Negative differential conductance and magnetoresistance oscillations due to spin accumulation in ferromagnetic double-island devices

0.16 MB·English

by  Ireneusz Weymann

Negative differential conductance and magnetoresistance oscillations due to spin
accumulation in ferromagnetic double-island devices
Ireneusz Weymann1,∗ and J´ozef Barna´s1,2
1Department of Physics, Adam Mickiewicz University, 61-614 Poznan´, Poland
2Institute of Molecular Physics, Polish Academy of Sciences, 60-179 Poznan´, Poland
6 (Dated: February 6, 2008)
0
0 Spin-dependentelectronic transport inmagnetic double-island devicesisconsidered theoretically
2 inthesequentialtunnelingregime. Electriccurrentandtunnelmagnetoresistance areanalyzedasa
functionofthebiasvoltageandspinrelaxationtimeintheislands. Itisshownthattheinterplayof
n
spinaccumulationontheislandsandchargingeffectsleadstoperiodicmodificationofthedifferential
a
J conductance and tunnel magnetoresistance. For a sufficiently long spin relaxation time, the mod-
ulations are associated with periodic oscillations of the sign of both the tunnel magnetoresistance
4
and differential conductance.

]
l PACSnumbers: 72.25.-b;73.23.Hk;85.75.-d
l
a
h
- Introduction – Spin-dependent transport in single- nanobridges, electronic transport between the external
s electron devices is attracting currently much attention leadsoccursasaresultofconsecutiveelectronjumps via
e
m from both fundamental and application points of view metallic grains, and the corresponding electronic paths
[1,2,3,4,5,6,7,8]. Ithasbeenshownthattheinterplay can include either one, two, or more grains. Transport
.
t of single-electron charging effects and spin dependence through a single grain and the associated TMR have al-
a
of tunneling processes (caused by ferromagnetism of ex- ready been analyzed [2], but the results cannot describe
m
ternal and/or central electrodes) in single-electron tran- the above mentioned data obtained on magnetic multi-
-
d sistors gives rise to novel interesting phenomena, such grain nanobridges. However, the experimental setup de-
n as periodic modulation of the tunnel magnetoresistance scribed above can be modelled by assuming only two
o (TMR)withincreasingeitherbiasorgatevoltages,quasi- grains involved in the transport processes [9]. Although
c oscillatorybehaviorofthe spinaccumulationonthe cen- the Coulomb steps have been accounted for by the ex-
[
tral electrode (referred to as an island in the following), istingtheoreticalmodelsbasedondiscretechargingwith

1 enhancement of TMR in the Coulomb blockade regime, single electrons, the origin of negative differential con-
v andothers. Some ofthe theoreticallypredictedphenom- ductance (NDC) as well as the oscillations in TMR with
9 ena have already been confirmed experimentally. alternatingsignremainsunclear. Therefore,thisproblem
6
Recently, transport characteristics of granular mag- is considered in this paper in more details.
0
netic nanobridges connecting two external magnetic Following the above discussion, we consider transport
1
0 and/ornonmagnetic electrodes were investigatedexperi- through a double-island device consisting of two metal-
6 mentally [9, 10, 11]. The nanobridges consisted of mag- lic grains attached to two external leads. The two is-
0 netic grains distributed in a nonconducting matrix. The lands are separated from each other and also from the
/ correspondingcurrent-voltage(I-V)curvesrevealedchar- external leads by tunnel barriers. In a general case all
t
a acteristicstepsduetodiscretechargingofthegrainswith the four metallic components ofthe device (two external
m
single electrons (Coulomb steps). Apart from this, oscil- electrodes and two islands) can be ferromagnetic [1, 5],
- lations ofthe TMReffect (associatedwiththe transition but in the following we will consider the situation with
d
from one magnetic configuration to the other) with the oneexternalleadbeingnonmagneticandtheotherthree
n
bias voltagehavealso beenobserved[11]. Moreover,two components being ferromagnetic. We also restrict our
o
c additional features of the transport characteristics were considerations to collinear [parallel (P) and antiparallel
: found, whose physical mechanism and origin needs fur- (AP)] alignments of the magnetic moments, as indicated
v
ther considerations. First, the differential conductance in the inset of Fig. 1(d).
i
X wasfoundtochangesignperiodicallywithincreasingbias The key role in our analysis is played by the nonequi-
r voltage [9, 10]. Second, the TMR was shown to oscillate libriumspinaccumulation,whichoccursforasufficiently
a
betweenpositiveandnegativevalueswithincreasingbias long spin relaxation time on the islands. Strictly speak-
voltage. Physicaloriginoftheseoscillations,however,re- ing, the accumulation may take place when the spin re-
mained unclear, although the role of spin accumulation laxation time is significantly longer than the time be-
was invoked following the results obtained for a single- tween successive tunneling processes. Assuming this is
island system [2]. the case, we show that the interplay of charging effects
For small lateral and vertical dimensions of the andspinaccumulationgivesrisetobotheffectsdescribed
above, i.e., to the NDC and periodic change of the sign
of TMR with increasing bias voltage. This behavior of
transport characteristics accounts for the experimental
∗Electronicaddress: [email protected] observations. We also show that the presence of NDC
2
depends onthe magnetic configurationof the device and neling rate from the left electrode to the first island is
by switching from one configuration to the other one then given by the formula
may change sign of the differential conductance (eg., by
1 ∆Eσ (n ,n )
an external magnetic field). This behavior may be of Γσ (n ,n )= L1 1 2 , (2)
L1 1 2 e2Rσ exp[∆Eσ (n ,n )/k T]−1
someinterestforpossiblefutureapplicationsinspintron- L L1 1 2 B
ics/magnetoelectronics devices. whereRσ isthespin-dependentresistanceoftheleftbar-
L
Model – The systemadaptedto modelthe phenomena rierand∆Eσ (n ,n )describesthechangeintheelectro-
L1 1 2
discussed above consists of the left and right leads and staticenergyofthe systemcausedbythe respectivetun-
two central islands, separated from each other and from nelingevent,∆Eσ (n ,n )=E(n +1,n )−E(n ,n )+
L1 1 2 1 2 1 2
the leads by tunnel barriers, as shown schematically in eV + ∆Eσ . (According to our notation the electron
L F1
the inset of Fig. 1(d). The bias voltage is applied to charge is −e with e>0.)
the external leads – VL to the left one and VR to the For the following discussion it is convenient to distin-
right one. In a general case one can attach capacitively guish between spin projection on the global quantiza-
gate voltages to both islands. However, to model the tion axis (σ =↑,↓) and on the local quantization axis
experimentalresultsdescribedabove,weneglectthegate (σ = +,−, with σ = + and σ = − corresponding to
voltagesinthefollowing. Weassumethatthesizeofeach spin-majority and spin-minority electrons, respectively).
island is sufficiently small to have the charging energies The spin asymmetry in tunneling processes follows from
oftheislandssignificantlylargerthanthethermalenergy spin-dependentdensityofstatesandspin-dependenttun-
kBT, but still large enough to neglect size quantization. neling matrix elements. Let us define the parameters
Apart from this, we assume that the barrier resistances β =D+/D− for the leads (r =L,R) and β =D+/D−
r r r j Ij Ij
aremuchlargerthanthequantumresistance,Rr ≫h/e2 fortheislands(j =1,2),whereD+(−) andD+(−) arethe
(r =L,M,R). Thesystemistheninawelldefinedcharge r Ij
appropriate densities of states for spin-majority (spin-
statedescribedbyn andn excesselectronsonthe first
1 2 minority) electrons in the leads (r = L,R) and islands
(left) and second (right) islands, respectively, and the
(j = 1,2), respectively. Thus, the Fermi level shifts due
electrostaticenergyofthesystemisgivenbytheformula
to spin accumulation on the islands obey the condition
[12, 13]
∆E+/∆E− = −β for the first (j = 1) and second
Fj Fj j
2 2 (j =2) islands.
C V C V
E(n ,n ) = E n − L L +E n − R R The spin asymmetry of the barrier resistances can be
1 2 C1(cid:18) 1 e (cid:19) C2(cid:18) 2 e (cid:19)
describedbytheparametersα =R↑/R↓forr =L,M,R.
r r r
+2E n − CLVL n − CRVR , (1) Assuming a constant (independent of energy) density of
CM(cid:18) 1 e (cid:19)(cid:18) 2 e (cid:19) statesandconstant(independentofenergyandspin)ma-
trix elements, one can write α = D↓D↓/D↑D↑ for the
where C is the capacitance of the left (right) junc- r r Ij r Ij
L(R) left (r = L, j = 1) and right (r = R, j = 2) barri-
tion, E and E denote the charging energies of the
C1 C2 ers, and α = D↓ D↓ /D↑D↑ for the central (r = M)
two islands, E = e2/(2C )[1 − C2/(C C )]−1, r I1 I2 I1 I2
C1(C2) 1(2) M 1 2 barrier. When the local and global spin quantization
and E is the energy of electrostatic coupling between
CM axescoincide(parallelconfiguration),thenonecanwrite
the islands, E = e2/(2C )(C C /C2 − 1)−1, with
CM M 1 2 M α = 1/(β β ) for the left (r = L, j = 1) and right
C being the total capacitance of the first (second) r r j
1(2) (r = R, j = 2) barriers, and α = 1/(β β ) for the cen-
island, C = C +C , and C denoting the ca- r 1 2
1(2) L(R) M M tral (r =M) barrier. The above formulas are also appli-
pacitance of the middle junction (the one between the
cabletothesituationwithmagneticmomentofaleador
islands). Thisallowsustoemploythequasi-classicalthe-
an island reversed (antiparallel configuration), but with
ory based on the master equation and the Fermi golden
the corresponding β replaced by 1/β.
ruleforthetunnelingrates[14]. Suchapproachdescribes
The probability P(n ,n ) that the system is in a
welltransportinthesequentialtunnelingregimeandcor- 1 2
charge state (n ,n ) can be determined in a recursive
responds to taking into account only the first-order tun- 1 2
way from the appropriate steady-state master equation
neling processes, which are exponentially suppressed in
[13]. The electric current flowing through the left junc-
theCoulombblockaderegimebutgivethedominantcon-
tion is then given by
tributiontoelectriccurrentwhenthe appliedvoltageex-
ceedsacertainthreshold. Moreover,wetakeintoaccount I =−e [Γσ (n ,n )−Γσ (n ,n )]P(n ,n ).(3)
L L1 1 2 1L 1 2 1 2
only non-spin-flip (spin-conserving) tunneling processes
Xσ nX1,n2
through the barriers.
TheassociatedshiftsoftheFermilevelcanbecalculated
WhenabiasvoltageV =V −V isappliedtothesys-
L R
in a self-consistent way from the relations [15]
temandspinrelaxationtimeontheislandsissufficiently
loonnge,acahniosnlaenqdu.iliLberituumsmdeangonteetitchmeocmorernestpcoanndainccgusmhuifltatoef 1 Iσ −Iσ − DIσ1(2)ΩI1(2)∆Eσ =0, (4)
e M(R) L(M) τ F1(2)
theFermilevelforelectronswithspin-σonthej-thisland (cid:16) (cid:17) sf,1(2)
by∆Eσ . Whenintheinitialstatethereweren andn where Ω is the volume of the island j, τ denotes the
Fj 1 2 Ij sf,j
excess electrons on the islands, the spin-dependent tun- spin-flip relaxationtime in the j-th island, and Iσ is the
r
3

V) 30 a D E+ lel and antiparallel configurations are defined. This sys-
me 20 F1 tem geometry corresponds to the situation studied ex-
( perimentally in Ref. [9]. The transport characteristics
vel shift -11000 D EF+2 panartiapllaerlallel httioamvseep.ibnTeeahncecocuubmrtruaeilnnattei-doinnfdo)urfcoterhdethslheimifstipstinoo-ffmtlohanejogFriestprymineilreleeclvtarexolan(tsdiouinne
e
mi l-20 bothislandsareshowninFig.1(a)forthePandAPcon-
Fer--4300 D EF+1 D EF+2 fifrgoumrartiigohntstaonldeffto).r Aponsiotnivzeerboiasspivnolatcacguem(uellaectitornonosccfluorws
in both magnetic configurations. In the P configuration
b
0.5 the shift of the Fermi level for spin-majority electrons is
negativeforbothislands,whereasintheantiparallelcon-
0.4
parallel figuration it is positive for the first island and negative
A) 0.3 antiparallel for the second one. Such a behavior can be accounted
n
( for by taking into account spin asymmetries of the tun-
I 0.2
neling processes through all the three barriers, similarly
0.1 as it was done in the case of a double-barrier system [2].
The Coulomb steps in the P and AP configurations
0.0
are significantly different [see Fig. 1(b)]. There are two
1.2 c parallel
reasons of this difference. First, the overall resistance
1.0 antiparallel
of a given barrier depends on the relative orientation of
)
V0.8
themagneticmomentsofadjacentferromagneticcompo-
/
A
n0.6 nents of the device. Second, the spin accumulations in
V (0.4 thePandAPconfigurationsarealsodifferent. Thelatter
d0.2 fact is of particular importance for the present analysis.
/
I
d0.0 ThedifferentialconductancecorrespondingtotheI−V
curvesfromFig.1(b)isshowninFig.1(c). InbothPand
-0.2
AP configurations the differential conductance changes
1.0 d
sign periodically with increasing bias. However,the bias
voltagerangeofNDC for the P configurationis different
0.5 LL RR from that for the AP one. The corresponding phase dif-
ference varies with the bias voltage. Furthermore, NDC
R
M 0.0 ismorepronouncedintheAPconfigurationanditsabso-
T lute magnitude increaseswith increasingbias voltage,as
shown in Fig. 1(c). This behavior of NDC is consistent
-0.5
with experimental observation reported in Ref. 9.
The TMR effect associated with the transition from
0.0 0.5 1.0 1.5 2.0
antiparallel to parallel configurations is visualized in
V (V)

FIG.1: Shiftsof theFermilevelsforspin-majority electrons
(a) and currents (b) in the P and AP configurations; differ- 1.0 antiparallel tsf = 10-9 s
t = 10-7 s
ential conductance in the AP configuration (c); and TMR sf
0.8 t = 10-4 s
(d) as a function of the bias voltage for τsf,1 = τsf,2 → ∞. ) sf
The other parameters are: T = 140 K, CL = 0.45 aF, V 0.6
CM = 0.2 aF, CR = 0.35 aF, β1 = β2 = βR = 0.2, and nA/
βL = 1. The total barrier resistances in the P configuration V ( 0.4
are: RL/3500=RM =RR =1MΩ. IntheAPconfiguration, d
RrA,P↑ = RrA,P↓ = (RrP,↑RrP,↓)1/2, for r = M,R. The parameters dI/ 0.2
correspond to theexperimental ones taken from Ref. 9. 0.0
limit of long
-0.2 spin relaxation limit of short spin relaxation
0.0 0.5 1.0 1.5 2.0
currentflowingthroughthebarrierr(r =L,M,R)inthe V (V)
spin channel σ.
Results and discussion – In Figure 1 we show trans- FIG.2: (coloronline)Differentialconductanceintheantipar-
port characteristicsof a device whose two islands as well allelconfigurationasafunctionofthebiasvoltagecalculated
as the right electrode are made of the same ferromag- for different values of the spin relaxation time τsf,1 =τsf,2 =
netic metal, whereas the left electrode is nonmagnetic, τsf and for DI+1ΩI1 = DI+2ΩI2 = 1000/eV. The other parame-
asshowninthe insetofFig.1(d), wherealsobothparal- ters are thesame as in Fig. 1.
4

0.20 effectsatlargervoltages. Thisisbecauseatsomevoltage
t = 10-9 s
0.15 limit of short tsf = 10-7 s there is a crossover from the fast to slow spin relaxation
0.10 spin relaxation tssff = 10-6 s limits. In fact such a behavior is consistent with experi-
t = 10-4 s mental data presented in Ref. 9.
sf
0.05
ThedisappearanceofNDCwithdecreasingspinrelax-
R
M0.00 ationtimeτsf isshownexplicitlyinFig.2,wherethebias
T dependence of differential conductance is presented for
-0.05
different values of the spin relaxation time. This figure
-0.10
clearlyshowsthatNDC disappearswhenspinrelaxation
-0.15 limit of long time decreases, in agreement with the above discussion.
spin relaxation
-0.20 In the limit of fast spin relaxation time, the differential
0.0 0.5 1.0 1.5 2.0
conductanceispositive,althoughitsperiodicmodulation
V (V)
still remains.
Similarly,periodic oscillationsofthe signofTMRalso
FIG. 3: (color online) The TMR effect as a function of the
disappear with decreasing spin relaxation time τ . This
bias voltage calculated for different values of spin relaxation sf
time τsf,1 = τsf,2 = τsf and for DI+1ΩI1 = DI+2ΩI2 = 1000/eV. behavior is shown in Fig. 3, where the bias dependence
The other parameters are the same as in Fig. 1. of TMR is shown for several values of τsf. First, the
transitions to negative TMR disappear with decreasing
τ . The TMR becomes then positive, although some
sf
Fig. 1(d). The TMR is periodically modulated with in- periodic modulations survive. Second, the phase of the
creasingbiasvoltage. Theinitialphaseoftheoscillations modulations shifts by about π when the spin relaxation
depends on the system parameters and can change from varies from fast to slow limits.
negative to positive. Moreover, the modulations are as- In conclusion, we have analyzed transport through
sociatedwithaperiodicchangeofTMRbetweenpositive double-island device and shown that the negative dif-
and negative values. ferential conductance measured experimentally is due to
Oscillations of the sign of TMR and differential con- nonequilibrium spin accumulation in the islands. Fur-
ductance result from spin accumulations in both islands thermore,spinaccumulationmayalsoleadtooscillations
and are absent in the limit of fast spin relaxation (no inTMRbetweennegativeandpositivevalues. Theeffect
spinaccumulation)onthe islands. The resultspresented ofNDCoccursinbothconfigurationsandtransitionfrom
in Fig. 1 have been calculated for the long spin relax- one configuration to the other may result in transition
ation limit. In such a limit some spin accumulation may frompositive to negativedifferentialconductance,which
occur even for a very small current flowing through the maybeofsomeimportanceforapplicationsinspintronics
system, giving rise to NDC and oscillations of the TMR devices.
sign for small bias voltages. However, both effects dis- The work was supported by the Polish State Com-
appearwhen the spin relaxationtime is shorterthanthe mittee for Scientific Research through the projects
time between successive tunneling events. Thus, for a PBZ/KBN/044/P03/2001 and 2 P03B 116 25, and by
finite relaxation time, one may expect no NDC and no EU through RTNNANO contract No MRTN-CT-2003-
TMRoscillationsforsmallvoltagesandanonsetofthese 504574.
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