06/M/ST3/00042. We also gratefully acknowl- + L↑ R↑ L↓ R↓ R↑ R↑ R↑ R↓
edge support by UAM-Santander collaborative project (Γ− +Γ− )(Γ Γ− +Γ Γ− )2
R↑ R↓ L↑ R↓ L↓ R↑
(2015/ASIA/04) as well as by the Spanish MINECO
2Γ Γ (Γ +Γ +Γ− +Γ− )(Γ− +Γ− )
(MAT2012-32743andMAT2015-66000-P)grantsandthe L↑ L↓ R↑ R↑ R↑ R↓ R↑ R↓ (A11)
ComunidaddeMadridthroughNANOFRONTMAG-CM − (ΓL↑Γ−R↓+ΓL↓Γ−R↑)3 #
9
and the explicit formula for shot noise S 2Γ− Γ− (Γ+ +Γ+ +Γ− +Γ− )(Γ− +Γ− )
2 L↑ L↓ R↑ R↓ R↑ R↓ R↑ R↓ . (A12)
S =2e2xΓ− Γ− Γ+ +Γ+ +Γ− +Γ− − (Γ−L↑Γ−R↓+Γ−L↓Γ−R↑)3 #
2 L↑ L↓ R↑ R↓ R↑ R↓
(cid:0) 2(Γ+ +Γ+ )+Γ−(cid:1)+Γ−
R↑ R↓ R↑ R↓
×"(Γ−R↑+Γ−R↓)(Γ−L↑Γ−R↓+Γ−L↓Γ−R↑)
2(Γ− Γ− +Γ− Γ− )(Γ+ +Γ+ +Γ− +Γ− )
L↑ R↑ L↓ R↓ R↑ R↓ R↑ R↓
+
(Γ− +Γ− )(Γ− Γ− +Γ− Γ− )2
R↑ R↓ L↑ R↓ L↓ R↑
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