Pressure-Induced $0-\pi$ Transitions and Supercurrent Crossover in Antiferromagnetic Weak Links

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by  H. Enoksen

Pressure-Induced0−πTransitionsandSupercurrentCrossoverinAntiferromagneticWeakLinks
Henrik Enoksen,1 Jacob Linder,1 and Asle Sudbø1
1DepartmentofPhysics,NorwegianUniversityofScienceandTechnology,N-7491Trondheim,Norway
(Dated:January17,2014)
We compute self-consistently the Josephson current in a superconductor-antiferromagnet-superconductor
junction using a lattice model, focusing on 0−π transitions occurring when the width of the antiferromag-
neticregionchangesfromaneventoanoddnumberoflatticesites.Previousstudiespredicted0−πtransitions
whenalternatingbetweenanevenandanoddnumberofsitesforsufficientlystrongantiferromagneticorder.
Westudynumericallythemagnitudeofthethresholdvalueforthistooccur,andalsoexplainthephysicsbehind
its existence in terms of the phase-shifts picked up by the quasiparticles constituting the supercurrent in the
4
antiferromagnet. Moreover,weshowthatthisthresholdvalueallowsforpressure-induced0−πtransitionsby
1
destroyingtheantiferromagneticnestingpropertiesoftheFermisurface,aphenomenonwhichhasnocounter-
0
partinferromagneticJosephsonjunctions,offeringanewwaytotunethequantumgroundstateofaJosephson
2
junctionwithouttheneedofmultiplesamples.

n
a PACSnumbers:73.20.-r73.40.Gk74.25.F-85.25.Cp
J

5
1 Introduction. Thestudyoftheinterplaybetweensupercon- antiferromagnetism in the first place. In this way, the pres-
ductivityandmagnetismhasbeenofconsiderableinterestin surecontrolsthemagnitudeofthestaggeredorderparameter

] condensed matter physics over the last decades. Phenomena whichtriggersa0−πtransitiononceitdropsbelowtheafore-
n
o suchasthe0−π-transition1inferromagneticJosephsonjunc- mentionedthresholdvalue. Thiseffecthasnocounterpartin
c tions has received much attention both from a fundamental conventional SFS junctions, since ferromagnetism does not
- quantumphysicspointofviewinadditiontobeingsuggested rely on Fermi-surface nesting. The antiferromagnetic order
r
p as a potential basis for qubits.2. While most of the focus in thus offers a novel mechanism for controlling the quantum
u the above context has been on ferromagnetic (F) order, an- mechanicalgroundstateofaJosephsonjunction.
s tiferromagnetic (AF) Josephson junctions are also of funda- Theory. OurapproachcloselyfollowsthatofRef. 10. The
.
t mentalinterest, duetothecloserelationshipbetweenthesu- system in question consists of an itinerant antiferromagnet
a
m perconducting (S) phase and the antiferromagnetic phase in sandwiched between two conventional s-wave superconduc-
forinstancehigh-temperaturecuprateandiron-pnictidesuper- tors. Theinterfacesarecutinthe(110)directionandarecon-
-
d conductors. Superconductivity and antiferromagnetism spin- sidered transparent. We model the antiferromagnetic region
n density wave states may even coexist in the superconduct- asconsistingoftwosquaresublatticesshowninFig.1,where
o ing pnictides.3 Similar to SFS junctions, antiferromagnetic theA-andB-latticehaveoppositelypreferredspindirections.
c
Josephson junctions (SAFS) have been predicted to display Themean-fieldHamiltonianthenreads(seeforinstanceRef.
[
0−π-transitions4. However, for SAFS these transitions dis- 4)

1 play a high sensitivity to the exact number of atomic layers (cid:16) (cid:17)
v (evenvs. oddnumber)intheantiferromagnet. Ref.5reported Hˆ =−t ∑ cˆ†iσcˆjσ+∑ ∆icˆ†i↑cˆ†i↓+H.c. −µ∑cˆ†iσcˆiσ
3
that an antiferromagnetic Josephson junction is in a π-state (cid:104)ij(cid:105)σ i iσ
76 for an odd number of layers, while it is in the 0-state for an +∑m (cid:16)cˆ†cˆ −cˆ†cˆ (cid:17), (1)
evennumberoflayersprovidedthattheantiferromagneticor- i i↑ i↑ i↓ i↓
3 i
. derismuchstrongerthanthesuperconductingorder.Aneven-
1 oddeffecthasalsobeenobservedinJosephsonjunctionswith wherecˆ† createsanelectronwithspinσonlatticesitei,µis
0 iσ
magneticimpuritiesinthemiddlelayer.6 thechemicalpotential,tdenotesthenearest-neighborhopping
4
1 In this Rapid Communication, we report on a novel as- integral,andmi and∆i arethemagneticandsuperconducting
: pect of antiferromagnetic Josephson junctions which allows orderparameters,respectively.
v
We use the Bogoliubov-Valatin transformations c =
i for control over 0−π transitions within a single sample in a (cid:16) (cid:17) iσ
X waywhichhasnocounterpartinSFSstructures.Wefirstcom- ∑n unσ(i)γnσ−σv∗nσ(i)γ†nσ¯ , where σ¯ = −σ, to obtain the
ar putenumericallythethresholdvaluefortheantiferromagnetic standardBogoliubov-deGennesequations(BdG)10
order parameter at which the even-odd effect occurs. Below
(cid:18) (cid:19)(cid:18) (cid:19) (cid:18) (cid:19)
this threshold, even and odd junctions behave qualitatively ∑ Hijσ ∆ijσ unσ(j) =E unσ(i) . (2)
similar, both displaying a monotonic decay of the supercur- j ∆∗jiσ −Hi∗jσ vnσ¯(j) n vnσ¯(i)
rent with superimposed small-scale oscillations, but without
any sign-change of the critical current. As a result of this, Here,H σ=−tδ −µδ +σmδ whereσ=±1forspin
ij (cid:104)i,j(cid:105) ij i ij
we show that it is possible to obtain pressure-induced 0−π up and down, and δ and δ are the Kroenecker deltas
ij (cid:104)i,j(cid:105)
transitionsinantiferromagneticJosephsonjunctions.Namely, for onsite and nearest-neighbor sites, respectively. The off-
applying pressure alters the Fermi level by moving it away diagonalblock∆ =−∆δ fors-wavesymmetry.
ijσ i ij
from the van Hove singularity of the system, which simulta- Due to the crystal periodicity along the interface, we can
neously destroys the Fermi surface nesting properties giving Fourier transform the BdG-equations to make the problem
2
0.35
100
0.3
0.25 10−1
mc
0.2 g
o
mc l10−2
0.15
0.1 10−3
FIG. 1: Left panel: A model of the Josephson junction in (110)- 101 logL 102
direction. Red(blue)dotsdenotetheA(B)-lattice. Thedashedver-
0.05
ticallineistheinterfaceforanevenjunction, whiletherightsolid
line is the interface for an odd junction. The left solid line is the 0
20 40 60 80 100 120 140 160
sameforbothevenandoddjunctions. Rightpanel: Criticalcurrent L
asafunctionoflengthLfordifferentvaluesofthemagneticcoupling
constantU. ThedataarenormalizedtotheL=4valuetosimplify
comparison.WehaveusedV =1.9. FIG. 2: (Color online) Critical value of the magnetic moment mc
asafunctionofjunctionwidthLforanantiferromagneticjunction
(bluecircles/diamonds)andaferromagneticjunction(redsquares).
Intheinset,theresultsplottedonalog-logscalefortheferromag-
effectively one-dimensional. Using the method described in
netic case fall on a straight line corresponding to 1/L. The results
Ref. 10, we get the one-dimensional BdG-equations for the
fortheantiferromagneticcase,however,saturateatlargeL. Forthe
(110)-direction10
ferromagneticcase,theresultsshowthatforawideenoughSFSjunc-
k √ (cid:104) (cid:105) tion,anyamountoforderingsufficestoproducea0−πtransitionin
−µuAi,σ(B)(ky)−2tcos√y2e±iky/ 2 uiB,σ(A)(ky)+uBi∓(A1,)σ(ky) thejunction.FortheantiferromagneticSAFScase,athresholdvalue
isneededforthistooccur,sinceonlyedgespinsareuncompensated.
+σmA(B)uA(B)(k )−∆A(B)vA(B)(k )=EuA(B)(k ), (3) WehaveusedV =1.9.
i i,σ y j i,σ¯ y i,σ y
k √ (cid:104) (cid:105)
µvA(B)(k )+2tcos√y e±iky/ 2 vB(A)(k )+vB(A) (k )
i,σ¯ y 2 i,σ¯ y i∓1,σ¯ y Wefixthephaseattheendofeachsuperconductor,obtain
+σmA(B)vA(B)(k )−∆A(B)∗uA(B)(k )=EvA(B)(k ). (4) self-consistent solutions from the above equations, and use
i i,σ¯ y j i,σ y i,σ¯ y
Eq. (7) to calculate the Josephson current for that particular
The A(B) denotes the A(B) sublattice while the order phase difference. We set µ=0 throughout, and t is used as
parameters are defined as ∆ = −V(cid:104)cˆ cˆ (cid:105) and mA(B) = theunitofenergyinallcalculations.
i i i↓ i↑ i
(U/2)((cid:104)nˆA(B)(cid:105) − (cid:104)nˆA(B)(cid:105)). These are determined self- Results and Discussion. In the right panel of Fig. 1 we
i i↑ i↓ showresultsforthecriticalcurrentasafunctionofthenum-
consistentlythrough10
ber of AF-layers in the junction, for various values of the
nA(B)= ∑(cid:104)|uA(B)(k )|2f(E ) magnetization coupling constantU. The main point to note
iσ n,i,σ y n,ky,σ is that the even-odd effect (sign-change for the current) ap-
n,ky pearsonlyprovidedthatU exceedsathresholdvalue. Below
(cid:105)
+|vA(B)(k )|2f(−E ) , (5) this threshold, the critical current displays no change of sign
n,i,σ y n,ky,σ
withincreasingL,althoughtherearesmalloscillationsinthe
(cid:18)βE (cid:19)
∆A(B)=−V ∑ u (k )v∗ (k )tanh n,ky,σ , (6) quantity superimposed on a monotonic decay. However, in-
i i n,i,σ y n,i,σ¯ y 2 creasing U sufficiently will induce 0−π-transitions for odd
n,ky,σ
junctions,whileonlydecreasingtheeffectiveinterfacetrans-
where f(E)=(1+exp(βE))−1 is the Fermi-Dirac distribu- parency for even junctions, as seen in Fig. 1. Note that for
tion. Here, Ui and Vi denote the spatially dependent cou- L=5, thethresholdvalueforobservinga0−π-transitionin
pling constants for the magnetization and superconducting the critical current is above U = 1.4, while for L = 7, the
pairing, respectively. We model the coupling contants as threshold value is below U =1.4. Since the magnitude of
U =UΘ(i−i )Θ(i −i),V =VΘ(i −i)+VΘ(i−i ),where the staggered magnetic order parameter increases monotoni-
i L R i L R
Θ(x) is the Heaviside step function, and i denote the left callywithU,thisindicatesthatlongeroddjunctionsrequirea
L,R
andrightinterfacecoordinates,respectively. weakermagnetizationtoundergo0−πtransitions.Thisresult
ThedcJosephsoncurrentisobtainedfrom4,5 hassomeresemblancetothecaseofaferromagneticjunction,
(cid:18) (cid:19) wheresuchtransitionscanoccurregardlessofthemagnitude
J=−8te ∑ cos √ky (cid:2)Im(u (x)u∗ (x(cid:48)))f(E ) oftheexchangefieldprovidedthatthejunctionissufficiently
h¯Nkn,ky,σ 2 nσ n,σ nσ long. Wewilldiscussthispointinmoredetailbelow.
+Im(v (x)v∗ (x(cid:48)))f(−E )(cid:3), (7) Inordertounderstandthephysicsbehindtheseresults,note
nσ n,σ nσ
that even junctions have a vanishing total magnetic moment
whereeistheelectroncharge,andxandx(cid:48)denoteneighboring as there is an equal number of spins with opposite direction.
pointsintheantiferromagnet. An even junction is essentially an SNS junction with the ex-
3
traeffectthatincreasedstaggeringincreasestheresistanceof cal magnetic moment is defined as the value where the crit-
thejunction. Thismaybeunderstoodinsimpletermsbynot- ical current changes sign for a given value of the junction
ingthatanincreasingstaggeringprovidesanincreasingspin- length L. Although the fits appear to be quite similar on a
dependentpotentialthatscattersthecurrent-carryingstatesin linear scale, they are significantly different when viewed on
theAF-junctionforlowvaluesofU. a log-log scale. The results for the SFS-junction follows the
Odd junctions, on the other hand, have a finite magnetic intuitiononegetsbasedonthebound-stateenergyandphase-
moment due to the uncompensated spin at one of the edges shiftsshownabove,namelythatthecriticalvalueofthemag-
of the AF region. The layers adjacent to the interfaces have netic moment density, which is proportional to a threshold
parallelspinsforoddjunctions,whileevenjunctionshavean- value of Eex in order to get a 0−π transition, scales as 1/L.
tiparallelspins. Thisresemblesthesituationin, forinstance, ThisisnotsofortheSAFS-case. Rather, sinceEex∼1/Lin
SFIFSjunctionswithparalleloranti-parallelalignmentofthe this case, one expects the critical value of the magnetic mo-
exchangefieldsofthetwoferromagnets,8 orSIFISjunctions ment density to be non-zero also for long junctions. This is
where the interfaces are spin-active.9 In odd junctions, the indeedseeninthelog-logplot,wheretheresultsfortheSFS-
subgapstatesarespin-split,whileforevenjunctions,thestates junctionfallsonalinearcurve,whiletheresultsfortheSAFS-
are spin degenerate. Naively, odd junctions are thus equiva- junctionreachesaconstantvalueasymptotically.
lenttoferromagneticjunctions,asthereisanetmagneticmo- Furthermore, we have considered the crossing of the
ment (albeit weak) if one averages over the junction length. energy-levels of the current-carrying states as the strength
Onenotabledifferenceis,however,thattheaveragemagnetic of the antiferromagnetic ordering, parametrized byU, is in-
momentdensityintheSAFS-casescaleswith1/L,whileitis creased. ThisisshowninFig. 3. Levelcrossingsofspin-up
constant in the SFS-case. Moreover, it is known in the SFS statesandspin-downstates,withoppositeφ-dispersion,occur
case that, provided the junctions are wide enough, any finite forlargeenoughvalueofU,i.e. asthemagnitudeofthemag-
amountofmagneticmomentdensitysufficestoinducea0−π neticorderingincreases. Suchlevelcrossingstendtoreverse
transition. FortheSAFScase,itisfarfromobviousthatone thesignofthecurrent,whichisessentiallydeterminedbythe
shouldgetthesamebehavior,duetotheeffective1/L-scaling φ-derivativeofthelevels. Thelevelsthatcontributemostsig-
of the average magnetic moment density. It is therefore of nificantlytothecurrentsareseentobethelevelsclosetozero
someimportancetoinvestigatethescalingwithLofthecriti- energy,asthelevelsfurtherawayareφ-independentandthere-
calmomentdensityrequiredtoinduce0−πoscillationsinthe fore carry little current. The main difference between such
SAFS-case,andcompareitwiththebehavioroftheSFS-case. spectrafortheSAFS-junctionsandthecorrespondingonesfor
Todoso,itisinstructivetoconsiderananalogywithafer- SFS-junctions, is that complete level-reversion of near-zero
romagnetic junction, where the Andreev-bound state energy energy states occurs much more easily in the SFS-case than
in the ballistic limit reads11: εσ(ϕ)=∆(cid:12)(cid:12)cos(cid:0)ϕ+2σα(cid:1)(cid:12)(cid:12). Here, the SAFS case, since the total magnetic moment scales with
α= 2EexL, where E is the ferromagnetic exchange energy, thewidthofthejunctionintheSFS-case,whileitdoesnotin
andvh¯viFstheFermievxelocity. Wehaveassumedthattheinter- theSAFS-case. Hence,theenergybandswithadefinitespin-
F
facetransparencyisperfectinordertouseasimpleanalytical content are much more susceptible to a Zeeman effect in the
expressionfortheensuingdiscussion. Now,forevenLtheto- SFS-case,comparedtotheSAFS-case. Inparticular,onefea-
talmagneticmomentoftheantiferromagnetiszero,hencewe ture of the SAFS-spectra shown below is that there is essen-
getno0−πtransitioninevenjunctions. Fortheoddjunction tiallyonlyonelevel-reversionbetweenspin-upandspin-down
in question we can relate the finite magnetic moment to an subgapstatesbeforethebandsflattenout,thusnolongercon-
effectiveferromagneticexchangefield. Thetwofactorscon- tributing to the currents. For the SFS-case (not shown here),
tributing to this extra phase is the magnetic moment and the there are several level-inversions of subgap-states with only
widthofthejunction,byanalogytoaneffectiveferromagnet minorflatteningofthe bandsasthemagnetizationincreases,
foroddL. Alongjunctioncandisplaya0−πtransitionfora thusprovidingamuchmoreefficientwayofreversingthesign
smallermagneticmomentmagnitudethanashortjunction,as ofthecurrentsoverthejunction.
seenintherightpanelofFig.1. ForU =1.4, L=5hasnot Itisclearthatantiferromagneticorderisnotinitselfsuffi-
undergonethetransitionyet,whileithasundergoneatransi- cient to induce 0−π transitions in the supercurrent. Ref. 4
tionforL=7andlongerjunctions.This,superficiallyatleast, showedthatforasufficientlystrongstaggeredorderparame-
seemstoindicatethatthebehaviorissimilartothatofanSFS- ter,thecurrent-phaserelationofanantiferromagneticJoseph-
junction. Note however, that in the SFS-case, the parameter sonjunctionrevealeda0-orπ-statedependingonwhetherthe
α increases linearly with L and will exceed any prescribed antiferromagnethadanevenoroddnumberofatomiclayers.
threshold value, provided L is large enough. For an SAFS- Wehaveaboveexplicitlystudiedtheactualthresholdvalueof
junction,however,whereonenaivelyexpectsEex∼1/Lsuch themagnitudeofthestaggeredorderparameterwheretransi-
thatEexL∼1,athresholdvalueforthemagnitudeofthestag- tions cease to occur, regardless of whether the interlayer has
geredorderparameterwouldberequiredinordertoobservea an odd or even number of lattice sites. To observe the even-
0−πoscillation. oddeffect(anditsabsenceaspredictedhereforweakantifer-
To investigate this further, we have numerically compared romagnets),itwouldbenecessarytoexertsatisfactorycontrol
an SFS-junction with an odd SAFS-junction. Fig. 2 shows ofthejunctionlength(oforderlatticespacing). Itwouldcer-
thecriticalmagneticmomentasafunctionofjunctionwidth tainly be an experimental challenge to tailor the junctions in
for both kinds of junctions, SFS and SAFS. Here, the criti- thisway12. Nevertheless,previousstudieshavedemonstrated
4
1.0 values in candidate systems. That would require a more re-
fined approach including non-ideal effects such as disorder
0.5 both in terms of non-magnetic impurities and with regard to
∆ themagneticsublattices.
ε/ 0.0
The threshold value predicted here has an interesting con-
0.5
sequence, namely that it becomes possible to induce a 0−π
1.0 transitioninantiferromagnetsviapressure. Thereasoningis
asfollows. Foranodd-LjunctionwithsufficientlystrongAF
1.0
order,thejunctionisintheπ-state.Wereonetobringthemag-
0.5 nitude of the AF order back down below the threshold value
∆ for the 0−π transition, one would effectively have reversed
ε/ 0.0 thedcJosephson-currentacrossthejunction. Suppressionof
0.5 the AF order may effectively be obtained by destroying the
Fermi-surfacenesting,somethingthatcanbeachievedbyap-
1.0 plying pressure to the sample. The pressure may be induced
0 1 2 3 4 5 6 0 1 2 3 4 5 6
ϕ ϕ via an electric mechanism, such that one effectively is using
electric means to control magnetic order, and by that in turn
FIG.3:(Coloronline)Energy-levelsofcurrent-carryingstatesofthe
current-switching. This effect has no counterpart in an SFS-
systemasafunctionofsuperconductingphase-twistacrossthejunc-
junction,sinceferromagnetismdoesnotrelyonFermisurface
tion. Red lines denote spin-up states, blue lines denote spin-down
nesting. The combination of a threshold value for the stag-
states.Asthestrengthoftheantiferromagneticorderingincreases(U
gered order parameter at which one may have 0−π oscilla-
increases), spin-downstatesareloweredinenergybytheZeeman-
effectduetotheuncompensatedspin,whilespin-upstatesincrease tions,andthepossibilitytodestroynestingandthusAFMor-
in energy. Upper left panel: U =0.5. Upper right panelU =1.2. derviapressure,amountstoelectricallycontrolled0−πtran-
Lowerleftpanel:U=1.545. Lowerrightpanel:U=2.0Thecon- sitionsinasinglesample. ThissetstheresultsforanSAFS-
tributionstothecurrentareessentiallydeterminedbytheφ-derivative junctionapartfromwhatisfoundfortheferromagneticSFS-
ofeachofthecurves.Asthelevelscross,thelevelscontributingmost case,whereanyamountofmagneticorderinprinciplesuffices
significantlytothecurrentchangessign,leadingtothe0−πtransi- toinduce0−πtransitions.Suchtransitionstypicallyoccurby
tion. This requires a threshold value, unlike in the ferromagnetic
changingthejunctionlength, requiringmultiplesamples(al-
case.Otherparametervaluesare:V =1.9,µ=0,L=5.
though temperature-dependent transitions are also possible).
TheAFMorderthusoffersanovelmechanismforcontrolling
thequantummechanicalgroundstateofaJosephsonjunction.
thatitispossibletofabricateultrathinfilmswithatomic-scale
control over the thickness13,14, which would allow for a test H.E. thanks NTNU for financial support. JL and AS ac-
ofourpredictions. Itisdifficultwithintheframeworkofour knowledge support from the Norwegian Research Council,
model to make quantitative predictions for actual parameter throughGrants205591/V20and216700/F20.
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