Heat transport by Dirac fermions in normal/superconducting graphene junctions

0.14 MB·English

by  Takehito Yokoyama

Heattransportby Diracfermions innormal/superconducting graphene junctions
Takehito Yokoyama1, Jacob Linder2 and Asle Sudbø2
1 Department of Applied Physics, Nagoya University, Nagoya, 464-8603, Japan
2 DepartmentofPhysics, NorwegianUniversityofScienceandTechnology, N-7491Trondheim, Norway
(Dated:February2,2008)
Westudyheattransportinnormal/superconductinggraphenejunctions. Wefindthatwhilethethermalcon-
ductance displays the usual exponential dependence on temperature, reflecting the s-wave symmetry of the
superconductor, itexhibitsanunusualoscillatorydependenceonthepotentialheightorthelengthofthebar-
8
rierregion. Thisoscillatorydependencestemsfromtheemergentlow-energyrelativisticnatureoffermionsin
0
graphene,essentiallydifferentfromtheresultinconventionalnormalmetal/superconductorjunctions.
0
2

n Therecentprogressinpracticalfabricationtechniquesfora Topgate Superconductingelectrode
a monoatomic layer of graphite, called graphene, has allowed
J Graphenelayer
for experimental studies of this system, which in turn has
16 triggereda tremendousinterest1,2,3,4,5,6,7. Grapheneis a two- VoltageU
dimensionalsystemofcarbonatoms,andthelow-energyelec-

trons in graphene are governed by Dirac equation. Up to
]
n now, intensive studies on graphene have been conducted for
o instancequantumHalleffect6,8,9,minimumconductivity7and Heatcurrent Bottomgate
c
bipolarsupercurrent10.
-
r From applied physics point of view, graphene is also an
p FIG. 1: (color online) The proposed experimental setup to
u importantmaterial. Grapheneexhibitshighmobilityandcar- measure heat transport by Dirac fermions in a graphene nor-
s rierdensitycontrollablebygatevoltage,whichmakesitwell mal/superconductor proximity structure. The top and bottom gate
t. suitedforachievingdeviceapplications.5,6,11,12Inordertoap- allowforthechemicalpotentialinthemiddleregiontobeadjusted.
a
ply graphene to electric devices, it is an important issue to
m
clarifycharacteristicsoftransportphenomenaingraphene.
d- In conventionalnormal metal/superconductorjunctions, it withH = vF(kxσx+kyσy). Thesuperconductingorderpa-
n is known that electric and thermal conductances reflect the rameterreads
o magnitudeorsymmetryofthegapofthesuperconductor.13,14
∆=∆(T)eiφΘ(x−L), (2)
c Whileconductanceinnormal/superconductorgraphenejunc-
[ tionhasbeenstudied,15,16,17thermalconductanceinthesame where Θ(x) is the Heaviside step function, while φ is the

1 junctionhasnotyetbeeninvestigated. Thestudyofthether- phase corresponding to the globally broken U(1) symmetry
v malconductanceinnormal/superconductorgraphenejunction in the superconductor. Also, v ≈ 106m/s is the energy-
6 will complement the study of the conductance in the same F
independent Fermi velocity for graphene, σ (i = x,y) de-
4 i
junction.
4 notesthePaulimatrices,Eistheexcitationenergy,anduand
2 In this paper, we study heat transport in nor- v denote the electron-like and hole-like exictations, respec-
. mal/superconducting graphene junctions. We find that tively,describedbythewave-function.ThePaulimatricesop-
1
the thermal conductance has an exponential dependence on erateonthetwotriangularsublatticespaceofthehoneycomb
0
8 temperature, which reflects the s-wave symmetry of the su- structure,correspondingtotheAandBatoms.Thelineardis-
0 perconductor.However,itdisplaysanoscillatorydependence persionrelationisareasonableapproximationevenforFermi
: on the potential height or the length of the barrier region. levelsashighas1eV,20suchthatthefermionsingraphenebe-
v
Thisoscillatorydependencestemsfromtherelativisticnature havelikemasslessDiracfermionsinthelow-energyregime.
i
X offermionsingraphene,anddiffersinanessentialwayfrom Let us consider an incident electron from the normalside
r the result in the conventional normal metal/superconductor ofthejunction(x<0)withenergyE. Forpositiveexcitation
a junctions. energiesE > 0,theeigenvectorsandcorrespondingmomen-
We briefly present the formalism to be used in this pa- tumoftheparticlesread
per, following Ref.17. Consider a two dimensional nor-
mal/insulating/superconducting graphene junction18 where ψ+e =[1,eıθ,0,0]Teıpecosθx, pe =(E+EF)/vF, (3)
the superconducting (normal) region is located in the semi-
for a right-moving electron at angle of incidence θ, while
infinite regionsx > L(x < 0). The proposedexperimental
a left-moving electron is described by the substitution θ →
setupofourmodelisshowninFig. 1. Byexploitingtheval-
π−θ. ThesuperscriptT denotesthetranspose. IfAndreev-
leydegeneracy,19theBogoliubov-deGennesequationforthe
reflectiontakes place, a left-movingholewith energyE and
junctioninthexy-planereads
angleofreflectionθ isgeneratedwithcorrespondingwave-
A
function
H −E ˆ1 ∆ˆ1 u u
∆†ˆ1F E ˆ1−H v =E v (1) ψh =[0,0,1,e−ıθA]Te−ıphcosθAx, ph =(E−E )/v ,(4)
(cid:18) F (cid:19)(cid:18) (cid:19) (cid:18) (cid:19) − F F
2
where the superscript e (h) denotes an electron-like (hole- Itisappropriatetoinserttherestrictionwhichwillbeused
like) excitation. Since translational invariance in the y- throughoutthe paper,namely∆ ≪ E′ . Since we areusing
F
directionholds, the correspondingcomponentofmomentum a mean-field approach to describe the superconducting part
is conserved. This conditionallows for determinationof the oftheHamiltonian,phase-fluctuationsoftheorderparameter
Andreev-reflection angle θ via phsinθ = pesinθ. From havetobesmall22.
A A
this equation, one infers that there is no Andreev-reflection We definethewavefunctionsinthenormal,insulatingand
(θA =±π/2)foranglesofincidenceabovethecriticalangle superconductingregionsbyψ,ψ˜I andΨ,respectively,with
θ =sin−1(|E−E |/(E+E )). (5)
c F F ψ =ψe +rψe +r ψh, (10)
+ − A −
Onthesuperconductingsideofthesystem(x > L),thepos- ψ˜ =t˜ψ˜e +t˜ψ˜e +t˜ψ˜h +t˜ψ˜h, (11)
I 1 + 2 − 3 + 4 −
siblewavefunctionsfortransmissionofaright-movingquasi-
Ψ=teΨe +thΨh. (12)
particlewithagivenexcitationenergyE >0reads + −
Ψe = u,ueıθ+,ve−ıφ,veı(θ+−φ) T The wavefunctionsψ˜differfrom ψ in that the Fermi energy
+ isshiftedbyanexternalpotential,suchthatE → E −U
F F
×eıqecosθ+x, q(cid:16)e =(E′ + E2−∆2)/v(cid:17), (6) whereU isthebarrierheight. Also,notethatthetrajectories
F F
of the quasiparticles in the insulating region, defined by the
Ψh = v,veıθ−,uep−ıφ,ueı(θ−−φ) T anglesθ˜andθ˜ ,differbythesamesubstitution:
− A
×eıqhcosθ−x, q(cid:16)h =(EF′ − E2−∆2)/v(cid:17)F. (7) sinθ˜/sinθ =(E+E )/(E+E −U), (13)
F F
Thecoherencefactorsaregivenbyp21 sinθ˜A/sinθ =(E+EF)/(E−EF +U). (14)
1 E2−|∆|2 Note thatthe subscript± onthe wavefunctionsindicatesthe
u= 1+ , (8) directionofmomentum,whichisingeneraldifferentfromthe
s2 E
p groupvelocitydirection.
(cid:16) (cid:17)
1 E2−|∆|2 Bymatchingthewavefunctionsatbothinterfaces,ψ|x=0 =
v = 1− . (9) ψ˜ | and ψ˜ | = Ψ| ,23 we obtain the following
s2 E I x=0 I x=L x=L
(cid:16) p (cid:17) expressions for the normal reflection coefficient r and the
Above,wehavedefinedθ+ =θe andθ− =π−θh.Thetrans- Andreev-reflectioncoefficientrA:17
S S
mission angles θ(i) for the electron-like and hole-like quasi-
S
particlesaregivenbyq(i)sinθ(i) =pesinθ,i=e,h. Notethat r =te(A+C)+th(B+D)−1, (15)
in allthewavefunctionslistedSabove,forclaritywehavenot rA =te(A′+C′)+th(B′+D′), (16)
includeda commonphase factor eıkyy which correspondsto
theconservedmomentuminthey-direction. wherethetransmissioncoefficientsread
t =2cosθ[e−ıθA(B′+D′)−(B′e−ıθ˜A −D′eıθ˜A)]ρ−1, (17)
e
t =t [eıθA(A′e−ıθ˜A −C′eıθ˜A)−A′−C′][B′+D′−eıθA(B′e−ıθ˜A −D′eıθ˜A)]−1, (18)
h e
ρ=[e−ıθA(B′+D′)−(B′e−ıθ˜A −D′eıθ˜A)][e−ıθ(A+C)+(Aeıθ˜−Ce−ıθ˜)]
−[(De−ıθ˜−Beıθ˜)−e−ıθ(B+D)][A′e−ıθ˜A −C′eıθ˜A −e−ıθA(A′+C′)] (19)
andwehaveintroducedtheauxiliaryquantities A′ =veı(q++p−−φ)[1+(eıθ+ −e−ıθ˜A)(2cosθ˜ )−1],
A
B′ =ueı(q−+p−−φ)[1+(eıθ− −e−ıθ˜A)(2cosθ˜ )−1],
A
C′ =veı(q+−p−−φ)(e−ıθ˜A −eıθ+)(2cosθ˜ )−1,
A
A=ueı(q+−p+)[1−(eıθ˜−eıθ+)(2cosθ˜)−1], D′ =ueı(q−−p−−φ)(e−ıθ˜A −eıθ−)(2cosθ˜ )−1.(21)
B =veı(q−−p+)[1−(eıθ˜−eıθ−)(2cosθ˜)−1], A
C =ueı(p++q+)(eıθ˜−eıθ+)(2cosθ˜)−1,
D =veı(p++q−)(eıθ˜−eıθ−)(2cosθ˜)−1, (20)
3
Here,wehavedefined withχ =LU/v . Thisindicatesthatthermalconductanceis
F
π-periodicwithrespecttoχinthislimit.
q+ =qecosθ+L, q− =qhcosθ−L,
p+ =p˜ecosθ˜L, p− =p˜hcosθ˜ L. (22)
A
Inthe thin-barrierlimitdefinedasL → 0andU → ∞, one
gets
θ˜→0, θ˜ →0, q →0,p →∓χ (23) Finally,thenormalizedthermalconductanceisgivenby
A ± ±
∞ π/2 cosθ E2
κ= dEdθcosθ(1−|r(E,θ)|2−Re( A)|r (E,θ)|2) (24)
cosθ A ∆ T2cosh2(E )
Z0 Z−π/2 0 2T
(cid:26)(cid:27)(cid:23)
withthegapatzerotemperature∆ ≡∆(0).
0
(cid:26)(cid:27)(cid:29)
(cid:5) (cid:6) (cid:26)(cid:27)(cid:22)
(cid:6)(cid:7)(cid:8)(cid:9)(cid:10)
(cid:6)(cid:7)(cid:8)(cid:9)(cid:3)(cid:10) (cid:26)(cid:27)(cid:28)
(cid:7)(cid:8)(cid:9)(cid:3)(cid:10)(cid:10)
(cid:4) (cid:26)(cid:27)(cid:21) (cid:30)(cid:31) !(cid:26)
(cid:30)(cid:31) !(cid:25)
(cid:26)(cid:27)(cid:25) )+* (cid:30)(cid:31) !(cid:21)
(cid:3) (cid:30)(cid:31) !(cid:29)
(cid:15)(cid:17)(cid:16)
(cid:26) (cid:21) (cid:22) (cid:23) (cid:24) (cid:25)(cid:26)
$"%#
(cid:26)(cid:27)(cid:29)
(cid:5)
(cid:6)
(cid:6)(cid:7)(cid:8)(cid:9)(cid:10) (cid:26)(cid:27)(cid:22)
(cid:4) (cid:6)(cid:7)(cid:7)(cid:8)(cid:8)(cid:9)(cid:9)(cid:3)(cid:3)(cid:10)(cid:10)(cid:10) (cid:26)(cid:27)(cid:28)
(cid:3) (cid:26)(cid:27)(cid:21) ),* --..//(cid:31)(cid:31)!!(cid:26)(cid:25)
(cid:18)(cid:20)(cid:19) (cid:26)(cid:27)(cid:25) -./(cid:31)!(cid:21)
-./(cid:31)!(cid:29)
(cid:10) (cid:10)(cid:11)(cid:4) (cid:10)(cid:11)(cid:12) (cid:10)(cid:11)(cid:13) (cid:10)(cid:11)(cid:14) (cid:3) (cid:26) (cid:25)(cid:26) (cid:21)(cid:26)
’ (
(cid:2)(cid:0)(cid:2)(cid:1) &
FIG. 3: (Color online) (a) Thermal conductance as a function of
FIG.2: (Coloronline)ThermalconductanceasafunctionofT/TC U/EF forvariouskFLwithT/TC =0.5andEF =EF′ =100∆0.
for various kFL with U/EF = 10 and EF′ = 100∆0 at EF = (b) Thermal conductance as a function of kFL for various U/EF
100∆0in(a)andEF =10∆0in(b). withT/TC =0.5andEF =EF′ =100∆0.
We next present our results for the normalized thermal
conductance. Figure 2 (a) shows thermal conductance as a symmetryofthesuperconductor. However,thelengthdepen-
function of T/T for various k L with U/E = 10 and dence of the thermal conductance is nonmonotonic (oscilla-
C F F
E = E′ = 100∆ . Here T is the transition temper- tory) and thus essentially different from that in the conven-
F F 0 C
ature and k ≡ E /v . From Fig. 2 (a), an exponen- tionalnormalmetal/superconductorjunctions. Asimilarplot
F F F
tialdependenceofthethermalconductanceontemperatureis forE = 10∆ isshownin Fig. 2 (b). We also find anex-
F 0
seen,similartotheconventionalnormalmetal/superconductor ponentialtemperaturedependence, but the dependenceon L
junctions.13 Thisexponentialdependencereflectsthes-wave getsweaker. Therefore,themagnitudeoftheoscillationwith
4
respect to k L gets reduced with the increase of the Fermi variationof the superconductinggapnear the interface. The
F
wavevectormismatch. suppression of the order parameter near the interface is ex-
Figure 3 (a) depicts thermal conductance as a function pectedtobeleastpronouncedwhenthesharpedgecriteriais
of U/E for various length k L with T/T = 0.5 and satisfied and there is a large Fermi-vector mismatch. In the
F F C
E = E′ = 100∆ . An oscillatory dependence of the present case, this is precisely so, whence we do not expect
F F 0
thermalconductanceonU/E isseen. Theperioddecreases our qualitative results to be affected by taking into account
F
with k L. Figure 3 (b) displays thermal conductances as a the reductionof the gap near the interface. Finally, we have
F
function of k L for various U/E with T/T = 0.5 and assumedthatthereisnolatticemismatchattheinterfacesand
F F C
E = E′ =100∆ . Wealsofindanoscillatorydependence thatthesearesmoothandimpurity-free4. Amorerefinedpic-
F F 0
on k L. The period also decreases with U/E . These fea- ture couldbe obtainedbyusing more realistic modelsof the
F F
turesstemfromtheπ-periodicityofthethermalconductance variationofthechemicalpotential,i.e. acontinuousslopein-
with respect to χ = k LU/E in the limit of U ≫ E steadofastep-likevariation.
F F F
and k L ≪ 1 similar to the junction conductance.16,17,24 In In summary, we have studied heat transport in nor-
F
other words, the damped oscillatory behavior of the thermal mal/superconductinggraphene junctions. We found that the
conductance is a direct manifestation of the relativistic low- thermalconductancehasanexponentialdependenceontem-
energy Dirac fermions. Also, the presence of the insulating peraturewhichreflectsthes-wavesymmetryofthesupercon-
regionisessentialfortheoscillatorybehavior. ductor but oscillatory dependence on the potential height or
Since we have assumed a homogeneous chemical poten- the lengthof the barrierregion. Thisoscillatorydependence
tial in each of graphene regions, the experimental observa- stemsfromtherelativisticnatureoffermionsingraphene,es-
tion of the predicted effects require charge homogeneity of sentially differentfromthe resultin the conventionalnormal
the graphene samples. This is a challenge, since electron- metal/superconductorjunctions.
holepuddlesingrapheneimagedbyascanningsingleelectron T.Y.acknowledgessupportbytheJSPS.J.L.andA.S.were
transistor device25 suggest that such chargeinhomogeneities supported by the Research Council of Norway, Grants No.
play an important role in limiting the transport characteris- 158518/431 and No. 158547/431 (NANOMAT), and Grant
ticsofgraphene26. Inaddition,wehaveneglectedthespatial No. 167498/V30(STORFORSK).
1 T.Ando,J.Phys.Soc.Jpn.74,777(2005). 4515(1982).
2 M.I.KatsnelsonandK.S.Novoselov,SolidStateCommun.143, 15 C.W.J.Beenakker,Phys.Rev.Lett.97,067007(2006).
3(2007). 16 S. Bhattacharjee and K. Sengupta, Phys. Rev. Lett. 97, 217001
3 A. H.CastroNeto, F.Guinea, N. M. R.Peres, K.S.Novoselov (2006); S.Bhattacharjee,M.Maiti,andK.Sengupta,Phys.Rev.
andA.K.Geim,arXiv:0709.1163v1. B76,184514(2007).
4 C.W.J.Beenakker,arXiv:0710.3848. 17 J. Linder and A. Sudbø, Phys. Rev. Lett. 99, 147001 (2007);
5 K.S.Novoselov,A.K.Geim,S.V.Morozov,D.Jiang,Y.Zhang, arXiv:0712.0831.
S. V. Dubonos, I. V. Grigorieva and A. A. Firsov, Science 306, 18 We underline that the notation ”insulator” in this context refers
666(2004). toanormalsegmentofgrapheneinwhichoneexperimentallyin-
6 Y.Zhang,Y.-W.Tan,H.L.StormerandP.Kim,Nature438,201 ducesaneffectivepotentialbarrier.
(2005). 19 A.F.MorpurgoandF.Guinea,Phys.Rev.Lett.97,196804(2006).
7 K.S.Novoselov,A.K.Geim,S.V.Morozov,D.Jiang,M.I.Kat- 20 P.R.Wallace,Phys.Rev.71,622(1947).
snelson,I.V.Grigorieva,S.V.Dubonos,andA.A.Firsov,Nature 21 K.Fossheimand A.Sudbø, Superconductivity: Physicsand ap-
438,197(2005). plications,JohnWiley&SonsLtd.,Ch.5(2004).
8 K.S.Novoselov, E.McCann, S.V.Morozov, V.I.Fal’ko, M.I. 22 A.K.NguyenandA.Sudbø,Phys.Rev.B60,15307(1999);H.
Kastenelson, U. Zeitler, D. Jiang, F. Schedin, and A. K. Geim, Kleinert,Phys.Rev.Lett.,84286(2000).
Nat.Phys.2,177(2006). 23 Notethattheseconditionsareequivalenttovˆxψ|x=0=vˆxψ˜I|x=0
9 K.Yang,SolidStateCommun.143,27(2007). and vˆxψ˜I|x=L = vˆxΨ|x=L with velocity operator vˆx =
10 H.B.Heersche,P.Jarillo-Herrero,J.B.Oostinga,L.M.K.Van- ∂H/∂kx = vFσx and hencethecurrent isconserved atthein-
dersypenandA.F.Morpurgo,Nature446,56(2007). terfaces.
11 J.ScottBunch,Y.Yaish,M.Brink,K.Bolotin,andP.L.McEuen, 24 M.I.Katsnelson,K.S.NovoselovandA.K.Geim,NaturePhys.
NanoLett.5,287(2005). 2,620(2006).
12 C.Berger,Z.Song,X.Li,X.Wu,N.Brown,C.Naud,D.Mayou, 25 J.Martin,N.Akerman,G.Ulbricht,T.Lohmann,J.H.Smet,K.
T.Li,J.Hass,A.N.Marchenkov, E.H.Conrad, P.N.First,and vonKlitzingandA.Yacoby,arXiv:0705.2180v1.
W.A.deHeer,Science312,1191(2006). 26 E.-A.KimandA.H.CastroNeto,arXiv:cond-mat/0702562v2.
13 A.F.Andreev,Sov.Phys.JETP19,1228(1964).
14 G.E.Blonder,M.Tinkham,andT.M.Klapwijk,Phys.Rev.B25,