Correspondence between Andreev reflection and Klein tunneling in bipolar graphene

Correspondence between Andreev reflection and Klein tunneling in bipolar graphene

Beenakker, C.W.J.; Akhmerov, A.R.; Recher, P.; Tworzydlo, J.

Citation

Beenakker, C. W. J., Akhmerov, A. R., Recher, P., & Tworzydlo, J. (2008). Correspondence between Andreev reflection and Klein tunneling in bipolar graphene. Physical Review B, 77(7), 075409. doi:10.1103/PhysRevB.77.075409

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Correspondence between Andreev reflection and Klein tunneling in bipolar graphene

C. W. J. Beenakker,¹A. R. Akhmerov,¹P. Recher,¹and J. Tworzydło²

1Instituut-Lorentz, Universiteit Leiden, P.O. Box 9506, 2300 RA Leiden, The Netherlands

2Institute of Theoretical Physics, Warsaw University, Hoża 69, 00-681 Warsaw, Poland 共Received 26 January 2008; published 13 February 2008兲

The Andreev reflection at a superconductor and the Klein tunneling through an n-p junction in graphene are two processes that couple electrons to holes—the former through the superconducting pair potential⌬ and the latter through the electrostatic potential U. We derive that the energy spectra in the two systems are identical at low energies ␧Ⰶ⌬ and for an antisymmetric potential profile U共−x,y兲=−U共x,y兲. This correspondence implies that bipolar junctions in graphene may have zero density of states at the Fermi level and carry a current in equilibrium, analogous to the superconducting Josephson junctions. It also implies that nonelectronic sys- tems with the same band structure as graphene, such as honeycomb-lattice photonic crystals, can exhibit pseudosuperconducting behavior.

DOI:10.1103/PhysRevB.77.075409 PACS number共s兲: 73.23.Ad, 73.23.Ra, 73.40.Lq, 74.45.⫹c

I. INTRODUCTION

Tunneling through an n-p junction in graphene is called the Klein tunneling^1–3with reference to relativistic quantum mechanics, where it represents the tunneling of a particle into the Dirac sea of antiparticles.⁴ The Klein tunneling in graphene共see Fig.1兲 is the tunneling of an electron from the conduction band into hole states from the valence band—

which plays the role of the Dirac sea. Several recent experiments^5–7 have investigated this unusual coupling of electronlike and holelike dynamics.

In the course of an analysis of these experiments, a curi- ous similarity was noticed⁸between negative refraction^3,9at an n-p junction and the Andreev retroreflection¹⁰ at the in- terface between a normal metal 共N兲 and a superconductor 共S兲. As illustrated in Figs.2共a兲and2共b兲, the trajectories at an

n-p junction and at a normal-metal–superconductor 共NS兲 junction are related by mirroring x哫−x at the interface 共taken at x=0兲. Here, we show that the similarity is not lim- ited to classical trajectories, but it extends to the fully quan- tum mechanical wave functions and energy spectra. This im- plies that quantum effects associated with superconductivity, such as the proximity effect and the Josephson effect, have analogs in an n-p junction.

We have found a precise mapping between the Dirac Hamiltonian¹¹ of an n-p junction and the Dirac- Bogoliubov–de Gennes Hamiltonian¹²of an NS junction un- der the condition that the electrostatic potential U in the n-p junction is antisymmetric, U共−x,y兲=−U共x,y兲, with respect to the interface. The Fermi level is chosen at zero energy, symmetrically between the n and p regions. Such a symmet- ric n-p junction turns out to have the same excitation spec-

FIG. 1.共Color online兲 Conical band structure in graphene at two sides of a potential step共height 2␮, width d兲, forming an n-p junc- tion. In equilibrium, all states below the Fermi level共indicated in blue兲 are filled and all states above are empty. The Klein tunneling is the interband tunneling of an electron from the conduction band in the n region共blue ball at the right兲 into the valence band of the p region共blue ball at the left兲. In this work, we show that the low- energy excitation spectrum of a symmetric n-p junction is the same as that of an NS junction, obtained by replacing the region x⬍0 by a superconductor.

FIG. 2.共Color online兲 Periodic orbits 关panel 共a兲兴 in an n-p junc- tion and关panel 共b兲兴 in a NS junction at ␧=0 in the case of an abrupt interface.共Solid and dashed lines distinguish electronlike and hole- like trajectories.兲 Negative refraction in the n-p junction maps onto the Andreev retroreflection in the NS junction upon mirroring in the interface at x = 0. Destructive interference of the electronlike and holelike segments of the periodic orbit suppresses the density of states at the Fermi level. Panels共c兲 and 共d兲 show alternative geom- etries that exhibit a suppression of the local density of states in an unbounded system.

trum as an NS junction for excitation energies␧ small com- pared to the superconducting gap ⌬. After presenting the mapping in its mathematical form, we consider the two ma- jor physical implications: zero density of states at the Fermi level and persistent current flow in equilibrium. A compari- son with computer simulations of a tight-binding model of graphene is presented at the end of the paper.

II. DERIVATION OF THE MAPPING

The correspondence between the Klein tunneling and the Andreev reflection consists of a mapping of an eigenstate⌿ of the Dirac Hamiltonian H of a symmetric n-p junction onto electron and hole eigenstates⌿eand⌿h in the normal part x⬎0 of the NS junction. The Dirac Hamiltonian is given 共in the valley-isotropic representation兲 by

H =v关共p + eA兲 ·␴兴丢␶0+ U␴0丢␶0, 共1兲 with p = −iប共⳵/⳵^{x ,}⳵/⳵y兲 the momentum operator in the x-y plane of the graphene layer, A = Bxyˆ the vector potential of a perpendicular magnetic field B, andv the electron velocity.

The Pauli matrices␴iand␶iact, respectively, on the sublat- tice and valley degree of freedom 共with␴0 and␶0 a 2⫻2 unit matrix兲. We introduce the time-reversal operator T=

−共␴y丢␶y兲C, with C the operator of complex conjugation, and the parity operator P=i共␴x丢␶0兲R, with R the operator of reflection 共x哫−x兲. The key property of the Dirac Hamil- tonian that we need, in order to map the symmetric n-p junc- tion onto an NS junction, is the anticommutation relation,

TPH = − HTP, 共2兲

satisfied for any B when U共−x,y兲=−U共x,y兲.

Starting from a solution H⌿=␧⌿ of the Dirac equation in the n-p junction, we now construct an eigenstate in the NS junction at the same eigenvalue␧ by means of the transfor- mations

⌿e共x,y兲 = ⌿共x,y兲, ⌿h共x,y兲 = P⌿共x,y兲. 共3兲 According to Refs. 12 and 13, the electron and hole wave functions ⌿e,⌿h in the normal part of the NS junction should satisfy

H⌿e=␧⌿e, −THT⌿h=␧⌿h, x⬎ 0, 共4兲 with a boundary condition at the NS interface that for 兩␧兩 Ⰶ⌬ takes the form

⌿h共0,y兲 = i共␴x丢␶0兲⌿e共0,y兲 ⬅ P⌿e共0,y兲. 共5兲 The proof of the mapping now follows by inspection. First, Eq.共4兲 results directly from the transformation 关Eq. 共3兲兴 with the anticommutation relation 关Eq. 共2兲兴. Second, since ⌿ is continuous at x = 0, the boundary condition关Eq. 共5兲兴 is auto- matically satisfied.

The applicability of the mapping extends to the crystallo- graphic edges of the graphene layer in the following way:

The edges of the n-p junction are described by the boundary condition⌿共r兲=M共r兲⌿共r兲 for r at the edge.^14,15 The map- ping to an NS junction still holds, provided that M commutes withP, which requires

共␴x丢␶0兲M共x,y兲 = M共− x,y兲共␴x丢␶0兲. 共6兲 For example, an armchair edge parallel to the x axis共with M⬀␴xindependent of x兲 satisfies the requirement 关Eq. 共6兲兴, but a zigzag edge parallel to the x axis共M ⬀␴z兲 does not. A pair of zigzag edges at x =⫾W 关with M共⫾W,y兲= ⫾␴z丢␶z兴, on the other hand, do satisfy the requirement关Eq. 共6兲兴. An infinite mass boundary condition 关with M共⫾W,y兲= ⫾␴y 丢␶z兴, likewise, satisfies this requirement.

III. SUPPRESSION OF THE DENSITY OF STATES We have calculated the density of states␳共␧兲 by solving the Dirac equation in the n-p junction of Fig.2共a兲. The Fermi level共taken at ␧=0兲 is separated from the Dirac point by the energy⫾␮in the n and p regions. We take an abrupt inter- face 共width d small compared to the Fermi wavelength ␭F

= hv/␮兲 and wide and long n and p regions 共width WⰇ␭F, length LⰇW兲. The precise choice of boundary condition at x =⫾W does not matter in this regime, as long as it preserves the symmetry of the geometry.

The calculation for the bipolar junction follows step by step the analogous calculation for the Josephson junction in Ref.16. The dispersion relation共smoothed over rapid oscil- lations兲 is given by

␧m共q兲 =␲^ET

共

^{m +}2¹

兲 ^冑

^{1 −共បvq/␮兲2, 兩␧兩 Ⰶ␮, 共7兲}

with m = 0 ,⫾1, ⫾2,... the mode index and បq the momen- tum parallel to the n-p interface.共The energy ET=បv/2W is the Thouless energy, which isⰆ␮^{for W}Ⰷ␭F.兲 The resulting density of states␳共␧兲=共4/␲兲兺m兩⳵␧m/⳵^q兩⁻¹ is plotted in Fig.

3. It vanishes linearly as

␳共␧兲 =␳0兩␧兩/ET 共8兲

for small 兩␧兩, with ␳0=共2␮/␲兲共បv兲⁻² the density of states 共per unit area and including spin plus valley degeneracies兲 in the separate n and p regions. This suppression of the density of states at the Fermi level by a factor ␧/ET is precisely analogous to an NS junction, where the density of states is suppressed by the superconducting proximity effect 共com- pare, for example, our Fig. 3 with Fig. 8 of Ref. 16兲. In particular, the peaks in␳共␧兲 at ␧=␲^ET

共

m +¹₂

兲

are analogous

2 0 2 4 6 8 10 12

0 1 2 3 4 5

ET

Ρ
Ρ0

FIG. 3. Density of states in the n-p junction of Fig.2共a兲, calcu- lated from Eq.共7兲. The dotted line is the value in the isolated n and p regions, which is energy independent for兩␧兩Ⰶ␮. The density of states vanishes at the Fermi level共␧=0兲, according to Eq. 共8兲.

BEENAKKER et al. PHYSICAL REVIEW B 77, 075409共2008兲

075409-2

to the de Gennes–Saint James resonances in Josephson junctions.¹⁷

In a semiclassical description, the suppression of the den- sity of states in the n-p junction can be understood as de- structive interference of the electronlike and holelike seg- ments of a periodic orbit关solid and dashed lines in Fig.2共a兲兴.

At the Fermi level, the dynamical phase shift accumulated in the n and p regions cancels, and what remains is a Berry phase shift of ␲ from the rotation of the pseudospin of a Dirac fermion.^18,19

IV. PERSISTENT CURRENT

If the n and p regions enclose a magnetic flux⌽, as in the ring geometry of Fig.4共inset兲, then the Berry phase shift can be compensated and the suppression of the density of states can be eliminated. The resulting flux dependence of the ground state energy E =A兰_−⬁⁰ ␳共␧兲␧d␧ 共with A the joint area of the n and p regions兲 implies that a current I=dE/d⌽ will flow through the ring in equilibrium at zero temperature, as in a Josephson junction.²⁰According to Eq.共8兲, the order of magnitude,

I₀=共e/ប兲ET2/␦⁼共e/ប兲NET, 共9兲 of this persistent current is set by the level spacing ␦

=共A␳0兲⁻¹and by the Thouless energy E_T=បv/␲^{r = N}␦^{in the} ring geometry 共of radius r and width wⰆr, supporting N

= 4␮w/␲បvⰇ1 propagating modes兲. Because of the macro- scopic suppression of the density of states, this is a macro- scopic current—larger by a factor N than the mesoscopic persistent current in a ballistic metal ring.^20,21

We have calculated I共⌽兲 for a simple model of an abrupt n-p junction in an N-mode ring without intermode scattering, neglecting the effect of the curvature of the ring on the spec- trum and also assuming that the magnetic field is confined to the interior of the ring.共These approximations are reasonable for ␭FⰆwⰆr.兲 The slowly converging, oscillatory integral over␳共␧兲 was converted into a rapidly decaying sum over the Matsubara frequencies by the method of Ref. 22. The zero-temperature result is plotted in Fig.4共solid curve兲. The maximal persistent current is I_c⬇0.2I0. This is the same value, up to a numerical coefficient, as the critical current of a ballistic Josephson junction.²³ We have also included the

results at finite temperature, showing the decay when the thermal energy kBT⯝ET.

V. HOW TO OBSERVE PSEUDOSUPERCONDUCTIVITY To test our analytical predictions against a computer simulation, we have numerically solved a tight-binding Hamiltonian on a honeycomb lattice共lattice constant a兲. We took a symmetric n-p junction with zigzag boundaries at x

=⫾W 共with W/a=400兲 and calculated the density of states

␳共␧兲, smoothed by a Lorentzian 共width of 0.01ET兲 to elimi- nate the rapid oscillations. Results are shown in Fig. 5 for different Fermi wavelengths ␭F= hv/␮ and widths d of the n-p interface关potential profile U共x兲=−␮tanh共4x/d兲兴. A clear suppression of ␳共␧兲 is observed within an energy range ET

from the Fermi level at ␧=0. The suppression is somewhat smaller than predicted by Eq.共8兲 共black solid line兲, in par- ticular, for d⯝a 共red curve, when the Dirac equation no longer applies兲 and for dⲏ␭F 共blue curve, when the Klein tunneling happens only near normal incidence²兲.

As expected, the suppression is sensitive to perturbations of the reflection symmetry. For example, as shown in Fig.5 共yellow curve兲, a displacement of the n-p interface by ␭F

spoils the systematic destructive interference due to the Berry phase and thus eliminates the suppression of the global density of states.

We would still expect an effect on the local density of states if we could confine the carriers to the n-p interface.

This might be achieved by means of the saddle point poten- tial U =␮sgn共xy兲 of Fig.2共c兲or by means of the nonuniform magnetic field B = B₀x of Fig. 2共d兲. Destructive interference of the periodic orbits in each of these unbounded geometries will suppress the local density of states near the interface by the same mechanism as in the confined geometry of Fig.

2共a兲. Because of disorder, the suppression will be limited to a mean free path or corrugation length from the n-p interface.

Since the predicted suppression of the density of states at the Fermi level happens at a large energy separation␮^{from the} Dirac point 共see Fig. 1兲, it should be distinguishable in a FIG. 4. Persistent current through a ring containing an abrupt

n-p interface as a function of the magnetic flux through the ring.

The solid curve is for zero temperature T = 0, the dashed curve for T = E_T/4kB, and the dotted curve for T = E_T/2kB.

FIG. 5.共Color online兲 Same as Fig.3but now calculated from a tight-binding model of graphene共lattice constant a, W/a=400兲. The colors distinguish different values of␭Fand d, corresponding to an abrupt interface 共␭F/a=65, d/a=12兲, a smooth interface 共␭F/a

= 12, d/a=12兲, and an atomically sharp interface 共␭F/a=12, d/a

⯝1兲. The suppression of the density of states vanishes if the reflec- tion symmetry is broken by displacing the interface共yellow curve,

␭F/a=65, d/a=12, displacement=65a兲.

local measurement共for example, by a tunneling probe兲 from any features associated with the conical singularity in the band structure at the Dirac point.

From a different perspective, the correspondence derived here offers the intriguing opportunity to observe supercon- ducting analogies in nonelectronic systems governed by the same Dirac equation as graphene. An example would be a two-dimensional photonic crystal on a honeycomb or trian- gular lattice,^24,25in which the analog of an n-p junction has been proposed recently.²⁶The detrimental effects of disorder should be relatively easy to avoid in such a metamaterial.

ACKNOWLEDGMENTS

We acknowledge discussions with J. Nilsson and R. A.

Sepkhanov. This research was supported by the Dutch Sci- ence Foundation NWO/FOM.

APPENDIX: CALCULATION OF THE PERSISTENT CURRENT AND COMPARISON WITH SUPERCURRENT

In this appendix, we present the calculation leading to the persistent current through the bipolar junction plotted in Fig.

4. We follow closely the analogous calculation for the super- current through a Josephson junction of Ref.22and compare the two systems at the end. For the sake of this comparison, it is convenient to work with the density of states ^˜␳

=共A/2兲␳per spin direction, integrated over the areaA of the system. We will likewise, in this appendix, count the number of propagating modes N˜ =N/2 per spin direction.

1. Persistent current

The persistent current I = dF/d⌽ at temperature T is given by the derivative of the free energy F with respect to the flux

⌽ enclosed by the ring containing the n-p junction. This can be expressed as an integral over the density of states,

I = − 2k_BT d

d⌽

冕

_−⬁^{⬁ d␧˜␳}共␧兲ln关2 cosh共␧/2kBT兲兴. 共A1兲 We have set the Fermi energy at zero and used the electron- hole symmetry^˜␳共␧兲=^˜␳共−␧兲. The factor of 2 in front accounts for the two spin directions共which are not counted separately in^˜␳兲.

Since the spectrum of the ring is discrete, the density of states^˜␳共␧兲=兺i␦共␧−␧i兲 consists of delta functions at the so- lutions of the equation

F共␧兲 ⬅ F0共␧兲

兿

_i ^{共␧ − ␧i兲 = 0. 共A2兲}

共The index i counts spin-degenerate levels once.兲 The func- tion F0 is ⬎0 and even in ␧ but can otherwise be freely chosen. The density of states is then written as

␳

˜共␧兲 = −1

␲ d

d␧ Im lnF共␧ + i0⁺兲, 共A3兲 with 0⁺a positive infinitesimal.

Substitution of Eq.共A3兲 into Eq. 共A1兲 gives, using again the electron-hole symmetry,

I =2kBT

␲ⁱ d

d⌽

冕

_−⬁+i0^⬁+i0++d␧ ln关2 cosh共␧/2kBT兲兴d

d␧ lnF共␧兲.

共A4兲 The expression for the persistent current becomes, upon par- tial integration,

I = − 1

␲ⁱ d

d⌽

冕

−⬁+i0⁺

⬁+i0⁺

d␧ tanh共␧/2kBT兲ln F共␧兲. 共A5兲

We close the contour in the upper half of the complex plane.

We assume thatF0is chosen such that lnF has no singulari- ties for Im␧⬎0. The only poles of the integrand in Eq. 共A5兲 then come from the hyperbolic tangent, at the Matsubara frequencies, i␻n=共2n+1兲i␲^kBT. Summing over the residues, we arrive at the expression²²

I = − 4kBT d d⌽

兺

n=0

⬁

lnF共i␻n兲. 共A6兲

In our model of an N˜ -mode ring without intermode scat- tering, we can calculate separately the contribution to I from each propagating mode, with transverse momentum qm. The total current is then a sum over these contributions,

I = − 4kBT

兺

m=1 N˜

d d⌽

兺

n=0

⬁

lnF共i␻n,qm兲. 共A7兲 The functionF共␧,q兲, which determines the energy levels in the bipolar junction for a given transverse mode, is the limit

⌬→⬁ of the analogous function in a Josephson junction.¹³ We find

F共␧,q兲 =␮²⁻␧²+共បvq兲²

␪+␪−E_T² sin␪+sin␪−

+ cos␪+cos␪−+ cos共e⌽/ប兲, 共A8兲

␪_⫾^{= E}T−1

冑

共␮⫾ ␧兲²−共បvq兲². 共A9兲 Substituting into Eq.共A7兲 gives the persistent current,

I = 4kBTe

ប sin共e⌽/ប兲

兺

m=1 N˜

兺

n=0

⬁ 1

F共i␻n,qm兲. 共A10兲 We wish to evaluate the expression 关Eq. 共A10兲兴 in the regime␮ⰇET, N˜ Ⰷ1. The sum over modes may be replaced by an integral, according to 兺_m=1^N˜ →共N˜/kF兲兰₀^kFdq, with kF

=␮/បv the Fermi wave vector. Since the sum over the Mat- subara frequencies converges exponentially fast for␻nⲏET, we can also assume␮Ⰷ␻n. In this large-␮ regime, we may approximate␪_⫾⬇␣⫾i⍀n, with

␣⁼共␮/ET兲关1 − 共q/kF兲²兴^1/2, 共A11兲

⍀n=共␻n/ET兲关1 − 共q/kF兲²兴^−1/2. 共A12兲 The functionF takes the form

BEENAKKER et al. PHYSICAL REVIEW B 77, 075409共2008兲

075409-4

F共i␻n,q兲 = X cos²␣^{+ Y sin2}␣^, 共A13兲

X = Z sinh²⍀n+ cosh²⍀n+ cos共e⌽/ប兲, 共A14兲

Y = Z cosh²⍀n+ sinh²⍀n+ cos共e⌽/ប兲, 共A15兲

Z = ␮²⁺共បvq兲²+␻n2

␮²−共បvq兲²+共ET⍀n兲²⬇1 +共q/kF兲²

1 −共q/kF兲². 共A16兲 The phase ␣ varies rapidly as a function of q, so we average 1/F first over this phase,

1 F→

冕

0

2␲d␣ 2␲

1

X cos²␣+ Y sin²␣⁼

冑

XY¹ . 共A17兲 We substitute Eq. 共A17兲 into Eq. 共A10兲 and evaluate it numerically to arrive at the curves of I versus ⌽ shown in Fig.4.

2. Comparison with supercurrent

The mapping between bipolar junctions and Josephson junctions is illustrated in Fig.6. Instead of a ring geometry, we may equivalently consider a planar superconductor–

normal-metal–superconductor 共SNS兲 junction, with a phase difference␾between the two superconducting reservoirs. In the absence of mode mixing, the two geometries carry the same supercurrent I_Jat the same number of transverse modes N˜ . 共The Thouless energy in the SNS junction is ET=បv/L, with L the separation of the two NS interfaces.兲

The supercurrent IJ through the Josephson junction is given by²⁷

IJ= − 2kBT2e ប

d d␾

冕

0

⬁

d␧␳J共␧兲ln关2 cosh共␧/2kBT兲兴, 共A18兲

with␾ the phase difference across the junction and ␳J the density of states per spin direction. The mapping relates

␾↔e⌽/ប 共see Fig.6兲 and␳J↔˜. Comparison of Eqs.␳ 共A1兲 and共A18兲 then shows that IJ共␾兲↔I共⌽兲. The bipolar junction and the Josephson junction therefore carry the same current in equilibrium.

The result in the literature^28–32for a ballistic SNS junction is a piecewise linear dependence of IJon␾at zero tempera- ture, close to but not identical to the solid curve in Fig.4. As we will now show, the difference is due to the presence or the absence of a step in the Fermi energy at the NS inter- faces.

On the one hand, the mapping between bipolar and the Josephson junctions relies on the boundary condition 关Eq.

共5兲兴 at the NS interface, which assumes that the Fermi energy

␮Sin the superconductor is much larger than the value␮in the normal region.¹³On the other hand, Refs.28–32assume

␮S=␮. The functionF共␧,q兲 is then given by

F共␧,q兲 = cos共␪+−␪−兲 + cos␾^, 共A19兲

resulting in

IJ共␾兲 = 4kBTe ប

兺

m=1 N˜

兺

n=0

⬁ sin␾

cosh 2⍀n共qm兲 + cos␾^{. 共A20兲}

The resulting supercurrent is plotted in Fig. 7 for different temperatures.

At T = 0, the sum over n reduces to an integral, 兺_n=0^⬁

→共2␲^kBT兲⁻¹兰0⬁d␻, which evaluates to FIG. 6.共Color online兲 Mapping of a bipolar ring containing two

n-p junctions共left panel兲 onto a Josephson ring containing two NS junctions 共right panels兲 by mirroring the holelike trajectories 共dashed兲 in the line through the interfaces. The persistent current I through the bipolar ring at the left maps onto a supercurrent I_J through a Josephson ring at the right. Because the enclosed flux⌽ is halved by the mapping, the h/e periodicity of I maps onto an h/2e periodicity of IJ. The flux enclosed by the Josephson ring in the upper right panel may be gauged away, with the introduction of a phase difference ␾=e⌽/ប between the order parameters at the two NS interfaces共lower right panel兲.

FIG. 7. Supercurrent through a ballistic Josephson junction as a function of the phase difference␾ between the two superconducting reservoirs, calculated from Eq.共A20兲. The solid curve is for zero temperature T = 0, the dashed curve for T = E_T/4kB, and the dotted curve for T = E_T/2kB. The difference with the analogous result for a bipolar junction in Fig. 4 arises because this figure is for equal Fermi energy ␮S=␮ in superconductor and normal metal, while Fig.4maps onto a Josephson junction with␮SⰇ␮.

IJ共␾兲 =2eET

␲ប

兺

m=1 N˜

关1 − 共qm/kF兲²兴^1/2

冕

0

⬁

d␻ ^sin␾ cosh 2␻^{+ cos}␾

=␾^eET

␲ប

兺

m=1 N˜

关1 − 共qm/kF兲²兴^1/2, 兩␾兩 ⬍␲^. 共A21兲

共The ␾ dependence is repeated periodically outside of the interval −␲⬍␾⬍␲.兲 We thus recover the piecewise linear␾ dependence of the supercurrent.^28–32

For N˜ Ⰷ1, the sum over modes may also be evaluated as an integral,兺_m=1^N˜ →共N˜/kF兲兰₀^kFdq, with the result

IJ共␾兲 =␾^{eN˜ ET}

4ប , 兩␾兩 ⬍␲^. 共A22兲 The critical current Ic=␲^{eN˜ E}T/4ប=共␲/8兲I0 is about two times larger than the maximal persistent current Ic⬇0.2I0

found in the bipolar junction because of the absence of a step in the Fermi energy at the NS interfaces.

Equation共A22兲 holds in a two-dimensional geometry. In three dimensions, the sum over modes becomes 兺_m=1^N˜

→共2N˜/kF2兲兰₀^kFqdq, resulting in

IJ共␾兲 =␾^{2eN˜ ET}

3␲ប , 兩␾兩 ⬍␲^, 共A23兲 in agreement with Refs.29 and 30. 共The numerical coeffi- cient in Ref. 28 is different.兲 In the one-dimensional case, N˜ =1 of a single spin-degenerate mode 共group velocity v_group=v关1−共q1/kF兲²兴^1/2兲, we find instead

IJ共␾兲 =␾^evgroup

␲^{L , 兩}␾兩 ⬍␲^, 共A24兲 in agreement with Ref. 31共up to a factor of 2, presumably because Ref.31does not account for the spin degeneracy of the mode兲.

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