Josephson effect in ballistic graphene
Titov, M.; Beenakker, C.W.J.
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Titov, M., & Beenakker, C. W. J. (2006). Josephson effect in ballistic graphene. Retrieved
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Josephson effect in ballistic graphene
M. Titov1and C. W. J. Beenakker2
1Department of Physics, Konstanz University, D-78457 Konstanz, Germany 2Instituut-Lorentz, Universiteit Leiden, P. O. Box 9506, 2300 RA Leiden, The Netherlands
共Received 23 May 2006; revised manuscript received 14 June 2006; published 5 July 2006兲 We solve the Dirac–Bogoliubov–de Gennes equation in an impurity-free superconductor–normal-metal– superconductor junction, to determine the maximal supercurrent Icthat can flow through an undoped strip of graphene with heavily doped superconducting electrodes. The result Ic⯝共W/L兲e⌬0/ប is determined by the superconducting gap⌬0and by the aspect ratio of the junction共length L small relative to the width W and to
the superconducting coherence length兲. Moving away from the Dirac point of zero doping, we recover the usual ballistic result Ic⯝共W/F兲e⌬0/ប, in which the Fermi wavelength Ftakes over from L. The product
IcRN⯝⌬0/ e of the critical current and normal-state resistance retains its universal value共up to a numerical prefactor兲 on approaching the Dirac point.
DOI:10.1103/PhysRevB.74.041401 PACS number共s兲: 73.23.Ad, 74.45.⫹c, 74.50.⫹r, 74.78.Na
While the Josephson effect was originally discovered in a tunnel junction,1 any weak link between two
superconduct-ors can support a dissipationless current in equilibrium.2The
current I共兲 varies periodically with the phase difference of the pair potential in the two superconductors, reaching a maximum Ic 共the critical current兲 which is characteristic of
the strength of the link. A measure of the coupling strength is the resistance RN of the junction when the
superconduct-ors are in the normal state. The product IcRN increases as
the separation L of the two superconductors becomes smaller and smaller, until it saturates at a value of order ⌬0/ e, determined only by the excitation gap ⌬0 in the
superconductors—but independent of the coupling strength. This phenomenology has been well established in a variety of superconductor–normal-metal–superconductor 共SNS兲 junctions3and forms the basis of operation of the Josephson
field-effect transistor.4,5
A new class of weak links has now become available for research,6 in which the superconductors are coupled by a
monatomic layer of carbon共graphene兲. The low-lying exci-tations in this material are described by a relativistic wave equation, the Dirac equation. They are massless, having a velocityv that is independent of energy, and gapless,
occu-pying conduction and valence bands that touch at discrete points共Dirac points兲 in reciprocal space.7Graphene thus
pro-vides a unique opportunity to explore the physics of the “relativistic Josephson effect” 共which had remained unex-plored in earlier work8 on relativistic effects in
high-temperature and heavy-fermion superconductors兲. We ad-dress this problem here in the framework of the Dirac– Bogoliubov–de Gennes共DBdG兲 equation of Ref.9.
The basic question that we seek to answer is what hap-pens to the critical current as we approach the Dirac point of zero carrier concentration. Earlier theories11–13 have found
that undoped graphene has a quantum-limited conductivity of order e2/ h, in the absence of any impurities or lattice
defects. We find that the critical current is given, up to nu-merical coefficients of order unity, by
Ic⯝
e⌬0
ប max共W/L,2W/F兲, 共1兲
in the short-junction regime LW, 共with =បv/⌬0 the
superconducting coherence length, W the width of the junc-tion, andFthe Fermi wavelength in the normal region兲. At
the Dirac point F→⬁, so the critical current reaches its
minimal value of共e⌬0/ប兲W/L. Since the normal-state
resis-tance has its maximal value RN⯝共h/e2兲L/W at the Dirac
point, the IcRNproduct remains of order⌬0/ e as the carrier
concentration is reduced to zero.
The system considered is shown schematically in Fig.1. A layer of graphene in the x-y plane is covered by supercon-ducting electrodes in the regions x⬍−L/2 and x⬎L/2. The normal region 兩x兩 ⬍L/2 has electron and hole excitations described by the DBdG equation,9,10
冉
H0− 0 0 − H0冊冉
⌿e ⌿h冊
=冉
⌿e ⌿h冊
. 共2兲Here H0= −iបv共xx+yy兲 is the Dirac Hamiltonian, ⬎0
is the excitation energy, and is the chemical potential or Fermi energy in the normal region共measured with respect to the Dirac point, so that = 0 corresponds to undoped graphene兲. The electron wave functions ⌿e and the hole
wave functions ⌿h have opposite spin and valley indices,
which are not written explicitly.共A fourfold degeneracy fac-tor will be added in the final results.兲 The Pauli matricesiin
H0 operate on the isospin index, which labels the two
lattices of the honeycomb lattice of carbon atoms.
Andreev reflection at a normal-metal–superconductor 共NS兲 interface couples ⌿e and ⌿h. This coupling may be
described globally by a scattering matrix, as was done in Ref.9 to determine the conductance of a NS junction. Here we follow a different approach, more suited to determine the energy spectrum共and therefrom the Josephson current兲. In this approach electrons and holes are coupled locally by means of a boundary condition on the wave function in the normal region.
We consider the energy range⬍⌬0below the excitation gap⌬0in the superconductor, where the spectrum is discrete. At a point r on the NS interface共with unit vector nˆ pointing from N to S, perpendicular to the interface兲, the boundary condition takes the form
⌿h共r兲 = U⌿e共r兲, 共3兲
U = 1
⌬共 − i
冑
兩⌬兩2−2nˆ ·兲 = e−i⌽−inˆ·. 共4兲 Here ⌬=⌬0ei⌽ is the complex pair potential in S, =共x,y兲 is the vector of Pauli matrices, and 
= arccos共/⌬0兲苸共0,/ 2兲.
The relation共3兲 follows from the DBdG equation,9,14
un-der three assumptions characterizing an “ideal” NS interface: 共I兲 The Fermi wavelength F
⬘
in S is sufficiently small thatF
⬘
,F, where F= hv / is the Fermi wavelength in Nand=បv/⌬0 is the superconducting coherence length; 共II兲
the interface is smooth and impurity-free on the scale of; 共III兲 there is no lattice mismatch at the NS interface, so the honeycomb lattice of graphene is unperturbed at the bound-ary. The absence of lattice mismatch might be satisfied by depositing the superconductor on top of a heavily doped re-gion of graphene. As in the case of a semiconductor two-dimensional electron gas,15,16 we expect that such an
ex-tended superconducting contact can be effectively described by a pair potential⌬ in the x-y plane 共even though graphene by itself is not superconducting兲.
The particle current density out of the normal region, given by jparticle=v⌿e * nˆ ·⌿e−v⌿h * nˆ ·⌿h, 共5兲
should vanish for⬍⌬0, because subgap excitations decay
over a lengthin S.共The possibility of a subgap excitation entering the superconductor at one point along the boundary and exiting at another point within a distanceis excluded by assumption II.兲 By substituting the boundary condition 共3兲
one indeed finds that jparticle= 0, since U is a unitary matrix
which commutes with nˆ ·.
In the SNS junction the normal region has two interfaces with the superconductor, one at x = −L / 2共with superconduct-ing phase⌽=/ 2 and outward normal nˆ = −xˆ兲 and another at
x = L / 2 共with ⌽=−/ 2 and nˆ = xˆ兲. The boundary condition
共3兲 at the points r±=共±L/2,y兲 thus takes the form
⌿h共r−兲 = U共兲⌿e共r−兲, ⌿h共r+兲 = U−1共兲⌿e共r+兲, 共6兲 U共兲 = e−i/2+ix, = arccos共/⌬
0兲. 共7兲
Since the wave vector ky parallel to the NS interface is
conserved upon Andreev reflection, we may solve the prob-lem for a given ky⬅q. The transfer matrix M共,q兲 relates
the states at the two ends of the normal region:
⌿e共r+兲 = M共,q兲⌿e共r−兲, ⌿h共r+兲 = M共− ,q兲⌿h共r−兲.
共8兲 共For ease of notation, the q dependence will not be written explicitly in what follows.兲 The condition for a bound state 共the Andreev level兲 in the SNS junction is that the transfer matrix for the round trip from r−to r+and back to r−has an
eigenvalue equal to unity. This condition can be written in the form of a determinant,
Det关1 − M−1共兲U共兲M共− 兲U共兲兴 = 0, 共9兲
which we have to solve for as a function of q and. The electron transfer matrix M共兲 is readily obtained from the Dirac equation,
M = ⌳eikLz⌳, 共10兲 ⌳ = ⌳−1=共2 cos␣兲−1/2
冉
e−i␣/2 e i␣/2 ei␣/2 − e−i␣/2冊
, 共11兲 ␣共兲 = arcsin冉
+បvq冊
, 共12兲 k共兲 = 共បv兲−1共 +兲cos␣共兲. 共13兲The angle␣is the angle of incidence of the electron, and k is its longitudinal wave vector.
Evaluation of the determinant共9兲 leads after some algebra
to the quantization condition
cos=
冉
cos+cos−+ sin+sin− cos␣+cos␣−冊
cos 2 +冉
sin+cos− cos␣+ −cos+sin− cos␣−冊
sin 2 − sin+sin−tan␣+tan␣−, 共14兲where we abbreviated␣±=␣共±兲,±= k共±兲L.
We introduce a finite width W to quantize the transverse wave vectors, q→qn, n = 0 , 1 , 2 , . . ., and denote byn共,兲
the density of states in mode n. The Josephson current at zero temperature is then given by
I共兲 = −4e ប d d
冕
0 ⬁ d兺
n=0 ⬁ n共,兲, 共15兲where the factor of 4 accounts for the twofold spin and val-ley degeneracies. To be definite we take “infinite mass” boundary conditions at y = 0 , W, for which12 q
n=共n
+ 1 / 2兲/ W. 共For WL the choice of boundary conditions becomes irrelevant.兲 At the Fermi level, the lowest N共兲 =W /បv modes are propagating 共real k兲, while the higher
modes are evanescent共imaginary k兲.
We analyze the Josephson effect in the experimentally M. TITOV AND C. W. J. BEENAKKER PHYSICAL REVIEW B 74, 041401共R兲 共2006兲
most relevant short-junction regime that the length L of the normal region is small relative to the superconducting coher-ence length. In terms of energy scales, this condition re-quires⌬0 បv/L. To leading order in the small parameter
⌬0L /បv we may substitute ␣±→␣共0兲, ±→k共0兲L in the
quantization condition共14兲. The solution is a single bound
state per mode,
n共兲 = ⌬0
冑1 −
nsin2共/2兲, 共16兲 n= kn2 kn2cos2共knL兲 + 共/ប v兲2sin2共knL兲 , 共17兲 with kn=关共/បv兲2− qn2兴1/2. This expression for the Andreev
levels in terms of a normal-state transmission probabilityn
has the usual form for a short SNS junction.17 Comparison
with Ref.12shows that nis indeed the transmission
prob-ability for a ballistic strip of graphene between two heavily doped electrodes in the normal state共⌬0= 0, F
⬘
F兲. Thenormal-state resistance RNis thus given by
RN−1=4e
2 h
兺
n=0⬁
n. 共18兲
Substitution of n共,兲=␦关−n共兲兴 into Eq. 共15兲 gives
the supercurrent due to the discrete spectrum,
I共兲 =e⌬0 ប n=0
兺
⬁ nsin 关1 −nsin2共/2兲兴1/2 . 共19兲Contributions to the supercurrent from the continuous spec-trum are smaller by a factor L /and may be neglected in the short-junction regime.18 For LW the summation over n
may be replaced by an integration. The resulting critical cur-rent Icand the IcRNproduct are plotted as a function ofin
Fig.2.
The limiting behavior at the Dirac point共 បv/L兲 for a short and wide normal region共LW,兲 is
I共兲 =e⌬0 ប 2W L cos共/2兲arctanh关sin共/2兲兴, 共20兲 Ic= 1.33 e⌬0 ប W L, IcRN= 2.08⌬0/e. 共21兲
These results for ballistic graphene at the Dirac point are formally identical to those of a disordered normal metal 共Fermi wave vector kF, mean free path l兲,17,19upon
substitu-tion kFl→1. This correspondence is consistent with the
find-ing of Ref. 12 that ballistic Dirac fermions have the same shot noise as diffusive nonrelativistic electrons.
In the opposite regime បv/L we have instead 共still for LW,兲 the result
Ic= 1.22
e⌬0
ប W
ប v, IcRN= 2.44⌬0/e. 共22兲
共We do not have a simple analytic expression for the de-pendence in this regime.兲 The critical current 共22兲 is about
half the ideal ballistic value17,20 I
c= 2Ne⌬0/ប, with N
=W /បv the number of propagating modes 共per spin and
valley兲. This reduction is due to the mismatch in Fermi wavelength at the NS interfaces. Equations 共21兲 and 共22兲
together contain the scaling behavior共1兲 with F= hv /
an-nounced in the introduction.
In conclusion, we have shown that a Josephson junction in graphene can carry a nonzero supercurrent even if the Fermi level is tuned to the point of zero carrier concentra-tion. At this Dirac point, the current-phase relationship has the same form as in a disordered normal metal—but without any impurity scattering. Instead of being independent of the length L of the junction, as expected for a short ballistic Josephson junction, the critical current Ic at the Dirac point
has diffusionlike scaling proportional 1 / L. Since the normal-state resistance RN⬀L, the IcRNproduct remains fixed at the
superconducting gap 共up to a numerical prefactor兲 as the Fermi level passes through the Dirac point. This unusual “quasidiffusive” scaling of the Josephson effect in undoped graphene should be observable in submicrometer scale junc-tions.
This research was supported by the Dutch Science Foun-dation NWO/FOM. We have benefited from discussions on the experimental implications of this work with A. Morpurgo, B. Trauzettel, L. M. K. Vandersypen, and other members of the Delft/Leiden focus group on Solid State Quantum Information Processing.
FIG. 2. Critical current Icand IcRNproduct of a ballistic
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M. TITOV AND C. W. J. BEENAKKER PHYSICAL REVIEW B 74, 041401共R兲 共2006兲
